Pricing a Basket Option When Volatility Is Capped Using Affine Jump-Diffusion Models

Pricing a basket option when volatility is capped using affine jump-diffusion models

DANIEL KREBS

Master of Science Thesis Stockholm, Sweden 2013

Pricing a basket option when volatility is capped using affine jump-diffusion models

DANIEL KREBS

Degree Project in Mathematical Statistics (30 ECTS credits) Degree Programme in Engineering Physics (270 credits) Royal Institute of Technology year 2013 Supervisor at KTH was Camilla Johansson Landén Examiner was Boualem Djehiche

TRITA-MAT-E 2013:27 ISRN-KTH/MAT/E--13/27--SE

Royal Institute of Technology School of Engineering Sciences

KTH SCI SE-100 44 Stockholm, Sweden

URL: www.kth.se/sci

Abstract

This thesis considers the price and characteristics of an exotic option called the Volatility-Cap-Target-Level(VCTL) option. The payoﬀ function is a simple Euro- pean option style but the underlying value is a dynamic portfolio which is comprised of two components: A risky asset and a non-risky asset. The non-risky asset is a bond and the risky asset can be a fund or an index related to any asset category such as equities, commodities, real estate, etc.

The main purpose of using a dynamic portfolio is to keep the realized volatility of the portfolio under control and preferably below a certain maximum level, denoted as the Volatility-Cap-Target-Level (VCTL). This is attained by a variable allocation between the risky asset and the non-risky asset during the maturity of the VCTL-option. The allocation is reviewed and if necessary adjusted every 15th day. Adjustment depends entirely upon the realized historical volatility of the risky asset.

Moreover, it is assumed that the risky asset is governed by a certain group of stochastic differential equations called affine jump-diffusion models. All models will be calibrated using out-of-the money European call options based on the Deutsche-Aktien-Index(DAX).

The numerical implementation of the portfolio diffusions and the use of Monte Carlo methods will result in different VCTL-option prices. Thus, to price a nonstandard product and to comply with good risk management, it is advocated that the financial institution use several research models such as the SVSJ- and the Sepp- model in addition to the Black-Scholes model.

Keywords: Exotic option, basket option, risk management, greeks, aﬃne jump- diﬀusions, the Black-Scholes model, the Heston model, Bates model with log- normal jumps, the Bates model with log-asymmetric double exponential jumps, the Stochastic-Volatility-Simultaneous-Jumps(SVSJ)-model, the Sepp-model.

Acknowledgements

I would like to thank my supervisor at Royal Institute of Technology, Dr. Camilla Landén for excellent guid- ance in the world of derivatives and superb feedback. I would also like to acknowledge Dr. Artur Sepp, Vice President Equity Derivatives Analytics, Bank of America Merrill Lynch, for fruitful discussions about aﬃne jump-diﬀusions and Professor Kenneth Holmström, Tomlab for insightful comments about calibration. I’m also grateful to Dubravko Salcic, Tom Andersson, Nela Cekredzi and Robert Axelsson at SEB, for highlight- ing the practical aspects of structured derivatives. Finally, I would like to thank my mentor Dr. Christian von Ledebur, and my family and friends for their valuable support and encouragement.

Daniel Krebs ([email protected])

Contents

1 Introduction 1

2 The dynamic portfolio 3 2.1 The self-ﬁnancing portfolio ...... 3 2.2 The VCTL-option ...... 4 2.3 Mathematics behind the relative weights ...... 4 2.4 The annualized realized historical volatility and the discretized portfolio ...... 6 2.5 A volatility cap target level and the corresponding allocation table ...... 7

3 Using Fourier transform for option pricing where the risky asset satisfies different affine jump- diffusions 9 3.1 Introduction ...... 9 3.2 The general and transformed price dynamic for the risky asset S ...... 9 3.2.1 The option price formula using the inverse and forward Fourier transform ...... 10 3.2.2 The characteristic formula: Explicit expressions corresponding to the Black-Scholes model ...... 11 3.2.3 The characteristic formula: Explicit expressions corresponding to the Heston model . 12 3.2.4 The characteristic formula: Explicit expressions corresponding to the Bates model . . 12 3.2.5 The characteristic formula: Explicit expressions corresponding to the SVSJ model . . 13 3.2.6 The characteristic formula: Explicit expressions corresponding to the SEPP model . . 14

4 Model calibration 15 4.1 Introduction ...... 15 4.2 A multi-variate function and estimation procedure ...... 15 4.2.1 Several error measures and the choice of error functional ...... 16 4.2.2 The market data ...... 16 4.3 Boundaries and estimated parameter values ...... 17

5 Simulation of the transformed portfolio aﬃne jump-diﬀusions 19 5.1 Introduction ...... 19 5.2 Simple Euler discretization of the transformed portfolio dynamic: The risky asset evolves according to the Black-Scholes model ...... 20 5.3 Full truncation Euler discretization of the transformed portfolio dynamics: The risky asset evolves according to the Heston model ...... 21 5.4 Full truncation Euler discretization of the transformed portfolio dynamics: The risky asset evolves according to the BatesLN model ...... 22 5.5 Full truncation Euler discretization of the transformed portfolio dynamics: The risky asset evolves according to the BatesLDE model ...... 23 5.6 Full truncation Euler discretization of the transformed portfolio dynamics: The risky asset evolves according to the SVSJ model ...... 24 5.7 Full truncation Euler discretization of the transformed portfolio dynamics: The risky asset evolves according to the Sepp model ...... 25

6 The distributional features of the portfolio 27 6.1 Introduction ...... 27 6.2 The mean of the portfolio X ...... 27

vi 6.2.1 The mean versus volatility ...... 27 6.2.2 The mean versus VCTL ...... 29 6.3 The variance of the portfolio X ...... 30 6.3.1 The standard deviation of log(X) versus maturity for diﬀerent volatilities ...... 30 6.4 The standard deviation of log(X) versus VCTL ...... 32 6.5 Measuring skewness and kurtosis of the Volatility Cap Portfolio ...... 32 6.6 The kurtosis and skewness of the Volatility Cap Portfolio ...... 33

7 The VCTL-option price 37 7.1 The VCTL-option price versus VCTL under the BS-model ...... 37 7.2 The VCTL-option price versus VCTL under the Heston-, BatesLN-,BatesLDE-,SVSJ- and Sepp-model ...... 38 7.3 The VCTL-option price versus volatility when VCTL=15% ...... 41 7.4 The VCTL-option price versus volatility when VCTL=10% ...... 44 7.5 Concluding remarks ...... 48

8 The sensitive measures of the VCTL-option 49 8.1 The Greeks ...... 49 8.2 Greeks for a European call option when the risky asset S evolves according to the Black- Scholes model ...... 50 8.2.1 Delta ...... 50 8.2.2 Gamma ...... 51 8.2.3 Vega ...... 51 8.3 Greeks for the VCTL-option, when the risky asset S evolves according to the Black-Scholes model ...... 51 8.4 Greeks for the VCTL-option, when the risky asset S evolves according to the Heston-model . 53 8.5 Greeks for the VCTL-option, when the risky asset S evolves according to the BatesLN-, BatesLDE-,SVSJ- and Sepp-model ...... 53 8.6 Delta for VCTL-option versus volatility ...... 54 8.7 Vega for VCTL-option versus volatility ...... 54

9 Implied volatility and skew eﬀects 55 9.1 Implied volatility versus the volatility of volatility ε ...... 56 9.2 Implied volatility versus the correlation ρ ...... 57

10 Conclusions 59

A Appendix 61 A.1 Allocation table and relative weights ...... 61 A.2 Standard deviation of log(S) versus maturity ...... 63 A.3 The drift (risky asset S) versus volatility ...... 64 A.4 The standard variation of log(X) versus VCTL, maturity 1 year ...... 64 A.5 The standard variation of log(X) versus VOLATILITY. Maturity 1 year. VCTL=15% . . . . . 64 A.5.1 Kurtosis and skewness for log(S) ...... 64 A.6 The standard variation of log(X) versus VCTL, maturity 1 year ...... 65 A.7 Moments for log(X) versus VCTL ...... 66 A.8 Greeks for the VCTL-option, when the risky asset S evolves either according to the BatesLN- model and BatesLDE-model ...... 68 A.9 Greeks for the VCTL-option, when the risky asset S evolves according to the SVSJ-model . . 69 A.10 Greeks for the VCTL-option, when the risky asset S evolves according to the Sepp-model . . 69 A.11 Greeks for a simple European call option under affine jump-diffusions ...... 70 A.12 Moments for different ε ...... 71 A.13 Implied volatility, 3-dimensional, different rho:s ...... 72

Bibliography 75 Chapter 1

Introduction

An investor interested in emerging markets can, instead of investing directly in an emerging market’s fund, buy a Structured Investment Product (SIP) based on this fund. A SIP typically involves various exposures to fixed income markets and various derivatives. Depending on the exposure, it can attract both conservative and risk-tolerant investors. By combining an investment in the fixed income market together with the Volatility- Cap-Target-Level(VCTL)-option, the financial institution(FI) can offer an investor a Principal Protected Note. Hence, the investor is offered the opportunity to invest in the same emerging market, but now with a guaranteed minimum return equal to the investor’s initial investment (the principal amount) i.e the risk of losing all money is now zero.

Some FIs bundle up the SIP themselves but buy the contingent claim from other FIs. Hence, the number of derivatives which makes up for a certain portion of the principal, is conditional upon the option price. This price is in turn dependent on which model governs the risky asset. The asset price is normally assumed to evolve in accordance with the Black-Scholes(BS) model [4]. However, empirical evidence suggests that log-returns are not normally distributed, as the BS-model predicts, but exhibit other characteristics typically assigned to more advanced models [24] [23]. One type of asset price dynamics which can reproduce some of these characteristics are affine jump-diffusions. They can incorporate stochastic volatility and jumps in different constellations. Another shortcoming of the BS-model is its inability to reproduce implied volatility surfaces seen in the option markets. While the BS-model predicts a flat surface i.e the same volatility independently of strike and maturity, the implied volatility surface of the market shows signs of variation such as a volatility smile. Hence, the use of a more realistic model is almost imperative. A perfect pricing model should at least fulfill two conditions: It should have a perfect replication of the option payoff, and be able to perfectly reproduce the implied volatility surface. But to find a model which can reproduce all peculiarities of the financial market will prove impossi- ble. In terms of reproducing the implied volatility, some affine jump-diffusion models can be regarded as better models than the BS-model, but they fail in terms of replicating.

More speciﬁcally, the VCTL-option has a payoﬀ function using a simple European option style, but the underlying value is a dynamic portfolio which is comprised of two components: A risky asset and a non-risky asset. The non-risky asset is a bond and the risky asset can be a fund or an index related to any asset category such as equities, commodities, real estate, etc. The main purpose of using a dynamic portfolio is to keep the realized volatility of the portfolio under control and preferably below a certain maximum level, denoted as the Volatility-Cap-Target-Level (VCTL). This is attained by a variable allocation between the risky asset and the non-risky asset during the maturity of the VCTL-option. The allocation is reviewed and if necessary adjusted every 15th day. Adjustment depends entirely upon the realized historical volatility of the risky asset.

To obtain the VTCL-option price we will discretize using a full truncation Euler scheme together with Monte Carlo methods. In order to get the risk-neutral parameters, the stochastic diﬀerential equations (SDEs) satisﬁed by the risky asset S, are calibrated to the market data using OTM European call options of DAX.

This thesis is organized as follows. In Chapter 2, some intuitions are given and the mathematical description of the dynamic portfolio are presented. Some basics about the Fourier transform, its use and the resulting Characteristic formula for options with payoff functions of European option style are discussed in Chapter 3. Hence, this general formula provides model prices for plain European options, for a wide number of affine jump-diffusion models. Issues relating to calibration and the use of a suitable solver for the optimization

1 CHAPTER 1. INTRODUCTION problem are described in Chapter 5. The numerical implementation where the full truncation Euler scheme is applied to the logarithm of the portfolio, is described in Chapter 6. The distributional features such as variance and skewness of the dynamic portfolio, for varying volatility and VCTLs are given in Chapter 7. The sensitivity measures of the VCTL-option, commonly known as the greeks are calculated and analyzed in Chapter 8. In Chapter 9, the input parameters, volatility of volatility and correlation, are adjusted in order to study the eﬀect on the VCTL-option price via the implied volatility. The conclusions are presented in Chapter 10.

2 Chapter 2

The dynamic portfolio

Suppose that there is a financial market consisting of the non-risky asset and the risky asset with price processes B and S respectively. The task is to use these assets to create a self-financing portfolio denoted as the dynamic portfolio or portfolio X. The purpose of constructing X is to keep the realized portfolio volatility below a certain annualized historical volatility level denoted as VCTL. Theoretically, it means that given an outcome of the realized volatility (of the risky asset S) and the VCTL, the relative weights can be adjusted accordingly. The relative weights of the portfolio are assigned different constant values conditional upon a predefined allocation table with volatility ranges. This will be done every 15th market day, until the expiration date. The relative weight corresponding to the risky asset is assigned values in the range of [0.05,1] and will act as a dampening factor on the price process S. Since the non-risky asset does not contribute neither with volatility nor with the realized volatility, the realized portfolio volatility will be equal to or lower than the desired VCTL.

In practise, the portfolio X will be a basket consisting of an index and a bond. The option will have a payoﬀ function of a European call option style and together with the underlying portfolio X, this constitutes the VCTL-option.

2.1 The self-ﬁnancing portfolio

In terms of time t and relative portfolio weights u(t) = (u1, u2, ..., uN ), the general dynamics of a self-ﬁnancing portfolio X with no consumption or dividend yield can be expressed as:

∑N dS (t) dXh(t) = Xh(t) u (t) i (2.1) i S (t) i=1 i ∑ N = where S i(t) is the price of one asset of type i, ui(t) is the relative weight of type i and i=1 ui(t) 1. Next, consider a ﬁnancial market consisting of only two assets, a bond and an index with price processes B and S respectively. The dynamic of the risk free asset with constant interest rate is given by:

dB(t) = rB(t)dt (2.2)

Furthermore, the price of the index can be assumed to follow a geometric Brownian motion with constant drift or local rate of return α and diﬀusion parameter σ.

dS (t) = αS (t)dt + σS (t)dW P(t) (2.3) where W P(t) is a Wiener process under the probability measure P. If the models of the assets are substituted into the self-ﬁnancing portfolio it yields: ( dB(t) dS (t) ) dXh(t) = Xh(t) (u (t) + u (t) ) 1 B(t) 2 S (t) (( ) ) (2.4) h P = X (t) u1(t)α + u2(t)r dt + u1σdW (t)

3 CHAPTER 2. THE DYNAMIC PORTFOLIO

This portfolio is free of arbitrage if and only if the local rate of return α equals the short rate r [3]. Hence, we have the Black-Scholes model which governs the risky asset. The portfolio diﬀusion under this martingale or Q-measure is then given by:

( dB(t) dS (t) ) dXh(t) = Xh(t) (u (t) + u (t) ) 1 B(t) 2 S (t) (( ) ) = h + + σ Q (2.5) X (t) r (|u 1 ( t ){z u 2 ( t})) dt u1 dW (t) 1 h h Q = rX (t)dt + u1σX (t)dW (t)

Using aﬃne jump-diﬀusions instead of the Black-Scholes model will yield more complex expressions for the portfolio, but the local rate or return still has to equal the short rate r, in order to prevent arbitrage possibilities.

2.2 The VCTL-option

We proceed and define Πt as the price process of an option, where the portfolio has value X. Together with the payoff function Φ(X(T)), this indicates that the arbitrage free price can be written as: [ ] −r(T−t) Q Πt = e E Φ(X(T))|Ft (2.6) where the Q-dynamic of X is Eq. (2.5), when the risky asset is governed by the Black-Scholes model. The σ-algebra Ft contains all information about the market that is known up to time t. For the VTCL-option, it is assumed that the payoff function is of a European option style i.e Φ(X(T)) = max[φ[X(T) − K, 0]], where the binary values φ for a European call option and a European put option are +1 and -1 respectively. Hence, the general arbitrage free price of the VTCL-option, using affine jump-diffusion models, can be expressed as: [ ] −r(T−t) Q Πt = e E max[φ[X(T) − K, 0]|Ft] (2.7) for the Q-dynamic(s) of X.

2.3 Mathematics behind the relative weights

In order to allow for variable allocation, which is conditional upon the realized volatility of the risky asset, the random variable δ has to be deﬁned and inserted. This produces the following stochastic processes uS (t, δ) and , δ S uB(t ). More formally, if the risky asset S has generated information up to time t, Ft then this also implies X ≥ that the information about the portfolio X is known i.e Ft . For all t 0 the relative weights are adapted to the { S } { X} , δ , δ S ﬁltrations Ft t≥0 and Ft t≥0. Thus, the stochastic processes uS (t ) and uB(t ) are not only Ft -measurable X but also Ft -measurable.

Normally, when the portfolio value only depends on the today’s date t and today’s value of some asset price S(t), it could be said that the portfolio has a Markovian property. But since δ will depend upon some past price trajectory {S (u); u ≤ t}, the portfolio no longer exhibits the Markovian property. Thus, the portfolio is non-Markovian but since it exhibit the martingale property, this will still guarantee that an arbitrage free VCTL-option price can be obtained.

Moreover, the weights are defined for all t but they will not be continuous (see graph below). They sum up to one and will be assigned different values depending on the result for δ in the time-period before the weights re-balance. More in detail, the relative weights will be assigned to constant values in accordance with the pre-defined allocation table. The graph below is a special case, showing the δ and the values that the relative weights are assigned to accordingly.

4 2.3. MATHEMATICS BEHIND THE RELATIVE WEIGHTS

RELATIVE WEIGHTS

1,0 u s (t, ) 0,8 u B (t, ) 0,6 u s (t, ) 0,4 0,2 CONTINUOUS 0 u B (t, ) TIMES (t)

1 u B (t, 1) 3 u B (t, 3)

Figure 2.1. A special case of the relative weights in continuous time. δX is a functional of the S-trajectory in every time- period. This will generate the relative weights according to the pre-deﬁned volatility scheme. δ0 or uS (t, δ0) = 1, uB(t, δ0) = 0 are given in advance and δ2, δ4 are omitted in the ﬁgure. The form of the transformations at the reallocation dates are similar to the Heaviside step function in discrete form. In this example, δ1 = δ3 i.e uB(t, δ1) = uB(t, δ3) = 0.2. In the last step the relative weights have changed places due to the δ4 omitted.

Given the relative weights uS (t, δ) and uB(t, δ), the resulting self-ﬁnancing portfolio under the BS-model can be expressed as:

dS (t) dB(t) dX(t) = uS (t, δ)X(t) + uB(t, δ)X(t) (S (t) B()t) rS (t)dt + σS (t)dW(t) rB(t)dt = u (t, δ)X(t) + u (t, δ)X(t) S S (t) B B(t) = u (t, δ)X(t)rdt + u (t, δ)X(t)σdW(t) + u (t, δ)X(t)rdt (2.8) ( S S) B = , δ + , δ + , δ σ |uS ( t ) {zu B ( t }) X(t)rdt uS (t )X(t) dW(t) 1

= rX(t)dt + σuS (t, δ)X(t)dW(t)

In the following section, we will try to find out whether a known probability density function for X is attainable or not. If known, this would facilitate greatly, since knowing the cumulative distribution function allows us to use an analytical approach. Hence, instead of using numerical methods to approximate the option price, we could obtain an exact price of the VCTL-option. To gain insight, this task will be performed by investigating three different types of relative weights: A constant relative weight (uS ), a nice (regarding integrability) deterministic relative weight (uS (t)) and the stochastic relative weight (uS (t, δ)). Before looking further into the relative weights, a variable transformation Zt = lnXt is made first and then the Itos¯ formula is applied:

= 1 + 1 − 1 2 dZt dXt ( 2 )(dXt ) Xt 2 Xt = 1 + σ , δ + 1 − 1 σ2 2 , δ 2 (rXtdt uS (t )XtdWt) ( 2 )( uS (t )Xt dt) Xt 2 Xt 1 = rdt + σu (t, δ)dW − (σ2u2 (t, δ))dt S t 2 S 1 = (r − σ2u2 (t, δ))dt + σu (t, δ)dW 2 S S t

5 CHAPTER 2. THE DYNAMIC PORTFOLIO

Integrating gives us: ∫ ∫ ∫ T T 1 T dZ = (r − σ2u2 (h, δ))dh + σu (h, δ)dW h 2 S S h t t ∫ t ∫ T T = + − 1 σ2 2 , δ + σ , δ ZT Zt (r uS (h ))dh uS (h )dWh t 2 t

ZT Substituting using XT = e : ∫ ∫ T T = − 1 σ2 2 , δ + σ , δ XT Xtexp( (r uS (h ))dh uS (h )dWh) t 2 t

It is assumed that the diﬀusion term is constant (σuS ), thus making the SDE a Geometric Brownian Motion:

dX(t) = rX(t)dt + σuS X(t)dW(t)

Moreover, with the Wiener process, it is known that the random variable U belongs to a Gaussian distribution. This means that the stochastic variable XT will, although the linear transformation of U, be log-normally distributed:

= − 1 σ2 2 − + σ − XT Xtexp((r uS )(T t) uS (|W T {z W }t)) 2 (2.9) | {z U } L

∈ − 1 σ2 2 − σ2 2 − where L N((r uS )(T t), uS (T t)). Knowing the probability density function of L and ultimately | 2 {z } | {z } µ δ2 Q XT, enables us to evaluate the expected value E [Φ(X(T))|Ft] for a payoﬀ function Φ of European option style. Moving over to the nice deterministic diﬀusion term (σuS (t)) will not change the distribution of XT i.e it will still be a log-normally distributed variable: ∫ ∫ T T = − 1 σ2 2 + σ XT Xtexp( (r uS (h))dh uS (h)dWh) | t 2 {z t } (2.10) Y ∫ ∫ T T Y∈ N r − 1 σ2u2 h dh, σ2u2 h dh where ( t ( 2 S ( )) t S ( ) ). The expected value can be calculated once again purely by analytical means. So under the assumptions of a constant (uS ) or a deterministic (uS (t)) relative weight, an option price can be calculated just by using an analytical approach.

Finally, looking at the stochastic relative weight (uS (t, δ)), it is seen that it is deﬁned for all t but not continuous at the reallocation dates (recall Figure 2.1): ∫ ∫ T T = − 1 σ2 2 , δ + σ , δ XT Xtexp( (r uS (h ))dh uS (h )dWh) | t 2 {z t } (2.11) Z

It cannot be said with certainty what the distribution for XT looks like, since it is a function of a stochastic variable Z with an unknown distribution. That is, due to the relative weights the stochastic variable Z is not normally distributed. However, this is not equivalent to say that there does not exist a distribution, it does but what it looks like cannot be foretold. Thus, the use of numerics to generate an approximative solution is required.

2.4 The annualized realized historical volatility and the discretized portfolio

The adjustment of the reallocation is contingent on the value of the annualized realized historical volatility of the risky asset S. Practically, the volatility is calculated by annualizing the realized historical volatility of the

6 2.5. A VOLATILITY CAP TARGET LEVEL AND THE CORRESPONDING ALLOCATION TABLE risky index using the past 15 business days, under the assumption of 256 business days a year. We consider the mathematical deﬁnition for realized historical volatility [26] and let the discretized version of the realized historical δ volatility be deﬁned asσ ˜ (tK ). Together with the yield of the risky asset r(th), we have:

( S (th) ) r(th) = ln , h = 1, 2..., 255, S (t0), ..., S (t255) S (th−1) ∑K 1 r¯(t ) = r(t ), K = 15, 30, ..., 255 K 15 i (2.12) i=K−14 tv ∑K 256 σ˜ (t ) = (r(t ) − r¯(t ))2 K 14 i K i=K−14

where tK is the re-balancing date for K=15,30,...255 when maturity is 1 year. More specifically, the variable r(th) is the yield of the risky asset, calculated as the log of the risky asset, between two consecutive days th and th−1. Using the yield and the geometric average of 15 returnsr ¯(tK ) leads to a discretized annualized realized historical volatilityσ ˜ (tK ). Thus, the discretized version of δ will beσ ˜ (tK ) abbreviated asσ ˜ K . We will assume that for K=15 (the first fifteen days), the relative weights (uS (t0, σ˜ 15), ..., uS (t14, σ˜ 15)) all equal one and hence the relative weight for the non-risky asset will equal zero. The consecutive relative weights will be assigned to values in accordance with a predefined allocation table.

We recall Eq. (2.8) where the portfolio X satisﬁed the following SDE: dS (t) dB(t) dX(t) = u (t, δ)X(t) + u (t, δ)X(t) (2.13) S S (t) B B(t) To numerically implement this equation, we have to move from continous time to discrete time. Hence, the time notation th where ∆th = th − th−1 and time partition t0 = 0 < tt < ... < tn = T will be used. This allows us to discretize the SDE as:

S (th) − S (th−1) B(th) − B(th−1) X(th) − X(th−1) = uS (th−1, σ˜ K )X(th−1) + uB(th−1, σ˜ K )X(th−1) S (th−1) B(th−1) ( ) S (th) − S (th−1) B(th) − B(th−1) X(th) = X(th−1) 1 + uS (th−1, σ˜ K ) + uB(th−1, σ˜ K ) S (th−1) B(th−1)

Given that the maturity is 1 year, we could express the discretized version as:

∏255 ( ) S (th) − S (th−1) B(th) − B(th−1) X(t255) = X(t0) 1 + uS (th−1, σ˜ K ) + uB(th−1, σ˜ K ) S (t − ) B(t − ) h=1 h 1 h 1 where K = 15, ..., 255, i = K − 14, ..., K and h = 1, 2..., 255,

uS (th−1, σ˜ K ) + uB(th−1, σ˜ K ) = 1 ∀h, K

f or K = 15, uS (t0, σ˜ 15), ..., uS (t14, σ˜ 15) = 1

2.5 A volatility cap target level and the corresponding allocation table

The next question is; What particular values will the relative weights uS (th, σ˜ k) and uB(th, σ˜ k) be assigned to? The allocation table is created conditional upon the VCTL. One condition that the VCTL must meet, when using back testing results is that the minimum ratio, between the VCTL divided by the average realized volatility of the fund/index, should be around 1.25-1.30 [2]: If n numbers are given with each number deﬁned as ARHV j then it can be written as ARHV1, ARHV2, ..., ARHVn. The average realized volatility is given by the arithmetic mean:

∑n 1 ARHV[ = ARHV (2.14) n j j=1

7 CHAPTER 2. THE DYNAMIC PORTFOLIO

Then, if the desired VCTL is divided by the constant arithmetic mean, it should meet the following requirement: VCTL ≈ 1.25 − 1.30 (2.15) ARHV[ It has been proposed to use a VCTL of 15%, with the following volatility ranges:

Volatility range Allocation VCTL uS (th, σ˜ k) uB(th, σ˜ k) 0.0-15.0% 100% 0% 15% >15.0-20.0% 75% 25% 15% >20.0-25.0% 60% 40% 15% >25.0-37.5% 40% 60% 15% >37.5-50.0% 30% 70% 15% >50.0-75.0% 20% 80% 15% >75.0% 5% 95% 15%

Given the above allocation table, we can also formalize the indicator variables as:

(Ω, F, Q), Y ∈ F

IY : Ω −→ R, is deﬁned by

IY (σ ˜ k) = 1

ifσ ˜ k ∈ Y, otherwise

IY (σ ˜ k) = 0

Using the subsets A=[0,0.15), B=[0.15,0.20), C=[0.20,0.25), D=[0.25,0.375),E=[0.375,0.50), F =[0.50,0.75) and G =[0.75,∞) enable us to write down the relative weights as:

us(th−1, σ˜ k) = IA(σ ˜ k) + 0.75IB(σ ˜ k) + 0.60IC(σ ˜ k) + 0.40ID(σ ˜ k) + 0.30IE(σ ˜ k) + 0.20IF (σ ˜ k) + 0.05IG(σ ˜ k)

uB(th−1, σ˜ k) = 0 + 0.25IB(σ ˜ k) + 0.40IC(σ ˜ k) + 0.60ID(σ ˜ k) + 0.70IE(σ ˜ k) + 0.80IF (σ ˜ k) + 0.95IG(σ ˜ k) (2.16)

The high-end of every volatility range will never exceed the VCTL in the calculations. However, it is noted that the last volatility (>75%) could theoretically exceed this value, but since realized volatility of this magnitude is extremely rare, it is allowed. The reallocation process is repeated until the date of maturity. As can be seen, the condition uS (th, σ˜ k) + uB(th, σ˜ k) = 1 is always valid. To further clarify the idea with variable allocation, an example is given below:

Example: Calculating the value 24% for the variableσ ˜ (tK ), corresponds to the relative weights uS (th, σ˜ k)=60% and uB(th, σ˜ k)=40% in the volatility table. Since the volatility of the bond is zero, the annualized realized portfolio volatility equals:

0.24 × 60% + 0 × 40% = 14.4% < VCTL = 15%.

8 Chapter 3

Using Fourier transform for option pricing where the risky asset satisfies different affine jump-diffusions

3.1 Introduction

To obtain the risk-neutral parameters, both market prices of vanilla options and model option prices are needed. In case the BS-model is given, the model prices can easily be obtained by using the Black-Scholes formula. For more advanced models such as aﬃne jump-diﬀusions, semi-analytical model prices can be obtained. It has been shown that by using the Fourier transform, the Characteristic formula and Feynman-Kaˇc, an explicit expression can be achieved. In this chapter we follow Sepp [7][22][23] and Kou [24].

3.2 The general and transformed price dynamic for the risky asset S

To solve the pricing problem for European-style contingent claims, Sepp suggests a general SDE to start from. By doing so, several option-pricing formulas can be obtained by simplifying the general option-pricing formula. The SVSJ-model needs some special consideration, but the corresponding option-price formula is derived in a similar manner [22]. The general dynamics which governs the risky asset S are given by: √ dS (t) = (r − d − λ(t)m)S (t)dt + V(t)S (t)dW s(t) + (eJ − 1)S (t)dN(t), S (0) = S √ dV(t) = κ(θ − V(t))dt + ε V(t)dWv(t) + JvdNv(t), V(0) = V √ (3.1) λ = κ θ − λ + ε λ λ , λ = λ d (t) λ( λ (t))dt |λ ({zt)dW ( t}) (0) 0 The last intensity-term can be set to zero, since in this thesis affine jump-diffusion with stochastic jump intensity will not be used. To facilitate, a variable change from S(t) to Xe(t)=ln S(t) will be made. Then the the Itô formula is applied to Xe(t) directly under the martingale measure Q:

V(t) √ dXe(t) = (r − d − λ(t)m − )dt + V(t)dW s(t) + JdN(t), X(0) = X √2 (3.2) dV(t) = κ(θ − V(t))dt + ε V(t)dWv(t) + JvdNv(t), V(0) = V dλ(t) = κλ(θλ − λ(t))dt, λ(0) = λ

The parameters for the log stock price dynamics are: r=constant risk-free interest rate, d=constant continuous dividend yield, assumed to be zero in the models, W s(t) and Wv(t) are correlated Wiener processes with constant correlation ρ, N s(t)=Poisson process with intensity λ(t), J=random jump size in the logarithm∫ of the asset price with probability density function(PDF)ω ¯ (J) and the average jump amplitude m = (eJ − 1)ω ¯ (J)dJ = EQ(eJ ) − 1.

For the variance dynamics, the parameters are as follows: κ=constant mean-reverting rate, θ=constant long-term variance, ε=constant volatility of volatility, Nv(t)=Poisson

9 CHAPTER 3. USING FOURIER TRANSFORM FOR OPTION PRICING WHERE THE RISKY ASSET SATISFIES DIFFERENT AFFINE JUMP-DIFFUSIONS process with constant intensity λv, Jv=random jump size in variance with PDF ψ¯(Jv).

The variables for the jump intensity dynamics are given as: κλ=constant mean-reverting rate, θλ=constant long-term intensity.

For the SVSJ-model, the jump term i asset price (eJ − 1)S (t)dN(t) is replaced with (eJs − 1)S (t)dN s(t). Moreover, the Poisson process Nv(t) is zero for all the aﬃne jump-diﬀusion models, except for the SVSJ- model where Nv(t) and the N s(t) are i.i.d.

3.2.1 The option price formula using the inverse and forward Fourier transform The payoﬀ function for a European call (φ = +1) or a put(φ = +1) is normally deﬁned as max{φ[S − K, 0]}. e Given the variable change from S to Xe(t)=ln S(t) we instead have max{φ[eX − K, 0]}. Then the risk-neutral valuation formula suggests that the value of a European call or put option, can be written as follows: [ ] −r(T−t) Q X˜(T) F(t, x˜(t)) = e E max[φ(e − K, 0)|Ft] [ ] 1 + φ 1 − φ = e−r(T−t) EQ[eX˜(T)|F ] + EQ[K|F ] − EQ[min(eX˜(T), K)|F ] 2 | {z }t 2 t t eX˜(T) Q |F = x˜(t) (3.3) rewrite asE [ − − t] e |e ( r {zd)(T }t) mg 1 + φ 1 − φ = ex˜(t)−(T−t)d + e−r(T−t)K − e−r(T−t)EQ[min(eX˜(T), K)|F ] 2 2 | {z }t f (x ˜,V,λ,t) To proceed with the derivation of the last term f (x ˜, V, λ, t), a change of variables, from t to τ = T − t is made. Then we have:

−rτ Q X˜(τ+t) f (x ˜, V, λ, τ) = e E [min(e , K)|Ft] (3.4) If we let the variable τ equal zero then we have an initial condition:

−r∗0 Q X˜(0+t) Q X˜(t) x˜ f (x ˜, V, λ, 0) = e E [min(e , K)|Ft] = E [min(e , K)|Ft] = min(e , K) (3.5) Then Feyman-Kac tells us that the option value function f (x ˜, V, λ, τ) Eq. (3.4), is the solution to a certain partial integro diﬀerential equation(PIDE) with initial condition min(ex˜, K) Eq. (3.5) , where the stochastic process X˜ evolves according to Eq. (3.2). To take care of the last term in Eq. (3.3) we once again look at the general risk neutral pricing formula ∫ +∞ −r(T−t) Q |F = 1 izi −izx˜ ˆ e E [ f (X˜(T)) t]. This together with the inverse Fourier transform f (x ˜) π e f (z)dz and the 2 izi−∞ Q izX˜(T) characteristic function of X˜(T), ϕT (z) = E [e ], enables us to derive the Characteristic formula as follows: [ ∫ +∞ ] [ ∫ +∞ ] 1 izi e−r(T−t) izi f (t, x˜) = EQ[e−r(T−t) f (X˜(T))] = e−r(T−t)EQ e−izX˜(T) fˆ(z)dz = EQ[e−izX˜(T)] fˆ(z)dz 2π −∞ 2π −∞ [ ∫ ] izi izi −r(T−t) izi+∞ = e ϕ − ˆ π T ( z) f (z)dz 2 izi−∞ (3.6)

Moreover, we need to apply the forward transform to the initial condition min(ex˜, K) Eq. (3.5): ∫ ∞ Kiz+1 eizlnK K fˆ z = eizx˜min ex˜, K dx = ... = = (3.7) ( ) ( ) 2 2 −∞ z − iz z − iz [ ] − − ∫ +∞ e r(T t) izi ϕ − ˆ We recall the ﬁnal formula π T ( z) f (z)dz , the connection between the characteristic function and 2 izi−∞ moment generating function ϕ(−k) = G(−ik) together with the transformed initial condition fˆ(z) Eq. (3.7) .

10 3.2. THE GENERAL AND TRANSFORMED PRICE DYNAMIC FOR THE RISKY ASSET S

This enables us to express the option value function as: ∫ +∞ [ ] Ke−rτ izi G(−iz, x˜, V, λ, τ) f (x ˜, V, λ, τ) = eizlnK dz (3.8) π 2 − 2 izi−∞ z iz Using the substitution z=k+i/2, k ∈ R together with the integrand as a symmetric function ﬁnally gives us:

∫ [ − − + / ] Ke−rτ ∞ e ( ik 1 2)lnKG(−ik + 1 , x˜, V, λ, τ) f (x ˜, V, λ, τ) = R 2 dk (3.9) π 2 + 1 0 k 4 Φ = − + 1 Φ, , , λ, τ If we substitute ik 2 we have G( x˜ V ). Next, we consider the moment generating function G(Φ, x˜, V, λ, τ) of the log of terminal asset price X˜(τ) = lnS (τ):

G(Φ, x˜, V, λ, τ) = EQ[eΦx˜(τ)] = e−rτEQ[erτeΦx˜(τ)]

If we let τ = 0, then we have the initial condition G(Φ, x˜, V, λ, 0) = e−r∗0EQ[er∗0eΦx˜(0)] = eΦx˜. Under the jump-diﬀusion Eq. (3.2), the Feyman-Kac theorem implies that G(Φ, x˜, V, λ, τ) solves the following PIDE:

1 1 1 2 −Gτ + (r − d − V − λm)G + VG + κ(θ − V)G + ε VG + ρεVG 2 x˜ 2 x˜x˜ V 2 VV xV˜ ∫ ∞ 1 2 (3.10) +κλ(θλ − λ)Gλ + ελλGλλ + λ [G(x ˜ + J) − G]ω ¯ (J)dJ = 0 2 −∞ G(Φ, x˜, V, λ, 0) = eΦx˜ It can be shown that this PIDE Eq. (3.10) can be solved using the method of indetermined coefficients by using G = eA(τ)+B(τ)+C(τ)λ. The solution has an affine-form which entails that the models are referred to as affine jump-diffusions. The full solution to the PIDE above is given by:

G(Φ, x˜, V, λ, τ) = ex˜Φ+(r−d)τΦ+A(Φ,τ)+B(Φ,τ)V+C(Φ,τ)+D(Φ,τ)λ where the variables A(Φ, τ), B(Φ, τ), C(Φ, τ) and D(Φ, τ) have different expressions depending on the affine jump-diffusion in question. The SVSJ-model requires certain care with regard to the moment generating function, but the PIDE is derived in a similar manner as above. Hence, the solution to that PIDE, where X˜ evolves according to the SVSJ-model is given by:

G(Φ, x˜, V, λ, τ) = ex˜Φ+(r−d)τΦ+A(Φ,τ)+B(Φ,τ)V+∆(Φ,τ) where the explicit expressions for A(Φ, τ),B(Φ, τ) and ∆(Φ, τ) will be seen in Chapter 3.2.5. , , , λ, τ = −(−ik+1/2)lnK − + 1 , , , λ, τ We proceed and simplify Eq. (3.9) by setting Q(k x˜ V ) e G( ik 2 x˜ V ) which ﬁnally yields a general option formula: ∫ [ ] −rτ ∞ 1 + φ − − 1 − φ − − Ke Q(k, x˜, V, λ, τ) F(t, x˜) = ex˜(t) (T t)d + e r(T t)K − R dk (3.11) 2 2 π 2 + 1 0 k 4 Hence by using Fourier transforms and the characteristic function, we have obtained a general option formula Φ = − + 1 which only contains a one-dimensional integration along the real axis. By substituting ik 2 and A(Φ, τ), B(Φ, τ), C(Φ, τ), ∆(Φ, τ) and/or D(Φ, τ)λ, corresponding to a certain SDE, we will obtain the model option prices for that particular SDE. The integral can easily be evaluated using one of the built-in functions in Matlab, either the Gauss-Lobatto- or Gauss-Konrad-quadrature. In the subsequent sections, we will state the explicit expressions for Q(k, x˜, V, λ, τ), corresponding to the following diﬀusions: The Black-Scholes-, the Heston-, the BatesLN-, the BatesLDE-, the SVSJ- and the Sepp- model.

3.2.2 The characteristic formula: Explicit expressions corresponding to the Black-Scholes model The Black-Scholes model with the Q-dynamic below is the simplest of the models and assumes that the price of the underlying asset follows a geometric Brownian motion with constant drift and diﬀusion parameters:

dS (t) = (r − d)S (t)dt + σS (t)dW s(t), S (0) = S (3.12)

11 CHAPTER 3. USING FOURIER TRANSFORM FOR OPTION PRICING WHERE THE RISKY ASSET SATISFIES DIFFERENT AFFINE JUMP-DIFFUSIONS

The explicit expressions for Q(k, x˜, V, λ, τ) = e(−ik+1/2)X˜+A(k,τ)+B(k,τ)V+C(k,τ)+D(k,τ)λ under the BS-dynamic are:

A(k, τ) ≡ 0, B(k, τ) = −(k + 1/4)(T − t)/2

C(k, τ) ≡ 0, D(k, τ) ≡ 0 with X˜ = ln(S/K) + (r − d)τ.

3.2.3 The characteristic formula: Explicit expressions corresponding to the Heston model The main drawback with the BS-model is that it assumes that the volatility is constant, which contradicts empirical evidence. The Heston model [12] assumes that the volatility is stochastic, a feature which reproduces the implied volatility surface much better, especially for longer maturities [9]. The Q-dynamics are given by: √ dS (t) = (r − d)S (t)dt + V(t)S (t)dW s(t), S (0) = S √ (3.13) dV(t) = κ(θ − V(t))dt + ε V(t)dWv(t), V(0) = V

The explicit expressions for Q(k, x˜, V, λ, τ) = e(−ik+1/2)X˜+A(k,τ)+B(k,τ)V+C(k,τ)+D(k,τ)λ corresponding to the Heston- dynamics are: [ ] ( −ζ − ) −ζ − κθ ψ−+ψ+e (T t) 2 1−e (T t) A(k, τ) = − ψ+(T − t) + 2ln , B(k, τ) = −(k + 1/4) −ζ − ε2 2ζ ψ−+ψ+e (T t) √ ψ + = + ρε + ζ, ζ = 2ε2 − ρ2 + ρε + 2 + ε2/ , = κ − ρε/ + (u ik ) (k (1 ) 2ik u u 4 u 2

C(k, τ) ≡ 0, D(k, τ) ≡ 0 with X˜ = ln(S/K) + (r − d)τ.

3.2.4 The characteristic formula: Explicit expressions corresponding to the Bates model Although the Heston-model produces a much better fit than the simple BS-model, it is difficult for it to handle short-term smiles properly. Hence, supported by empirical studies, a stochastic volatility model with the addition of jumps in the asset price, would be an even better fit since it produces more skew for shorter maturities [6][7] [9] . The Bates model has the following Q-dynamics: √ dS (t) = (r − d − λm)S (t)dt + V(t)S (t)dW s(t) + (eJ − 1)S (t)dN(t), S (0) = S √ (3.14) dV(t) = κ(θ − V(t))dt + ε V(t)dWv(t), V(0) = V The jumps can be drawn from a normal distribution[19], with the following probability density function:

2 1 − (J−ν) ω¯ (J) = √ e 2δ2 2πδ2 where λ=constant jump intensity, ν=jump mean and δ=jump variance. The jumps can also be drawn from an asymmetric double exponential distribution, which provides an even better ﬁt [24]:

−η1 J η2 J ω¯ (J) = pη1e 1{J≥0} + qη2e 1{J<0}, η1 > 1, η2 > 0 where λ=constant jump intensity, 1 is the mean of positive jumps, 1 =mean of negative jumps, p=probability η1 η2 of positive jumps, and q=1-p=probability of negative jumps. Given the normal jump size distribution, the ν+ 1 δ2 average jump amplitude is m=e 2 − 1. Given the Double Exponential jump size distribution, the average η η jump amplitude m is equal to p 1 + q 2 − 1. η1−1 η2+1 Hence, the explicit expressions for Q(k, x˜, V, λ, τ) = e(−ik+1/2)X˜+A(k,τ)+B(k,τ)V+C(k,τ)+D(k,τ)λ corresponding to the Bates-dynamics are:

12 3.2. THE GENERAL AND TRANSFORMED PRICE DYNAMIC FOR THE RISKY ASSET S

[ ] ( −ζτ ) −ζτ κθ ψ−+ψ+e 2 1−e A(k, τ) = − ψ+τ + 2ln , B(k, τ) = −(k + 1/4) −ζτ ε2 2ζ ψ−+ψ+e √ ψ + = + ρε + ζ, ζ = 2ε2 − ρ2 + ρε + 2 + ε2/ , = κ − ρε/ + (u ik ) (k (1 ) 2ik u u 4 u 2

C(k, τ) ≡ 0, D(k, τ) = τΛ(k)

Given that the jumps are taken from the normal distribution then:

Λ(k) = e−ik(ν+δ2/2)−(k2−1/4)δ2/2+ν/2 − 1 − (−ik + 1/2)(eν+δ2/2 − 1) or if the jumps are taken from the double-exponential distribution then:

Λ(k) = p + q − 1 − (−ik + 1/2)( p + q − 1) 1−(−ik+1/2)ηu 1+(−ik+1/2)ηd 1−ηu 1+ηd

1 1 where X˜ = ln(S/K) + (r − d)(T − t) and = η1, = η2. ηu ηd Depending on which jump size distribution we choose, normal or asymmetric double-exponential, we sep- arate the models simply by denoting them as the BatesLN-model and BatesLDE-model, respectively.

3.2.5 The characteristic formula: Explicit expressions corresponding to the SVSJ model

In general, there are four types of a stochastic volatility jump (SVJ) models:

1) Jumps in the asset price (e.g the Bates model). 2) Jumps in the variance coming from an exponential distribution. 3) Jumps in the the asset price and independent jumps in the variance. 4) Correlated jumps in the price and variance.

In addition to the Bates model, the last alternative, correlated jumps in the price and variance, also denoted as the Stochastic Volatility Simultaneous Jumps (SVSJ)-model is investigated [7]. Jumps in the volatility significantly affect the skew and moreover, when a big jump occurs in the asset price it often entails a big jump in volatility [9] [17]. According to several empirical studies, the model "provides a remarkable fit to observed volatility surfaces" [7][8]. As mentioned earlier, due to the jump term in asset price, we cannot obtain this model by simplifying the general SDE Eq. (3.1). Hence, the Q-dynamics of the SVSJ-model must be stated explicitly:

√ s dS (t) = (r − d − λm)S (t)dt + V(t)S (t)dW s(t) + (eJ − 1)S (t)dN s(t), S (0) = S √ (3.15) dV(t) = κ(θ − V(t))dt + ε V(t)dWv(t) + JvdNv(t), V(0) = V

The marginal distribution of the jump size in the variance is given by:

Jv ∼ exp( 1 ) η j

v where η j is the volatility jump mean and conditional on a realization of J . The jump variable in the asset price Js is normally distributed and given by:

s v v 2 J |J ∼ N(ν + ρJ J , δ ) where ν=jump mean, ρ j=volatility jump correlation and δ=jump volatility.

We recall the moment generating function(MGF) corresponding to the SVSJ-process, G(Φ, x˜, V, λ, τ) = eΦx˜+(r−d)τΦ+A(Φ,τ)+B(Φ,τ)V+Λ(Φ,τ) Φ − + 1 and also substitute the variable with ik 2 . Thus, the parameters in the MGF, corresponding to the SVSJ-

13 CHAPTER 3. USING FOURIER TRANSFORM FOR OPTION PRICING WHERE THE RISKY ASSET SATISFIES DIFFERENT AFFINE JUMP-DIFFUSIONS model, are as follows:

[ −ζτ ] κθ ψ− + ψ+e A(Φ, τ) = − ψ+τ + 2ln( ) ε2 2ζ [ ] − e−ζτ Φ, = − Φ − Φ2 1 B( t) ( ) −ζτ ψ− + ψ+e

    ( )   ν+ δ2   η [ ψ − η ]   e 2   νΦ+ 1 δ2Φ2 ψ− 2 jU +L jU −ζτ ∆ Φ, τ = λ −Φ  −  −  τ + λe 2 τ − − − e ( )   1 1 2 2 ln 1 (1 ) 1 − ρ jη j ψ−L + η jU (ζL) − (ML + η jU) 2ζL where U = Φ − Φ2, M = κ − ρεΦ,

L = 1 − ρ jη jΦ ψ + = + (κ − ρεΦ) + ζ √ ζ = (κ − ρεΦ)2 + ε2(Φ − Φ2)

3.2.6 The characteristic formula: Explicit expressions corresponding to the SEPP model The model that provides the best fit to the implied volatility surface [22] is an affine jump-diffusion model with stochastic volatility, deterministic jump rate intensity where jumps are taken from the asymmetric double exponential distribution. It is simply denoted as the Sepp-model where the Q-dynamics are given by: √ dS (t) = (r − d − λ(t)m)S (t)dt + V(t)S (t)dW s(t) + (eJ − 1)S (t)dN(t), S (0) = S √ dV(t) = κ(θ − V(t))dt + ε V(t)dWv(t) V(0) = V (3.16) dλ(t) = κλ(θλ − λ(t))dt λ(0) = λ

As for the BatesLDE-model, the jumps are taken from the asymmetric double exponential distribution:

−η1 J η2 J ω¯ (J) = pη1e 1{J≥0} + qη2e 1{J<0}, η1 > 1, η2 > 0

Hence, the explicit expressions for Q(k, x˜, V, λ, τ) = e(−ik+1/2)X˜+A(k,τ)+B(k,τ)V+C(k,τ)+D(k,τ)λ under the Sepp-dynamics are: [ ] ( −ζτ ) −ζτ κθ ψ−+ψ+e 2 1−e A(k, τ) = − ψ+τ + 2ln , B(k, τ) = −(k + 1/4) −ζτ ε2 2ζ ψ−+ψ+e √ ψ + = + ρε + ζ, ζ = 2ε2 − ρ2 + ρε + 2 + ε2/ , = κ − ρε/ + (u ik ) (k (1 ) 2ik u u 4 u 2

−κ τ θλΛ(k) −κ τ Λ(k) , τ = κλτ − + λ , , τ = − λ C(k ) ( 1 e ) κλ D(k ) (1 e )( κλ ) Λ(k) = p + q − 1 − (−ik + 1/2)( p + q − 1) 1−(−ik+1/2)ηu 1+(−ik+1/2)ηd 1−ηu 1+ηd with X˜ = ln(S/K) + (r − d)τ

14 Chapter 4

Model calibration

4.1 Introduction

The main reason for creating the VCTL-option is to offer an opportunity to invest in an emerging market. But for some emerging markets, there are no plain vanilla options due to the problem to adequately cover the volatility risk i.e the extra charge for taking the volatility risk is too high. But this reasoning puts us in an awk- ward position when it comes to pricing the VCTL-option for that particular market. To price the VCTL-option consistently with no arbitrage, it is required to use risk-neutral parameters. That is, parameters corresponding to some martingale measure, which is a Catch 22, if we do not have market prices of any options. Hence, the financial institutions are theoretically exposed to the risk of creating arbitrage possibilities when pricing the VCTL-option, if they do not use the correct market price of risk. One way to overcome this problem is to look at a similar market which in fact has traded options and market prices. The word "similar" refers to similar features of the financial market. However by doing so, we inevitability use a martingale measure, corresponding to a market price of risk for some other market.

There are in general two possibilities to estimate the required parameters. If we think the future will mimic the past, we can estimate parameters for a chosen risky asset process by using historical option data [6] [7] [8]. Or we can "back out" the parameters implied by the current market option prices together with the model prices for which formulas in the previous chapter were derived. The second approach is normally used by the market makers since they want to model their option consistently with the market’s martingale measure, but calibrating parameters using historical time series, may be of interest, if someone is searching for arbitrage possibilities [22]. The main concern of this thesis lies within the pricing issue, therefore the latter approach was chosen.

When the optimization problem is to be solved, we have to choose whether to use a local optimization solver or a global optimization method. However, to really ensure that we obtain an arbitrage-free option price, we have to ﬁnd the risk-neutral parameters corresponding to the global minimum i.e choose a global optimization solver. However, by choosing a global solver, the computation time may increase quite substantially if the computer capacity is low.

4.2 A multi-variate function and estimation procedure

We also have to choose an error metric but this is subordinated to the choice of the solver for the optimization problem. Fortunately, there exists a global optimization method which eﬃciently backs out the risk-neutral parameter estimates: A global solver which is based on the "DIRECT" algorithm where a numerical implementation can be found via the external company Tomopt and the Matlab toolbox Tomlab. The advantage with this "DIRECT" method is that it carries out simultaneous searches using all possible constants i.e operates on both the global and local level, whereas other global methods put higher emphasis on the global search which leads to slow convergence [13]:

It is guaranteed to converge to the global optimal function value, if the objective function f is continuous or at least continuous in the neighborhood of a global optimum.This could be guaranteed since, as the number of iterations goes to inﬁnity, the set of points sampled by DIRECT from a dense subset of the unit hypercube.

15 CHAPTER 4. MODEL CALIBRATION

[13].

However infinitely many iterations means an infinite time period, but practically a finite amount of iterations (approximately 60000) will do just fine. More formally, the "DIRECT" algorithm is able to solve unconstrained global optimization problems of the following type:

min f (x) x

s.t − ∞ < xL ≤ x ≤ xU < ∞

n where x,xL,xU ∈ R , f (x) ∈ R.

Given an error functional of the following type f(Θ), we are then interested in obtaining the optimized parameter vector Θ˜ for each model. However, we need to be careful since the model price equations involves integrands and exponentials so we need to set appropriate upper and lower bounds for the vector so the computations do not crash. It is also good if we can insert linear and/or nonlinear constraints to restrict the hypercube. Since we have stochastic volatility in all aﬃne jump-diﬀusions and want to assure the variance process V(t) is strictly positive, we will indeed implement a nonlinear condition: The Feller condition where 2κθ ≥ ε2.

4.2.1 Several error measures and the choice of error functional There are several possible error functionals or metrics and although all metrics constitute a multi-variate function, there are diﬀerences between them in terms of accuracy and time needed for computer calculations. Detlefsen and Härdle [16] review some of the most common error metrics. We omit to state these explicitly, but a thorough examination of the advantages and disadvantages between the error measures with respect to the diﬀerent SDEs could be a subject for further research.

In general, the market and the model quantities of the claims can be measured by a root mean squared error or just the squared error metric given by:

∑n 1 (QModel − QMarket)2 n i i i=1

Model ΠModel σModel Market where Qi refers to either, the model price i , or the implied model volatility i and Qi refers ΠMarket σMarket to either, the market price i , or the implied market volatility i . The subscript "i" runs over all "n" observations corresponding to the total number of market price data.

In this thesis a simple error functional, sometimes denoted as the Absolute-Price-error metric is chosen:

∑n ( )2 min f (Θ) = ΠModel(K, T, Θ) − ΠMarket(K, T) Θ i i i=1

ΠModel , , Θ Θ where i (K T ) refers to the model price of option i with strike K, maturity T and parameter vector . ΠMarket , i (K T) refers to the corresponding market price of option i with strike K and maturity T.

4.2.2 The market data Next in line is to choose an appropriate set of market price data. To ensure that the set of market price data fulfills the word "appropriate", we need to go through some definitions. Recall the payoff functions for Euro- pean calls and European puts, max[S(T)-K,0] and max[K-S(T),0 ]respectively. Normally the price of an option consists of two components, its intrinsic value and extrinsic value (or time value) i.e the extrinsic value is the difference between the option value and intrinsic value. More specifically, an option’s extrinsic value captures the price impact relating to variables other than the price of underlying security and strike price i.e the price impact due to time to expiration, interest rates, volatility and dividends(if applicable). So in order to gain as much information as possible, prices from out-of-the-money European call options are used. This ensures that the option’s entire premium consists of pure extrinsic value and no intrinsic value [18].

16 4.3. BOUNDARIES AND ESTIMATED PARAMETER VALUES

Consequently, the data set consists of market prices for 36 out-of-the-money European call option prices, where the DAX (The Deutscher Aktienindex) is the underlying asset. The DAX is a stock market index consisting of the 30 major German companies trading on the Frankfurt Stock Exchange. Considerations whether this financial market really has similar features with an emerging market is justified but since the main issue is to obtain the risk-neutral, it is disregarded. The market data is from 19/10-2012 using three maturities: 20, 45 and 105 business days. Hence, corresponding to different maturities, we could also assume different interest rates but for simplicity we have used an ad hoc interest rate of 3,5%, and also zero dividend yield.

4.3 Boundaries and estimated parameter values

As mentioned before, a global optimization solver based on the "Direct" algorithm will be used. The advantage with this solver besides the high possibility of finding the global minimum is that an initial start vector does not have to be declared. Instead we have to define the upper and lower bounds for the parameters. This has been done in accordance with the values in the Table 1:Estimated parameters and in-sample fit [22]. Thus, we have used the following boundaries:

V(0) κ θ ε ρ λ κλ θλ ν δ η1 η2 p(= 1 − q) η j ρ j

Lower boundary 0 0 0 0 -1 0 0 0 -1 0 0 0 0 0 -1

Upper boundary 1 5 1 1 1 1 1 1 1 1 20 10.5 1 1 1

We conclude by looking at the estimated parameters on the next page. There we see that the volatility parameter is almost the same for all i.e around 13%. But a word of caution is required for the SVSJ-model. In order to capture the risk-neutral property of the jump variable in variance, we would have to use Volatility Index(VIX)-options. However to simplify the calibration, only OTM European call option prices were used.

17 CHAPTER 4. MODEL CALIBRATION

Table 4.1. The risk neutral parameters Black Scholes Heston Bates LN Bates LDE SVSJ Sepp

V(0) 0.016950 0.018315 0.018805 0.019196 0.017248 0.019128

VOL % 13.019216 13.53329 13.71313 13.85496 13.13 13.8304

κ 1.351152 1.666662 1.419795 1.879414 1.424707

θ 0.043436 0.052127 0.086428 0.129630 0.086496

ε 0.269779 0.333336 0.333342 0.698039 0.333410

ρ -0.555561 -0.777780 -0.777795 -0.777829 -0.777930

λ 0.11110 0.111103 0.037012 0.111035

κλ 0.543591

θλ 0.085124

ελ

ν 0.0 -0.000013

δ 0.1111 0.055543

ηu 0.005555 0.005552

ηd 0.008209 0.008203 p(=1-q) 0.925917 0.925850

η j 0.000094

ρ j -0.066669

18 Chapter 5

Simulation of the transformed portfolio aﬃne jump-diﬀusions

5.1 Introduction

By using the Feynman Kaˇc formula, it was previously shown that a particular option value function solved a certain PIDE, where the stochastic process satisfied a cumbersome SDE. Hence, it would be desirable if we could do something similar i.e derive a similar partial integro differential equation where the portfolio X satisfied the portfolio affine jump-diffusion. Then we could obtain a semi-analytical VCTL-option price since the PIDE could be solved using the method of undetermined coefficients. But this is neither recommended nor feasible, since it was concluded in Chapter 2 that the distribution of the stochastic portfolio weight was unknown. Instead of a numerical implementation on any semi-analytical option pricing formula, we need to numerically implement an approximation of the portfolio diffusions and calculate the VCTL-option price via Monte Carlo simulations.

" A Monte Carlo calculation is typically of the following structure: carry out the same procedure many times, take into account all the individual results, and summarize them into an overall approximation to the problem in question".[15]

When it comes to the Black-Scholes model and European-style options, it is preferable to use the Black- Scholes formula to directly calculate the prices. But a simple Euler scheme could be applied to the BS-model and thereafter the option price using Monte Carlo is calculated. However when we encounter the affine jump- diffusion and stochastic volatility, another scheme called the full truncation Euler scheme will be applied. This will eliminate the probability of negative variance and also minimize the discretization bias [11] [21]. In order to run the simulations, the so called full truncation Euler scheme will be applied on the logarithm of the portfolio diffusion (under the different models). The portfolio affine jump-diffusions will look cumbersome, but it will actually facilitate our computer calculations later on, when the features of the portfolio’s distribution are analyzed. [ ] −r(T−t) Q We recall the arbitrage free price of the contingent claim, Πt = e E max[φ[X(T) − K, 0]|Ft] , where the Q-dynamics of the portfolio X are given in the following sections. The discretization will be in terms of the log-price Z(t) = log(X(t)) and since only the payoff function of European call option-type will be used, the price of the VCTL-option can be directly stated as:

−r(T−t) Q Z(T) Πt = e E [max(e − K, 0)|Ft]

Q Z(T) We can then estimate E [max(e − K, 0)|Ft] by using the following algorithm [10]: for i=1,...,m generate and set Zi(T) = .... −r(T−t) Zi(T) set Ci = e max(e − K, 0) set Cˆm = (C1 + ... + Cm)/m

19 CHAPTER 5. SIMULATION OF THE TRANSFORMED PORTFOLIO AFFINE JUMP-DIFFUSIONS

Where m is the number of simulations. It is important that a reasonable number is chosen in order to get an accurate result, yet it cannot be too big with regards to computer capacity and calculation time. There will always exist a discretization bias regardless of the number of simulations. To choose an optimal number of simulations, we will compare the implied volatility surface using analytical model prices and the implied volatility surface using Euler model prices, under the Sepp-model. Thus, to choose m = 105 simulations seems reasonable as can be seen, in Figure 5.1 below (the last picture).

0.4 1 0.4 (M,T) (M,T) (M,T) σ σ σ

0.2 0.5 0.2

0 0 0 1.5 1.5 1.5 Implied Volatility 4 Implied Volatility 4 Implied Volatility 4 1 2 1 2 1 2 Moneyness M=S/K 0.5 0 Time to maturity T Moneyness M=S/K 0.5 0 Time to maturity T Moneyness M=S/K 0.5 0 Time to maturity T

0.4 0.4 0.4 (M,T) (M,T) (M,T) σ σ σ

0.2 0.2 0.2

Figure 5.1. First ﬁgure, upper left, the implied volatility surface using analytical model prices. In the consecutive ﬁgures the simulations ranges from 101 up to 105

In the following sections, we go through the discretization process in detail by applying the simple Euler- and the full truncation Euler scheme. However when the transformed portfolio affine jump-diffusions is discretized, a special attention needs to be paid to the part containing the jumps [10]. We will use the variable 2 transformation Zt = logXt = f (Xt, t) together with fx = 1/x, fxx = −1/x and then Itô’s lemma as defined by Jondeau [20]. However we need to redo the final derivation so it contains the jump-term ct(Jt − 1) (and not ct(Jt + 1)). Thus, the final dynamic is given by: ( ) 1 2 dZ = a f + b f + f dt + b f dW + [ f (X − + c (J − 1), t) − f (X −, t−)]dN(t) (5.1) t t x 2 t xx t t x t t t t t

Where Wt, is a Brownian motion and Nt is a pure Poisson process, independent of Wt. Jt is the random variable corresponding to a jump realization.

5.2 Simple Euler discretization of the transformed portfolio dynamic: The risky asset evolves according to the Black-Scholes model

We recall from Chapter 2, the money market account dB(t) = rB(t)dt and the portfolio diffusion dX(t) = , δ dS (t) + , δ dB(t) ffi uS (t )X(t) S (t) uB(t )X(t) B(t) . Hence, the appearance of the portfolio will change when the a ne jump- diffusion for the risky asset S changes. We begin with the simplest model, the Black-Scholes model: √ dS (t) = (r − d)S (t)dt + σ2S (t)dW(t),

Together with the portfolio dynamic above, we derive and simplify the given formula as: √ dX(t) = (u (t, δ)(r − d) + u (t, δ)r))X(t)dt + u (t, δ) σ2X(t)dW(t) S B √ s = − , δ + , δ σ2 (|r u s ( {zt ) d ) X ( t}) dt |us ( t ){z X ( t}) dW(t)

at bt

20 5.3. FULL TRUNCATION EULER DISCRETIZATION OF THE TRANSFORMED PORTFOLIO DYNAMICS: THE RISKY ASSET EVOLVES ACCORDING TO THE HESTON MODEL

2 Recall the variable transformation, Z(t) = logX(t) = f (X(t)), fx = 1/x, fxx = −1/x and Itˆo’s lemma: ( 1 ) dZ = a f + b2 f + f dt + b f dW t t x 2 t xx t t x t ( 1 ) √ = r − u (t, δ)d − u2(t, δ)σ2 dt + u (t, δ) σ2dW(t) s 2 s s Finally, the discretized version of the portfolio dynamics using a simple Euler scheme is then given by: ( ) = + − , σ − 1 2 , σ σ2 ∆ + ... Z(th) Z(th−1) r us(th−1 ˜ K )d us (th−1 ˜ K ) th √ √ 2 (5.2) 2 S ... + us(th−1, σ˜ K ) σ ∆thW s With h=1,...n where ∆th = th − th−1, and the initial value Z(t0). W ∈ N(0, 1)

Trajectories of the portfolio X and the risky asset S under the BS−model 120

110

100

80 0 50 100 150 200 250 300 Number of days

Figure 5.2. Trajectories for X(black) and S(blue) under the BS-model using a simple Euler scheme. The initial value is 100 for both the portfolio and risky asset. VCTL=15%

5.3 Full truncation Euler discretization of the transformed portfolio dynamics: The risky asset evolves according to the Heston model

We proceed with a model which takes stochastic volatility into consideration, the Heston model: √ dS (t) = (r − d)S (t)dt + V(t)S (t)dW s(t), √ dV(t) = κ(θ − V(t))dt + ϵ V(t)dWv(t), The portfolio dynamics: dS (t) dB(t) dX(t) = u (t, δ)X(t) + u (t, δ)X(t) S S (t) B B(t)

√ dX(t) = (u (t, δ)(r − d) + u (t, δ)r)X(t)dt + u (t, δ)X(t) V(t)dWS (t) s b √s = − , δ + , δ S (|r u s ( {zt ) d ) X ( t}) dt |us ( t ) X{z(t) V ( t}) dW (t)

at bt 2 Recall the variable transformation, Z(t) = logX(t) = f (X(t)), fx = 1/x, fxx = −1/x and Itˆo’s lemma: ( 1 ) dZ = a f + b2 f + f dt + b f dW(t) t t x 2 t xx t t x ( 1 ) √ dZ(t) = r − u (t, δ)d − u2(t, δ)V(t) dt + u (t, δ) V(t)dW(t) s 2 s s the discretized version of the portfolio dynamics is then given by: ( ) = + − , σ − 1 2 , σ + ∆ + ... Z(th) Z(th−1) r us(th−1 ˜ K )d us (th−1 ˜ K )V(th−1) th √ √ 2 + S (5.3) ... + u (t − , σ˜ ) V(t − ) ∆t W s h 1 K h 1 h √ √ + + V(th) = V(th−1) + κ(θ − V(th−1) )∆th + ε V(th−1) ∆thX

21 CHAPTER 5. SIMULATION OF THE TRANSFORMED PORTFOLIO AFFINE JUMP-DIFFUSIONS

= ∆ = − + = , With h 1,...n where th th th−1 and V(th) max(V(th) 0). The initial√ values are Z(t0) and V(t0). If we use the Cholesky decomposition, we can substitute WS and WV with ρX + (1 − ρ2)Y and X respectively, where X ∈ N(0, 1) and Y ∈ N(0, 1) are independent random variables.

Trajectories of the portfolio X and the risky asset S under the Heston−model 105

100

85 0 50 100 150 200 250 300 Number of days

Figure 5.3. Trajectories for X(black) and S(blue) under the Heston-model using a full truncation Euler scheme. The initial value is 100 for both the portfolio and risky asset. VCTL=15%

5.4 Full truncation Euler discretization of the transformed portfolio dynamics: The risky asset evolves according to the BatesLN model

If we add a jump part to the Heston-model and assume that the jumps are distributed in a certain way, we obtain the BatesLN-model. Hence, the discretization procedure resembles the one for the Heston-model but as mentioned before, special consideration has to be made for the jump part. We will discretize the jump part using ﬁxed dates as described by Glasserman [10]: √ dS (t) = (r − d − λm)S (t−)dt + V(t)S (t−)dW s(t) + (eJ − 1)S (t−)dN(t), √ dV(t) = κ(θ − V(t))dt + ϵ V(t)dWv(t), 2 1 − ( j−ν) δ2 fJ( j) = √ e 2 2πδ2 The portfolio dynamics: dS (t) dB(t) dX(t) = u (t, δ)X(t) + u (t, δ)X(t) S S (t) B B(t)

√ S J dX(t) = (us(t, δ)(r − d − λm) + ub(t, δ)r)X(t)dt + us(t, δ)X(t) V(t)dW (t) + us(t−, δ)Xt−(e − 1)dN(t)

√ = − , δ + λ + , δ S + , δ J − |(r u s ( t )( {zd m )) X ( t}) dt |us ( t ) X{z(t) V ( t}) dW (t) |us ( {zt ) X }t(e 1)dN(t)

at bt ct 2 Recall the variable transformation, Z(t) = logX(t) = f (X(t)), fx = 1/x, fxx = −1/x and Itˆo’s lemma: ( ) 1 2 S J dZ = a f + b f + f dt + b f dW (t) + [ f (X − + c (e − 1), t) − f (X −, t−)]dN(t) t t x 2 t xx t t x t t t ( 1 ) √ dZ(t) = r − u (t, δ)(λm + d) − u2(t, δ)V(t) dt + u (t, δ) V(t)dWS (t) + ... s 2 s s J ... + [log(Xt− + us(t−, δ)Xt−(e − 1)) − log(Xt−)]dN(t) ( 1 ) √ = r − u (t, δ)(λm + d) − u2(t, δ)V(t) dt + u (t, δ) V(t)dWS (t) + ... s 2 s s J ... + log(1 + us(t−, δ)(e − 1))dN(t)

22 5.5. FULL TRUNCATION EULER DISCRETIZATION OF THE TRANSFORMED PORTFOLIO DYNAMICS: THE RISKY ASSET EVOLVES ACCORDING TO THE BATESLDE MODEL the discretized version of the portfolio dynamics is then given by: ( ) 1 2 + Z(t ) = Z(t − ) + r − u (t − , σ˜ )(λm + d) − u (t − , σ˜ )V(t − ) ∆t + ... h h 1 s h 1 K 2 s h 1 K h 1 h √ √ N∑(th) ... + , σ + ∆ S + + , σ Ji − us(th−1 ˜ K ) V(th−1) thW log(1 us(th−1 ˜ K )(e 1)) (5.4) i=N(t − )+1 √ h 1 √ + + V V(th) = V(th−1) + κ(θ − V(th−1) )∆th + ε V(th−1) ∆thW

= ∆ = − + = , With h 1,...n where th th ∫ th−1 and V(th) max(V(th) 0). The initial values are Z(t0) and V(t0). The J − = ν+ 1 δ2 − average jump amplitude equals [e 1] fJ( j)d j or more explicitly m e 2 1 for√ Normal distributed jumps [22]. Once again, we use the Cholesky decomposition and substitute WS = (ρX + (1 − ρ2)Y) and WV = X, where X ∈ N(0, 1) and Y ∈ N(0, 1) are independent. Since the jump J belongs to a Normal distribution, they are easily drawn from this distribution using a pre-deﬁned function in Matlab.

Trajectories of the portfolio X and the risky asset S under the BatesLN−model 110

100

70 0 50 100 150 200 250 300 Number of days

Figure 5.4. Trajectories for X(black) and S(blue) under the BatesLN-model using a full truncation Euler scheme. The initial value is 100 for both the portfolio and risky asset. VCTL=15%

5.5 Full truncation Euler discretization of the transformed portfolio dynamics: The risky asset evolves according to the BatesLDE model

Once again we add jumps to the Heston model but instead of assuming that the logarithms of the jumps are normally distributed, it is assumed that the jumps belongs to asymmetric double exponential distribution. The BatesLDE-model is given by: √ dS (t) = (r − d − λm)S (t−)dt + V(t)S (t−)dW s(t) + (eJ − 1)S (t−)dN(t), √ dV(t) = κ(θ − V(t))dt + ϵ V(t)dWv(t),

−η1 j η2 j fJ ( j) = p · η1e 1{ j≥0} + q · η2e 1{ j<0}, η1 > 1, η2 > 0, p, q ≥ 0, p + q = 1 Where the jump notation from the paper of Kou [24] has been used. If we insert the BatesLDE-model into the portfolio dynamic, we arrive at the same discretization scheme as that for the BatesLN-model. Hence, it is simpliﬁed by stating the scheme directly: ( ) 1 2 + Z(t ) = Z(t − ) + r − u (t − , σ˜ )(λm + d) − u (t − , σ˜ )V(t − ) ∆t + ... h h 1 s h 1 K 2 s h 1 K h 1 h √ √ N∑(th) ... + , σ + ∆ S + + , σ Ji − us(th−1 ˜ K ) V(th−1) thW log(1 us(th−1 ˜ K )(e 1)) (5.5) i=N(t − )+1 √ h 1 √ + + V V(th) = V(th−1) + κ(θ − V(th−1) )∆th + ε V(th−1) ∆thW

With∫ h=1,...n where∆th = th − th−1 and the initial values Z(t0) and V(t0). The average jump amplitude is J [e − 1] fJ( j)d j which equals

23 CHAPTER 5. SIMULATION OF THE TRANSFORMED PORTFOLIO AFFINE JUMP-DIFFUSIONS

( η η ) = · 2 + · 1 − m q η + p η − 1 for asymmetric double exponential jumps. We use the same Cholesky decomposition 2 1 1 1 √ as before i.e we substitute WS = (ρX + (1 − ρ2)Y) and WV = X with X ∈ N(0, 1) and Y ∈ N(0, 1).

For the BatesLN-model we could sample the jump variable by using a pre-deﬁned function in Matlab. However this cannot be done for the asymmetric double exponential distribution. Hence, to implement the jump distribution, the inverse transformation method [10] is used. It is a basic method for generating sample numbers at random from any probability distribution given its cumulative distribution function(CDF). Thus, we need to derive the inverse CDF (seen in Appendix A.1) and implement this into Matlab, before random numbers can be generated. Below, we present the resulting algorithm which will generate the random numbers:

1.Generate a random number u from standard uniform distribution i.e U(0,1). 2.Compute the value y such that F(y)=u or y = F−1(u). Where: −1 1 u 1 (1−u) F (u) = ln( )1{u

Trajectories of the portfolio X and the risky asset S under the BatesLDE−model 110

100

70 0 50 100 150 200 250 300 Number of days

Figure 5.5. Trajectories for X(black) and S(blue) under the BatesLDE-model using a full truncation Euler scheme. The initial value is 100 for both the portfolio and risky asset. VCTL=15%

5.6 Full truncation Euler discretization of the transformed portfolio dynamics: The risky asset evolves according to the SVSJ model

One model that "provides a remarkable ﬁt to observed volatility surfaces" is the SVSJ-model. It incorporates simultaneous correlated jumps in price and variance [7] [8]: √ s dS (t) = (r − d − λm)S (t−)dt + V(t)S (t−)dW s(t) + (eJ − 1)S (t−)dN s(t), √ dV(t) = κ(θ − V(t))dt + ϵ V(t)dWv(t) + JvdNv(t),

v ∼ 1 , s| v ∼ ν + ρ v, δ2 J exp( η ) J J N( j J )

The portfolio dynamic yields: √ S Js dX(t) = (u (t, δ)(r − d − λm) + u (t, δ)r)X(t)dt + u (t, δ)X(t) V(t)dW (t) + u (t−, δ)X −(e − 1)dN(t) s b √s s t = − , δ + λ + , δ S + , δ Js − (r us(t )(d m))X(t)dt us(t )X(t) V(t)dW (t) |us ( {zt ) X }t(e 1)dN(t)

2 Recall the variable transformation, Z(t)=log X(t)=f(X(t)), fx = 1/x, fxx = −1/x and Itˆo’s lemma. Hence, the resulting transformed portfolio dynamics are given by: ( 1 ) √ dZ(t) = r − u (t, δ)(λm + d) − u2(t, δ)V(t) dt + u (t, δ) V(t)dWS (t) + ... s 2 s s Js ... + [log(Xt− + us(t−, δ)Xt−(e − 1)) − log(Xt−)]dN(t) ( ) √ = − , δ λ + − 1 2 , δ + , δ S + + −, δ Js − r us(t )( m d) us (t )V(t) dt us(t ) V(t)dW (t) log(1 us(t )(e 1))dN(t) √ 2 dV(t) = κ(θ − V(t))dt + ϵ V(t)dWv(t) + JvdNv(t),

24 5.7. FULL TRUNCATION EULER DISCRETIZATION OF THE TRANSFORMED PORTFOLIO DYNAMICS: THE RISKY ASSET EVOLVES ACCORDING TO THE SEPP MODEL

If we apply the full truncation Euler scheme, we receive the following discretized portfolio dynamics:

( ) 1 2 + Z(t ) = Z(t − ) + r − u (t − , σ˜ )(λm + d) − u (t − , σ˜ )V(t − ) ∆t + ... h h 1 s h 1 K 2 s h 1 K h 1 h √ √ N∑(th) + S Js ... + u (t − , σ˜ ) V(t − ) ∆t W + log(1 + u (t − , σ˜ )(e i − 1)) s h 1 K h 1 h s h 1 K (5.6) i=N(th−1)+1 √ √ N∑(th) = + κ θ − + ∆ + ε + ∆ + v V(th) V(th−1) ( V(th−1) ) th V(th−1) thX Ji i=N(th−1)+1

= ∆ = − With∫h 1,...n where th th th−1 and the initial values Z(t0) and V(t0). The average jump amplitude is s v v 1 2 J |J ν+ρ j J + δ m = [e − 1] fJs|Jv ( j)d j which equals m = e 2 − 1. For simultaneously correlated jumps, we have ≡ v ≡ s ∈ λ · ∆ identically distributed variables√ i.e N N N Po( th). If we use the Cholesky decomposition, we substitute WS = (ρX + (1 − ρ2)Y) and WV = X where X ∈ N(0, 1) and Y ∈ N(0, 1).

Trajectories of the portfolio X and the risky asset S under the SVSJ−model 120

110

100

80 0 50 100 150 200 250 300 Number of days

Figure 5.6. Trajectories for X(black) and S(blue) under the SVSJ-model using a full truncation Euler scheme. The initial value is 100 for both the portfolio and risky asset. VCTL=15%

5.7 Full truncation Euler discretization of the transformed portfolio dynamics: The risky asset evolves according to the Sepp model

Last but not least is the Sepp-model which is very similiar to the BatesLDE-model but with the assumption that the jump intensity is deterministic and not constant. Sepp has showed that this model provides the closest ﬁt to the implied volatility surface [22].

√ dS (t) = (r − d − λ(t)m)S (t−)dt + V(t)S (t−)dW s(t) + (eJ − 1)S (t−)dN(t), √ dV(t) = κ(θ − V(t))dt + ϵ V(t)dWv(t), dλ(t) = κλ(θλ − λ(t))dt

Inserting this into the portfolio dynamic yields:

√ S dX(t) = (us(t, δ)(r − d − λ(t)m) + ub(t, δ)r)X(t)dt + us(t, δ)X(t) V(t)dW (t) + ... J u (t−, δ)X −(e − 1)dN(t) s t √ = − , δ − λ + , δ S + , δ J − |(r u s ( t )( d{z ( t ) m )) X ( t}) dt |us ( t ) X{z(t) V ( t}) dW (t) |us ( {zt ) X }t(e 1)dN(t)

at bt ct

25 CHAPTER 5. SIMULATION OF THE TRANSFORMED PORTFOLIO AFFINE JUMP-DIFFUSIONS

Together with the variable transformation and Itˆo’s lemma, the logarithm of portfolio that satisﬁes the following SDE is obtained: ( 1 ) √ dZ(t) = r − u (t, δ)(λ(t)m + d) − u2(t, δ)V(t) dt + u (t, δ) V(t)dWS (t) + ... s 2 s s J ... + [log(Xt− + us(t−, δ)Xt−(e − 1)) − log(Xt−)]dN(t) ( 1 ) √ = r − u (t, δ)(λ(t)m + d) − u2(t, δ)V(t) dt + u (t, δ) V(t)dWS (t) + ... s 2 s s J ... + log(1 + us(t−, δ)(e − 1))dN(t)

Thus, the discretized version of the portfolio dynamics is given by: ( ) 1 2 + Z(t ) = Z(t − ) + r − u (t − , σ˜ )(λ(t − )m + d) − u (t − , σ˜ )V(t − ) ∆t + ... h h 1 s h 1 K h 1 2 s h 1 K h 1 h √ √ N∑(th) + S Ji ... + us(th−1, σ˜ K ) V(th−1) ∆thW + log(1 + us(th−1, σ˜ K )(e − 1)) i=N(th−1)+1

√ √ + + V(th) = V(th−1) + κ(θ − V(th−1) )∆th + ε V(th−1) ∆thX

λ(th) = λ(th−1) + κλ(θλ − λ(th−1))∆th

With∫ h=1,...n where∆th = th − th−1 and the initial values Z(t0),V(t0) and λ(t0). The average jump amplitude is ( η η ) J − = · 2 + · 1 − [e 1] fJ ( j)d j which equals m q η + p η − 1 for asymmetric double exponential jumps. If we 2 1 1 1 √ use the Cholesky decomposition, we substitute WS = ρX + (1 − ρ2)Y and WV = X where X ∈ N(0, 1) and Y ∈ N(0, 1) are independent random variables. The jumps are generated at random using the same algorithm as that of the BatesLDE-model. Until now, the procedure has been identical to the BatesLDE-model. However one thing remains and that is, instead of the constant jump intensity λ, we now have a deterministic (non- negative) function of time λx(t). Hence, N(t) is called an inhomogeneous Poisson process. This means that increments over diﬀerent time intervals of equal length can have diﬀerent means [10].∫ The number of jumps in th the interval (t − , t ] is Poisson-distributed with mean Λ(t )-Λ(t − ) where Λ(t ) = λ (u)du. Thus, the mean h 1 h h h 1 h 0 x Λ(th) − Λ(th−1) is given by: ∫ th ( λ2 ) κ ( ) (u) th λ 2 2 κλ(θλ − λ(u))du = κλ θλu − = κλ(θλ∆th) − λ (th) − λ (th−1) . th−1 th−1 2 2 ( ( )) κλ 2 2 ∈ κλ θλ∆ − λ − λ − Hence, N Po ( th) 2 (th) (th 1) .

Trajectories of the portfolio X and the risky asset S under the Sepp−model 130

120

110

100

90 0 50 100 150 200 250 300 Number of days

Figure 5.7. Trajectories for X(black) and S(blue) under the Sepp-model using a full truncation Euler scheme. The initial value is 100 for both the portfolio and risky asset. VCTL=15%

26 Chapter 6

The distributional features of the portfolio

6.1 Introduction

In this chapter, we will investigate the distributional properties of the portfolio X under diﬀerent dynamics for the underlying risky asset S. Any distribution can be characterized by a number of moments: The mean, variance, skewness and kurtosis.

Knowing that the market has a 70% probability of going up and a 30% probability of going down may ap- pear helpful if you rely on normal distributions. However, if you were told that if the market goes up, it will go up 2% and if it goes down, it will go down 10%, then you could see the skewed returns and make a better informed decision. [25]

Knowledge about the moments of the portfolio will later on, facilitate the analysis of the characteristics of the VCTL-option. The different moments of the risky asset S depends only on the distributional properties of S but this is not the case for the portfolio X, since it also depends on the stochastic relative weights us(th, σ˜ k) and ub(th, σ˜ k) and their covariation with the asset S. We recall from Chapter 2 and the definition of the relative weights, that they are dependent upon the realized annualized historical volatilityσ ˜ k of the risky asset S, and the pre-defined VCTL. Where the realized volatilityσ ˜ k is in turn dependent upon the input volatility and finally, the variable VCTL, which depends upon the volatility scheme. Hence, to study the effects on the distribution of X, the initial volatility as well as the VCTL will be adjusted by creating six new volatility schemes. We will let the VCTL-values range from 5% to 25% in intervals of 2.5%. The number of simulations will be set to 105.

6.2 The mean of the portfolio X

The ﬁrst moment of the portfolio X at maturity, is given by:

Q Gx(t; T) = E [XT |Ft] where Q represents the martingale measure which implies that we have used the optimized parameters from Chapter 4. The variable Ft, represents all available information up to time t. However, instead of quoting the mean directly, the relative values will be quoted via the logarithm by using the compounded implied risk-neutral drift δ˜:

δ˜(T−t) Gx(t; T) = X(t)e

1 (G (t; T) ) δ˜ = log x T − t X(t)

6.2.1 The mean versus volatility Given the different dynamics of the underlying risky asset S and by quoting the drift δ˜, we illustrate how the mean depends on the volatility for each portfolio. For the Heston-model, we have the option to not only adjust the spot variance V0, but also the long term variance θ. For the affine jump-diffusions, we can also, in addition

27 CHAPTER 6. THE DISTRIBUTIONAL FEATURES OF THE PORTFOLIO to the variance variables, shift the diﬀerent jump parameters. But we will restrict ourselves√ to simultaneous upward adjustment for only the aforementioned variables. Hence, the initial volatility V0 will take on the following values:

√ Table 6.1. The range of the initial volatility, V0 for all models

1% 10% 15% 17.5% 20% 25% 27.5% 30%

The Feller condition imposes a lower limit on the long term variance. If we recall the risk-neutral parameters in Chapter 4, we note that for the SVSJ-model we have to choose θ > 0.13 in order for the condition to be fulfilled. For the remaining models, it is sufficient to have θ > 0.04. Thus, the different values assigned to the long-term variance θ are given as:

Table 6.2. The range of the long term variance, θ for the Heston-model, BatesLN-, BatesLDE-, Sepp- and the SVSJ-model

0.04 0.05 0.06 0.07 0.08 0.09 0.10 0.12

SVSJ 0.13 0.135 0.14 0.145 0.15 0.155 0.16 0.165

As a comparison, we have a graph showing what happens with the drift δ˜ for the risky asset S, when we increase the short term- and long term-variance as above. These figures can be seen in the Appendix (A.3) and as expected, the drift is just pending back and forth i.e the diffusion parameter does not affect the drift. Below are the figures of drift of the portfolios, where a completely different pattern emerges.

Drift of porftolio X under the BS−model Drift of porftolio X under the Heston−model 3.9 5.5

3.8 5 3.7 (%) (%) δ δ 3.6

Drift Drift 4.5 3.5

3.4 4 0 5 10 15 20 25 30 0 5 10 15 20 25 30 Volatility (%) Volatility (%)

Figure 6.1. The drift versus volatility under the BS-model and Heston. Time to maturity 1 year. VCTL=15%

Drift of porftolio X under the BatesLN−model Drift of porftolio X under the BatesLDE−model 6.5 6.5

6 6 (%) (%) δ 5.5 δ 5.5 Drift Drift 5 5

4.5 4.5 0 5 10 15 20 25 30 0 5 10 15 20 25 30 Volatility (%) Volatility (%)

Figure 6.2. The drift versus volatility under the BatesLN-model and BatesLDE. Time to maturity 1 year. VCTL=15%

28 6.2. THE MEAN OF THE PORTFOLIO X

Drift of porftolio X under the SVSJ−model Drift of porftolio X under the Sepp−model 8.2 6.5

8.1 6 8 (%) (%) δ δ 5.5 7.9 Drift Drift 5 7.8

7.7 4.5 0 5 10 15 20 25 30 0 5 10 15 20 25 30 Volatility (%) Volatility (%)

Figure 6.3. The drift versus volatility under the SVSJ-model and Sepp. Time to maturity 1 year. VCTL=15%

The portfolios under the BS-, Heston-, BatesLN-,BatesLDE-,SVSJ- and Sepp-model all experience the same qualitative feature. The drift increases and thereafter it levels off. To describe the behavior, we also compute the expected mean of the relative weight uS (t, δ), for each increase in overall volatility. Intuitively the realized volatility will increase as input volatility increases and the mean of the weights will decrease. As seen in the mean of the relative weight, it decreases slowly but steadily as volatility increases. If we recall the pre-defined volatility chart in Chapter 2, the portfolio construction restricts the influence of the risky asset S in the long-run, as input volatility increases. The influence of the drift of the bond increases, as volatility amplifies up to around 25% in all graphs, which corresponds to around 50% in the expected mean for the relative weight. Thus the influence of both assets is equally big.

6.2.2 The mean versus VCTL

The volatility parameters are now left intact and instead we evaluate the mean, using the drift δ˜, when we make adjustments to the VCTL. We will increase each step with 2.5%, and let VCTL vary between 5% to 20% as given:

Table 6.3. Volatility cap target levels

5% 7.5% 10% 12.5% 15% 17.5% 20%

Using diﬀerent VCTLs implies that we have to change the constant values in the volatility table. We also need to bear in mind, that when we create these six new allocation tables, the realized volatility multiplied with the relative weights, should always be, either equal to or less than the VCTL. All six tables can be seen in Appendix A.1.

Drift of porftolio X under the BS−model Drift of porftolio X under the Heston−model 3.7 5.2

3.65 5

3.6 4.8 (%) (%) δ δ 3.55 4.6 Drift Drift 3.5 4.4

3.45 4.2 5 10 15 20 5 10 15 20 Volatility cap target level (%) Volatility cap target level (%)

Figure 6.4. The drift versus VCTL under the BS-model and Heston. Time to maturity 1 year.

29 CHAPTER 6. THE DISTRIBUTIONAL FEATURES OF THE PORTFOLIO

Drift of porftolio X under the BatesLN−model Drift of porftolio X under the BatesLDE−model 6 6.5

6 5.5 (%) (%) δ δ 5.5

Drift 5 Drift 5

4.5 4.5 5 10 15 20 5 10 15 20 Volatility cap target level (%) Volatility cap target level (%)

Figure 6.5. The drift versus VCTL under the BatesLN-model and BatesLDE. Time to maturity 1 year.

Drift of porftolio X under the SVSJ−model Drift of porftolio X under the Sepp−model 8 6.5

7.5 6 7 (%) (%) δ δ 5.5 6.5 Drift Drift 5 6

5.5 4.5 5 10 15 20 5 10 15 20 Volatility cap target level (%) Volatility cap target level (%)

Figure 6.6. The drift versus VCTL under the SVSJ-model and Sepp. Time to maturity 1 year.

We recall the parameter values from the Appendix and see that the short-term volatility, for all models is more or less 13%. The convex-shaped curvature can be seen in all the graphs. Hence, there is an increase in drift for volatility cap target levels up to 12.5% whereas a decrease in drift occurs for target levels above that.

6.3 The variance of the portfolio X

Standard deviation is a preferable measure since we will adjust the volatility and the VCTL. Firstly, we investigate the dispersion of log(X) versus volatility, and secondly the dispersion of log(X) versus VCTLs.

6.3.1 The standard deviation of log(X) versus maturity for different volatilities The standard deviation will definitely reveal whether the portfolio construction really does its job to restrict the volatility i.e does the standard deviation really equal the volatility cap target level in the long run for all models? We will use the default value VCTL=15% and use three different input volatilities, the same for all underlying models.√ As in the previous section, we will do a simultaneous upward shift. The ranges of the initial volatility, V0 and long term variance, θ are 10%,20%, 30% and 1%, 5%, 10% respectively. Until now, we have only performed Monte Carlo simulations for a maturity of 1 year, but for the upcoming analysis it is extended to include the following maturities:

Table 6.4. Maturities T (years)

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

For comparison, we will also measure the standard deviation for log(S) with varying volatility under the aﬃne jump-diﬀusions. What will be expected when the volatility is increased? Intuitively, if the volatility for the risky asset is increased, the standard deviation should also increase. As expected, this is really the case. We show the graphs for log(S) under the BatesLN- and BatesLDE-model. The qualitative feature is the same for the other models, which can be seen in the Appendix A.2:

30 6.3. THE VARIANCE OF THE PORTFOLIO X

Standard deviation of log(S) under the BatesLN−model Standard deviation of log(S) under the BatesLDE−model 60 80

50 60 40 40 30 20 20 Standard deviation (%) Standard deviation (%) 10 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Time to maturity (years) Time to maturity (years)

Figure 6.7. Standard deviation of log(S) versus maturity for the volatilities 10%(red star), 20%(blue circle) and 30%(black square), under the BatesLN- and BatesLDE-model

As we recall, the main purpose of using a dynamic underlying value is to keep the realized volatility of the portfolio under control and preferably below a certain maximum level, the VCTL. To accomplish this, we vary the allocation between the risky asset and the non-risky asset in a predetermined way. As a result, the realized volatility of the portfolio will remain under the so-called volatility cap target level. Hence, the volatility of the portfolio is capped, independently of any adjustment of the short term- and long term- volatility. Compared with the risky asset S, where the standard deviation increases as time goes by, the standard deviation for X levels oﬀ around the VCTL, as expected:

Standard deviation of log(X) under the BS−model Standard deviation of log(X) under the Heston−model 16 16

14 14

12 12

10 10

8 8 Standard deviation (%) Standard deviation (%) 6 6 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Time to maturity (years) Time to maturity (years)

Figure 6.8. Standard deviation of log(X) versus maturity for the volatilities 10%(red star), 20%(blue circle) and 30%(black square), under the BS-model and Heston. VCTL=15%

Standard deviation of log(X) under the BatesLN−model Standard deviation of log(X) under the BatesLDE−model 20 16

14 15 12

10 10 8 Standard deviation (%) Standard deviation (%) 5 6 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Time to maturity (years) Time to maturity (years)

Figure 6.9. Standard deviation of log(X) versus maturity for the volatilities 10%(red star), 20%(blue circle) and 30%(black square), under the BatesLN-model and BatesLDE. VCTL=15%

Standard deviation of log(X) under the SVSJ−model Standard deviation of log(X) under the Sepp−model 20 16

14 15 12 10 10 5 8 Standard deviation (%) Standard deviation (%) 0 6 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Time to maturity (years) Time to maturity (years)

Figure 6.10. Standard deviation of log(X) versus maturity for the volatilities 10%(red star), 20%(blue circle) and 30%(black square), under the SVSJ-model and Sepp. VCTL=15%

31 CHAPTER 6. THE DISTRIBUTIONAL FEATURES OF THE PORTFOLIO

6.4 The standard deviation of log(X) versus VCTL

Next in line is to investigate what happens with the standard deviation when we adjust the volatility cap target level. From the previous section, we know that using the default value of VCTL=15% indeed resulted in almost the same value for the standard deviation. Hence, conditional on a long maturity, we should expect the same type of behavior independently of the choice of VCTL. Below, we show what happens with the standard deviation for diﬀerent volatility cap target levels when we have a maturity of 5 years.

Standard deviation of log(X) under the BS−model Standard deviation of log(X) under the Heston−model 20 20

15 15

10 10 Standard deviation (%) Standard deviation (%) 5 5 5 10 15 20 5 10 15 20 Volatility cap target level (%) Volatility cap target levels (%)

Figure 6.11. Standard deviation of log(X) versus VCTL, under the BS- and Heston-model. Maturity 5 years.

Standard deviation of log(X) under the BatesLN−model Standard deviation of log(X) under the BatesLDE−model 20 20

15 15

10 10 Standard deviation (%) Standard deviation (%) 5 5 5 10 15 20 5 10 15 20 Volatility cap target levels (%) Volatility cap target levels (%)

Figure 6.12. Standard deviation of log(X) versus VCTL, under the BatesLN- and BatesLDE-model. Maturity 5 years.

Standard deviation of log(X) under the SVSJ−model Standard deviation of log(X) under the Sepp−model 20 20

15 15

10 10 Standard deviation (%) Standard deviation (%) 5 5 5 10 15 20 5 10 15 20 Volatility cap target levels (%) Volatility cap target levels (%)

Figure 6.13. Standard deviation of log(X) versus VCTL, under the SVSJ- and Sepp-model. Maturity 5 years.

We note that the portfolio construction or the VCTL-strategy eﬃciently encapsulates the realized volatility, which entails that the standard deviation almost equals the choice of VCTL.

6.5 Measuring skewness and kurtosis of the Volatility Cap Portfolio

Computing the higher moments skewness and kurtosis, becomes interesting from the point of view of simpli- ﬁcation. Simpliﬁcation in terms of simplifying the portfolio distribution; Can we treat the portfolio X as a log-normally distributed asset? In case we could do so, this would facilitate, since we could price the VCTL- option and calculate sensitivity measures just by analytical means, instead of using cumbersome Monte Carlo simulations.

Skewness is a statistical parameter which quantiﬁes the amount by which the curve is shifted about the center. If we have an excessive number of large positive spikes that lengthen the positive tail, then we have a positively

32 6.6. THE KURTOSIS AND SKEWNESS OF THE VOLATILITY CAP PORTFOLIO skewed curve. Conversely, if we have an excessive number of large negative spikes we have a negative skew. As a comparison, a symmetrical distribution such as the normal distribution does not have any skewness i.e it equals zero.

Kurtosis is higher than 3 if we have an increased number of very small and very large price changes, ac- companied by a decreased number of intermediate price changes. Conversely, we have a kurtosis less than 3, if we have very few small or large price changes but an increased number of intermediate price changes. That is, kurtosis reveals whether the peak is tall and thin or short and ﬂat or has a peak that corresponds to the peak of a normal distribution (kurtosis equals 3).

Skewness and kurtosis are deﬁned as:

∑n 1 m = (x − X¯)2 2 n i i=1 ∑n 1 m = (x − X¯)3 3 n i i=1 ∑n 1 m = (x − X¯)4 4 n i i=1 m S kewness = √3 m2 m2 = m4 − Kurtosis 2 3 m2

where xi is data number i and X¯ is the average price change.

6.6 The kurtosis and skewness of the Volatility Cap Portfolio

As mentioned previously, the kurtosis and skewness tells us something about the peak and tails of the distribution. By looking at the histograms for log(S), we see that all distributions (except for the BS-model), are more outlier-prone i.e they have sharper peaks and longer fatter tails. Calculating the kurtosis clearly indicates this, since all the values are higher than 4, which can be seen in Appendix A.5.1. Below, quantile-quantile plots for log(X) and log(S) are shown, under the Black Scholes-, Heston-, BatesLN-,BatesLDE-, SVSJ- and Sepp-model, respectively:

33 CHAPTER 6. THE DISTRIBUTIONAL FEATURES OF THE PORTFOLIO

Log(X) under the BS−model QQ Plot, log(X) under the BS−model Log(S) under the BS−model QQ Plot, log(S) under the BS−model 4000 5.5 3500 5.5

3500 3000

3000 2500 5 5 2500 2000 2000 1500 1500 4.5 4.5 Quantiles of log(X) 1000 Quantiles of log(S) 1000

500 500

0 4 0 4 4 4.5 5 5.5 −5 0 5 4 4.5 5 5.5 −5 0 5 Standard Normal Quantiles Standard Normal Quantiles

Figure 6.14. Time to maturity 1 year. VCTL is in default i.e 15%

Log(X) under the Heston−model QQ Plot, log(X), under the Heston−model Log(S) under the Heston−model QQ Plot, log(S) under the Heston−model 4000 5.4 5000 5.5

3500 5.2 4000 5 3000 5

2500 4.8 3000 4.5 2000 4.6

1500 4.4 2000 4 Quantiles of log(X) Quantiles of log(S) 1000 4.2 1000 3.5 500 4

0 3.8 0 3 3.5 4 4.5 5 5.5 −5 0 5 3 4 5 6 −5 0 5 Standard Normal Quantiles Standard Normal Quantiles

Figure 6.15. Time to maturity 1 year. VCTL is in default i.e 15%

Log(X) under the BatesLN−model QQ Plot, log(X), under the BatesLN−model Log(S) under the BatesLN−model QQ Plot, log(S) under the BatesLN−model 3500 5.5 7000 5.5

3000 6000 5

2500 5000 5 4.5 2000 4000 4 1500 3000 4.5 3.5 1000 Quantiles of log(X) 2000 Quantiles of log(S)

500 1000 3

0 4 0 2.5 4 4.5 5 5.5 −5 0 5 2 3 4 5 6 −5 0 5 Standard Normal Quantiles Standard Normal Quantiles

Figure 6.16. Time to maturity 1 year. VCTL is in default i.e 15%

Log(X) under the BatesLDE−model QQ Plot, log(X), under the BatesLDE−model Log(S) under the BatesLDE−model QQ Plot, log(S) under the BatesLDE−model 3500 5.5 6000 6

3000 5000 5.5

2500 5 5 4000 2000 4.5 3000 1500 4 4.5 2000 1000 Quantiles of log(X) Quantiles of log(S) 3.5

500 1000 3

0 4 0 2.5 4 4.5 5 5.5 −5 0 5 2 3 4 5 6 −5 0 5 Standard Normal Quantiles Standard Normal Quantiles

Figure 6.17. Time to maturity 1 year. VCTL is in default i.e 15%

34 6.6. THE KURTOSIS AND SKEWNESS OF THE VOLATILITY CAP PORTFOLIO

Log(X) under the SVSJ−model QQ Plot, log(X), under the SVSJ−model Log(S) under the SVSJ−model QQ Plot, log(S) under the SVSJ−model 3500 5.5 8000 6

3000 7000 5 6000 2500 5 5000 4 2000 4000 1500 3000 3 4.5 1000 Quantiles of log(X) Quantiles of log(S) 2000 2 500 1000

0 4 0 1 4 4.5 5 5.5 −5 0 5 0 2 4 6 −5 0 5 Standard Normal Quantiles Standard Normal Quantiles

Figure 6.18. Time to maturity 1 year. VCTL is in default i.e 15%

Log(X) under the Sepp−model QQ Plot, log(X), under the Sepp−model Log(S) under the Sepp−model QQ Plot, log(S) under the Sepp−model 3500 5.4 6000 6

5.2 3000 5000 5.5 5 2500 5 4000 4.8 2000 4.5 4.6 3000 1500 4 4.4 2000 1000 Quantiles of log(X) Quantiles of log(S) 3.5 4.2 1000 500 4 3

0 3.8 0 2.5 3.5 4 4.5 5 5.5 −5 0 5 2 3 4 5 6 −5 0 5 Standard Normal Quantiles Standard Normal Quantiles

Figure 6.19. Time to maturity 1 year. VCTL is in default i.e 15%

The histograms for the affine jump-diffusions also indicates that the data are spread out more to the left of the mean than to the right i.e the left tail is longer. When we compute the skewness for the different data-sets, we see that they hover around -1 or higher, a negative value which implies that they are indeed skewed to the left:

Table 6.5. Kurtosis and skewness for log(S) Black Scholes Heston Bates LN Bates LDE SVSJ Sepp

Kurtosis 3.0041 4.5354 5.1275 4.8612 6.8947 4.8163

Skewness -0.0061 -0.82 -1.0986 -1.0275 -1.5186 -1.0151

This is also one of the reasons for choosing an affine jump-diffusion instead of the simple Black-Scholes model. The affine jump-diffusions have asymmetric leptokurtic features, high peaks and skew to the left, which resembles the features seen in many return distributions [24].

If we look at the histograms and the QQ-plots of log(X), we see that the distributions, in general, have become more "normalized". From the pictures and calculations of the kurtosis and skewness, we can deduce that this is really the case:

35 CHAPTER 6. THE DISTRIBUTIONAL FEATURES OF THE PORTFOLIO

Table 6.6. Kurtosis and skewness for log(X) when VCTL=15% Black Scholes Heston Bates LN Bates LDE SVSJ Sepp

Kurtosis: 3.0014 3.0474 3.0221 2.8896 2.9231 2.9118

Skewness: 0.0104 -0.2655 -0.3573 -0.1876 -0.2767 -0.1872

But this is not that surprising, since this is what the VCTL-strategy is supposed to do. It efficiently restricts observations from populating the extremes since the relative weight will dampen the influence of the risky asset. Hence, more observations cluster near the average but also make observations to spread more symmet- rically around the mean, thus making the distribution less skewed and the peak more flatter.

The ﬁgures also indicate that log(X), under the BS-model, follows a normal distribution. It is also implied by the kurtosis and skewness of 3.0 and 0 respectively. The results suggest that at least under the BS-model, the portfolio can be treated as a log-normally distributed asset. This will be of great help when we look at it from the pricing point of view. Hence, we could assume that X evolves in accordance with the Black Scholes model and compute VCTL-option prices using the Black-Scholes formula, greatly improving the calculation time.

Conclusively, under the Heston-,BatesLN-, BatesLDE-,SVSJ- and Sepp-model, the logarithms of the portfolios are not distributed normally, although we are led to believe this after a quick glance at the kurtosis for Heston- and BatesLN-model. They are still a bit too negatively skewed. Choosing a diﬀerent VCTL for the portfolio and looking at the implications in terms of higher/lower peaks and thinner/fatter tails, could be a subject for future research.

36 Chapter 7

The VCTL-option price

There are two types of stock options, the first is standardized options sometimes referred to as plain vanilla options, typically call options or put options traded on a stock exchange. The second is referred to as non- standardized options or exotic options. They are options that come with special conditions, sometimes very complex, which make them more flexible and better suited for individual investor needs and they are normally traded Over-The-Counter (OTC). The VCTL-option is a typical exotic option. Although we will use a payoff function of plain European call type, the claim depends on some portfolio value which in turn depends on other assets’ price processes. It is also noted that the portfolio depends upon some past price trajectory of the risky asset via the use of relative weights. Hence, the VCTL-option can be regarded as a combination of an Asian option (which protects an investor from volatility risk) and a basket option, a contingent claim whose underlying asset depends on a group of securities, in this case, one risky asset and one non-risky asset.

In this chapter, we will try to assess the model risk involved in terms of the VCTL-option price. When pricing exotic options such as the VCTL-option, there is usually a considerable model risk involved. Hence the price of the VCTL-option will possibly vary depending on which stochastic differential equation the underlying asset S satisfies. Although the different models can approximately reproduce market prices of the vanilla options, there are major differences, especially when it comes to the ability to replicate the implied volatility surface.

We will consider a VCTL-option with the following properties: Maturity 1 year, strike price 95 and spot price 100. All risk-neutral parameters for the respective model are those seen in Chapter 4. Beginning with different VCTLs and the impact on the option price, we continue and let the VCTL be static as we adjust the input volatility. To achieve a good accuracy, we maintain the same number of Monte Carlo simulations i.e 105. The different VCTLs will be assigned to the following values: 5%,7.5%,10%,12.5%,15%,17.5% and 20%. Details about the VCTLs, corresponding volatility ranges and the allocation weights can be seen√ in the Appendix A.1. For the affine jump-diffusions we will use the same values for the short term volatility V0 i.e 1%, 10%,15%,17.5%, 20%, 25%, 27.5% and 30%. However, we will not do an upward simultaneous shift in the long term variance θ, since the qualitative behavior of the option-price will remain the same, independently of any shift in this variable.

7.1 The VCTL-option price versus VCTL under the BS-model

In the left figures, we have plotted the VCTL-option prices for different VCTLs (red stars). As a comparison, we have also inserted the VCTLs into the Black Scholes formula to obtain the corresponding European call option prices (black circles). The right figures contain the same VCTL-option prices as the left ones, but instead of inserting VCTLs, we have now inserted the standard deviations of log(X) at maturity, for example Figure 7.1 (the right picture). However, we have to be aware that this standard deviation is actually the realized volatility or historical volatility and not the volatility implied by the option price. But we allow this procedure in order to clarify the possible mispricing of the VCTL-option that would occur, if we were to simplify and use the BS-formula instead of Monte Carlo simulations.

37 CHAPTER 7. THE VCTL-OPTION PRICE

VCTL option price under the BS−model A measure of mispricing 13 10.5

12 10

11 9.5

Price 10 Price 9

9 8.5

8 8 5 10 15 20 5 10 15 20 Volatility cap target level (%) Volatility cap target level (%)

Figure 7.1. Left ﬁgure: The VCTL-option price versus VCTL, under the BS-model (red star). European call option price, Black Scholes formula (black circle). Analytical VCTL-option price with drift correction (blue triangle). Right ﬁgure: The VCTL-option price versus VCTL, under the BS-model (red star). European call option price, Black Scholes formula using the standard deviations for log(X) under the BS-model(black circle).

Before we analyze the VCTL-prices in the Black Scholes framework, we recall Chapter 6 and the QQ-plot of log(X) under the BS-model, Figure 6.6. From that we could conclude that log(X) is a normally distributed asset. We could therefore replace this particular portfolio diffusion with the simple BS-model and use the analytical approach to price the VCTL-option. But in order to do that, we are required to modify the interest rate, so that the mean at maturity stays the same. Given VCTL=15% and no dividend yield, we carry out a correction of the interest rate, by multiplying it with 0.97. Although the simplification facilitates the pricing, the drift correction is contingent upon the current VCTL. Thus, a different VCTL may imply a different correction of the interest rate in a non-intuitive way. Moreover, the drift-correction is also a subordinate of the choice of picking an asset price model and obtaining a better option premium. That is, to choose a model which provides a better resemblance of the real world, (than the BS-model), is more important for the FIs to get a more accurate (higher) price for the VCTL-option, than to compute the price very quickly using the BS-formula. We have performed the drift-correction for VCTL=15% and as expected, the analytical price of the VCTL-option (blue triangle in the left picture in Figure 7.1) equals the option price obtained with Monte Carlo simulations.

If we look at the picture, Figure 7.1, we note that when VCTL=5%, the VCTL-option price is around 8.5. We proceed to VCTL=12.5% and observe that the black and red lines still aligns, meaning that the two option prices are identical. Thus, up to VCTL=12.5% we could calculate the VCTL-option price using the analytical Black Scholes formula without doing any drift-correction.

What would the implications be if the VCTL-option was priced using the Black-Scholes formula without performing any drift corrections? From the ﬁnancial institution’s point of view, this would not be a problem. They would receive a higher option premium , for VCTLs higher than 12.5% and the same option price (as the European call option price) for lower VCTLs. Hence, the individual investor would pay the same price for VCTLs up to 12.5% but a higher price for VCTLs above that, compared with the VCTL-option price obtained with Monte Carlo methods.

In Figure 7.1, we have inserted the standard deviations or historical volatility (see Appendix A.7), into the Black Scholes formula. It then becomes obvious that log(X) under the BS-model, could be regarded as a normally distributed asset, since the option prices are almost identical. The reason for this is that, when we use the historical volatility as input volatility, it generates an option price, and when we use this option price to "back-out" the implied volatility, they equal each other if and only if the portfolio is log-normally distributed.

7.2 The VCTL-option price versus VCTL under the Heston-, BatesLN-,BatesLDE-,SVSJ- and Sepp-model

If we use the Heston-model to generate the option prices, the price will increase for all VCTLs. We note that the mean for the portfolio is actually decreasing when we increase the VCTL , but overall the mean is

38 7.2. THE VCTL-OPTION PRICE VERSUS VCTL UNDER THE HESTON-, BATESLN-,BATESLDE-,SVSJ- AND SEPP-MODEL higher (compared with the BS-model) and the standard deviation for each VCTL has increased. Hence, the probability that the portfolio value at maturity will be farther away from the strike price is higher resulting in a higher VCTL-option price. Under the Heston model, it becomes evident that we cannot use the simple Black Scholes formula to obtain any analytical VCTL-option prices. At least not for VCTL under 17.5% since the FI would face the risk of selling the VCTL-option too cheap. If we insert the realized volatility of the portfolio under the Heston-model, it once again becomes obvious that the portfolio is not log-normally distributed.

VCTL option price under Heston A measure of mispricing 13 12

12 11 11 10 Price 10 Price 9 9

8 8 5 10 15 20 5 10 15 20 Volatility cap target level (%) Volatility cap target level (%)

Figure 7.2. Left ﬁgure: The VCTL-option price versus VCTL, under the Heston-model (red star). European call option price, Black Scholes formula (black circle). Right ﬁgure: The VCTL-option price versus VCTL, under the Heston-model (red star). European call option price, Black Scholes formula using the standard deviations for log(X) under the Heston-model(black circle).

In addition to stochastic volatility, we could also have a jump term in the price. For the BatesLN-model this will generate an overall increase in the mean as well as an increased variation in the standard deviation. This results in a higher VCTL-option price independently of the VCTL. We also note that since log(X) under the BatesLD-model is not a normally distributed asset, the price curves do not coincide in the right ﬁgure below.

VCTL option price under BatesLN A measure of mispricing 13 13

12 12

11 11

Price 10 Price 10

9 9

8 8 5 10 15 20 5 10 15 20 Volatility cap target level (%) Volatility cap target level (%)

Figure 7.3. Left ﬁgure: The VCTL-option price versus VCTL, under the BatesLN-model (red star). European call option price, Black Scholes formula (black circle). Right ﬁgure: The VCTL-option price versus VCTL, under the BatesLN-model (red star). European call option price, Black Scholes formula using the standard deviations for log(X) under the BatesLN- model(black circle).

Next, we assume that the jumps belong to the assymetric double exponential distribution. Then we note that the VCTL-option prices under the BatesLDE-model are almost identical to those under the BatesLN-model, conditional that VCTL is under 12.5%. We also recall that the input volatilities 13.71% and 13.85% are almost the same for both models. Up to VCTL=12.5% , the mean and standard deviation are almost the same for both models. For higher VCTLs, we see that the mean has increased for the BatesLDE. In addition to that,

39 CHAPTER 7. THE VCTL-OPTION PRICE the standard deviation has increased and this has resulted in a higher VCTL-option price. Thus, the impact of choosing the asymmetric double distribution becomes evident only for higher VCTL(>15%), due to the high input volatility.

VCTL option price under BatesLDE A measure of mispricing 14 14

12 12 Price Price 10 10

8 8 5 10 15 20 5 10 15 20 Volatility cap target level (%) Volatility cap target level (%)

Figure 7.4. Left ﬁgure: The VCTL-option price versus VCTL, under the BatesLDE-model (red star). European call option price, Black Scholes formula (black circle). Right ﬁgure: The VCTL-option price versus VCTL, under the BatesLDE-model (red star). European call option price, Black Scholes formula using the standard deviations for log(X) under the BatesLDE- model(black circle).

The VCTL-option prices under the Sepp- and BatesLDE-model are almost identical. This is also reﬂected in the mean and standard deviation of the portfolio at maturity. Thus, the assumption of deterministic jump intensity is superﬂuous.

VCTL option price under Sepp A measure of mispricing 14 14

12 12 Price Price 10 10

8 8 5 10 15 20 5 10 15 20 Volatility cap target level (%) Volatility cap target level (%)

Figure 7.5. Left ﬁgure: The VCTL-option price versus VCTL, under the BatesLDE-model (red star). European call option price, Black Scholes formula (black circle). Right ﬁgure: The VCTL-option price versus VCTL, under the BatesLDE-model (red star). European call option price, Black Scholes formula using the standard deviations for log(X) under the BatesLDE- model(black circle).

However, the assumption to include simultaneous jumps in the variance will prove not to be pointless. The mean has increased for all VCTLs under the SVSJ-model but the standard deviation has decreased compared with the Sepp-model for VCTLs above 12.5%. The input volatility is 13.13% and hence the eﬀect of using simultanous jumps becomes visible only for VCTL higher than that. When the FI chooses the analytical approach, they will not earn as much as they would do if they priced the VCTL-option using Monte Carlo methods, independently of the VCTL.

40 7.3. THE VCTL-OPTION PRICE VERSUS VOLATILITY WHEN VCTL=15%

VCTL option price under SVSJ A measure of mispricing 16 16

14 14

12 12 Price Price

10 10

8 8 5 10 15 20 5 10 15 20 Volatility cap target level (%) Volatility cap target level (%)

Conclusively, the model risk becomes apparent when we depict the VCTL-option prices under the diﬀerent models. If we do not use a suitable model which incorporates more of the real world (read more risk), the FI faces the risk of selling the VCTL-option too cheap to the individual investor. For example, if we compare the VCTL-option prices under the BS-model and the SVSJ-model for VCTL=12.5%, the prices are approximately 10 and 13.8 respectively. This shows an increase in price of almost 40%. Hence, the question which underlying model the FI should use to price the VCTL-option, is deﬁnitely a very important issue to consider.

7.3 The VCTL-option price versus volatility when VCTL=15%

Next in line is to investigate how the VCTL-option price behaves compared with a European call option price, when we increase the volatility i.e when volatility equals 1%,10%,15%,17.5%,20%,25%,27.5% and 30%. We begin with VCTL=15%, where the corresponding allocation table yields:

41 CHAPTER 7. THE VCTL-OPTION PRICE

VCTL option price under the BS−model 18 40

16 30 14 20 Price 12 10 10

8 Increase in VCTL−price(%) 0 0 10 20 30 0 10 20 30 Volatility (%) Volatility(%)

Figure 7.6. Left ﬁgure: The VCTL-option price versus volatility, under the BS-model (red star). European call option price versus volatility, Black Scholes formula (black circle). Right ﬁgure: Increase in percentage for VCTL-price versus volatility

It is not surprising the VCTL-option price follows the European call option price since we concluded in the previous section that we could substitute the VCTL-option with a European call option for lower volatilities. The effect of the VCTL-strategy becomes very clear when we increase the volatility. It efficiently restricts the realized volatility, which is also seen in the calculations of the standard deviation of log(X). Hence, an increase in volatility will have less impact on the VCTL-option price compared with the European call option price, which is seen in Figure (7.6) (the left picture). In the right figure, we illustrate the increase in percentage for the VCTL-option price, when we adjust the volatility. The increase is below 40%, whilst the increase in the European call option price is almost 100%, more than twice as much.

VCTL option price under the Heston−model 18 20

16 15 14 10 Price 12 5 10

8 Increase in VCTL−price(%) 0 0 10 20 30 0 10 20 30 Volatility (%) Volatility (%)

Figure 7.7. Left ﬁgure: The VCTL-option price versus volatility, under the Heston-model (red star). European call option price versus volatility, Black Scholes formula (black circle).Right ﬁgure: Increase in percentage for VCTL-price versus volatility

In the Heston-framework, we note that the entry price for the VCTL-option is now higher compared with the BS-model. But that the gradient of the curve is lower than for the BS-model’s curve. We expect that the mean has risen and that the variation in standard deviation is less for Heston. When analyzing the computations of these moments, we can deduce that this is really the case (see Appendix A.7). The reason for the bigger variation in standard deviation for the BS-model is that an increase in input volatility counts for the entire impact. In the Heston model, where volatility itself is stochastic, there are in addition to the input volatility other variables to take into account.√ The result is a higher VCTL-option price from the very beginning, but where the eﬀect of the increase in V0, will be less prominent. Due to this, the slope of the VCTL-option price curve is less than for the BS-model.

42 7.3. THE VCTL-OPTION PRICE VERSUS VOLATILITY WHEN VCTL=15%

VCTL option price under the BatesLN−model 18 20

16 15 14 10 Price 12 5 10

8 Increase in VCTL−price(%) 0 0 10 20 30 0 10 20 30 Volatility (%) Volatility(%)

Figure 7.8. Left ﬁgure: The VCTL-option price versus volatility, under the BatesLN-model (red star). European call option price versus volatility, Black Scholes formula (black circle).Right ﬁgure: Increase in percentage for VCTL-price versus volatility

The addition of jumps to the Heston model becomes quite distinguishable in the VCTL-option price. The mean, under the BatesLN-model, has increased even more which is reflected in the entry price. It has risen a little bit to 11.8, compared with the VCTL-option price 10.8 (using the Heston-model). Also, the standard deviation has decreased, which can be seen in Figure(7.8)(on the right). It reveals that in addition to stochastic volatility we also have the jumps that need to be taken into consideration when calculating the realized volatility. Thus, the impact of increasing the input volatility will be less for an affine jump-diffusion.

VCTL option price under the BatesLDE−model 18 10

16 8

14 6

Price 12 4

10 2

8 Increase in VCTL−price(%) 0 0 10 20 30 0 10 20 30 Volatility (%) Volatility (%)

Figure 7.9. The VCTL-option price versus volatility, under the BatesLN-model (red star). European call option price versus volatility, Black Scholes formula (black circle).VCTL=15%

When we change the jump distribution from a normal distribution to an asymmetric double exponential distribution, we have the BatesLDE-model. This change of jump-distribution causes the option price to increase just slightly but still noticeable. Although the mean has risen, it levels out for higher volatilities. We suspect that the variation in the realized volatility will be less, which is conﬁrmed both by the right ﬁgure and calculations.

43 CHAPTER 7. THE VCTL-OPTION PRICE

VCTL option price under the SEPP−model 18 10

16 8

14 6

Price 12 4

10 2 Increase in price(%)

8 0 0 10 20 30 0 10 20 30 Volatility (%) Volatility (%)

Figure 7.10. The VCTL-option price versus volatility, under the BatesLN-model (red star). European call option price versus volatility, Black Scholes formula (black circle).VCTL=15%

The substitution of constant jump intensity to deterministic, does not change the price if we look at the left figures for the BatesLDE- and Sepp-model, Figure 7.9-7.10. This can also be seen in the computations of the mean and standard deviation. There are some minor differences in the values, but due to the discretization bias we cannot regard these as significant. Thus, the VCTL-option prices are (almost surely) equal under the BatesLDE- and Sepp-model.

VCTL option price under the SVSJ−model 18 8

16 6 14 4 Price 12 2 10 Increase in price(%)

8 0 0 10 20 30 0 10 20 30 Volatility (%) Volatility (%)

Figure 7.11. The VCTL-option price versus volatility, under the SVSJ-model (red star). European call option price versus volatility, Black Scholes formula (black circle).VCTL=15%

Last but not least, the VCTL-option prices under the SVSJ-model. We now see that the VCTL-option prices, for all volatilities, have increased substantially. The variation in standard deviation is the smallest but the mean is the highest, which entails a higher but almost straight option price curve. When input volatility is 10% the VCTL-option prices under the BS- and SVSJ-model are 9 and 14 respectively. Thus, an increase in price of over 50%.

7.4 The VCTL-option price versus volatility when VCTL=10%

The volatility range is the same as before, 1%,10%,15%,17.5%,20%,25%,27.5% and 30%, but now VCTL equals 10%. The corresponding allocation table is given by:

44 7.4. THE VCTL-OPTION PRICE VERSUS VOLATILITY WHEN VCTL=10%

Volatility range Allocation VCTL uS (th, σ˜ k) uB(th, σ˜ k) 0.0-10.0% 100% 0% 10% >10.0-15.0% 66% 34% 10% >15.0-20.0% 50% 50% 10% >20.0-25.0% 40% 60% 10% >25.0-37.5% 26% 74% 10% >37.5-50.0% 20% 80% 10% >50.0-75.0% 13% 87% 10% >75.0% 5% 95% 10%

Now we have limited the realized portfolio volatility even more, by imposing stricter restrictions on the risky relative weight. Intuitively, if we restrict the realized volatility, this should of course affect the VCTL-option price. Since then, the probability for the portfolio value to be further away from the strike price at maturity is lower. The price of the VCTL-option should still increase as we increase the volatility, but at a lower rate (compared with VCTL=15%). Thus, the effect of the VCTL-strategy becomes especially apparent for input volatilities higher than 10%, which are around 13% for all affine jump-diffusions.

VCTL option price under the BS−model 18 25

16 20

14 15

Price 12 10

10 5

8 Increase in VCTL−price(%) 0 0 10 20 30 0 10 20 30 Volatility (%) Volatility(%)

Figure 7.12. Left ﬁgure: The VCTL-option price versus volatility, under the BS-model (red star). European call option price versus volatility, Black Scholes formula (black circle). Right ﬁgure: Increase in percentage for VCTL-price versus volatility

We noted in the previous section that, the biggest increase in option price versus volatility occurred using the BS-model, Figure 7.6 (the right picture). This was simply due to the fact that input volatility accounted for the entire impact on the realized volatility. Though we can expect an increase in option price under the BS-model when VCTL=10%, it has almost been halved when input volatility equals 30% (and VCTL equaled 15%), Figure 7.12 (on the right).

45 CHAPTER 7. THE VCTL-OPTION PRICE

VCTL option price under the Heston−model 18 10

16 8

14 6

Price 12 4

10 2

8 Increase in VCTL−price(%) 0 0 10 20 30 0 10 20 30 Volatility (%) Volatility (%)

Figure 7.13. Left ﬁgure: The VCTL-option price versus volatility, under the Heston-model (red star). European call option price versus volatility, Black Scholes formula (black circle).Right ﬁgure: Increase in percentage for VCTL-price versus volatility

The same effect is also seen for the Heston-model, the price curve is much flatter. Now the VCTL-strategy is even more effective when increasing the input volatility, since the impact of other volatility parameters accounts for bigger part of the realized volatility. For stochastic volatility models, increased input volatility will have less impact when the VCTL is decreased.

VCTL option price under the BatesLN−model 18 6

16 4 14

Price 12 2 10

8 Increase in VCTL−price(%) 0 0 10 20 30 0 10 20 30 Volatility (%) Volatility(%)

Figure 7.14. Left ﬁgure: The VCTL-option price versus volatility, under the BatesLN-model (red star). European call option price versus volatility, Black Scholes formula (black circle).Right ﬁgure: Increase in percentage for VCTL-price versus volatility

In addition to the volatility-term in the Heston-model, there is a jump-term which accounts for a portion of the realized volatility. Hence the price-curve should be even ﬂatter. The increase in VCTL-option price for both the BatesLN-model (Figure 7.14) and the BatesLDE-model (Figure 7.15) has once again been more than halved, compared to when VCTL=15% and input volatility equaled 30% (Figure 7.8-7.9).

46 7.4. THE VCTL-OPTION PRICE VERSUS VOLATILITY WHEN VCTL=10%

VCTL option price under the BatesLDE−model 18 4

16 3 14 2 Price 12 1 10

8 Increase in VCTL−price(%) 0 0 10 20 30 0 10 20 30 Volatility (%) Volatility (%)

Figure 7.15. The VCTL-option price versus volatility, under the BatesLN-model (red star). European call option price versus volatility, Black Scholes formula (black circle).VCTL=15%

VCTL option price under the SEPP−model 18 4

16 3 14 2 Price 12 1 10 Increase in price(%)

8 0 0 10 20 30 0 10 20 30 Volatility (%) Volatility (%)

Figure 7.16. The VCTL-option price versus volatility, under the BatesLN-model (red star). European call option price versus volatility, Black Scholes formula (black circle).VCTL=15%

VCTL option price under the SVSJ−model 18 3

16 2 14

Price 12 1 10 Increase in price(%)

8 0 0 10 20 30 0 10 20 30 Volatility (%) Volatility (%)

For the Sepp- and SVSJ-model, we observe two almost straight price curves. When VCTL equals 10%, the curve is almost not aﬀected by increased input volatility. This is because of the volatility parameters and the jump term(s) that accounts for almost all the impact on the realized portfolio volatility. Hence increasing the

47 CHAPTER 7. THE VCTL-OPTION PRICE volatility will almost make no difference. From a sell side option trader’s point of view, this is quite appealing since it does not matter if the input volatility is misspecified, the VCTL-option price will almost stay the same. Hence, to price the VCTL-option using the SVSJ- and Sepp-model will actually facilitate, compared to if the BS-model is used. Since then more effort has to be put on how to "tweak" the input volatility. Thus, by using a more advanced model to price the VCTL-option, the sell-side option trader does not only receive a higher price for the VCTL-option but also decreases the risk to misspecify the input volatility.

7.5 Concluding remarks

There are two types of model risk. One is that the model will give the wrong price at the time a product is bought or sold. The other risk concerns hedging; if you use the wrong model and calculate the greeks which the hedges are based on, they are liable to be wrong. Both these risks can cause serious damage for the FI, both in terms of money and reputation. To come up with a single "almost perfect" model, will probably be the ﬁrst the FI will think of. However this approach may be even more dangerous than to continue to use the old BS-model (since then they at least know that the model has its limitations).

If a corporation does not have the in-house capability to value an instrument, it should not trade it. [14]

Furthermore, the price for the VCTL-option using a more advanced model is considerably higher compared with the simple BS-model. There is a signiﬁcant increase in the VCTL-price when we include the feature of stochastic volatility but also an additional increase in price, whether we chose to have simultaneous jumps in both asset price and variance, or to include only one jump term in the asset price. This indicates that the use of only the BS-model is very risky. Although the SVSJ-model generates the highest prices independently of any increase in the volatility or volatility cap target level, the model may not be ﬂexible enough when the so called regime shifts occur [14]. This enforces the use of at least one more model, for example the Sepp-model, which generates the second highest prices.

A ﬁnancial institution should not rely on a single model for pricing nonstandard products. Instead it should, whenever possible, use several diﬀerent models. This leads to a price range for the instrument and a better understanding of the model risks being taken. -John C. Hull, Professor of Derivatives and Risk Management

Conclusively, instead of just relying on the Black Scholes-model, we advocate the FI to use at least two more research models, the SVSJ- and the Sepp-model: Both in order to produce a price range for the instrument in question, but also to obtain a range of sensitivity measures or greeks, which the hedge is based upon.

48 Chapter 8

The sensitive measures of the VCTL-option

8.1 The Greeks

In order for traders and alike to keep profitable trades, it is often of vital importance, to have a thorough understanding of how for example the value of a portfolio, will change given a change in the underlying asset price. This could be due to a change in the underlying value, but also due to misspecified model parameter, such as volatility. Hence, to limit or lower the financial risk, traders could set up different types of hedges depending on their attitude towards risk.

To improve our understanding about how the price of the VCTL-option will change, given a change in the underlying value, the different risk measures must be calculated, regardless of whether it is due to a parameter that has been adjusted or due to a parameter that has been misspecified. Also, the fact that we have not only used the general Black-Scholes model, but five other affine jump- models will help build intuition about the risks the models themselves incorporate into the sensitivity measures of the VCTL-option. For example, we assume that the VCTL-option has delta of 0.6 under the SVSJ-model but only 0.3 under the BS-model. If the underlying value is £10 and makes a jump and increases with £5, then the VCTL-option value will increase with 5*0.6=£3 and 5*0.3=£1.5 under the SVSJ- and BS-model respectively. Hence, the increase in the VCTL-option price will be twice as much under the SVSJ-model. Thus, depending on which model the risky asset satisfies, the VCTL-option value may be more sensitive to small changes in the underlying value. The greeks used in this thesis, will only take one variable into consideration, so possible cross-greeks are omitted and left for future research. We are confined to just analyze three of the most commonly known sensitivity measures: Delta, Gamma and Vega.

"Like with other option valuation methods, there is one fallback technique for the calculation of the Greeks. We can always redo the entire valuation with varied inputs reflecting the potential change in the underlying asset, and use an explicit finite differencing approach to compute the Greek we are after." [15]

The methods for estimating sensitivity measures can be separated into two categories; methods that involve finite difference approximations, as cited above, and methods that do not, for example the pathwise method and the likelihood ratio method [10]. The finite-difference method poses an advantage in terms of understanding, but lacks in terms of accuracy since it produces biased estimates. Hence, to produce unbiased estimates, we will use the pathwise method. Besides the advantage of producing unbiased estimates, the computational speed is increased since the number of Monte Carlo simulations is reduced: "...a single simulation can be used to estimate multiple derivatives along with a security’s price" [5]. Thus, there is no need for re-simulations.

Furthermore, the first and second order of central finite difference approximations will be used to calculate the greeks. The first order is given below:

f (x + ε) − f (x − ε) f ′(x) ≈ (8.1) 2ε

49 CHAPTER 8. THE SENSITIVE MEASURES OF THE VCTL-OPTION and second order ﬁnite diﬀerence:

f (x + ε) − 2 f (x) + f (x − ε) f ′′(x) ≈ (8.2) ε2

An initial simulation is run for the function f(x), without any perturbation ε, to determine a reference value. Thereafter the parameter of interest, whether it is the volatility σ or price x, is perturbed. The size of the perturbation is small, normally around 1% of x or even less when the volatility-parameter is perturbed [1]. Whereas the finite difference method requires one or two re-simulations, the direct method uses the same random numbers from the first simulation to calculate both f (x + ε) and f (x − ε). Thus, the direct method only requires information from the initial simulation run to calculate two or more of the option-values, greatly reducing the number of Monte Carlo simulations.

The following data has been used throughout the calculations: Strike price K=95, time to maturity T=0.1176 years (when time to maturity is constant) and VCTL=15%.

8.2 Greeks for a European call option when the risky asset S evolves according to the Black-Scholes model

Let Πt denote the pricing function at time t for the VCTL-option. The VCTL-option has a pay-off function of European call option type, with the portfolio as its underlying asset, where the risky asset S satisfies one of the six different affine jump-diffusions. If we recall the QQ-plots from Chapter 7, we noticed that the distribution of the log of the portfolio X were in one case, normally distributed. Hence, we can make an interchange of variables, X=S and use Delta, Gamma and Vega for a European call option where the underlying asset S evolves in accordance with the Black-Scholes model.

We will use a three-dimensional view and a two-dimensional view, with regards to both the spot price and maturity. In that way, it becomes a bit clearer what influence the portfolio construction has on the sensitivity measure in question. But also in terms of model-risk i.e what effect will the different affine jump-models have on the sensitivity measures compared with the ordinary Black-Scholes model.

8.2.1 Delta

Delta for a European call

0.5 Delta

0 1 110 0.5 100 90 0 80 Time T (years) Price

Figure 8.1. Delta for an ordinary European call option under the BS-model.

The delta of the VCTL-option is deﬁned as:

∂Π Π | = +ε − Π | = −ε ∆ = t ≈ t x0 x t x0 x (8.3) x ∂x 2ϵ

Hence, it measures the rate of change of the VCTL-option value with respect to changes in the underlying portfolio value which is also referred to as the "hedge ratio". The ﬁgure shows delta for a simple European call option.

50 8.3. GREEKS FOR THE VCTL-OPTION, WHEN THE RISKY ASSET S EVOLVES ACCORDING TO THE BLACK-SCHOLES MODEL 8.2.2 Gamma

Gamma for a European call

0.2

0.1 Gamma 0 1 110 0.5 100 90 0 80 Time T (years) Price

Figure 8.2. Gamma for an ordinary European call option under the BS-model.

Mathematically, gamma is the ﬁrst derivative of delta and it is deﬁned for the VCTL-option as:

∂2Π Π | = +ε − 2Π | = + Π | = −ε Γ = t ≈ t x0 x t x0 x t x0 x (8.4) x ∂x2 ϵ2

When the underlying portfolio value changes, delta for the VCTL-option also changes. Gamma tells us how sensitive our delta is to the underlying price change. If gamma is high, we have to re-balance often. Hence it is preferable to have a portfolio which is also gamma neutral.

8.2.3 Vega

Vega for a European call

20 Vega

0 1 110 0.5 100 90 0 80 Time T (years) Price

Figure 8.3. Vega for an ordinary European call option under the BS-model.

Vega shows how much the value of the VCTL-option changes when a small change occurs in the volatility. For the VCTL-option it is deﬁned as given:

∂Π Π |σ =σ+ε − Π |σ =σ−ε ν = t ≈ t 0 t 0 (8.5) ∂σ 2ϵ

Vega is an important Greek to monitor for an option trader, especially in volatile markets, since the value of some option strategies can be particularly sensitive to changes in volatility. When a 1% change occurs in the volatility of the underlying asset, the size of vega represents how much the value of the VCTL-option will change.

8.3 Greeks for the VCTL-option, when the risky asset S evolves according to the Black-Scholes model

Delta for VCTL, under the BS−model 1 0.4

0.8 1.5 0.3 1 0.6 0.2 Delta 0.5 Delta 0.4 Delta 0 0.1 1 0.2 110 0.5 100 90 0 0 0 80 80 90 100 110 0 0.5 1 Time T (years) Spot price Spot price Time T (years)

Figure 8.4. Left ﬁgure: Delta for the VCTL-option under BS-model. Right ﬁgure: Delta of the VCTL-option vs. spotprice and time (starred red line). Delta of an ordinary European call option under the BS-model (black line)

51 CHAPTER 8. THE SENSITIVE MEASURES OF THE VCTL-OPTION

Gamma for VCTL, under the BS−model 0.1 0.04

0.08 0.2 0.03 0.06 0 0.02

Gamma 0.04 Gamma Gamma −0.2 0.01 1 0.02 110 0.5 100 90 0 0 0 80 80 90 100 110 0 0.5 1 Time T (years) Spot price Spot price Time T (years)

Figure 8.5. Left ﬁgure: Gamma for the VCTL-option under BS-model. Right ﬁgure: Gamma of the VCTL-option vs. spotprice and time (starred red line). Gamma of an ordinary European call option under the BS-model (black line)

Vega for VCTL, under the BS−model 15 40

40 30 10 20 20 Vega 0 Vega Vega 5 −20 10 1 110 0.5 100 90 0 0 0 80 80 90 100 110 0 0.5 1 Time T (years) Spot price Spot price Time T (years)

Figure 8.6. Left ﬁgure: Vega for the VCTL-option under BS-model. Right ﬁgure: Vega of the VCTL-option vs. spotprice and time (starred red line). Vega of an ordinary European call option under the BS-model (black line)

We recall the QQ-plot for the risky asset X, when S evolved according to the BS-model. Since it is a log- normally distributed asset, it is expected that the shape of delta and gamma is identical with those for the ordinary European call option. As can be seen in the Figure 8.4-8.5 the graphs coincides almost perfectly.

20 Vega

−20 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Time T (years)

Figure 8.7. Vega for diﬀerent input volatilities vs. time for the VCTL-option under BS-model. 13%(red star), 18.5%(blue- cross-hair) and 22.5%(magenta-plus).

However for the two-dimensional view, vega versus maturity, we notice a slight deviation compared with the vega for a European call option. This requires some additional analysis, and as depicted in Figure 8.7, this behavior becomes even more pronounced when the input volatility is increased when we use longer maturities. The red starred line corresponds to the calibrated volatility of around 13%. We then increase this to 18.5% (blue-cross-hairs line) and end with the magenta-plus line which corresponds to an input volatility of around 22.5%.

If we look at the vega for an ordinary European call option, we see that the more time we have for an option to expire, the higher the vega. This is because of the extrinsic value or time value which makes up for a larger proportion of the premium for longer term options. Thus, it is the time value that is sensitive to changes in volatility. From the ﬁgures, it can be deduced that the behavior of vega for the VCTL-option is the opposite, if very short maturities are disregarded. The more time remaining to option expiration, the lower the vega and it becomes even lower when input volatility increases (see Figure 8.7). How can this be? We recall the previous chapter where it was shown that when we increased the volatility, the VCTL-option price rose, but it rose less than for the price of the European call option. We also have the VCTL-strategy which makes up for a big part of the behavior. If we increase input volatility, the realized volatility will increase and will hit or reach the VCTL much quicker. Causing the relative weights of the risky asset to decrease which can bee seen when we analyze the mean of the relative weights. If the relative weight of the risky asset decreases to almost zero, we will only have inﬂuence of the non-risky asset, which is immune to any change of volatility. The time value will be less sensitive to changes in volatility and vega will decrease.

52 8.4. GREEKS FOR THE VCTL-OPTION, WHEN THE RISKY ASSET S EVOLVES ACCORDING TO THE HESTON-MODEL 8.4 Greeks for the VCTL-option, when the risky asset S evolves according to the Heston-model

Delta for VCTL, under the Heston−model 1 0.5

1 0.8 0.4

0.6 0.3 0.5 Delta Delta 0.4 Delta 0.2 0 1 0.2 0.1 110 0.5 100 90 0 0 0 80 80 90 100 110 0 0.5 1 Time T (years) Spot price Spot price Time T (years)

Figure 8.8. Left ﬁgure: Delta for the VCTL-option under Heston-model. Right ﬁgure: Delta of the VCTL-option vs. spotprice and time (starred red line). Delta of an ordinary European call option under the BS-model (black line)

Gamma for VCTL, under the Heston−model 0.1 0.05

0.2 0.08 0.04

0.06 0.03 0.1 Gamma Gamma 0.04 Gamma 0.02 0 1 0.02 0.01 110 0.5 100 90 0 0 0 80 80 90 100 110 0 0.5 1 Time T (years) Spot price Spot price Time T (years)

Figure 8.9. Left ﬁgure: Gamma for the VCTL-option under Heston-model. Right ﬁgure: Delta of the VCTL-option vs. spotprice and time (starred red line). Delta of an ordinary European call option under the BS-model (black line)

Vega for VCTL, under the Heston−model 15 40

30 30 10 20 20 Vega 10 Vega Vega 5 0 10 1 110 0.5 100 90 0 0 0 80 80 90 100 110 0 0.5 1 Time T (years) Spot price Spot price Time T (years)

Figure 8.10. Left ﬁgure: Vega for the VCTL-option under Heston-model. Right ﬁgure: Delta of the VCTL-option vs. spotprice and time (starred red line). Delta of an ordinary European call option under the BS-model (black line)

If we take a quick look at Appendix A.11 for delta and gamma for a European call option, where S follows the Heston model, we see that the qualitative features are almost the same as that for delta and gamma under the portfolio. Delta is slightly higher for longer maturities, compared with the European call, meaning that a small change in the underlying price process will have more impact on the VCTL-option price. When we increase the maturity, this will be even more distinguishable.

Vega 20

0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Time T (years)

Figure 8.11. Vega for diﬀerent input volatilities vs. time for the VCTL-option under Heston-model. 13%(red star), 18.5%(blue-cross-hair) and 22.5%(magenta-plus).

8.5 Greeks for the VCTL-option, when the risky asset S evolves according to the BatesLN-, BatesLDE-,SVSJ- and Sepp-model

The qualitative features are the same when the risky asset is governed by any of the affine jump-diffusions. Hence, the corresponding figures can be seen in the Appendix A.8-A.10.

53 CHAPTER 8. THE SENSITIVE MEASURES OF THE VCTL-OPTION

8.6 Delta for VCTL-option versus volatility

Delta BS−model Delta Heston−model 1 1

0.8 0.8

0.6 0.6

Delta 0.4 Delta 0.4

0.2 0.2

0 0 80 85 90 95 100 105 110 80 85 90 95 100 105 110 Price Price

Delta BatesLN−model Delta BatesLDE−model 1 1

0.8 0.8

0.6 0.6

Delta 0.4 Delta 0.4

0.2 0.2

0 0 80 85 90 95 100 105 110 80 85 90 95 100 105 110 Price Price

Delta SVSJ−model Delta SEPP−model 1 1

0.8 0.8

0.6 0.6

Delta 0.4 Delta 0.4

0.2 0.2

0 0 80 85 90 95 100 105 110 80 85 90 95 100 105 110 Price Price

In the above figures we have calculated Delta for different affine jump-diffusions. The almost straight line is when we double the initial volatility from ∼ 13% up to ∼ 27%. Then the delta increases for out-of-the-money VCTL-options and conversely for in-the-money VCTL-options. Hence when volatility rises, then the extrinsic or time value goes up, making OTM options more sensitive and ITM option less sensitive to changes in the underlying portfolio value.

8.7 Vega for VCTL-option versus volatility

Vega BS−model Vega Heston−model 15 15

10 10 5 Vega Vega 5 0

−5 0 80 85 90 95 100 105 110 80 85 90 95 100 105 110 Price Price

Vega BatesLN−model Vega BatesLDE−model 15 15

10 10 Vega Vega 5 5

0 0 80 85 90 95 100 105 110 80 85 90 95 100 105 110 Price Price

Vega SVSJ−model Vega SEPP−model 20 15

15 10 10 Vega Vega 5 5

0 0 80 85 90 95 100 105 110 80 85 90 95 100 105 110 Price Price

We proceed in the same way with vega and once again double the input volatility. The curvy line is for the intitial volatility, so when we double the initial volatility, vega increases for both OTM- and ITM-options. Hence, the extrinsic or time value goes up, making both OTM- and ITM- options more sensitive to changes in the volatility.

54 Chapter 9

Implied volatility and skew eﬀects

As stated before, if the models are calibrated using the market’s IV, the current market price of risk will be obtained. If we know the current market price of risk and have obtained the prevailing martingale measure then we are able to calculate prices of whatever option type we can think of, since the martingale measure assures us that the resulting option price will be free of arbitrage. But the IV surface can also be looked upon in another way. Most professional option traders prefer to quote the option’s relative value with IV. Based on current option prices, they can deduce the market’s view, which is an indicator of the current sentiment of the market. The realized volatility or historical volatility only shows where the volatility has been in the past but it also indicates the expected volatility or trading range of the market. Options that have high IV compared to their past are sometimes called ’expensive’. Options that have low IV compared to their past are said to be ’cheap’. Thus, the option trader will employ diﬀerent types of selling and buying strategies, in order to exploit these diﬀerences.

Implied volatility for VCTL−option under the BatesLN−model

0.4 (M,T) σ

0.2

0 1.6

Implied Volatility 1.4 3 1.2 2 1 1 0.8 0 Moneyness M=S/K Time to maturity T Implied volatility for VCTL−option under the BatesLDE−model Implied volatility for VCTL−option under the Sepp−model

0.4 0.4 (M,T) (M,T) σ σ

0.2 0.2

0 0 1.6 1.6

Implied Volatility 1.4 3 Implied Volatility 1.4 3 1.2 2 1.2 2 1 1 1 1 0.8 0 0.8 0 Moneyness M=S/K Time to maturity T Moneyness M=S/K Time to maturity T Implied volatility for VCTL−option under the SVSJ−model

0.4 (M,T) σ

0.2

0 1.6

Implied Volatility 1.4 3 1.2 2 1 1 0.8 0 Moneyness M=S/K Time to maturity T

Figure 9.1. Implied volatility for the VCTL-option using the risk-neutral parameters, under the Heston-, BatesLN- and BatesLDE-model. VCTL=15%

However, since the VCTL-option is an OTC-option, there are no market values available for calculating the

55 CHAPTER 9. IMPLIED VOLATILITY AND SKEW EFFECTS

IV, as for vanilla options on a stock exchange. Nonetheless, we could still use the IV to see how it reacts when we adjust certain variables. This could also be regarded as a type of misspeciﬁcation. What happens with the IV and ultimately the VCTL-option price if we think that one of these variables is actually too low and needs to be increased or vice versa?

To compute the IV surfaces as above, we first calculate the VCTL-option prices, for different strikes and maturities using the risk-neutral parameters. Thereafter, to obtain the IV, we need to invert the Black-Scholes formula using the Newton-Raphson’s method. We could also use a pre-defined function in Matlab which directly gives the IV. The variables (correlation between the Wiener processes ρ or volatility of volatility ε) will be assigned three different values which will result in three different IV surfaces (seen in the Appendix). However, to further clarify possible effects on the IV versus strikes, we will instead use a two-dimensional view.

Depending on the shape of the IV curve, it is denoted by diﬀerent names. In case we have a U-shaped curve, resembling a smile, we simply denote it as a volatility smile. Another skew pattern is the forward skew. Then the IV for options at the lower strikes are lower than the IV at higher strikes; a common pattern for options in the commodities market. Last but not least, we could have the reverse skew also denoted as the volatility smirk. A pattern which typically appears for longer term equity options and index options. This skew pattern did not show up until after the market crash of 1987. Since then, investors are more worried about crashes and buy OTM puts to hedge themselves. Thus, a greater demand causes the put option-price to increase and consequently the IV. Also, a change in market sentiment can cause this aﬀect. In the next pages, we easily recognize the shape of all the curves as reverse skew curves or volatility smirks.

Not only the skew but also information about the convexity or the curvature of the skew can be valuable. In case an option trader is involved in an option strategy called a butterﬂy spread (e.g long a 90% and a 120% strike call option), he is interested in the convexity of the IV skew. If the skew becomes more convex the IVs have increased, hence the butterﬂy spread becomes more valuable.

We could also analyze the effect on the IV versus different maturities, using the variables κ, θ and input volatility V0. But also various choices of jump parameters seen in the affine jump-diffusions. However, we omit to adjust these variables and possible effects on the term structure of volatility and leave it for future research. We confine ourselves to study the IV versus strike prices, and the effect on the skew for the varying variables ε and ρ [9]. To facilitate the analysis, we will also investigate what happens with the portfolio X at maturity when we vary the aforementioned variables.

9.1 Implied volatility versus the volatility of volatility ε

We begin to investigate what happens with the implied volatility when we vary the variable ε. However,√ we need to be careful since Feller’s condition imposes an upper bound on the variable, via the condition ε ≤ 2κθ. Regardless of the underlying model, the diﬀerent ε will be as given:

Table 9.1.

ε 0.05 0.2 0.3

To obtain the 90%-100% skew(slope) for the Heston-model corresponding to ε = 0.05 ε = 0.3, we enlarge the ﬁgure below and look at the corresponding values on the y-axis and the x-axis for strikes 90 and 100. The − − slopes are approximately y2 y1 = 15−18.5 = −0.35 and y2 y1 = 15−16.1 = −0.11 and often quoted as positive x2−x1 100−90 x2−x1 100−90 numbers since the equity (reverse) skew is always negative i.e 0.35 and 0.11 for the red and black curve respectively. This entails that the curve for ε = 0.3 is more than twice as steep than for ε = 0.05. This behavior is seen in all ﬁgures regardless of the underlying model.

56 9.2. IMPLIED VOLATILITY VERSUS THE CORRELATION ρ

Implied volatility for different ε under the Heston−model Implied volatility for different ε under the BatesLN−model 0.25 0.35

0.3 0.2 0.25

0.2 0.15

Implied Volatility Implied Volatility 0.15

0.1 0.1 80 85 90 95 100 105 110 115 120 80 85 90 95 100 105 110 115 120 125 Strike (K) Strikes (K) Implied volatility for different ε under the BatesLDE−model Implied volatility for different ε under the Sepp−model 0.35 0.35

0.3 0.3 0.25 0.25 0.2 0.2 Implied Volatility Implied Volatility 0.15

0.1 80 85 90 95 100 105 110 115 120 125 80 85 90 95 100 105 110 115 120 125 Strikes (K) Strike (K) Implied volatility for different ε under the SVSJ−model 0.35

0.3

0.25

0.2

Implied Volatility 0.15

0.1 80 85 90 95 100 105 110 115 120 125 Strike (K)

Figure 9.2. Implied volatility for the VCTL-option under the BatesLDE-, the Sepp- and the SVSJ-model. Maturity 1 year. VCTL=15%. ε=0.05 (black), ε=0.2 (blue), ε=0.3 (red).

What about the curvature of the implied volatility skew i.e its convexity? To obtain these values we need to pick three values instead of two, for example a combination of implied volatilities with strikes at 90, 100 and 110 and then use the second order derivative via the finite central difference approximation. However, we do not show this explicitly by calculating, but if we choose for example the BatesLDE-model, we can conclude directly by ocular investigation that the curve for ε = 0.3 has more convexity than for ε = 0.05. Hence, to increase ε causes the curvature of the implied volatility to increase. We also note that when we increase ε, the mean of the portfolio will increase and hence the VCTL-option price. The BatesLDE-model will have the biggest mean for ε=0.3 causing the IV to be the highest in comparison with the other models. For higher strikes, the IV’s curves will eventually change places, except for the SVSJ-model. When we increase the volatility of volatility this will cause the realized standard deviation to decrease for all models except the SVSJ-model (which will increase). The Heston-model has the biggest variation in standard deviation and BatesLN-model is the next. Hence, the IV-curves for different ε will change place sooner for the Heston-model compared with the other models.

9.2 Implied volatility versus the correlation ρ

The variable ρ can be interpreted as the correlation between asset price and the volatility. Empirical evidence suggests that in stock markets, the asset price and the related volatility are negatively correlated and the term "leverage eﬀect" refers to this well-established relationship i.e when stock prices fall, the volatility will increase. Thus spreading the left tail and squeezing the right tail of the distribution, aﬀecting the skewness [9]. We will analyze what happens with the IV when the correlation variable ρ is assigned to the following values:

Table 9.2.

ρ -0.9 -0.5 0

57 CHAPTER 9. IMPLIED VOLATILITY AND SKEW EFFECTS

If we look at the ﬁgures below, we can deduce directly that all slopes are negative. When ρ decreases for lower strikes, the IV increases. The convexity or the curvature of the IV curve can easily be obtained via the approximation of the second order derivative but once again, we simplify and just perform an ocular analysis. The convexity increases when ρ becomes more negative. All this suggests that the VCTL-option price increases as ρ decreases, except for the Heston- and BatesLN-model over strikes of 115 and 117 respectively, which is due to the larger variation in standard deviation but smaller variation in the mean, compared with the other models.

Given the SVSJ-model and low strikes, we observe that this model produces the highest IV when ρ = −0.9. Whereas the mean for the Heston-model is the lowest, resulting in a lower IV-curve.

Implied volatility for different ρ under the Heston−model Implied volatility for different ρ under the BatesLN−model 0.35 0.35

0.3 0.3

0.25 0.25

0.2 0.2

Implied Volatility 0.15 Implied Volatility 0.15

0.1 0.1 80 85 90 95 100 105 110 115 120 80 85 90 95 100 105 110 115 120 Strike (K) Strike (K)

Figure 9.3. Implied volatility for the VCTL-option under the Heston- and the BatesLN-model. VCTL=15%. ρ=-0.9(black), ρ=-0.5(blue), ρ=0(red)

Implied volatility for different ρ under the BatesLDE−model Implied volatility for different ρ under the Sepp−model 0.35 0.35

0.3 0.3

0.25 0.25

0.2 0.2

Implied Volatility 0.15 Implied Volatility 0.15

0.1 0.1 80 85 90 95 100 105 110 115 120 80 85 90 95 100 105 110 115 120 Strike (K) Strike (K) Implied volatility for different ρ under the SVSJ−model 0.5

0.4

0.3

0.2 Implied Volatility

0.1 80 85 90 95 100 105 110 115 120 Strike (K)

Figure 9.4. Implied volatility for the VCTL-option under the BatesLDE-model.VCTL=15%. ρ=-0.9(black), ρ=-0.5(blue), ρ=0(red)

58 Chapter 10

Conclusions

The development of improved models of asset prices is gaining much interest. The hunt for a "better" or "more realistic" model has been ongoing ever since Black-Scholes-Merton published their ground-breaking paper in 1973. Although the report was published 40 years ago, the BS-model is still the most widely used model amongst option traders. The reason is that option traders like simplicity, a simple model such as the BS-model has only one unobservable parameter, the volatility. A parameter which can be easily grasp and intuitively felt.

More complex models are regarded as black boxes and very diﬃcult to develop intuition about. Although the BS-model fails to imitate many things seen in the ﬁnancial market, the option traders knows this in advance and try to compensate for this when they tweak the volatility parameter. For more complex models, we are faced with questions like: Which parameter should be adjusted and especially, by how much? Thus, these models are looked upon with scepticism which sometimes is well founded.

A danger in model building is so called overpameterization. The FI may for example find out that the SVSJ- model is an improvement over the BS-model. This can be, until a regime shift occurs [14]. A fundamental change in the behavior of market variables. Models of financial markets are ultimately models of human behavior, and the way people behave changes as opposed to physical science, where those kind of models or rather the physical behavior it models stays the same. Hence, what seemed to be a better model than the BS-model may turn out to be even worse, since it does not have the flexibility to cope with changing market conditions.

What then, is the hunt for a better model and endless chase for something unattainable? Yes, if the FI think that it should try to find one ultimate model. No, if the FI accept that all models have limitations but if they are used jointly, the same effect as having the "perfect" model can be achieved. Instead of looking after just one single model, the FI should use several models. This practise is advocated if the FI is aiming to obtain a realistic range for pricing and to understand the accompanying model risk. It is only then the FI fulfill good risk management. However, before implementing the new models, the FI needs to do a thorough analysis, for example the robustness. That is, how will the research models behave when the parameters are adjusted? Will a small tweak in any of the parameters cause a big shift in price or not?

It is also important to have in mind that when discretizing the models to perform Monte Carlo simulations, the total bias for the aﬃne jump-models is larger compared with the discretization bias for the BS-model. The discretization bias is not only introduced in the asset price but also in the volatility and the jumps.

In order to "back-out" the real risk-neutral parameters, or to perform the calibration, a suitable optimization solver, suitable meaning a global one is to be used. Theoretically, every time the market prices changes, the calculations should be remade in order to really capture the true market price of risk. However to use an eﬃ- cient global solver, possibly several times a day, requires a lot in terms of computer capacity.

Also, when we perform simulations using an aﬃne jump-model to obtain the price, it requires more computer capacity, since more variables needs to be generated and calculated. Two decades ago, this would probably not be doable within a reasonable time-frame. But nowadays the speed of the processors and memory capacity have increased enormously. Conclusively, the joint use of research models such as the SVSJ- and the SEPP-model in addition to the Black-

59 CHAPTER 10. CONCLUSIONS

Scholes model, is advocated if the FI wants to adhere to good risk management but also to obtain price- and hedge-ranges for traded OTC-products. This approach may decrease the risk even further and at the same time yield better proﬁts.

60 Appendix A

Appendix

A.1 Allocation table and relative weights

Volatility range Allocation VCTL Volatility range Allocation VCTL uS (th, σ˜ k) uB(th, σ˜ k) uS (th, σ˜ k) uB(th, σ˜ k) 0.0-5.0% 100% 0% 5% 0.0-7.5% 100% 0% 7.5% >5.0-10.0% 50% 50% 5% >7.5-10.0% 75% 25% 7.5% >10.0-15.0% 33% 66% 5% >10.0-15.0% 50% 50% 7.5% >15.0-20.0% 25% 75% 5% >15.0-20.0% 37.5% 62.5% 7.5% >20.0-25.0% 20% 80% 5% >20.0-25.0% 30% 70% 7.5% >25.0-37.5% 13% 87% 5% >25.0-37.5% 20% 80% 7.5% >37.5-50.0% 10% 90% 5% >37.5-50.0% 15% 85% 7.5% >50.0-75.0% 7% 93% 5% >50.0-75.0% 10% 90% 7.5% >75.0% 5% 95% 5% >75.0% 5% 95% 7.5%

Volatility range Allocation VCTL Volatility range Allocation VCTL uS (th, σ˜ k) uB(th, σ˜ k) uS (th, σ˜ k) uB(th, σ˜ k) 0.0-10.0% 100% 0% 10% 0.0-12.5% 100% 0% 12.5% >10.0-15.0% 66% 34% 10% 12.5-15.0% 83% 17% 12.5% >15.0-20.0% 50% 50% 10% >15.0-20.0% 63% 37% 12.5% >20.0-25.0% 40% 60% 10% >20.0-25.0% 50% 50% 12.5% >25.0-37.5% 26% 74% 10% >25.0-37.5% 33% 67% 12.5% >37.5-50.0% 20% 80% 10% >37.5-50.0% 25% 75% 12.5% >50.0-75.0% 13% 87% 10% >50.0-75.0% 17% 83% 12.5% >75.0% 5% 95% 10% >75.0% 5% 95% 12.5%

Volatility range Allocation VCTL Volatility range Allocation VCTL uS (th, σ˜ k) uB(th, σ˜ k) uS (th, σ˜ k) uB(th, σ˜ k) 0.0-15.0% 100% 0% 15% 0.0-17.5% 100% 0% 17.5% >15.0-20.0% 75% 25% 15% >17.5-20.0% 87.5% 12.5% 17.5% >20.0-25.0% 60% 40% 15% >20.0-25.0% 70% 30% 17.5% >25.0-37.5% 40% 60% 15% >25.0-37.5% 47% 53% 17.5% >37.5-50.0% 30% 70% 15% >37.5-50.0% 35% 65% 17.5% >50.0-75.0% 20% 80% 15% >50.0-75.0% 23% 77% 17.5% >75.0% 5% 95% 15% >75.0% 5% 95% 17.5%

61 APPENDIX A. APPENDIX

Volatility range Allocation VCTL uS (th, σ˜ k) uB(th, σ˜ k) 0.0-20.0% 100% 0% 20% >20.0-25.0% 80% 20% 20% >25.0-37.5% 53% 46% 20% >37.5-50.0% 40% 60% 20% >50.0-75.0% 27% 73% 20% >75.0% 5% 95% 20%

The density function

−η1y η2y fY (y) = p · η1e 1{y≥0} + q · η2e 1{y<0}, η1 > 1, η2 > 0, p, q ≥ 0, p + q = 1 { −η · η 1y ≥ = p 1e y 0 fY (y) η2y q · η2e y < 0

The cumulative distribution function(CDF)

∫ ∫ y y −η1u η2u y ≥ 0, F(y) = f (u)du = (p · η1e 1{u≥0} + q · η2e 1{u<0})du = ... −∞ −∞ ∫ ∫ y 0 y 0 −η η −η η −η 1u 2u 1u 2u 1y 0 p · η1e du + q · η2e = −pe + qe = −(pe − pe ) + q = ... 0 −∞ 0 −∞ −η −η = (1 − p) + p − pe 1y = 1 − pe 1y

∫ ∫ y y −η1u η2u y < 0, F(y) = f (u)du = (p · η1e 1{u≥0} + q · η2e 1{u<0})du = ... ∫ −∞ ∫−∞ y y y η η η 2u 2u 2u q · η2e du = q · η2e du = qe = ... −∞ −∞ −∞ η −η ∞ η = qe 2y − qe 2 = qe 2y, y < 0

{ −η 1 − pe 1y y ≥ 0 F(y) = η qe 2y y < 0

( ) −η1y η2y F(y) = 1 − pe 1{y≥0} + qe 1{y<0}

62 A.2. STANDARD DEVIATION OF LOG(S) VERSUS MATURITY

The inverse CDF When y=0, then : { −η ∗ 1 − pe 1 0 = q F(y = 0) = η ∗ qe 2 0 = q

Hence, the value q represents the border line when y=0

−η −1 − 1 − y A. y = 1 − pe 1 F (y) =⇒ −η F 1(y) = ln( ) 1 p 1 (1 − y) =⇒ F−1(y) = − ln( ) η1 p

η −1 − y B. y = qe 2 F (y) =⇒ η F 1(y) = ln( ) 2 q 1 y =⇒ F−1(y) = ln( ) η2 q

− −1 1 y 1 (1 y) F (y) = ln( )1{y

1.Generate a random number u from standard uniform distribution i.e U(0,1).

2.Compute the value y such that F(y)=u or y = F−1(u). Where: −1 1 u 1 (1−u) F (u) = ln( )1{u

A.2 Standard deviation of log(S) versus maturity

Standard deviation of log(S) under the BS−model Standard deviation of log(S) under the Heston−model Standard deviation of log(S) under the BatesLN−model 80 60 60

50 50 60 40 40 40 30 30 20 20 20 Standard deviation (%) Standard deviation (%) Standard deviation (%) 0 10 10 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Time to maturity (years) Time to maturity (years) Time to maturity (years) Standard deviation of log(S) under the BatesLDE−model Standard deviation of log(S) under the SVSJ−model Standard deviation of log(S) under the Sepp−model 80 100 80

80 60 60 60 40 40 40 20 20 20 Standard deviation (%) Standard deviation (%) Standard deviation (%) 0 0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Time to maturity (years) Time to maturity (years) Time to maturity (years)

Figure A.1. Standard deviation of log(S) versus maturity for the volatilities 10%(red star), 20%(blue circle) and 30%(black square)

63 APPENDIX A. APPENDIX

A.3 The drift (risky asset S) versus volatility

Drift of S under the BS−model Drift of S under the Heston−model Drift of S under the BatesLN−model 3.7 3.65 3.6

3.6 3.55 3.6 3.55 3.5 (%) (%) (%) δ 3.5 δ δ 3.5 3.45 Drift Drift Drift 3.4 3.45 3.4

3.3 3.4 3.35 0 5 10 15 20 25 30 0 5 10 15 20 25 30 0 5 10 15 20 25 30 Volatility (%) Volatility (%) Volatility (%) Drift of S under the BatesLDE−model Drift of S under the SVSJ−model Drift of S under the Sepp−model 3.6 3.6 3.6

3.55 3.55 3.55

3.5 3.5 3.5 (%) (%) (%) δ δ δ 3.45 3.45 3.45 Drift Drift Drift 3.4 3.4 3.4

3.35 3.35 3.35 0 5 10 15 20 25 30 0 5 10 15 20 25 30 0 5 10 15 20 25 30 Volatility (%) Volatility (%) Volatility (%)

Figure A.2. The drift of asset S versus volatility. Time to maturity 1 year.

A.4 The standard variation of log(X) versus VCTL, maturity 1 year

Standard deviation of log(X) under the BS−model Standard deviation of log(X) under the Heston−model Standard deviation of log(X) under the BatesLN−model 14 15 20

12 15 10 10 10 8 5 6 Standard deviation (%) Standard deviation (%) Standard deviation (%) 4 5 0 5 10 15 20 5 10 15 20 5 10 15 20 Volatility cap target level (%) Volatility cap target levels (%) Volatility cap target levels (%) Standard deviation of log(X) under the BatesLDE−model Standard deviation of log(X) under the SVSJ−model Standard deviation of log(X) under the Sepp−model 20 20 20

15 15 15

10 10 10 Standard deviation (%) Standard deviation (%) Standard deviation (%) 5 5 5 5 10 15 20 5 10 15 20 5 10 15 20 Volatility cap target levels (%) Volatility cap target levels (%) Volatility cap target levels (%)

Figure A.3. The standard variation of log(X) versus VCTL. Time to maturity 1 year.

A.5 The standard variation of log(X) versus VOLATILITY. Maturity 1 year. VCTL=15%

Standard deviation of log(X) under the BS−model Standard deviation of log(X) under the Heston−model Standard deviation of log(X) under the BatesLN−model 20 16 16

15 15 15 14 14 10 13 13 5 12 12 Standard deviation (%) Standard deviation (%) Standard deviation (%) 0 11 11 0 5 10 15 20 25 30 0 5 10 15 20 25 30 0 5 10 15 20 25 30 Volatility (%) Volatility (%) Volatility (%) Standard deviation of log(X) under the BatesLDE−model Standard deviation of log(X) under the SVSJ−model Standard deviation of log(X) under the Sepp−model 16 16 16

15 15 15

14 14 14

13 13 13 Standard deviation (%) Standard deviation (%) Standard deviation (%) 12 12 12 0 5 10 15 20 25 30 0 5 10 15 20 25 30 0 5 10 15 20 25 30 Volatility (%) Volatility (%) Volatility (%)

Figure A.4. The standard variation of log(X) versus VCTL. Time to maturity 1 year.

A.5.1 Kurtosis and skewness for log(S)

Table A.1. Kurtosis and skewness for log(S) Black Scholes Heston Bates LN Bates LDE SVSJ Sepp

Kurtosis 3.0041 4.5354 5.1275 4.8612 6.8947 4.8163

Skewness -0.0061 -0.82 -1.0986 -1.0275 -1.5186 -1.0151

64 A.6. THE STANDARD VARIATION OF LOG(X) VERSUS VCTL, MATURITY 1 YEAR

A.6 The standard variation of log(X) versus VCTL, maturity 1 year

Standard deviation of log(X) under the BS−model Standard deviation of log(X) under the Heston−model Standard deviation of log(X) under the BatesLN−model 14 15 20

15 15 15

Figure A.5. Standard deviation of log(X) versus VCTL. Maturity 1 year.

65 APPENDIX A. APPENDIX

A.7 Moments for log(X) versus VCTL

Table A.2. Moments for log(X) versus VCTL .

Mean BS 4.6387 4.6378 4.6360 4.6336 4.6327 4.6309 4.6309

Stdev 0.0548 0.0751 0.0947 0.1181 0.1258 0.1294 0.1291

Skewness 0.0028 -0.0061 0.0025 -0.0165 0.0109 -0.0087 -0.0044

Kurtosis 3.0244 3.0311 3.0146 3.0413 3.0236 2.9926 2.9922

Mean Heston 4.6463 4.6495 4.6490 4.6482 4.6443 4.6392 4.6372

Stdev 0.0566 0.07717 0.0950 0.1147 0.1281 0.1409 0.1488

Skewness -0.1306 -0.1037 -0.1529 -0.1899 -0.2765 -0.3396 -0.4234

Kurtosis 3.0019 2.9989 2.9812 2.9684 3.0699 3.1357 3.3225

Mean BatesLN 4.6506 4.6562 4.6574 4.6569 4.6536 4.6484 4.6438

Stdev 0.0587 0.0790 0.0967 0.1152 0.1292 0.1437 0.1523

Skewness -0.1919 -0.1512 -0.1865 -0.2566 -0.3581 -0.4587 -0.5257

Kurtosis 3.3031 3.0577 2.9410 2.9334 2.9848 3.1179 3.2249

Mean BatesLDE 4.6495 4.6544 4.6574 4.6589 4.6565 4.6530 4.6499

Stdev 0.0589 0.080 0.0992 0.1212 0.1381 0.1555 0.1667

Skewness -0.1446 -0.0688 -0.0808 -0.1117 -0.1809 -0.26595 -0.3397

Kurtosis 3.05693 2.9885 2.9364 2.8988 2.9169 2.9167 3.0070

Mean 4.649 4.6545 4.6571 4.6585 4.6566 4.6529 4.6492

Stdev Sepp 0.0592 0.0798 0.099 0.1211 0.1385 0.1557 0.1668

Skewness -0.1520 -0.0648 -0.0837 -0.1188 -0.1841 -0.2589 -0.3214

Kurtosis 3.0354 2.9735 2.9207 2.9411 2.9097 2.9130 2.9820

Mean 4.6579 4.6674 4.6722 4.6758 4.67461 4.67172 4.6687

Stdev SVSJ 0.06043 0.08080 0.0987 0.1178 0.1340 0.1497 0.1622

Skewness -0.3367 -0.1779 -0.1737 -0.2186 -0.2879 -0.3176 -0.3859

Kurtosis 3.2830 3.0376 2.9201 2.9153 2.9590 2.9517 3.0047

VCTL 5% 7.50% 10%66 12.5% 15% 17.5% 20% A.7. MOMENTS FOR LOG(X) VERSUS VCTL

Table A.3. Moments for log(X) versus VOLATILITY. VCTL=15%

Mean BS 4.6399 4.6346 4.6317 4.6311 4.6298 4.6304 4.6309 4.6300

Stdev 0.0099 0.0996 0.1370 0.1453 0.1495 0.1527 0.1538 0.1552

Skewness 0.0014 -0.0013 0.0015 0.0014 0.0066 -0.0020 0.0047 -0.0023

Kurtosis 3.0234 2.9939 3.0133 3.0183 3.0456 3.0487 3.0482 3.0304

Mean Heston 4.6421 4.6441 4.6448 4.6442 4.6463 4.6460 4.6452 4.6444

Stdev 0.1105 0.1214 0.1300 0.1355 0.1394 0.14729 0.1511 0.1543

Skewness -0.3582 -0.3116 -0.26572 -0.2289 -0.2002 -0.1323 -0.1107 -0.0921

Kurtosis 3.2517 3.1388 3.0639 3.0211 2.9887 2.9804 2.9632 2.9431

Mean BatesLN 4.6504 4.6521 4.6539 4.6555 4.6553 4.6553 4.6554 4.6539

Stdev 0.1161 0.1239 0.1311 0.1350 0.1390 0.1467 0.1501 0.1544

Skewness -0.4201 -0.3947 -0.3301 -0.3151 -0.2765 -0.2355 -0.1909 -0.1667

Kurtosis 3.1001 3.0468 2.9736 2.9712 2.9586 2.9034 2.8775 2.9206

Mean BatesLDE 4.6548 4.6565 4.6565 4.6576 4.6574 4.6560 4.6558 4.6535

Stdev 0.1279 0.1337 0.1392 0.1422 0.1453 0.1511 0.1548 0.1572

Skewness -0.2214 -0.2109 -0.1711 -0.1689 -0.1347 -0.1076 -0.0952 -0.0612

Kurtosis 2.9119 2.8998 2.8888 2.9056 2.8773 2.9149 2.9120 2.9278

Mean Sepp 4.6543 4.6560 4.6572 4.6571 4.6578 4.6563 4.6551 4.6533

Stdev 0.1279 0.1333 0.1395 0.1417 0.1454 0.1513 0.1543 0.1577

Skewness -0.2286 -0.1999 -0.1862 -0.1613 -0.1537 -0.1059 -0.0813 -0.0747

Kurtosis 2.9242 2.9035 2.8915 2.9021 2.9086 2.9015 2.8955 2.9482

Mean SVSJ 4.6736 4.6742 4.6747 4.6747 4.6745 4.6737 4.6737 4.6726

Stdev 0.1278 0.1316 0.1356 0.1378 0.1407 0.1469 0.1493 0.1528

Skewness -0.2756 -0.2867 -0.2872 -0.2698 -0.2523 -0.2312 -0.2292 -0.2042

Kurtosis 2.9049 2.9466 2.9571 2.9203 2.9201 2.8917 2.9034 2.8745

VOLATILITY 1% 10% 15% 17.5% 20% 25% 27.5% 30%

67 APPENDIX A. APPENDIX

A.8 Greeks for the VCTL-option, when the risky asset S evolves either according to the BatesLN-model and BatesLDE-model

Delta for VCTL, under the BatesLN−model Gamma for VCTL, under the BatesLN−model 1 0.5

1 0.8 0.4 0.2

0.6 0.3 0.5 0.1 Delta Delta Delta

0.4 0.2 Gamma 0 0 1 0.2 0.1 1 110 110 0.5 100 0.5 100 90 0 0 90 0 80 80 90 100 110 0 0.5 1 0 80 Time T (years) Stock price Stock price Time T (years) Time T (years) Stock price Vega for VCTL, under the BatesLN−model 0.1 0.06 15 40

0.08 40 30 0.04 10 0.06 20 20 Vega Vega Vega

Gamma 0.04 Gamma 0.02 5 0 10 0.02 1 110 0.5 100 0 0 90 0 0 80 90 100 110 0 0.5 1 0 80 80 90 100 110 0 0.5 1 Stock price Time T (years) Time T (years) Stock price Stock price Time T (years)

Figure A.6.

Delta for VCTL, under the BatesLDE−model Gamma for VCTL, under the BatesLDE−model 1 0.5

1 0.8 0.4 0.2

0.6 0.3 0.5 0.1 Delta Delta Delta

0.4 0.2 Gamma 0 0 1 0.2 0.1 1 110 110 0.5 100 0.5 100 90 0 0 90 0 80 80 90 100 110 0 0.5 1 0 80 Time T (years) Spot price Spot price Time T (years) Time T (years) Spot price Vega for VCTL, under the BatesLDE−model 0.1 0.05 15 40

0.08 0.04 40 30 10 0.06 0.03 20 20 Vega Vega Vega

Gamma 0.04 Gamma 0.02 5 0 10 0.02 0.01 1 110 0.5 100 0 0 90 0 0 80 90 100 110 0 0.5 1 0 80 80 90 100 110 0 0.5 1 Spot price Time T (years) Time T (years) Spot price Spot price Time T (years)

Figure A.7.

Vega 20

0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Time T (years)

Figure A.8. Vega for diﬀerent input volatilities vs. time for the VCTL-option under BatesLN-model. 13%(red star), 18.5%(blue-cross-hair) and 22.5%(magenta-plus).

Vega 20

0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Time T (years)

Figure A.9. Vega for diﬀerent input volatilities vs. time for the VCTL-option under BatesLDE-model. 13%(red star), 18.5%(blue-cross-hair) and 22.5%(magenta-plus).

68 A.9. GREEKS FOR THE VCTL-OPTION, WHEN THE RISKY ASSET S EVOLVES ACCORDING TO THE SVSJ-MODEL A.9 Greeks for the VCTL-option, when the risky asset S evolves according to the SVSJ-model

Delta for VCTL, under the SVSJ−model Gamma for VCTL, under the SVSJ−model 1 0.8

0.8 1 0.6 0.2 0.6 0.5 0.4 0.1 Delta Delta Delta

0.4 Gamma 0 0.2 0 1 0.2 1 110 110 0.5 100 0.5 100 90 0 0 90 0 80 80 90 100 110 0 0.5 1 0 80 Time T (years) Spot price Spot price Time T (years) Time T (years) Spot price Vega for VCTL, under the SVSJ−model 0.2 0.06 20 40

0.15 60 15 30 0.04 40 0.1 10 20 Vega 20 Vega Vega Gamma Gamma 0.02 0.05 0 5 10 1 110 0.5 100 0 0 90 0 0 80 90 100 110 0 0.5 1 0 80 80 90 100 110 0 0.5 1 Spot price Time T (years) Time T (years) Spot price Spot price Time T (years)

Figure A.10.

Vega 20

0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Time T (years)

Figure A.11. Vega for diﬀerent input volatilities vs. time for the VCTL-option under SVSJ-model. 13%(red star), 18.5%(blue-cross-hair) and 22.5%(magenta-plus).

A.10 Greeks for the VCTL-option, when the risky asset S evolves according to the Sepp-model

Delta for VCTL, under the Sepp−model Gamma for VCTL, under the Sepp−model 1 0.5

1 0.8 0.4 0.2

0.6 0.3 0.5 0.1 Delta Delta Delta

0.4 0.2 Gamma 0 0 1 0.2 0.1 1 110 110 0.5 100 0.5 100 90 0 0 90 0 80 80 90 100 110 0 0.5 1 0 80 Time T (years) Spot price Spot price Time T (years) Time T (years) Spot price Vega for VCTL, under the Sepp−model 0.1 0.06 15 40

0.08 40 30 0.04 10 0.06 20 20 Vega 0 Vega Vega

Gamma 0.04 Gamma 0.02 5 −20 10 0.02 1 110 0.5 100 0 0 90 0 0 80 90 100 110 0 0.5 1 0 80 80 90 100 110 0 0.5 1 Spot price Time T (years) Time T (years) Spot price Spot price Time T (years)

Figure A.12.

Vega 20

0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Time T (years)

Figure A.13. Vega for diﬀerent input volatilities vs. time for the VCTL-option under Sepp-model. 13%(red star), 18.5%(blue- cross-hair) and 22.5%(magenta-plus).

69 APPENDIX A. APPENDIX

A.11 Greeks for a simple European call option under aﬃne jump-diﬀusions

Delta, under the BS−model Vega, under the BS−model 1 0.5

1.5 0.8 0.4 50

1 0.6 0.3 0 Vega Delta 0.5 Delta 0.4 Delta 0.2 0 −50 1.5 0.2 0.1 1.5 1 110 1 110 100 100 0.5 90 0 0 0.5 90 0 80 80 90 100 110 0 0.5 1 1.5 0 80 Time T (years) Spot price Spot price Time T (years) Time T (years) Spot price Gamma, under the BS−model 0.08 0.04 20 40

0.06 0.03 0.2 15 30

0.04 0.02 0.1 10 20 Vega Vega Gamma Gamma Gamma 0.02 0.01 0 5 10 1.5 1 110 100 0 0 0.5 90 0 0 80 90 100 110 0 0.5 1 1.5 0 80 80 90 100 110 0 0.5 1 1.5 Spot price Time T (years) Time T (years) Spot price Spot price Time T (years) Delta, under the Heston−model Gamma, under the Heston−model 1 0.5

1 0.8 0.4 0.2

0.6 0.3 0.5 0.1 Delta Delta Delta

0.4 0.2 Gamma 0 0 1.5 0.2 0.1 1.5 1 110 1 110 100 100 0.5 90 0 0 0.5 90 0 80 80 90 100 110 0 0.5 1 1.5 0 80 Time T (years) Stock price Stock price Time T (years) Time T (years) Stock price Vega, under the Heston−model 0.08 0.05 20 50

0.04 40 0.06 60 15 0.03 40 30 0.04 10 Vega 20 Vega Vega

Gamma Gamma 0.02 20 0.02 0 5 0.01 1.5 10 1 110 100 0 0 0.5 90 0 0 80 90 100 110 0 0.5 1 1.5 0 80 80 90 100 110 0 0.5 1 1.5 Stock price Time T (years) Time T (years) Stock price Stock price Time T (years) Delta, under the BatesLN−model Gamma, under the BatesLN−model 1 0.8

0.8 1 0.6 0.2 0.6 0.5 0.4 0.1 Delta Delta Delta

0.4 Gamma 0 0.2 0 1.5 0.2 1.5 1 110 1 110 100 100 0.5 90 0 0 0.5 90 0 80 80 90 100 110 0 0.5 1 1.5 0 80 Time T (years) Spot price Spot price Time T (years) Time T (years) Spot price Vega, under the BatesLN−model 0.1 0.06 20 60

0.08 60 15 0.04 40 0.06 40 10 Vega 20 Vega Vega

Gamma 0.04 Gamma 0.02 20 0 5 0.02 1.5 1 110 100 0 0 0.5 90 0 0 80 90 100 110 0 0.5 1 1.5 0 80 80 90 100 110 0 0.5 1 1.5 Spot price Time T (years) Time T (years) Spot price Spot price Time T (years) Delta, under the BatesLDE−model Gamma, under the BatesLDE−model 1 0.8

0.8 1 0.6 0.2 0.6 0.5 0.4 0.1 Delta Delta Delta

0.06 100 15 60 0.04

0.04 50 10 40 Vega Vega Vega Gamma Gamma 0.02 0.02 0 5 20 1.5 1 110 100 0 0 0.5 90 0 0 80 90 100 110 0 0.5 1 1.5 0 80 80 90 100 110 0 0.5 1 1.5 Spot price Time T (years) Time T (years) Spot price Spot price Time T (years) Delta, under the SVSJ−model Gamma, under the SVSJ−model 1 0.8

0.8 1 0.6 0.2 0.6 0.5 0.4 0.1 Delta Delta Delta

0.4 Gamma 0 0.2 0 1.5 0.2 1.5 1 110 1 110 100 100 0.5 90 0 0 0.5 90 0 80 80 90 100 110 0 0.5 1 1.5 0 80 Time T (years) Spot price Spot price Time T (years) Time T (years) Spot price Vega, under the SVSJ−model 0.1 0.06 20 80

0.08 100 15 60 0.04 0.06 50 10 40 Vega Vega Vega

Gamma 0.04 Gamma 0.02 0 5 20 0.02 1.5 1 110 100 0 0 0.5 90 0 0 80 90 100 110 0 0.5 1 1.5 0 80 80 90 100 110 0 0.5 1 1.5 Spot price Time T (years) Time T (years) Spot price Spot price Time T (years) Delta, under the Sepp−model Gamma, under the Sepp−model 1 0.8

0.8 1 0.6 0.2 0.6 0.5 0.4 0.1 Delta Delta Delta

0.4 Gamma 0 0.2 0 1.5 0.2 1.5 1 110 1 110 100 100 0.5 90 0 0 0.5 90 0 80 80 90 100 110 0 0.5 1 1.5 0 80 Time T (years) Spot price Spot price Time T (years) Time T (years) Spot price Vega, under the Sepp−model 0.08 0.06 20 80

0.06 100 15 60 0.04

Figure A.14. The Bates LDE-model, the SVSJ-model and the Sepp-model when rho=-0.9, -0.5, 0.

70 A.12. MOMENTS FOR DIFFERENT ε

A.12 Moments for diﬀerent ε

Table A.4. Diﬀerent ε

Skewness Kurtosis Standard deviation Mean

Heston, ε=0.05 -0.02 3.03 0.1408 4.6344 Heston, ε=0.2 -0.19 2.96 0.1388 4.6417 Heston, ε=0.3 -0.31 3.14 0.1260 4.6452

BatesLN, ε=0.05 -0.04 3.01 0.1443 4.6347 BatesLN, ε=0.2 -0.15 2.89 0.1389 4.6475 BatesLN, ε=0.3 -0.30 2.98 0.1320 4.6527

BatesLDE, ε=0.05 0.014 3.00 0.1446 4.6359 BatesLDE, ε=0.2 -0.04 2.943 0.1435 4.6490 BatesLDE, ε=0.3 -0.145 2.896 0.1396 4.6556

Sepp, ε=0.05 0.0126 3.0199 0.1447 4.6355 Sepp, ε=0.2 -0.038 2.9768 0.1436 4.6486 Sepp, ε=0.3 -0.1568 2.9154 0.1395 4.6551

SVSJ, ε=0.05 0.012 3.0417 0.1432 4.6367 SVSJ, ε=0.2 0.009 3.0563 0.1443 4.6471 SVSJ, ε=0.3 -0.0040 3.0214 0.1445 4.6534

Once again, we calculate the moments of log(X) for varying ρ:

Table A.5.

Skewness Kurtosis Standard deviation Mean

Heston, ρ=-0.9 -0.5018 3.014 0.1251 4.6519 Heston, ρ=-0.5 -0.2456 3.1039 0.1288 4.6437 Heston, ρ=0 -0.0126 3.1484 0.1290 4.6321

BatesLN, ρ=-0.9 -0.4250 2.9651 0.1282 4.6576 BatesLN, ρ=-0.5 -0.2072 3.0649 0.1317 4.6459 BatesLN, ρ=0 -0.0150 3.1380 0.1331 4.6327

BatesLDE, ρ=-0.9 -0.2395 2.8654 0.1380 4.6600 BatesLDE, ρ=-0.5 -0.1140 3.0008 0.1383 4.6483 BatesLDE, ρ=0 0.0098 3.0470 0.1392 4.6317

Sepp, ρ=-0.9 -0.2417 2.8725 0.1378 4.6610 Sepp, ρ=-0.5 -0.1014 2.9821 0.1385 4.6478 Sepp, ρ=0 -0.0097 3.0365 0.1391 4.6305

SVSJ, ρ=-0.9 -0.3505 2.8968 0.1336 4.6813 SVSJ, ρ=-0.5 -0.1620 3.0451 0.1350 4.6592 SVSJ, ρ=0 0.0015 3.0571 0.1357 4.6326

71 APPENDIX A. APPENDIX

A.13 Implied volatility, 3-dimensional, diﬀerent rho:s

Implied volatility for VCTL−option under the Heston−model Implied volatility for VCTL−option under the Heston−model

0.4 0.25

0.2 0.2 0.15

Implied Volatility 0 Implied Volatility 0.1 120 120 110 3 110 3 100 2 100 2 90 1 90 1 80 0 80 0 Strike (K) Time to maturity T Strike (K) Time to maturity T Implied volatility for VCTL−option under the Heston−model Implied volatility for VCTL−option under the BatesLN−model

0.2 0.4

0.1 0.2 Implied Volatility

0 Implied Volatility 0 120 120 110 3 110 3 100 2 100 2 90 1 90 1 80 0 80 0 Strike (K) Time to maturity T Strike (K) Time to maturity T Implied volatility for VCTL−option under the BatesLN−model Implied volatility for VCTL−option under the BatesLN−model

0.4 0.2

0.2 0.1

Implied Volatility 0 Implied Volatility 0 120 120 110 3 110 3 100 2 100 2 90 1 90 1 80 0 80 0 Strike (K) Time to maturity T Strike (K) Time to maturity T

Figure A.15. The Heston model and the Bates LN-model when rho=-0.9, -0.5, 0.

72 A.13. IMPLIED VOLATILITY, 3-DIMENSIONAL, DIFFERENT RHO:S

Implied volatility for VCTL−option under the BatesLDE−model Implied volatility for VCTL−option under the BatesLDE−model

0.4 0.4

0.2 0.2

Implied Volatility 0 Implied Volatility 0 120 120 110 3 110 3 100 2 100 2 90 1 90 1 80 0 80 0 Strike (K) Time to maturity T Strike (K) Time to maturity T Implied volatility for VCTL−option under the BatesLDE−model Implied volatility for VCTL−option under the SVSJ−model

0.4 1

0.2 0.5

Implied Volatility 0 Implied Volatility 0 120 120 110 3 110 3 100 2 100 2 90 1 90 1 80 0 80 0 Strike (K) Time to maturity T Strike (K) Time to maturity T

Implied volatility for VCTL−option under the Sepp−model Implied volatility for VCTL−option under the Sepp−model

0.4 0.4

0.2 0.2

0.2

0.1

Implied Volatility 0 120 110 3 100 2 90 1 80 0 Strike (K) Time to maturity T

Figure A.16. The Bates LDE-model, the SVSJ-model and the Sepp-model when rho=-0.9, -0.5, 0.

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TRITA-MAT-E 2013:27 ISRN-KTH/MAT/E—13/27-SE