Forward Implied Volatility

Farshad Behvand Christ Church Oxford University

January 4, 2010

A thesis submitted in partial fulﬁllment of the requirements for the MSc in Mathematical Finance Abstract

We look at the pricing of structures that depend on forward volatility, such as globally floored cliquets. We start off by analysing market standard models, which prescribe the dynamics of volatility as either deterministic or instantaneous. We then move onto to analysing the Discrete Stochastic Implied Volatility (DSIV) model. This model specifies the dynamics of forward implied volatility directly at discrete times and as a result has more realism of and control over the dynamics. For each of these models we explore in detail the implied forward volatility dynamics and how to price exotic cliquet options. We then compare these prices against Totem marks, which represent actual market price information.

Acknowledgements

I would like to thank Jochen Theis for all his recommendations, inputs, support and patience. He has really helped me in developing a greater understanding of this topic. I would also like to thank my lecturers at Oxford university and colleagues at Markit, in particular Lukasz Dobrek for their support and input on various issues and topics in mathematical finance. Finally, my family and fiancéefor standing by me.

1 Contents

1 Introduction 4

2 Cliquet Options Market 6

3 Local Volatility 8

3.1 Model Outline and Dynamics ...... 8

3.2 Model Setup and Pricing ...... 9

4 Stochastic Volatility 15

4.1 Heston Stochastic Volatility Model ...... 15

4.2 Time Dependent Heston Stochastic Volatility Model ...... 21

4.2.1 Characteristic Function for the Time Dependent Heston Model 21

4.2.2 European & Forward Starting Option Prices ...... 23

4.2.3 Calibration ...... 26

4.2.4 Monte Carlo Numerical Implementation ...... 30

4.3 Exotic Cliquet Pricing ...... 32

5 Discrete Stochastic Implied Volatility Model 35

5.1 ModelOutline...... 36

5.2 Model Restrictions ...... 39

5.3 Arbitrage Freeness ...... 40

5.3.1 Volatility Conditions ...... 40

5.3.2 Existence of Risk-Neutral Measure ...... 42

2 5.3.3 Model Completeness ...... 44

5.4 Pricing...... 45

5.5 Calibration ...... 47

5.6 Exotic Cliquet Pricing ...... 49

6 Summary 51

A Appendix 54

A.1 SABR Model ...... 54

A.2 Implied Forward and Cliquet Volatility ...... 54

A.3 Totem Consensus ...... 56

A.4 Exotic Cliquet Totem Results for Heston Model ...... 57

A.5 Monte Carlo Results for Time Dependent Heston Model ...... 60

A.6 Matlab Code ...... 61

3 1 Introduction

In this thesis we address the issue of volatility modelling for derivatives that depend on “forward volatility”, such as reverse cliquets, Napoleons, accumulators and struc- tured notes with embedded forward starting features. Traditionally, smile models have been assessed according to how well they ﬁt the vanilla/spot implied volatility surface (e.g. the Local Volatility model). However, the pricing of exotic structures which have forward starting features embedded within them are more dependent on the assumptions made on the forward implied volatility dynamics than on the price of today’s vanilla options prices. A good volatility model not only has to be able to calibrate to the forward starting options but also to the spot implied volatilities. For example, on the reset date t1 a forward starting option turns into a European option. Since such options are liquidly traded, the model must be able to internally produce future implied volatility surfaces that are consistent with historic behaviour.

Many researchers and practitioners have tried to address this issue with stochastic volatility models. In the literature it is frequently stated that models of this type are well suited to price forward starting contracts, such as cliquets. However, this is not easily verified in the markets: figure 1, shows the cliquet implied volatility error (Ap- pendix A.2 has further details on calculation and table descriptions) when calibrating the Heston Model (4.1) to the vanilla options market and then pricing quarterly and annually resetting vanilla resetting cliquets of different maturities (see equation (1) for payoff definition). Clearly, the model does not match the market prices of the cliquets. Furthermore, this model with this calibration is clearly not suitable to price exotic cliquet options as it does not fit the forward implied volatility skew observed in the market.

We investigate this issue by looking at the dynamics of several diﬀerent volatility models in this thesis. We assess the quality of the models on the following criteria [8]:

• realism of and control over volatility dynamics

• ease of calibration to vanilla and forward starting options

• continuity in time - ability to price all structures for all times

4 • speed and suitability for practical use

In practice, to the author’s understanding no volatility model has all of these properties. Generally, something needs to give.

This thesis is organised as follows. In sections 3 and 4 we look at three continuous time models: the Local Volatility Model [10], the Heston Model [19] and the Time dependent Heston Model [11]. In section 5 we construct the Discrete Stochastic Implied Volatility Model (DSIV Model) [1], a risk-neutral model for the stock price such that all available information on current and future smiles can be incorporated. For each of these models we:

• describe the forward volatility dynamics implied by the model

• discuss in detail how to calibrate and price with the model

• compare the model prices of exotic cliquets against Totem market consensus

Finally, we end with a summary of our ﬁndings and outline possible future research.

(a) Annual resetting cliquet

(b) Quarterly resetting cliquet

Figure 1: S&P500 Implied cliquet volatility error

(σ ˆSV − σˆMkt)

5 2 Cliquet Options Market

Cliquets are a form of “volatility product” within the equity markets. They are contracts whose payoﬀ depends in some way on the performance of an asset over future time periods. Cliquets are often referred to as a forward volatility product and forward implied volatility surfaces can be backed out of vanilla cliquet prices (Appendix A.2). Here are the payoﬀs of the cliquet options we are going to price in this thesis [24], [2]:

• Vanilla call cliquet:

n + X Sti+1 − α (1) S i=1 ti • Napoleon:

M + X Sti+1 Cj + min − 1, 0 (2) ∀ti∈[tj−1,tj ] S j=1 ti

• Reverse cliquets:

( n )+ X Sti+1 C + min − 1, 0 (3) S i=1 ti • Globally ﬂoored cliquet:

( n )+ X Sti+1 r ∨ − 1 ∧ r (4) − S + i=1 ti

for C > 0 and α > 0

where S is the stock price, α a constant strike (moneyness), C and Cj are constant coupons or ﬁxed returns, r+ and r− are a cap and ﬂoor respectively, t0, ..., tn are the

reset dates and the ﬁnal payment is made at tn (it is also possible for the structure to pay at every reset date). The pricing and risk management of such products is not straightforward and the market has not yet settled on an agreed reference model.

6 Resetting cliquets on equity indices are liquidly traded as products in their own right; they are not traded on model marks. It is not common to have cliquets on single stock equity, however they do exist and pose further problems on modelling the discrete dividends1 within the exotic models. In this thesis we work with major equity indices and as a result have market data on both forward starting and spot implied volatility surfaces. Figure 2 shows the complete range of parameters for which we have calibration data available for our examples, which are typical of the market. Here SpotVol refers to the vanilla option data and 12MFwdVol, 6MFwdVol, etc, refers to the cliquet option data with resetting frequency of twelve months, six months, etc. Note that moneyness is K and for vanilla options this is based on the spot level. Sti

Figure 2: Instruments for which we have calibration data available

To hedge exotic cliquets we can use a combination of forward starting options, resetting cliquets, vanilla options and the underlying option. Also, to hedge our forward starting options we use a combination of vanilla options and the underlying. There- fore, it is vital that our model is able to reproduce the prices of the calibration instruments to prevent slippage.

1There is no market standard model to factor in discrete dividends into exotic option pricing; one way is to model the stock process as the pure stock process (Hans Buehler [6])

7 3 Local Volatility

The first model we review is the Local Volatility Model. It was one of the first volatility models used in practice to price resetting cliquets. Most financial institutions moved away from using this model with cliquets once they understood the dynamics of forward volatilities it produces. We start by outlining the model dynamics and the associated problems with Local Volatility Models. We then show to price with such models and use the model to price vanilla cliquets and compare these prices against the market as represented by the Totem consensus.

3.1 Model Outline and Dynamics

The local volatility model describes the stochastic evolution of the underlying stock

price St by means of a volatility term that is a deterministic function of the stock price:

dSt = (rd(t) − q(t))Stdt + σ(St, t)Stdz (5)

where rd is the instantaneous domestic interest rate and q the continuous dividend yield. The local volatility function σ(S, t) depends locally on both the time and the value of the underlying asset. The distribution of the asset price at some time t is thus no longer log-normal; rather, the goal is to determine the underlying process from the market-observed smile by determining the local volatility function. Dupire [10] showed that a unique local volatility surface can be implied by forward induction from vanilla option prices2. The implied volatility of an option with strike K expiring at T,σ ˆ(K,T ), can be seen as the average over all local volatilities that the underlying may have as time evolves until the maturity date.

Local volatility models are easy to implement and theoretically self consistent, as it is possible to perfectly replicate all vanilla option prices and to hedge every product with vanilla options. As a result of this, these models are widely used in practice to price path dependent options, such as barrier and Asian options. This comes at the cost of having unrealistic spot volatility dynamics; Hagan [26] deduced that the the dynamic behaviour of spot volatility smile and skews predicted by local volatility

2This is a special application of Gy¨ongy’s theorem [21]

8 models is exactly opposite to the behaviour observed in the market place: when the price decreases, local vol models predict that the smile shifts to higher prices; when the price increases, these models predict that the smile shifts to lower prices. In re- ality, asset prices and market smiles move in the same direction. This contradiction leads to incorrect hedges and hence incorrect pricing for exotic options, which can be arbitraged.

Local volatility models are deterministic models and produce sharply flattened future volatility smiles and skews as seen in figure 3, which shows the three month forward implied volatility skew at different reset dates, where 0M refers to the vanilla three-month option and 60M refers to an option with three months’ expiry starting in five years’ time. We can see as we go out further in time that the vol skew and curvature decays. This behaviour is not observed in the market.

A more general criticism of Dupire’s Local Volatility Model was raised by E.Ayache [12]; he points out that the model gives no explanation for the dynamics of how volatilities evolve, but is a way of tweaking the diﬀusion coeﬃcient in the Black- Scholes model to match a given set of vanilla option prices. Furthermore, he argues that it is more realistic and informative to explain the evolution of volatility through jumps or stochastic volatility, rather than having a model which changes volatility locally at a given point in time to match today’s option prices.

3.2 Model Setup and Pricing

No analytic solution for either vanilla or exotic option exists for the Local Volatility Model; numerical schemes such as ﬁnite diﬀerence methods or Monte Carlo must be adopted. Therefore, in this section we give a brief outline of how to price with this model in practice.

Fitting to vanilla volatility surfaces

The implementation of Dupire’s Local Volatility Model can be numerically delicate

∂CK,T (S,t0) because of the density-related term, ∂K2 , in the denominator. For values of the stock price very far away from the money this quantity becomes very small, and

9 (a) Local Volatility

(b) Heston Model

Figure 3: S&P500 3M implied forward volatility skew

10 the local volatility therefore remains ﬁnite and well-behaved only if the numerator approaches zero at the ‘right’ speed. This can lead to serious numerical problems. Fortunately, it is possible to recast Dupire’s Local Volatility Model in terms of implied volatilities and their derivatives [30]:

2 ∂σˆK,T ∂σˆK,T σˆ + 2τσˆK,T + 2(rd − q)KτσˆK,T σ(K,T )2 = K,T ∂T ∂K (6) 2 2 2 √ ∂σˆK,T 2 ∂ σˆK,T ∂σˆK,T 1 + Kd1τ ∂K + K τσˆK,T ∂K2 − d1 ∂K τ with

S 1 2 log K + (rd − q + 2 σˆK,T )τ d1 = √ σˆK,T τ

where τ is the time to maturity (T − t0). In practice option prices are given at a discrete set of points and in order to use (6) for calibrating the model to market data, we need to use interpolation and extrapolation in both time and strike. The local volatility model is arbitrage free given arbitrage free vanilla option prices for every expiry and maturity, so improper interpolation and extrapolation techniques may create arbitrage opportunities within the model. To avoid this we interpolate and extrapolate implied volatilities between strikes using the SABR model (Appendix A.1) and interpolate and extrapolate implied volatilities between maturities using cu- bic spline interpolation on the SABR parameters.

Cliquet pricing

We use Monte Carlo to price all cliquet structures with the Local Volatility Model. To price structures with Monte Carlo we make use of the fundamental theorem of risk neutral pricing [13]; which states that under the risk-neutral measure Q the derivative price process satisﬁes:

h R T i − rd(s)ds V (t) = EQ e t V (F ) | Ft (7) or, in other words, that the discounted derivative price process is a martingale under Q.

11 We calculate the expectation on the RHS of (7) by Monte Carlo. To do this, we simulate a discretisation of the local volatility process in time into piecewise Gaussian increments. As a derivative contract terminates with well-determined cash payments depending only on the stock price at a number of observation times, we can use (7) directly to calculate the value of the derivative.

The Monte Carlo algorithm we use to calculate the examples in this thesis works as follows:

• To avoid negative stock prices, we apply Itˆo’s lemma to log(S) and apply Euler discretisation (we only require weak convergence):

1 √ S = S exp r (t) − q(t) − σ(S , t)2 ∆t + σ(S , t) ∆t i.i.d∼ N(0, 1) t+∆t t d 2 t t i i

• For our example calculations we simulate daily on business days and perform 20,000 simulations.

• To improve performance we pre-compute the local volatility function for each time step as a piecewise linear interpolator function on the state domain from 1% to 400% of the forward (ﬂat extrapolation).

• We generate a simulated time series of stock prices, which is then passed into a generic payoﬀ to get our ﬁnal option price.

To test our implementation, we price a range of vanilla options and compare the results against the analytic prices. This will ensure that we do not have any implementation or discretisation errors.

Totem Testing

Please refer to A.3 for details on the Totem consensus, definition of % of Totem spread and table descriptions. Figure 5 shows the results of comparing the model prices against the market prices for resetting cliquets on the S&P500 (Appendix A.2). We can clearly see that the model significantly underprices vanilla cliquet options (figure 5a); we also see that only 16 out of 100 options pass totem consensus (red

12 indicates fail in figure 5b).3 Conversely, the model overprices in-the-money cliquet options, as the skew on market quoted cliquets is negative whereas the model implied skew is almost flat (see also figure 3a).

This model clearly fails to match the market and should not be used in practice to price any structure that depend on forward volatility. We see this again clearly in figure 4 where we compare the model prices for globally floored cliquet options against the Totem consensus, where red shade indicates a fail. The model pass rate is approximately 8% compared to 85% of the Time Dependent Heston model (“TD model” in figure 4).

Figure 4: S&P500 Quarterly resetting globally ﬂoored cliquet Totem test results

3To pass totem consensus the volatility diﬀerence must fall between half the total spread observed in the market.

13 (a) Implied cliquet volatility diﬀerence (ˆσLV − σˆMkt)

(b) Diﬀerence as a percentage of Totem spread

Figure 5: S&P500 resetting cliquet Totem test results

14 4 Stochastic Volatility

Stochastic volatility models are a class of models that use an additional stochastic process to drive instantaneous variance. They naturally exhibit implied volatility skew and smiles. Moreover, stochastic volatility models assume more realistic dynamics than local volatility models for the underlying; they allow for fat tails, high central peak and volatility clustering, which are all know empirical properties of ﬁnancial data.

One of the most commonly used stochastic volatility models is the Heston Model [19]. Its popularity is mainly due to the existence of a semi-analytic solution for European option prices, making the calibration fast and practical. We look at two versions of the Heston Model in this section, one with ﬂat parameters and another with time-dependent parameters.

4.1 Heston Stochastic Volatility Model

Heston [19] proposed a ﬁve parameter stochastic volatility model of the form:

p 1 dSt = (rd − q)Stdt + VtStdWt p 2 dVt = κ(θ − Vt)dt + σ VtdWt (8) √ where hdW 1, dW 2i = ρdt. In this model, κ is the speed of mean reversion, θ the t t √ long term volatility, σ is the volatility of variance, ρ correlation and V0 the short volatility. We can interpret the parameters σ and ρ as being responsible for the skew, the volatility of variance controlling the curvature and the correlation the tilt. The other three parameters control the term structure of the model; where the mean reversion controls the skewness of the curve from the short vol level to the long vol level. Note that the eﬀects of the parameters on the skew are strongly interdependent and it is very diﬃcult to put constraints and structure on these parameters such that the dynamics of implied and forward implied volatility produced can be controlled.

To analyse the forward volatility dynamics we follow the work of Bergomi [4]. In this article Bergomi analysed the structural features and dynamics of traditional volatility

15 models. He deduced the following results for the at-the-money forward volatility and skew via perturbation of the pricing equation:

• T τ, at order zero in T:

p dσˆ ρσ σˆF = V0, = √ (9) d ln K F 4 V0

1 • T τ, at order 1 in T : √ √ ρσ θ V0 − θ ρσ V0 − 3θ dσˆ ρσ σˆF = θ 1 + + + , = √ (10) 4κ 2κT θ 4κ θ d ln K F 2κ θ

1 where τ is the cutoﬀ between the short and long maturities, and is given by κ .

The first observation we make is that the short dated skew is inversely proportional to the at-the-money volatility (9), this is observed in figure 6a. We see in figure 6a that the observed convexity with respect to the spot implied volatility is larger for in-the-money strikes than out-of-the money strikes, also we notice the at-the-money level is lower, too. Furthermore, these effects get more pronounced as the forward start date increases. We do expect forward implied volatility smiles to be more convex than today’s4, however, these effects are overemphasized in the Heston model relative to actual market data. These low at-the-money and high skew scenarios effect significantly the pricing of short term resetting cliquets, as shown in figures 6b and 6c. These figures clearly show that the forward volatilities produced are significantly lower than the market except for high strikes as expected.

For short term volatilities, the volatility of variance has two diﬀerent objectives [4]:

• Static - create skewness in the distribution of ln S so as to match market smiles

• Dynamic - drive the dynamics of implied volatilities in a way which is consistent with their historical behaviour.

Bergomi observes that the calibrated vol of vol parameter in his examples is a factor of two larger than the historical value, and calibration of the model can not fulﬁll

4The price of a call option is an increasing and convex function of implied volatility, therefore uncertainty in the value of the future price increases the option value

16 (a) Forward 1-month volatility skew

(b) 1 Year, 1-month resetting cliquet volatility smile

Figure 6: Heston short term dynamics for S&P500

17 both objectives. This may be a reason why the model dynamics appear not to match the implied forward volatility dynamics as illustrated in ﬁgure 7. Figure 7 shows the one year, three month implied forward skew when calibrating the Heston model to the vanilla options and the quarterly resetting cliquet options. This ﬁgure illustrates that the volatility of variance is lower when calibrating to the cliquet options; there is less convexity and the at-the-money volatility is close to today’s implied at-the-money volatility level.

(a) One year, three month implied volatility skew

(b) Calibrated Heston parameters

Figure 7: S&P500 3-Month implied volatility skew

For long-dated options Bergomi states that all stochastic volatility models with mean reversion have increments in ln(S) that are stationary and independent at long times. 1 From (10), we expect the the skewness to decay proportionally to T . We can see this behaviour in ﬁgure 9a; the forward implied volatility skew looks very similar to today’s smile, there is no excessive convexity and lower at-the-money volatilities. Fur- thermore, 9b and 9c, show that the implied cliquet smile matches the market better

18 and we are much closer to the Totem consensus5.

Another drawback of the Heston model is that poor fits are observed when calibrating to the vanilla options; with shallow smiles for short and extremely long maturities. This is illustrated in figure 8 (Appendix A.2): the table and graph show the implied volatility calibration error for the Euro Stoxx 50. The results show significant implied volatility differences for short dated options maturing up to six months; with the model implying much higher at-the-money volatilities and lower skews. This behaviour can be avoided by putting more weight on the at-the-money volatilities during calibration, however this will make the fitting of the wings much worse, which are already several volatility points under the market. This can lead to arbitrage opportunities as the model can not reprice the hedging instruments, and thus the hedging portfolio correctly. One way to resolve this problem is to incorporate jumps or to make the parameters time dependent. We pursue the time dependent approach in the next section.

Figure 8: Euro Stoxx 50 Heston calibration error

5The results are obtained for the Heston model are at ﬁrst order in σ and are relevant for equity smiles. If ρ is small, as is the case for currency smiles, the contribution from terms of order σ2 dominates, altering the conclusions

19 (a) Forward 12-month volatility

(b) 5 Year, 12-month resetting cliquet volatility smile

Figure 9: Heston long term dynamics for S&P500

20 4.2 Time Dependent Heston Stochastic Volatility Model

We saw in the previous section that the Heston model does not give conclusive fits to the vanilla option market and stated that this could be resolved by making the parameters time dependent. Piterbarg [27] points out that “the need to relax the assumption of time-stationarity is real and serious, as models with constant coefficients are generally incapable of fitting market prices across option expiries. Calibration to options of multiple expiries is required for consistency if exotics are to be priced. This is so because exotics do not only depend on the distribution of the underlying at a particular point in time, but on its dynamics through time”. Therefore, this motivates us to look at the Time Dependent Heston Model:

√ 1 dSt = (rd(t) − q(t))Stdt + vtStdWt √ 2 dvt = κ(t)(θ(t) − vt)dt + σ(t) vtdWt (11)

1 2 where hdWt , dWt i = ρ(t)dt. We follow closely the work of Elices [11] on how to calibrate and price within this model assuming piecewise constant parameter functions κ, θ, σ and ρ. In his paper Elices outlines a general methodology to derive the

characteristic function for the distribution of (St, vt) at a time horizon where the parameters of the underlying process can change and is applied directly to the Heston model to derive analytic solutions for vanilla and forward starting options. We start by outlining this method and apply his work to price resetting cliquets and general forward starting options using Monte Carlo. We then extend the calibration to fit jointly the vanilla and resetting cliquet options, review the fits of both the Heston and Time Dependent Heston Models and finally price exotic cliquet options.

4.2.1 Characteristic Function for the Time Dependent Heston Model

Let xt is the logarithm of the stock price and vt, by [11] the time dependent characteristic function is given by ϕtntn+1 (X,V | xtn , vtn ) = exp Ctntn+1 (X,V ) + Dtntn+1 (X,V )vtn + iXxtn (12) where

21 ! κθ 1 − ge˜ −dt 2 C(X,V ) = i(r (t) − q(t))Xτ + ln + (κ − ρσXi − d)τ + C d σ2 1 − g˜ 0 κ − ρσXi + d g − ge˜ −dt D(X,V ) = σ2 1 − ge˜ −dt κ − ρσXi − d − V σ2 g˜(X,V ) = κ − ρσXi + d − V σ2 κ − ρσXi − d g(X,V ) = κ − ρσXi + d q d(X,V ) = (κ − ρσXi)2 + σ2X(i + X) (13)

with τ = ti+1 − ti. This representation is slightly diﬀerent from Heston’s original formulation [19]. Firstly, d appears with the sign change; Albrecher et al [18] prove that these representations are identical and this representation avoids all of the numerical problems associated with branches of the analytic functions appearing in (13).

The characteristic function ϕtntn+1 (X,V | xtn , vtn ) is given by the Fourier transform of the transition density function ftntn+1 ,

i(X,V )·(xtn+1 ,vtn+1 ) ϕtntn+1 (X,V | xtn , vtn ) = E e | xtn , vtn Z i(X,V )·(xtn+1 ,vtn+1 ) = e ftntn+1 (xtn+1 , vtn+1 | xtn , vtn )dxtn+1 dvtn+1 R2

The transition density functions can be reconstructed by inverse Fourier transform:

1 Z ∞ Z ∞ −i(xtn+1 ,vtn+1 )·(X,V ) ftntn+1 (xtn+1 , vtn+1 | xtn , vtn ) = e ϕtntn+1 (X,V | xtn , vtn )dXdV 2π −∞ −∞

As the transition density functions combine as

Z ft0tn+1 (xtn+1 , vtn+1 | xt0 , vt0 ) = ft0tn (xtn , vtn | xt0 , vt0 )ftntn+1 (xtn+1 , vtn+1 | xtn , vtn )dxtn dvtn R2

we can calculate ϕt0tn+1 from ϕt0tn and ϕtntn+1 using (12):

22 ϕt0tn+1 (X,V | xt0 , vt0 ) Z i(X,V )·(xtn+1 ,vtn+1 ) = e ft0tn+1 (xtn+1, vtn+1 | xt0 , vt0 )dxtn+1 dvtn+1 2 ZR Z i(X,V )·(xtn+1 ,vtn+1 ) = ft0tn (xtn , vtn | xt0 , vt0 ) e ftntn+1 (xtn+1, vtn+1 | xtn , vtn )dxtn+1 dvtn+1 dxtn dvtn 2 2 ZR R

= ft0tn (xtn , vtn | xt0 , vt0 )ϕtntn+1 (X,V | xtn , vtn )dxtn dvtn 2 ZR = ft0tn (xtn , vtn | xt0 , vt0 ) exp Ctntn+1 (X,V ) + Dtntn+1 (X,V )vtn + iXxtn dxtn dvtn R2 Z −1 i(X,i Dtntn+1 (X,V ))·(xtn ,vtn ) = Ctntn+1 (X,V ) e ft0tn (xtn , vtn | xt0 , vt0 )dxtn dvtn R2 −1 = Ctntn+1 (X,V )ϕt0tn X, i Dtntn+1 (X,V ) | xt0 , vt0 −1 −1 = exp Ctntn+1 (X,V ) + Ct0tn (X, i Dtntn+1 (X,V )) + Dt0tn (X, i Dtntn+1 (X,V ))vt0 + iXxt0

In summary the joint characteristic function is given by (12), where

−1 Ct0tn+1 (X,V ) = Ctntn+1 (X,V ) + Ct0tn (X, i Dtntn+1 (X,V )) −1 Dt0tn+1 (X,V ) = Dt0tn (X, i Dtntn+1 (X,V )) (14)

4.2.2 European & Forward Starting Option Prices

In this section we follow the work of Elices [11] to derive semi-analytic prices for European and forward starting options.

Semi-Analytic European Option Price:

The price of a European option can be broken down into two expectations,

+ C(K,T ; t0) = P (t0,T )E (ST − K) | Ft0

xT = P (t0,T ) E e 1{xT >ln K} | Ft0 − KE 1{xT >ln K} | Ft0

where P (t0,T ) is the discount factor, K is the strike and T is the maturity. To determine these expectations the marginal distribution of xT given xt0 and vt0 must

23 be determined. No analytic expression exists for this distribution, however Heston [19] derived an analytic expression for the characteristic function of this distribution. Elices [11] generalised this approach such that the methodology can be applied to the

joint distribution of xT and vT given xt0 and vt0 , from which the marginal characteristic function xT can be determined by evaluating the joint characteristic function at V = 0

iXxT ϕt0T (X | xt0 , vt0 ) = E[e | xt0 , vt0 ]

i(X,0)(xT ,vT ) = E[e | xt0 , vt0 ]

= ϕt0,T (X, 0 | xt0 , vt0 )

where the characteristic function is given by (14). The Fourier inversion for the this characteristic function is well known and given by,

1 1 Z ∞ ϕ(X)e−iX ln(K) P(xT > ln K) = + Re dX (15) 2 π 0 iX

We can use this result to calculate the second expectation, E 1{xT >ln K} | Ft0 . To xT calculate the ﬁrst expectation, E e 1{xT >ln K} | Ft0 , we deﬁne a new probability measure:

exT PeT = PT E[exT ] then

xT xT E e 1{xT >ln K} | xt0 vt0 = E[e ]Ee 1{xT >ln K} | xt0 , vt0

xT = E[e ]Pe (xT > ln K)

= FT Pe (xT > ln K)

where Pe (xT > ln K) is given by (15) and the characteristic function of PeT deﬁned by

iXxT +xT E e | xt0 , vt0 ϕ˜t0T (X | xt0 , vt0 ) = xT E [e | xt0 , vt0 ]

24 Now, we have a way of calculating both expectations the ﬁnal call option price is given by

n o C(K,T ; t0) = P (t0,T ) FT Pe (xT > ln K) − KP (xT > ln K) (16)

Semi-Analytic Forward Starting Option Price:

Let Cfwd(α, tn+1; tn) be the price of a forward starting call option, with strike α and reset date tn expiring on tn+1. This price can be given by:

xtn+1 xtn + Cfwd(α, tn+1; tn) = P (t0, tn+1)E (e − αe ) |Ft0

xtn xtn+1 −xtn + = P (t0, tn+1)E[e |Ft0 ]E (e − α) |Ft0 h i xtn x˜t ˜ = P (t0, tn+1)E[e |Ft0 ] E[e ]P(˜xt > α) − αP(˜xt > α) F tn+1 ˜ = P (t0, tn+1)Ftn P(˜xt > α) − αP(˜xt > α) (17) Ftn

where,x ˜t = xtn+1 − xtn when t > tn else equal to zero and Ft is the forward price at t.

To determine the above probabilities, the distribution of (˜xt, vt) given (xt0 , vt0 ) must be deﬁned. We calculate its characteristic function,

Z i(X,Ve )·(˜xt,vt) ˜ ϕ(X,Ve | 0, vt0 ) = e f(˜xt, vt | 0, vt0 )dx˜tdvt 2 ZR Z i(X,Ve )·(˜xt,vt) = ft0tn (vtn | xt0 , vt0 ) e ftntn+1 (˜xt, vt | 0, vtn )dx˜tdvtdvtn + 2 ZR R

= ft0tn (vtn | xt0 , vt0 )ϕt0,tn (X,Ve | 0, vtn )dvtn + R Z Ctn,tn+1 (X,Ve ) i(−iDtn,tn+1 (X,Ve ))vtn = e ft0tn (vtn | xt0 , vt0 )e dvtn R+ Ctn,tn+1 = e ϕt0,tn (0, −iDtn,tn+1 (X,Ve ))) n o = exp C(X,Ve ) + D(X,Ve )vt0 (18)

where C(X,Ve ) and D(X,Ve ) are given by

25 C(X,Ve ) = Ctn,tn+1 (X,Ve ) + Ct0,tn (0, −iDtn,tn+1 (X,Ve )) (19)

D(X,Ve ) = Dt0,tn (0, −iDtn,tn+1 (X,Ve ))

Now that we have the modiﬁed characteristic function, we can use the same pricing procedure and results from above to price the forward starting option. Note, we can also use this result to price vanilla options, by setting the ﬁrst reset date to t0. This is a very useful and simple test to check that our pricing algorithm is correct. Fur- thermore, we can price vanilla resetting cliquets as a sum of forward starting options:

N X CRC (α, tn+1; t0) = Cfwd(α, ti−1, ti)P (ti, t0) (20) i=1

4.2.3 Calibration

In the above sections we outlined semi-analytic solutions for both vanilla and forward starting options; this makes the calibration process simple, eﬃcient and practical. We can either calibrate to vanilla options, the forward starting options or a mixture of both. The model does not have suﬃcient degrees of freedom in order for us to perform a global calibration to all available market data. Therefore, we calibrate locally to the market data which is relevant to our hedging strategy. E.g. to price a semi-annual resetting reverse cliquet option, we could calibrate to the semi-annual resetting cliquets or a combination of cliquets and vanilla options, if vanilla options are used as part of the hedging strategy.

Calibration routine

The parameters in the time-dependent Heston model are not changing continuously but piecewise constant between different parameter sets (calibration node). To calibrate this model, we use a bootstrapping algorithm, where we fit each calibration node independently to all the selected vanilla and forward starting options in each period, starting from the first and iterating to the last node, solving a minimisation problem on the objective function:

26 ( N M ) X model market 2 X model market 2 min wi Cfwd (i) − Cfwd (i) + wi Cspot (i) − Cspot (i) i=1 i=1

The weights wi > 0 can be chosen freely, e.g. to give the highest priority to the at-the- money options. We keep this function constant at one in our example calculations. When the parameters of the first period has been calibrated, they are fixed and the parameters of the next calibration node are calibrated; this process continues to the last calibration node. Note that we fix the initial volatility at the first calibration node.

In our example computations we calibrate our model to three or four calibration nodes; the first is used to fit very short maturity options (1M - 6M), then next one or two to calibrate short to long dated options (12M to 60M) and the final one to calibrate very long dated options (84M-120M). We could have a calibration node for each period in our calibration instrument set, but then we would lose intuition of what our parameters imply on the dynamics of the model. To calibrate a flat parameter Heston model we simply set our calibration node to be the last maturity of all our selected calibration instruments.

Calibration Results

We define a good calibration fit as half a vol point difference or less; see Appendix A.2 for calculation details and table descriptions. Figures 10, 11 and 12 show the results of calibrating the flat and piecewise constant parameter Heston models to vanilla options, semi-annual resetting cliquets and a mixture of semi-annual resetting cliquets and at-the-money vanilla options on the FTSE 100 index. We can see, as expected, that the time dependent Heston model fits all the instruments significantly better, both for short and long term maturities.

Figure 10 shows the absolute calibration error, model implied volatility minus market implied volatility, to the vanilla options. The flat parameter model does not produce an adequate fit that could be used in practice; the differences are very significant and can lead to substantial slippage (hedging errors that arise from mispricing of the hedging instruments). However, we can see that the piecewise model with three sets of parameters produces a very good fit to the market which would be acceptable in

27 practice.

(a) Flat parameters

(b) Time dependent parameters

Figure 10: FTSE 100: Vanilla option calibration error

Figure 11 shows the absolute calibration error, model implied cliquet volatility minus market implied cliquet volatility, when fitting to the six monthly resetting cliquets. We again observe that the flat parameter model does not fit the market where the time-dependent model shows promising results. The strong fits to the vanilla cliquet market allow us to incorporate forward skew into our pricing and risk management of exotic cliquet structures.

Another desirable feature we wanted was the ability to calibrate to both vanilla and forward starting options. This is not achievable as shown in figure 12. Just adding the ATM spot volatilities has significant impact on the dynamics of the model and ultimately how well the model fits the market.

28 (a) Flat parameters

(b) Time dependent parameters

Figure 11: FTSE 100: 6M resiting cliquet calibration error

(a) Flat parameters

(b) Time dependent parameters

Figure 12: FTSE 100: 6M RC & ATM vanilla option calibration error

29 4.2.4 Monte Carlo Numerical Implementation

To price exotic cliquet structures with the Heston Models we use Monte Carlo simulation. Simulating the Heston Model using a simple discretisation scheme, such as the Euler scheme, can result in negative variance paths if the Feller condition 2κθ > σ2 is not satisﬁed; this leads to bias and ultimately mispricing. Various variance discretisation schemes have been proposed, Roger et al [28] analyse the bias associated for a range of these schemes. We follow the work of Leif Andersen [3] and use a quadratic- exponential (QE) scheme. The basic idea is to determine the distribution of the variance process V (t + ∆t) using moment matching and approximation techniques. It is well known that the variance process of the Heston model is proportional to a non-central chi-squared distribution with non-centrality parameter V (t) · n(t, t + ∆t), where n is independent of V (t) [9]. Andersen develops on this idea and states that the distribution of V (t + ∆t) can be approximated by a Gaussian variable for large values of V (t) and as a central chi-squared distribution for small values of V (t) (see [3] for full details).

We give a very brief outline of the discretisation of the stock and variance processes modifying results from [3] to incorporate time dependence. The discretisation of the variance process (QE scheme) is given by:

( a(t)(b(t) + )2 when Ψ(t) ≤ Ψ V (t + ∆t) = i c (21) −1 Ψ (ui; p(t), β(t)) else where

m(t) m(t)2 a(t) = , b(t)2 = 2Ψ(t)−1−1+p2Ψ(t)−1p2Ψ(t)−1 − 1 , Ψ(t) = 1 + b(t)2 s(t)2

with

−κ(t)(T −t) m(t) = E[V (T ) | V (t)] = θ(t) + (V (t) − θ(t))e V (t)σ(t)2e−κ(t)(T −t) s(t)2 = ar[V (T ) | V (t)] = 1 − e−κ(t)(T −t) + V κ 2 θ(t)σ(t) 2 1 − e−κ(t)(T −t) 2κ(t)

30 and ﬁnally

  0 0 ≤ ui ≤ p Ψ−1(u ; p(t), β(t)) = i −1 1−p(t) β(t) ln p ≤ ui ≤ 1  1−ui

with

Ψ(t) − 1 1 − p(t) u ∼ Uni[0, 1] , p(t) = , β(t) = , Ψ = 1.5 i Ψ(t) + 1 m(t) c

Ψc is the critical level controlling when to switch between the two distributions. An- dersen states that the choice of Ψc has very small relative eﬀects on the quality of the overall simulation scheme and is set to 1.5. We also see this in our numerical tests and get extremely good results using this scheme (see Appendix A.5 for results). For the stock discretisation we use the following discretisation scheme:

S(t + ∆t) = n p o S(t) exp K0 + K1V (t) + K2V (t + ∆t) + K3V (t) + K4V (t + ∆t) · i (22)

where

ρ(t)κ(t)θ(t) κ(t)ρ(t) 1 ρ(t) K = − ∆t , K = γ ∆t − − , 0 σ(t) 1 1 σ(t) 2 σ(t) κ(t)ρ(t) 1 ρ(t) K = γ ∆t − + ,K = γ ∆t(1 − ρ(t)2) , 2 2 σ(t) 2 σ(t) 3 1 2 K4 = γ2∆t(1 − ρ(t) )

Note that K0, ..., K4 are time dependent and γ1 = γ2 = 0.5. This deﬁnes our discretisation schemes for both the stock and variance process. For our examples we simulate daily on business days, using 20, 000 paths and pass our simulated market into our generic payoﬀ to get a price.

31 4.3 Exotic Cliquet Pricing

In this section we compare the model prices against the Totem consensus for a series of globally ﬂoored and reverse cliquets on the S&P500. To produce these results we calibrate the time dependent Heston model to the resetting cliquet options and the ﬂat parameter Heston model to the vanilla options and use the Monte Carlo scheme outlined in section 4.2.4 to calculate prices.

Figure 14 shows how far the model prices diﬀer from the Totem globally ﬂoored cliquet consensus (see Appendix A.3 for further details), where the red shade indicates a price which is more than half the total market spread (fail). We see that the time dependent Heston model passes nearly all the marks, only 13 marks out of a possible 312 fail (96% pass rate). The standard Heston model does well, too, especially for options with higher reset tenors, with a pass rate of 87.5%.

Figure 13 shows the results of the reverse cliquets. These results are surprising as the flat parameter Heston model has a pass rate of 75% compared to a pass rate of 12.5% for Heston model.6 We expected the time dependent Heston model to mach the market much closer, as this model fits the forward volatility skew observed in the market (Figure 6). A closer inspection of the individual contributions revealed that several contributors’ prices are consistently lower than the consensus and these marks are consistent with the time dependent Heston model prices. Furthermore, we notice that a majority of contributors prices are higher and in line with the flat parameter Heston model 7.

If we take a closer look at the payoﬀ deﬁnition of the reverse cliquet (3) and in particular the term

n X Sti+1 min − 1, 0 S i=1 ti we notice that this term will more negative if the volatility is higher. Therefore, any

6The Totem spreads are very tight for annual resetting cliquets, this is why we are observing such large diﬀerences. We believe the spreads in the market are wider than this as these structures are illiquid and traders will take provisions against potential risks. 7Conﬁdentiality of the data has prevented us from including these results in this section.

32 model which exhibits higher volatility will produce lower prices for reverse cliquets.

In section 4.1 we outlined that the standard Heston model calibrated to the vanilla options produces forward volatility smiles which have high skews with low at-the- money volatilities. This leads to the underpricing of resetting cliquet options with short reset tenors. As a result, this model will produce higher reverse cliquet prices. Moreover, a model which is correctly calibrated to the forward volatility smiles will produce lower reverse cliquet prices. Therefore, we believe the Totem consensus is split with the majority of the contributors pricing this structure using a stochastic volatility model calibrated to vanilla options and several sophisticated contributors incorporating forward volatility risk within their models. This is not observed in the globally ﬂoored cliquets as these options are less sensitive to extreme volatility movements.

Similar results where also found for the FTSE100 (ﬁgures 23 and 21) and Euro Stoxx 50 (ﬁgures 24 and 22) and can found in Appendix A.4.

Figure 13: S&P500 reverse cliquet Totem test results

33 (a) Monthly resetting

(b) Quarterly resetting

(d) Annual resetting

Figure 14: S&P500 globally ﬂoored cliquet Totem test results

34 5 Discrete Stochastic Implied Volatility Model

Standard stochastic volatility models such as the Heston Model prescribe dynamics for the instantaneous volatility, via stochastic diﬀerential equations. In such models, the level of forward skew is predominately determined by the correlation between spot

and volatility at t0. The DSIV model makes the implied volatility for each time step stochastic, therefore taking into account the eﬀects of stochastic future volatility and the joint distribution of the underlying prices for all time periods. This allows the DSIV model to have more control over the dynamics of future skew and smiles, i.e. it actually speciﬁes the dynamics for both the future-at-the-money volatility (FATMV) and future skew given the FATMV. However, the drawback of the model is that one gives up control of the behaviour of the model between periods. In particular, it is necessary to prove that there is no-arbitrage between periods.

As liquid price information for options is typically only available for particular time periods, the basic idea is to start with a discrete (ﬁnite) structure of times and to prescribe the propagation of the stock price between these times. This is similar to the idea of market models8 in interest rates, except that we have a single underlying asset, so there are more constraints to be satisﬁed. This will constrain our choices for the dynamics as we have to ensure that the model is arbitrage free.

To construct our model between our chosen times, we only need to deﬁne the transi-

tion kernel for those times, i.e. Pti,tj for i < j. To give a consistent transition kernel, these functions have to satisfy the composition rule (which follows from the Tower Law)

Z Ps,t(x, y)Pt,u(y, z)dy = Ps,u(x, z)

The simplest choice is to assume that the stock price and volatility level are Markov

variables for the model. Under this assumption, at each time ti the implied volatility i σ (S,K) for options of strike K expiring at ti+1 is a function of Sti and of the ATM vol level σti . Specifying this function is equivalent to specifying the transition kernel between the times ti - knowing the prices of options expiring at ti+1 for all strikes K

8In ﬁxed-income the Libor rates are the traded underliers over which options are written on, therefore models such as LMM control the dynamics of each Libor rate

35 given the stock price Sti and the at-the-money volatility σti allows us to determine the probability distribution of Sti+1 , σti+1 given (Sti , σti ), and hence the transition kernel Pti,ti+1 .

By the same argument all transition kernels from the initial time, Pt0,ti , are determined by the spot vanilla option prices. The composition rule prescribes that

Pt0,ti (x, y)Pti,ti+1 (y, z)dy = Pt0,ti+1 (x, z) (23)

9 This is not enough to fully determine Pti,ti+1 , but it places a constraint on it .

5.1 Model Outline

We will assume that we have a discrete set of times 0 = t0 < t1 < t2 < ... < tn−1 < tn = T for which we have information on spot-starting vanilla options and resetting cliquets; these will provide information on the spot and forward implied volatility skews. The basic set-up is to have two processes in the model, one for the stock and one for implied volatility for each time step. The model will be Markov with respect to these two processes: the level of the stock price and the level of the implied volatility fully determine the state of the model at any time.

Volatility Dynamics

To properly manage forward volatility risk we need a model which allows for more realism of and control over the implied forward volatility dynamics. The aim of the DSIV model is to address this issue, by allowing traders and risk managers to specify directly the implied forward volatility dynamics at each discrete step in the model. To have this ﬂexibility the at-the-money volatilities and volatility skew must be modelled.

We start by considering the at-the-money implied volatility dynamics ﬁrst. This

9If the transition kernel only depends on the spot price and we let the time step go to zero, i.e. assume that we have option prices for all expiries and all strikes, then the probability kernel becomes completely determined by spot volatilities, and we end up with a Dupire local volatility model (if the spot process is continuous).

36 is typically negatively correlated with relative stock price movements, mean revert- ing, positive and should not get too large; an exponential Gaussian process will have these properties. The following dynamics are proposed for the at-the-money volatility dynamics [1]:

1 −κ(ti+1−ti) σti+1 = σˆti+1 − 1 − e σˆti+1 − αti αti+1 αti γ p 1 2 Sti+1 σˆti+1 = αti+1 σti exp ν ti+1 − tiXi − ν (ti+1 − ti) (24) 2 Sti where,

• Xi ∼ N (0, 1),

• γ is the correlation between the stock price movement and at-the-money volatility, used to control the short term skew,

• αti is the mean reversion level of volatility process between ti and ti+1

• κ is a mean reversion speed, used to to control/ﬁt the long dated skew and curvature, and

• ν is the volatility of volatility

Each of the above parameters controls the dynamics of the at-the-money implied volatility for the period [ti+1, ti+2]. Note that the impact of γ and ν is conditional on the value of κ; if κ is large then this reduces the impact of γ and ν.

We still need to specify the dynamics of the implied volatility skew for each time step. We outlined earlier that local volatility models produce ﬂattening skews and smiles and stochastic volatility models produce very convex smiles with low at-the- money volatilities. One way to avoid this in the discrete setting is to impose time- homogeneity on the forward skews, i.e. to require the forward skew of a given tenor at any point to be similar to the spot skew. This can be achieved by making the

skew scale invariant, i.e. a function of moneyness. Let ∆i = ti+1 − ti and Fti+1 be the forward price at ti+1, we hence require:

σti+1 (x) = qti+1 (x, ∆i)σti+1 (0)

37 where K log F x = ti√+1 (25) σti+1 (0) ∆i

The volatility function skew qti+1 (x, ∆i) is deterministically given as an input σfwd(x, ∆i) σspot(x, ∆i) σfwd(x, ∆i) qti+1 (x, ∆i) = + − exp(−λti+1 ) (26) σfwd(0, ∆i) σspot(0, ∆i) σfwd(0, ∆i) where

• σfwd is the forward implied volatility taken from a model that is calibrated to the forward starting options, i.e. Heston or SABR.

• σspot is the spot implied volatility

• λti is a predetermined dampening factor.

This means that the transition kernels for different time steps are essentially the same as a function of moneyness. In general by fixing the forward volatility skew, it makes it difficult to simultaneous calibrate to implied volatilities, as there are not sufficient free parameters available.

This allows for full control of the the implied forward volatility dynamics at each time step. We can control the shape of the forward volatility skew and how much spot skew we would like to factor for. This level of ﬂexibility puts arbitrage constraints on the model, which we address in section 5.3. In our construction and examples, we do not know how to calibrate/determine λti+1 and therefore only work with forward volatility, this reduces qti+1 (x, ∆i) to:

σfwd(x, ∆i) qti+1 (x, ∆i) = (27) σfwd(0, ∆i)

Spot Dynamics

Once the implied volatility function σti+1 (K) is ﬁxed, it determines the transition kernel for the stock price from ti to ti+1 and allows us to determine the cumulative

38 distribution function from the market

Z ∞

C(K, ti+1; Sti , σti (K), ti) = (S − K)p(S, ti+1; Sti , σti (K), ti)dS (28) K Z ∞ dC(K, ti+1; Sti , σti (K), ti) = − p(S, ti+1; Sti , σti (K), ti)dS dK K

This gives:

Fti+1 (K; Sti , σti (K)) = P(Sti+1 < K) (29) dC(K, t ; S , σ (K), t ) = 1 + i+1 ti ti i dK

where C(K, ti+1; σt (K),St , ti) is the value of a call option expiring ti+1 and K = √ i i Fti exp{x · σti (0) ∆i} . Let us elaborate on this: at t0 we know σt0 (K) (implied

volatility) and St0 (spot), we can use these to solve (numerically) for the distribution

S1. Once we have the distribution for S1 we can determine the distribution of σt1 (K)

and repeat the process again to get the distribution of S2. By repeating this process we can evolve numerically the distribution of S for all periods. This completes our

deﬁnition of the model for the given times t0, ..., tn.

5.2 Model Restrictions

We construct the model at discrete times and model the implied volatility directly. This has the advantage of allowing more controllable and realistic dynamics for the volatility process, but comes at the cost of not being continuous in time. Consequently, this model is set up to price and risk manage a narrow range of exotic options, specif- ically exotic products that depend on forward volatility such as Napoleon options. It should not be used to price structures where the reset periods do not coincide with the discrete model periods. Therefore, to price within this model we must carry out a local calibration to the instruments which will be used in the hedging strategy and make sure that all reset and calibration periods are consistent.

One potential approach to avoiding this restriction is to extend the dynamics to

39 continuous time as in Sch¨onbucher [29]. Sch¨onbucher outlines a continuous stochastic implied volatility model by deriving the dynamics of the volatility and spot process from the no-arbitrage constraints the model has to satisfy. The problem with this approach is that the dynamics are limited and one loses the ability to control the volatility dynamics, which is the point of the DSIV model. In section 5.3 we explore how one can extend the dynamics of the DSIV model to continuous time and what constraints this imposes on the stock process.

5.3 Arbitrage Freeness

So far we have outlined the model and discussed its limitations. In this section we look at arbitrage freeness and model completeness.

To prove no-arbitrage we make use of the First and Second Fundamental Theorems of Asset Pricing [13]:

First Fundamental Theorem of Asset Pricing: A market model has a risk-neutral probability measure if, and only if it does not admit arbitrage.

Second Fundamental Theorem of Asset Pricing: An arbitrage free market model is complete if, and only if the risk-neutral probability measure is unique.

We show no arbitrage by extending the dynamics of the stock process to continuous time, and constructing a martingale measure. Subsequently we discuss model completeness.

A pre-condition for the existence of an arbitrage free extension of the model to continuous time is the absence of arbitrage in the discrete time model. As we model implied volatility this is not obvious and needs veriﬁcation. We start oﬀ by outlining the conditions that must be met by the volatility process to avoid arbitrage.

5.3.1 Volatility Conditions

For the implied volatility process not to admit arbitrage in discrete time, we need [14]:

40 • the at-the-money implied volatility to be non-negative: σti+1 > 0

• call and put spreads to be non-negative:

dC ∂P ≤ 0 , ≥ 0 dK ∂K • calls and puts to have positive convexity:

d2C d2P ≥ 0 , ≥ 0 dK2 dK2

For the ﬁrst no-arbitrage condition

σti+1 > 0 1 −κ∆i ⇔ σˆti+1 − 1 − e σˆti+1 − αti αti+1 > 0 αti 1 −κ∆i −κ∆i ⇔ σˆti+1 e + αti αti+1 1 − e > 0 αti 1 −κ∆i −κ∆i ⇔ σˆti+1 e > −αti+1 1 − e αti αt >0 i κ∆i ⇐⇒ σˆti+1 > −αti αti+1 e − 1 γ α >0 ti+1 p 1 Sti+1 ⇐⇒ σ exp ν ∆ X − ν2∆ > −α eκ∆i − 1 ti i i 2 i S ti ti | {z } <0

This is clearly satisﬁed if σti and Sti , Sti+1 are positive, which we can assume by induction on i. The second and third conditions limit the freedom in choosing the function qti+1 (x, ∆i) in (27). To construct qti+1 (x, ∆i), more speciﬁcally, σfwd(x, ∆i) and

σfwd(0, ∆i), we calibrate the Heston model to forward starting options with reset date

ti and expiry date ti+1. Once the model is calibrated we can compute σfwd(x, ∆i) from √ the model prices of forward starting options with strike K = Fti exp{x·σfwd(0, ∆i) ∆i}.

[23] and [14] show that stochastic volatility models such as the Heston model produce arbitrage free volatility smiles, so σfwd(x, ∆i) fulﬁls conditions one to three. Fur- σfwd(0,∆i) thermore, σti+1 (x), is given by scaling σfwd(x, ∆i) by a positive constant ; σti+1 (0)

therefore to show that σti+1 (x) meets conditions two and three, we must show that the scaled implied volatility surface is a valid implied volatility surface. Time constraints of this thesis have prevented us from doing so, but we have provided a numerical

41 illustration (ﬁgure 15) that σti+1 (x) is a valid volatility surface when qti+1 (x, ∆i) is represented by the Heston model. Figure 15 shows that even under extreme scaling, 10% and 300%, we satisfy conditions two and three.

dC d2C (a) dK ≤ 0 (b) dK2 ≥ 0

dP d2P (c) dK ≥ 0 (d) dK2 ≥ 0

Figure 15: No arbitrage conditions for scaled one year implied volatility surface

5.3.2 Existence of Risk-Neutral Measure

We start by extending the dynamics of the stock price process to a continuous process. The existence of a continuous extension can be shown in two diﬀerent ways:

(i) As the implied volatility is ﬁxed at time ti for expiry ti+1, we can construct a local volatility function σ (S , t; σ ), by extending the implied volatilities for ei t ti ti+1 to an implied volatilities surface for the period [ti, ti+1]. We can represent

St as a continuous time stochastic process:

42 dS t = σ (S , t; σ , ..., σ ) (30) e t t1 tn St

where σ (S , t; σ , ..., σ ) = σ (S , t; σ ) for t ≤ t < t . e t t1 tn ei t ti i i+1

The extension to an implied volatility surface, and hence the local volatility function is clearly not unique, but for our choice of the Heston model to generate forward smile we can use the Heston model to construct an arbitrage free volatility surface and thus obtain a natural extension.

We have omitted the drift for notational simplicity, this can easily be included by considering a numeraire-adjustment.

(ii) The other possibility is to the use Clark-Ocone formula [25]

Z ti+1 Sti+1 = E Sti+1 | Fti + E DtSti+1 | Ft dW (t) ti

where Dt is the Malliavin derivative. For simplicity we assume Sti+1 has zero drift, then E Sti+1 | Fti = Sti , we can now deﬁne the continuous process by

Z t St = E [Sti | Fti ] + E DtSti+1 | Ft dW (t) (31) ti

for ti ≤ t ≤ ti+1. Unfortunately the Malliavin derivative in (31) cannot be carried out analytically or even easily on a computer, so we do not pursue this any further within the constraints of this thesis.

1 R ti+1 Using (i) to construct the continuous stock price process and deﬁning Xi = √ dWs, ∆i ti we can re-write (24) and (30) as:

Z ti+1 1 Z ti+1 S = S exp σ (S , t; σ )dW 1 − σ (S , t; σ )2dt ti+1 ti ei t ti s ei t ti ti 2 ti 1 −κ∆i σti+1 = σˆti+1 − 1 − e σˆti+1 − αti αti+1 αti Z ti+1 2 γ 2 ν Sti+1 σˆti+1 = αti+1 σti exp ν dWs − ∆i ti 2 Sti

43 where dhW 1,W 2i = 0.

To construct the risk-neutral measure all derivative prices must be martingales [13]. This can be achieved by constructing a portfolio Π containing the derivatives, which

has price process V , a quantity Θ1 of the stock and a quantity Θ2 of another asset

whose value V1 depends with non-vanishing derivative on the implied volatility (forward starting options or resetting cliquet options):

Π = V − Θ1S − Θ2V1

dΠ = dV − Θ1dS − Θ2dV1

where dV dV 1 d2V dV 1 d2V d2V dV Itˆo= dt + dS + hdS, dSi + dσ + hdσ, dσi + hdS, dσi dt dS 2 dS2 dσ 2 dσ2 dSdσ

and dσ = dσti+1 for ti ≤ t < ti+1. We can explicitly construct our risk-neutral

process by equating the change in the portfolio to the risk free rate, dΠ = rdΠdt, dV and determining the market price of volatility risk [15] as long as dσ 6= 0. However, this is complicated to do analytically in our setup and can easily be done numerically during the calibration process. Eﬀectively, we assume that we are imputing the risk- neutral measure directly by ﬁtting the parameters of the process to the option prices [15]. This concludes the existence of the risk-neutral measure and from the First Fundamental Theorem of Asset Pricing that the model does not admit arbitrage.

5.3.3 Model Completeness

For the model to be complete, all the associated risks must be hedged within the model framework. The DSIV model has two factors of randomness; one on the stock process, the other on the implied volatility process. We can hedge the randomness on the stock price process using the stock and we need a form of volatility product to hedge out the randomness on the implied volatility process. This can be achieved by using either or a combination of vanilla options, forward starting options and cliquet options. So the choice of options used for hedging determines the risk-neutral measure used in pricing.

44 5.4 Pricing

So far we have outlined the model, but not discussed how to price with this model. No analytic solutions exist for vanilla or exotic options and the DSIV model must be implemented numerically. We opt for Monte Carlo due to time constraints. However, a calculation method based on representations of conditional expectation function such a tree or a grid should be viable as well and would be more appropriate for calibration purposes, as it should be considerably faster.

Monte Carlo Algorithm

Once the model is fully calibrated, the Monte Carlo algorithm works as follows. We start by setting the time grid (t0, t1, ..., tn) to exactly match the cliquet reset frequency, either monthly, quarterly, semi-annual or annual. At time t0; σt0 (K) (implied volatility) and St0 are known, from (29) we know the cumulative distribution of the stock price at time t1:

Ft1 (K; St0 ; σt0 (K)) = dC(K, t ; S , σ (K), t ) 1 + 1 t0 t0 0 = dK C(K + ∆K; t ; S , σ (K + ∆K), t ) − C(K − ∆K, t ; S , σ (K − ∆K), t ) 1 + 1 t0 t0 0 1 t0 t0 0 2∆K

To determine this distribution we generate a uniform random U1 and by inversion solve for St1 such that Ft1 (St1 ) = U1. Once we have St1 , we then generate a Gaussian random number to deduce σt1 (0)

1 −κ∆1 σt1 = σˆt1 − 1 − e (ˆσt1 − αt0 αt1 ) αt0 2 γ p ν St1 σˆt1 = αt1 σt0 exp ν ∆11 − ∆1 2 St0

and multiplying the at-the-money implied volatility by the forward skew qt1 (x, ∆1), we get σ1(K). We repeat the above process to determine S2 and carry on iteratively to generate the full path of the underlying.

45 Numerical Enhancements

To use Monte Carlo in the calibration process; the path generation must be highly eﬃcient for the calibration to be achievable in acceptable time. We immediately see that the most time consuming part of the algorithm is the computation of the distri-

bution of Sti+1 for each sample of the at-the-money volatility. This is not practical as we have to solve numerically for the distribution of S for each path and period. S Therefore, we pre-compute the distribution of ti+1 10 for all periods given a particular Fti+1 at-the-money level and then use interpolation to do the numerical inversion.

We assume the at-the-money volatility belongs to an interval [σmin, σmax] and dis- cretise this interval into N subsets. Once we know the at-the-money volatility level S we can compute the distribution of ti+1 : Fti+1

Sti+1 Feti+1 (K, σti (K)) = P < K Fti+1 dC(K, t ; σ (K), t ) = 1 + i+1 ti i dK ∂C(K, t ; σ (K), t ) ∂C(K, t ; σ (K), t ) ∂σ (K) = 1 + i+1 ti i + i+1 ti i ti ∂K ∂σ ∂K

p ∂σti (K) = N(−d ) + ∆ N 0(d ) 1 i 2 ∂K

where

1 2 ln(K) + 2 σti (K) ∆i d1 = √ σti (K) ∆i p d2 = d1 − σti (K) ∆i Z K 1 − 1 ξ2 N(ξ) = √ e 2 dξ −∞ 2π

where C(K, ti+1; σti (K), ti) is the Black call option price function; based on the forward which is equal to one in this setup.

10 If we pre-computed the distribution of Sti+1 , this will be dependent on today’s spot and forward level; but our forward is stochastic so we model the stock level relative to the forward and multiply back with the simulated forward.

46 We pre-compute the distribution by diving the probability interval [0, 1] into N subin- S tervals and then using a root ﬁnding method to solve for each ti+1 . We can then Fti+1 use interpolation to calculate intermediary points and multiplying with the simulated

Fti+1 to get Sti+1 . This will obviously lead to some numerical error, but improves signiﬁcantly the speed of calibration and makes the calibration of many reset dates practically possible.

5.5 Calibration

We get our market data from the vanilla call cliquets and spot vanilla options, see ﬁgure (2) for a full set of calibration instruments. Ideally, we will have market prices for forward starting options for each date on the time grid, we do not have this data and must imply the forward volatility dynamics from the cliquet option prices11. We calibrate each period independently to the cliquet options, starting from the ﬁrst to last solving a minimisation problem on the objective function:

( N ) X model market 2 min wi σfwd (i) − σfwd (i) i=1

The weights wi are chosen to give the highest priority to the at-the-money options. When the parameters of the ﬁrst period are calibrated, they are ﬁxed and the parameters of the next period are calibrated, this process continues until the last period.

This calibration algorithm is quite primitive as we ﬁt all parameters without any constraints to the data and parameters. Further improvements can be made by es- timating some of the parameters from the market data, this will reduce the dimen- sionality of the optimisation problem. Also, due to the complexity of this model and data requirements, we cannot calibrate the model on the ﬂy, it will have to be done once a day with traders having a view on how the parameters change intra-day.

In our examples we use 20,000 simulation paths in the calibration and 100,000 paths when pricing the exotic cliquet structures. Also, we only price structures up to three

years in maturity and the forward skew qti+1 (x, ∆i) only changes three times during

this period. The proﬁle of qti+1 (x, ∆i) changes as follows:

11We can compute forward starting options for annual resetting cliquets

47 • For implied forward volatility smiles less than one year in maturity we use the spot implied volatility skew. For example, for a quarterly resetting cliquet, we will ﬁx the three month implied volatility skew for all implied forward volatility skews less than one year of maturity, i.e. the six month, three month implied forward volatility skew will look like today’s implied volatility skew.

• For implied forward volatility skews greater than and equal to one year and less than two years in maturity we use the one year reset implied forward volatility skew (i.e one year, three month implied forward volatility skew).

• Finally for implied forward volatility smiles greater than equal to two years we use the two year reset implied forward volatility skew.

Calibration Results

Figures 16, 17 and 18 show the calibration error, model cliquet volatility minus implied cliquet volatility, for annual and semi-annual resetting cliquets. The calibration ﬁts seem good and would be acceptable in practice, although the ﬁts are not as good as the time dependent Heston model. This could be improved by increasing the number of simulations and changing qti+1 (x, ∆i) more often. Also, time limitations of the thesis have prevented us from looking at the joint calibration of forward and vanilla options and we leave this for future research.

(a) Annual resetting (b) Semi-annual resetting

Figure 16: S&P500 resetting cliquet calibration error

48 (a) Annual resetting (b) Semi-annual resetting

Figure 17: FTSE100 resetting cliquet calibration error

(a) Annual resetting (b) Semi-annual resetting

Figure 18: Euro Stoxx 50 resetting cliquet calibration error

5.6 Exotic Cliquet Pricing

Figures 19 and 20 show that the exotic cliquet prices for both the globally ﬂoored and reverse cliquets are in line with the Totem consensus. For globally ﬂoored cliquets we have a pass rate of 91% across the three major equity indices and no price being over a whole spread out. The reverse cliquets have a pass rate of 50%, which is satisfactory, however for higher reset frequency the results seem better with only two quotes being a whole spread out.

In summary this model is well suited to price such structures as indicated by the Totem tests. More research and testing is required to improve the calibration and performance of the model if it is to be used in practice.

49 (a) Quarterly resetting

(b) Semi-annual resetting

Figure 19: Globally ﬂoored cliquet Totem test results

50 Figure 20: Reverse cliquet Totem test results

6 Summary

In this thesis we have looked at modelling forward implied volatility by analysing three diﬀerent volatility models. We started oﬀ, by stating that the ideal model would give us control over volatility dynamics, calibrate to both forward and vanilla options, be continuous and practical. For each of the models, we discussed these criteria and found that none could satisfy all:

• The local volatility model: a continuous time deterministic volatility model which has the ideal quality of perfectly replicating the vanilla options. But this model incorrectly speciﬁes the dynamics of forward volatility and as a result misprices the resetting and exotic cliquets.

• The Time Dependent Heston Model: a continuous time model with an additional stochastic process to drive the instantaneous variance. This model gave good ﬁts to both the vanilla and resetting cliquets market data, but was unable to calibrate to both sets of data. Moreover, model prices for the exotic cliquet markets matched Totem concensus reasonably well. However, the limitation of this model is that we do not have control over the dynamics of forward volatility and are model implied.

51 • The DSIV model: models the implied volatility for each time step as stochastic; we specify the dynamics of the at-the-money implied volatility and have the ﬂexibility of specifying the volatility skew. This gives very good control over the dynamics of forward implied volatility. The model has the limitation that it can only price a very limited range of exotic structures as it is not continuous. In our implementation, we used Monte Carlo and this obviously has practical issues, but we believe a grid implementation would resolve these issues.

The DSIV model, like the time dependent Heston model, gave very good ﬁts to the resetting cliquet market data and performed well in Totem tests.

Our summary of the models above indicate that we have two potential models to use in pricing of exotic structures. To further test these models, we must look at the behaviour of the Greeks. These ultimately deﬁne our hedging strategy and how well suited the model is for risk managing an exotic structure.

There have also been recent developments in modelling volatility. H.Buehler [7], J.Gatheral [16] and L.Bergomi [5] have worked on a new breed of models, known as “Variance Swap Models”. Variance swaps are contracts on realised variance and inher- ently strike independent, which makes them natural instruments to hedge volatility products. Furthermore, forward variance swaps can be used as natural hedging instruments for cliquets and other forward starting exotic products. One particular model that we would like discuss brieﬂy at is Bergomi variance swap model [5].

Bergomi argues that variance swaps should be considered as a hedging instrument and have their own dynamics. He proposes a model that allows independently set the requirements for:

• The dynamics of variance swap volatilities.

• The level of short-term forward skew.

• The correlation between the underlying and the short and long variance swap volatilities.

The model setup is quite similar to DSIV model instead of modelling the implied i volatility it models the forward variance (ξt) at each time step as stochastic:

52 i i −k1(Ti−t) −k2(Ti−t) ξt = ξt(0) exp ω e Xt + θe Yt − ω2 e−2k1(Ti−t) [X2] + θ2e−2k2(Ti−t) [Y 2] + 2θe−(k1+k2)(Ti−t) [X Y ] (32) 2 E t E t E t t

i where ξt is a random process until t = ti+1, when t reaches ti+1 the Variance Swap i variance is known and is equal to ξt+1.

This model would have the same beneﬁts as the DSIV model but would have the advantage that variance swaps are more liquidly traded and are natural hedging instruments. Also, as realised variance rather than implied volatility is modelled the model is simpler to handle and no arbitrage conditions are easier to satisfy.

53 A Appendix

A.1 SABR Model

The SABR model [26] is a stochastic volatility model on the forward Ft:

β dFt = σtFt dWt

dσt = ασtdZt (33)

where Wt and Zt are Brownian motions such that hdWt, dZti = ρdt, α ≥ 0 is the volatility of volatility and β ∈ [0, 1] the skewness. Note that β can also be used to control the distribution of the forward process, i.e. if it is set to 1 we have a log- normal model.

The SABR model is a stochastic volatility model used in the interest rate and other derivative markets for pricing vanilla options. Its main application to date has been in the fitting and description of volatility skews, because it allows Black’s formula still to be used to value European options, but with a volatility value which depends on strike and the model’s other parameters. The advantage of the SABR model over other stochastic volatility models is that it is the simplest stochastic volatility model which is homogenous in the forward rate and the volatility. The SABR model can be used to accurately fit the implied volatility curves observed in the marketplace for any single exercise date. More importantly, the delta and vega risks are in line with market observations [26], making the SABR model an effective means to manage the smile risk in markets where each asset only has a single exercise date.

A.2 Implied Forward and Cliquet Volatility

If we work in the Black Scholes (BS) framework, we can easily derive an analytic solution for a forward starting call option:

S rd(ti+1−ti) ti+1 Cfwd(α, ti+1; ti), = e C , α, σti,ti+1 , ti+1; ti (34) Sti

54 where C is the Black call option price function and Sti+1 and Sti are given by the forward price at time ti+1 and ti. It is common practice to quote vanilla, option and cliquet prices in terms of volatility, as it is more intuitive to understand. The implied forward volatilityσ ˆfwd, is the ﬁtted forward volatility σti+1,ti+2 which matches forward starting option prices Cb fwd:

n o σˆfwd = min Cb fwd − Cfwd(σt ,t ) (35) σ i i+1

Similarly, we can also derive analytic resetting cliquet (RC) prices in the Black Sc- holes framework:

t Xn CRC = Cfwd(α, ti+1; ti) i=0

and the implied cliquet volatilityσ ˆRC is given by:

n o σˆRC = min Cb RC − CRC (σt0,tn ) (36) σ

The following terminology is used in Figures: 1, 5(a), 6(c), 8, 9(c), 10, 11, 12, 16, 17 and 18:

• Moneyness α is the ratio K . Sti • Term refers to is the expiry of the option in months.

• Implied vanilla/cliquet volatility error means the diﬀerence in volatility points

between the model price and the market:σ ˆMdl − σˆMkt.

• Two diﬀerent shades are used in the table to highlight any errors that are greater than half a volatility point. A light yellow shade for anything greater than 0.5% and a light blue shade for anything lower than (-0.5%). We believe any errors less than half a volatility point is reasonable.

55 A.3 Totem Consensus

The Totem consensus is a monthly service which provides the major market makers in OTC derivatives with deﬁnitive consensus market prices. More than ﬁfty of the leading banks, trading houses and other investment professionals use the Totem service as the key independent check of their trading book valuations.

The Totem service generates a mid market price and spread for a particular structure by collating quotes from leading investment banks that contribute to this service. Any outliers are removed and it is ensured that the market data which has gone into the pricing of these structures is consistent among all contributors; this avoids any ambiguity in the prices.

The conﬁdentiality of the data prevents us from disclosing actual contributor marks or the consensus and are limited to disclosing a measure called % of Totem spread:

|ModelPrice − TotemConsensus| % of Totem spread = (37) TotemSpread where TotemSpread = MaxAcceptedPrice − MinAcceptedPrice

If this measure is below 50% then the model price is within the Totem acceptance tolerance and can be considered a pass. If this measure is signiﬁcantly above 50%, then this is not satisfactory to pass the Totem consensus. The spreads of Totem are extremely tight for some exotic products and this may not be observed in actual quoted dealer prices as traders will take provisions against model and future risks.

The following terminology is used in Figures: 4, 5, 13, 14, 19, 20, 21, 22, 23 and 24:

• LV Model price - the local volatility model price.

• Flat model price - the ﬂat parameter Heston model price. This model is calibrated to the vanilla options market data.

• TD model price - the time dependent Heston model price. This model is calibrated to the resetting cliquets market data.

56 • A red shade indicates a point that falls outside Totem consensus.

A.4 Exotic Cliquet Totem Results for Heston Model

Note, to reduce the calibration error of the ﬂat parameter Heston Model, we do not include the four and ﬁve year vanilla option prices in the calibration.

Figure 21: FTSE100 reverse cliquet Totem test results

Figure 22: Euro Stoxx 50 reverse cliquet Totem test results

57 (a) Monthly resetting

(b) Quarterly resetting

(d) Annual resetting

Figure 23: FTSE100 globally ﬂoored cliquet Totem test results

58 (a) Monthly resetting

(b) Quarterly resetting

(d) Annual resetting

Figure 24: Euro Stoxx 50 globally ﬂoored cliquet Totem test results

59 A.5 Monte Carlo Results for Time Dependent Heston Model

Figure 25 shows the comparison of the semi-analytic price against the Monte Carlo price for a series of three month forward starting options on the S&P500. Using daily time steps and 20,000 simulations the Monte Carlo prices are in line with the semi- analytic prices (all within two Monte Carlo standard errors).

(a) Semi-analytic forward price

(b) Monte Carlo forward price

Figure 25: S&P500 three month forward numerical price comparison

60 A.6 Matlab Code

We have attached a sample of the main classes at the end of this document for reference purposes. For all the code used in this thesis please refer to supplementary CD.

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