Numerical advances in pricing forward sensitive equity derivatives

Fiodar Kilin Centre for Practical Quantitative Finance Frankfurt School of Finance & Management

A thesis submitted for the degree of Doktor rerum politicarum

Supervisor: Prof. Dr. Uwe Wystup

Submission date: July 7, 2009 Acknowledgements

I am deeply grateful to my supervisor, Professor Uwe Wystup for many helpful suggestions, important advice and constant encouragement during this work. I also wish to thank Professor Rolf Poulsen, Professor Wolfgang Schmidt, Professor Robert Tompkins, Professor Eckhard Platen, Dr. Bernd Engel- mann, Dr. Peter Schwendner, Dr. Michael Dirkmann, Dr. Matthias Fen- gler, Dr. Friedrich Hubalek, Dr. Antonis Papapantoleon, Morten Nalholm and Martin Keller-Ressel for useful comments and discussions. I wish to express my gratitude to the staff of the Frankfurt School of Finance & Management that have provided an excellent research environment. The financial support of Quanteam AG is gratefully acknowledged. Contents

List of Figures v

List of Tables vii

1 Introduction 1

2 Models 3 2.1 ...... 3 2.2 Bates model ...... 4 2.3 Barndorff-Nielsen&Shephard model with the Gamma-Ornstein-Uhlenbeck latent state ...... 5 2.4 Levy models with stochastic time ...... 6 2.5 Bergomi model ...... 7

3 Pricing exotic options in models 11 3.1 Model choice ...... 11 3.2 Calibration and pricing of vanilla options ...... 13 3.3 Monte-Carlo pricing of exotic options ...... 17 3.4 Analytical pricing of exotic options ...... 18

4 Accelerating the calibration of stochastic volatility models 21 4.1 Characteristic functions ...... 22 4.2 Pricing methods ...... 24 4.3 Caching technique ...... 28 4.4 Numerical experiment ...... 30

iii CONTENTS

5 Forward-start options in the Barndorff-Nielsen&Shephard model 35 5.1 Derivation ...... 36 5.2 Numerical examples ...... 39

6 On the cost of poor volatility modeling – The case of cliquets 51 6.1 Forward volatility and forward skew ...... 51 6.2 Cliquet options ...... 53 6.3 Price comparison ...... 56 6.4 Hedge performance ...... 58

7 Conclusion 65

Bibliography 67

iv List of Figures

5.1 Price of a forward-start call in the Barndorff-Nielsen&Shephard model, sensitivity to comovement ...... 40 5.2 Price of a forward-start call in the Barndorff-Nielsen&Shephard model, sensitivity to mean-reversion rate ...... 40 5.3 Price of a forward-start call in the Barndorff-Nielsen&Shephard model, sensitivity to exponential law parameter ...... 41 5.4 Price of a forward-start call in the Barndorff-Nielsen&Shephard model, sensitivity to Poisson intensity ...... 41 5.5 Price of a forward-start call in the Barndorff-Nielsen&Shephard model, sensitivity to initial latent state ...... 42 5.6 Price and theta of a forward-start call in the Barndorff-Nielsen&Shephard model ...... 42

6.1 Histogram of relative cumulative hedging errors of a call spread cliquet. Hedging with recalibration using delta and short-term vega ...... 62 6.2 Histogram of relative cumulative hedging errors of a call spread cliquet. Hedging with constant parameters using delta and short-term vega ... 63 6.3 Histogram of relative cumulative hedging errors of a call spread cliquet. Hedging with constant parameters using delta and parallel shift vega .. 64

v LIST OF FIGURES

vi List of Tables

3.1 Models examined and recommended in some of the existing literature on model risk...... 14

4.1 Comparison of three methods for pricing vanilla options in stochastic volatility models. Grid sizes that are needed to obtain some benchmark accuracy levels ...... 32 4.2 Comparison of three methods for pricing vanilla options in stochastic volatility models. Average calibration time ...... 33

5.1 Price of a forward-start call in the Barndorff-Nielsen&Shephard model, sensitivity to comovement ...... 44 5.2 Price of a forward-start call in the Barndorff-Nielsen&Shephard model, sensitivity to mean-reversion rate ...... 45 5.3 Price of a forward-start call in the Barndorff-Nielsen&Shephard model, sensitivity to exponential law parameter ...... 46 5.4 Price of a forward-start call in the Barndorff-Nielsen&Shephard model, sensitivity to Poisson intensity ...... 47 5.5 Price of a forward-start call in the Barndorff-Nielsen&Shephard model, sensitivity to initial latent state ...... 48 5.6 Price and theta of a forward-start call in the Barndorff-Nielsen&Shephard model ...... 49

6.1 Input parameters for pricing and hedging tests ...... 56 6.2 Input parameters for pricing and hedging tests ...... 56 6.3 Calibrated Heston parameters...... 57 6.4 Calibrated Barndorff-Nielsen&Shephard parameters...... 57

vii LIST OF TABLES

6.5 Calibrated VGSA parameters...... 57 6.6 Comparison of theoretical cliquet values. Scenario 1...... 58 6.7 Comparison of theoretical cliquet values. Scenario 2...... 58 6.8 Comparison of theoretical cliquet values. Scenario 3...... 59 6.9 Comparison of theoretical cliquet values. Scenario 4...... 59 6.10 Comparison of theoretical cliquet values. Scenario 5...... 59

viii GLOSSARY

Glossary

This glossary contains abbreviations for the models used in this thesis, the full name and a reference to a page, where they are first defined.

Bates 4 Bergomi 7 BNS (Barndorff-Nielsen&Shephard) 36 BNS-GOU (Barndorff-Nielsen&Shephard model with the Gamma-Ornstein-Uhlenbeck latent state) 5 CEV (Constant elasticity of variance) 8 Heston 3 NIG-CIR (Normal Inverse Gaussian with the Cox-Ingersoll-Ross stochastic clock) 6 NIG-GOU (Normal Inverse Gaussian with the Gamma-Ornstein-Uhlenbeck stochastic clock) 6 VG-CIR (Variance Gamma with the Cox-Ingersoll-Ross stochastic clock 6 VG-GOU (Variance Gamma with the Gamma-Ornstein-Uhlenbeck stochastic clock) 6 VGSA (Variance Gamma with stochastic arrival) 6

ix GLOSSARY

x 1

Introduction

The main purpose of this thesis is to describe new numerical and analytical methods for pricing equity derivatives in stochastic volatility models. We introduce an inno- vative method of high-speed calibration for this class of models. We also develop an efficient method of pricing forward-start options. Using this method we derive a ready- to-implement formula for this type of contracts. The described results are supported by numerical examples and description of thorough tests. Before describing the main results, we provide a literature overview that motivates introducing the new calibration and pricing techniques. We analyse applications of the introduced methods in affine stochastic volatility models (e.g., the models of Heston, Bates, Barndorff-Nielsen&Shephard) and the Levy models with stochastic time change. Furthermore, we illustrate the use of these methods for model risk analysis and evaluating hedging performance of stochastic volatility models. Among the contracts which we consider are forward start options, variance swaps, vola- tility swaps, options on realized variance, reverse cliquets, accumulators and Napoleons. The obtained results are important in both practical and academic applications. Attrac- tive benefits from using the described innovative methods in the quantitative analysis departments of investment banks are speed and accuracy of calculations, stability of calibrated model parameters in time direction, stability and reliability of . From the point of view of academic research the introduced techniques significantly acceler- ate experiments that require numerous recalculations of prices in different models and under different market conditions.

1 1. INTRODUCTION

The advantages of stochastic volatility models1 over models are the pos- sibility to produce a realistic forward smile, the leverage effect, the volatility clustering, the possibility to fit the whole market surface with five or six param- eters only, the possibility of hedging second order greeks with respect to volatility (i.e. volga and vanna). A natural question is often posed: Why are stochastic volatility models not used ex- tensively in equity derivatives modeling, although they possess a lot of theoretical ad- vantages? We point out following reasons for this situation: (1) In calm markets (e.g. 2004-2007), the local volatility model performs quite well and it is easier to handle. This statement is corroborated by tests described in Engelmann et al.(2006b). Mercurio & Morini(2009) also show that that the dynamic and hedging behaviors of local volatility and stochastic volatility models are not qualitatively as different as it is assumed in current market wisdom. (2) Application of stochastic volatility models often leads to calibration bias in cases of steep implied volatility skew. This bias negatively affects the quality of the valuation of down-and-out puts and makes static hedge strategies for reverse-knock-out options even more complicated. This problem is described in details in Engelmann et al.(2006a). (3) Greeks in a local volatility model are intuitive. Sen- sitivities in the stochastic volatility models are often not directly linked to observable market parameters. In giving these reasons we do not intend to advocate the local volatility model. However we would like to point out that practitioners still have not made their final choice between local volatility and stochastic volatility models. This fact motivates further model risk research which is also a component of this thesis. This thesis is organized as follows: In Chapter 2 we describe stochastic volatility models that are used in this thesis. In Chapter 3 we describe the steps needed to price exotic options in stochastic volatility models and provide an overview of existing numerical techniques typically used to accomplish these steps. Chapter 4 introduces an innovative method of accelerating the calibration of stochastic volatility models. A formula for pricing forward-start options in the Barndorff-Nielsen&Shephard model is derived in Chapter 5. An application of our numerical methods in assessment of model risk and hedging experiments is given in Chapter 6. Chapter 7 concludes.

1Some authors use the terms stochastic volatility model and Heston model as synonyms. In this thesis we we use the notion stochastic volatility for all stochastic volatility models, not only for the Heston model.

2 2

Models

In this chapter we describe stochastic volatility models that are used in this thesis. A further overview of the most promising stochastic volatility models can be found in Schoutens et al.(2004) and in Cont & Tankov(2003). The popular models we consider are representative of three different approaches to model volatility clustering, namely, diffusion models with stochastic volatility, non- Gaussian Ornstein-Uhlenbeck-based models and Levy models with a stochastic time- change. Specifically, we consider the models by Heston(1993), Bergomi(2005), Barndorff- Nielsen & Shephard(2001) and Carr et al.(2003). Here we outline the risk-neutral dy- namics in these models and describe the influence of each model parameter on the form and dynamics of the implied volatility surface.

2.1 Heston model

The risk-neutral dynamics in the Heston model is

dSt = rdt + σtdWt,S0 ≥ 0, (2.1) St where

2 2 ˜ dσt = κ(η − σt )dt + θσtdWt, σ0 ≥ 0, (2.2)

Cov[dWt, dW˜ t] = ρdt. (2.3)

The parameter κ denotes the mean-reversion speed. The reciprocal of this parameter 1 τ = κ separates short and long maturities in the sense that asymptotic expressions

3 2. MODELS for the ATM (at-the-money in the sense of strike being at the ) implied volatility and skew are valid for t ¿ τ (short-term asymptotics) and t À τ (long-term asymptotics). The long-run instantaneous variance η has a major impact on the long- term implied volatility surface1 - the long-term ATM implied volatility is approximately √ proportional to η, the long-term skew (the slope of the implied volatility curves) is ap- √ proximately inverse proportional to η. The volatility of variance θ creates convexity in the implied volatility curves for each and controls the dynamics of short-term implied volatilities. The correlation parameter ρ also has two different objectives in this model: it creates the skew for each maturity and measures the correlation between the spot and and the instantaneous variance. The initial value of the instantaneous volatility σ is σ0. The state variable σ is not an observable in theory. However, in prac- tice, this variable can be observed in liquid markets using extrapolation in the implied volatility surface to the zero maturity and ATM strike. Such an extrapolation is justified by the short-term asymptotics of the implied volatility surface generated by the Heston model.

2.2 Bates model

This extension of the Heston model introduces jumps in the underlying process. The model equation is

dSt = (r − λµJ )dt + σtdWt + JtdNt,S0 ≥ 0, (2.4) St where Nt is a Poisson process with intensity λ > 0. The process Nt is independent of

Wt and W˜ t in Equation (2.2). Jt denotes the percentage jump size. It is lognormally, identically and independently distributed over time µ ¶ σ2 ln(1 + J ) ∼ N ln(1 + µ ) − J , σ2 . (2.5) t J 2 J

The volatility process σt follows the SDE (2.2) and its driving W˜ t satisfies (2.3).

1The asymptotic expressions for the ATM implied volatility and skew in the Heston model can be found in Gatheral(2005) and Bergomi(2004).

4 2.3 Barndorff-Nielsen&Shephard model with the Gamma-Ornstein-Uhlenbeck latent state

2.3 Barndorff-Nielsen&Shephard model with the Gamma- Ornstein-Uhlenbeck latent state

This model introduces simultaneous up-jumps in the volatility and down-jumps in the underlying price. The risk-neutral dynamics of the log-spot is 1

2 d(ln St) = (r − λk[−ρ] − σt /2)dt + σtdWt + ρdJλt, ρ < 0, (2.6) with the latent state following the process

2 2 dσt = −λσt dt + dJλt, (2.7) where XNt Jt = xn. (2.8) n=1

Nt is a Poisson process with intensity a, xn is an i.i.d. sequence, each xn follows an exponential law with mean 1/b. The cumulant function of J1 is

−1 k(u) = ln E(exp(−uJ1)) = −au(b + u) . (2.9)

In contrast to the Heston model, the short-term skew in the Barndorff-Nielsen&Shephard model is not explained by the dependency between the underlying and the latent state processes. In this model the short-term skew is generated only by the possibility of a jump in the underlying. Therefore the short-term skew is controlled by a triplet of parameters {a, b, ρ}. Since the comovement parameter ρ is assumed to be negative, this model produces a negative short-term skew. The long-term skew in the Barndorff- Nielsen&Shephard model is generated by the superposition of two effects: jumps in the underlying and dependency between the two state processes St and σt. Therefore the long-term skew is more sensitive to the comovement parameter ρ than the short-term skew. The Poisson intensity parameter a has both static and dynamic objectives in this model. Its main static objective is to control the level of the long term ATM-volatility. 2 The reason for this fact is that the process σt is stationary and has a marginal law that follows a Gamma distribution with mean a and variance a/b. The dynamic objective

1Barndorff-Nielsen&Shephard model with the Gamma-Ornstein-Uhlenbeck latent state is a special case of the general Barndorff-Nielsen&Shephard model. The risk-neutral dynamics for the general case is given by (5.4)-(5.5)

5 2. MODELS of the parameter a is to control the dynamics of short-term implied volatilities. An intuitive comparison between the dynamic objectives of the Poisson intensity a in the Barndorff-Nielsen&Shephard model and the volatility of variance θ in the Heston model can be seen here. However, it should be taken into account that σt in the Barndorff-

Nielsen&Shephard model is not the only source of volatility since the term ρdJλt also affects the returns.

2.4 Levy models with stochastic time

This class of models introduces stochastic volatility effects by making the time stochas- tic. The risk-neutral underlying process is modeled as exp(rt) St = S0 exp(XYt ), (2.10) E[exp(XYt )|y0] where Xt is a Levy process, Yt is a business time Z t Yt = ys ds, (2.11) 0 and yt is the rate of time change. A possible choice for Xt is the Variance Gamma (VG) process or the Normal Inverse (NIG). Typical examples of the rate of time change yt are the CIR stochastic clock

1/2 dyt = κ(η − yt)dt + λyt dWt, (2.12) or the Gamma-Ornstein-Uhlenbeck process

dyt = −λytdt + dJλt, (2.13) where dJt is defined as in (2.8). A representative of this class of models is Variance Gamma with the CIR stochastic clock also known as Variance Gamma with Stochastic Arrival (VGSA). In this model

Xt is a Variance defined by three parameters: drift θ and volatility σ of the Brownian motion and the variance ν of the Gamma process that subordinates this Brownian motion. The VGSA short-term asymptotics is controlled by the three Variance Gamma parameters only. The short-term skew is mainly defined by the drift parameter1 θ. The subordinator variance ν creates convexity in the implied volatility

1Of course, this is only true if ν is positive. If ν is zero the Variance Gamma process becomes a Brownian motion, i.e. produces no skew.

6 2.5 Bergomi model curves for short maturities. The CIR stochastic clock prevents the long-term skew from flattening too quickly1. Therefore the lower the CIR-long-term-mean parameter η, the higher the similarity between the long- and the short-term skews. The reciprocal of the CIR-mean-reversion rate κ separates short and long maturities in the same sense as the corresponding parameter in the Heston model. The dynamics of short-term ATM implied volatilities is mainly controlled by the CIR-volatility λ.

2.5 Bergomi model

When pricing exotic options the shape of future implied volatility surfaces should be taken into account. Bergomi(2004) shows the importance of this issue for cliquet op- tions. The above described models do not accurately capture the market dynamics of the implied volatility surfaces. Motivated by these observations an option pricing model where the dynamics of the variance variances is modeled directly was suggested by Bergomi(2005). Here we outline the definition of this model.

The dynamics of forward variances is modeled for discrete time intervals [Ti,Ti+1], where Ti = t0 + i∆, i = 0, ..., N. The time step ∆ is typically equal to a reset period of a cliquet option2 that we need to price and hedge. A set of the forward variance processes is defined as

Ti+1 Ti (Ti+1 − t)V − (Ti − t)V ξTi (t) = t t , 0 ≤ t ≤ T , (2.14) ∆ i where V T is the implied variance observed at time t for maturity T . t n o Ti Initial values of the implied variance swap variances Vt , i = 0, ..., N are used as 0 © ª T an input to the Bergomi model. Initial forward variances ξ i (t0), i = 0, ..., N are calculated from this input using Equation (2.14). © ª The dynamics of each forward variance process ξTi (t), i = 0, ..., N is modeled as ³ ´ Ti Ti −k1(Ti−t) −k2(Ti−t) dξ (t) = ωξ (t) e dUt + θe dWt , (2.15)

Cov[dUt, dWt] = ρdt. (2.16)

1The main disadvantage of Levy models without stochastic arrival is that the implied volatility skew flattens too quickly. See, e.g., Chapter 13 in Cont & Tankov(2003). 2Cliquet options are discussed in Section 6.2.

7 2. MODELS

T ξ i (t) is a random process in the time interval [t0,Ti]. At time Ti the variance swap variance for the interval [Ti,Ti+1] is known to be

T V i+1 = ξTi (T ) . (2.17) Ti i

The solution of the SDE (2.15) is µ ¶ ω2 ξT (t) = ξT (0) exp ωA − B , (2.18) t 2 t where

−k1(T −t) −k2(T −t) At = e Xt + θe Yt, (2.19)

£ ¤ £ ¤ −2k1(T −t) 2 2 −2k2(T −t) 2 −(k1+k2)(T −t) Bt = e E Xt + θ e E Yt + 2θe E [XtYt] , (2.20)

dXt = −k1Xtdt + dUt, (2.21) and

dYt = −k2Ytdt + dWt. (2.22)

Over the interval [Ti,Ti+1] the risk-neutral dynamics of the underlying is dS = r dt + α Sβi−1dZ , (2.23) S t i t where r is the risk-free interest rate and the dividend yield is assumed to be zero. The parameters αi and βi are recalibrated when t reaches Ti so that q i CEV ξ (Ti) = σIV (STi ,Ti+1 − Ti, F, αi, βi) , (2.24)

q i CEV CEV χ+ν ξ (Ti) = σIV (STi ,Ti+1 − Ti, 1.05F, αi, βi)−σIV (STi ,Ti+1 − Ti, 0.95F, αi, βi) , (2.25)

r(Ti+1−Ti) CEV where F = STi e and σIV (STi , τ, K, αi, βi) is the Black-Scholes implied vo- latility calculated from the price of a in the CEV model (2.23) calculated at the time Ti. Here K and τ are strike and time to maturity of this option. The idea behind this approach is to have the model parameters χ and ν set by the responsible trader, where χ resembles a general level of the skew, which is called a in

8 2.5 Bergomi model the market and ν is inverse proportional to the overall volatility level. This way the skew becomes stochastic in addition to the volatility. The underlying process is correlated with both factors that drive the forward variance process via

Cov[dZt, dUt] = ρSX dt, (2.26)

Cov[dZt, dWt] = ρSY dt. (2.27)

As noted in Bergomi(2005), the model is currently difficult to use because of a lack of suitable calibration instruments. This will change if and once options on forward ATM or variance swap volatilities become standard products, thus qualitatively expanding the set of calibration instruments.

9 2. MODELS

10 3

Pricing exotic options in stochastic volatility models

Pricing exotic options involves several steps. First of all one has to choose a model that adequately reflects the risks of a particular . Typical problems caused by an inadequate model choice are mispricing, unsatisfactory hedging, errors in estimation of market and credit risk. The second step is the calibration. This step requires fast evaluation of vanilla option and possibly other calibration instruments. At the final stage Monte-Carlo simulation, solving differential equation or some analytical method is required to calculate the price of the exotic option for the calibrated model parameters.1 In this chapter we provide an overview of the techniques that can be used in practice to complete these steps. Section 3.1 deals with the model risk issues, Section 3.2 deals with vanilla pricing and calibration, Section 3.3 describes Monte-Carlo simulation in stochastic volatility models, Section 3.4 provides some formulae for analytical pricing of exotic options that are used in the subsequent parts of this thesis.

3.1 Model choice

Investment banks use mathematical models for pricing and hedging exotic equity deriva- tives. These models do not necessarily reflect the market dynamics adequately. There- fore the exotic contracts are exposed to model risk. The term model risk can be inter- preted as “the risk arising from the use of an inadequate model”(Hull & Suo(2002)).This

1In this thesis the valuation procedures for exotics are carried out via Monte Carlo simulation.

11 3. PRICING EXOTIC OPTIONS IN STOCHASTIC VOLATILITY MODELS definition can be explained as follows. Suppose an exotic option should be hedged with liquid hedging instruments. If the dynamics of market prices of the hedging instruments contradicts model assumptions, the initial hedging strategy cannot be completed. It turns out that the initial price of the exotic option was false because this price was calculated based on assuming the existence of a hedging strategy, which cannot be ex- ecuted in practice. Estimation of model risk for exotic options is not unambiguous. There are at least four approaches to accomplish this task. The first approach is to test whether modeled stochastic processes fit historical market data well. The second approach is to calibrate a model at one point in time and compare model and market prices at a later time. This approach has been used in Dumas et al.(1998), Gupta & Subrahmanyam(2000) and Driessen et al.(2000). The third approach consists of a straightforward compari- son of exotic option prices in different models. Schoutens et al.(2004) report significant price differences between the models of Heston(1993), Bates(1996), Barndorff-Nielsen & Shephard(2001) and Carr et al.(2003) for barrier options, cliquets and variance swaps. Eberlein & Madan(2007) also use this approach and provide numerical justification of high model risk for cliquets, options on realized variance and options on volatility. The fourth approach is to assume that the true data-generating process is produced by a complex model that takes into account specific risks of a particular class of ex- otic contracts. The pricing and hedging performance of this model can be compared with the pricing and hedging performance of the model being tested. This approach is used in Hull & Suo(2002), Andersen & Andreasen(2001), Longstaff et al.(2001) and we also used it in Chapter 6, where we deal with cliquet options and use the Bergomi model as a benchmark model to test model risk of the Heston model, the Barndorff- Nielsen&Shephard model and Variance Gamma model with stochastic arrival. Another possibility to deal with the model choice is to analyze and compare theoretical and heuristic features of different models and analytically identify factors that deter- mine the risk of using these models. Bergomi(2004) used this approach to show a large model risk when pricing cliquet options in the Heston and jump/Levy models. An obvious disadvantage of this approach is that it does not quantify the model risk. Such an analysis is, however, a good starting point for any of the four model risk estimation approaches described above. Indeed the theoretical reasoning in Bergomi(2004) is val- idated by a pricing experiment that we present in Chapter 6 of this thesis.

12 3.2 Calibration and pricing of vanilla options

It is necessary to point out that all these approaches do not guarantee the absence of model risk in case of positive test results. If the test results, however, show that the model prices or hedge performance are too far from the benchmark levels, we can be sure that there is a substantial model risk. Another important aspect of the model risk issue is the sensitivity of hedge strategies to model choice. This problem is analyzed in Nalholm & Poulsen(2006). They carry out a simulation study and compared performance of dynamic and static hedge strategies for barrier options in Black-Scholes, Constant Elasticity of Variance (CEV), Heston, Merton and Variance Gamma models. They show that the standard deviation of hedge errors produced by static hedge strategies are more model risk sensitive than those errors arising from delta hedging the options. Poulsen et al.(2007) analyse whether the performance of risk-minimizing hedge strate- gies1 is sensitive to model risk. They quantify four sources of error: wrong martin- gale measure, parameter uncertainty, wrong data-generating process and greeks which are not executable with market products. They have found that only one of these sources, namely wrong greeks, has considerably negative effect to the performance of risk-minimizing hedge. Some papers dealing with the model risk issues conclude with recommendations for model choice in practice. Table 3.1 summarizes their arguments in favor of chosen models. The main contribution of this thesis in the analysis of model risk issues is a recom- mendation for model choice for pricing cliquet structures. We recommend to use the Bergomi model, which will be supported by our analysis in Chapter 6.

3.2 Calibration and pricing of vanilla options

A typical input for the calibration of stochastic volatility models is a set of market prices of vanilla options. During the calibration procedure a number of recalculations of model prices for this set of options is required. These recalculations are the most time-consuming part of the calibration algorithm. Therefore it is important to be able to calculate model prices of vanilla options as fast as possible. A commonly used

1A locally risk-minimizing hedge is described in El Karoui et al.(1997) and Bakshi et al.(1997)

13 3. PRICING EXOTIC OPTIONS IN STOCHASTIC VOLATILITY MODELS

Compared Recommended Main Paper Instruments models models argument

Gatheral(2005) local volatility, barrier options, Heston dynamics of the Heston cliquets, volatility surface Cont& Heston, general analysis Bates flexibility of Tankov(2003) exponential Levy, term structure, Bates, BNS, accuracy of Levy with calibration stochastic clock Detlefsen(2005) Heston, Bates barrier options, Heston over- forward-start parameterization options of the Bates model Mercurio& local volatility, general analysis both models are hedge performance Morini(2009) SABR unapt to hedging Bergomi(2004) Heston, Levy, cliquets Bergomi, dynamics of the Levy with Levy with volatility surface stochastic clock, stochastic clock Bergomi

Table 3.1: Models examined and recommended in some of the existing literature on model risk.

14 3.2 Calibration and pricing of vanilla options method to accomplish this task is to use a semi-analytical formula. Raible(2000) shows that the price V of an option with payoff w(ST ) can be calculated as Z eζR−rT ∞ V (ζ) = eiuζ L [v](R + iu) φ (iR − u) du, (3.1) 2π −∞ where

v(x) = w(e−x), (3.2) ¡ rT ¢ ζ = − ln e S0 , (3.3)

St is the underlying price, T is the maturity of the option, r is the risk-free interest rate, the dividend yield is assumed to be zero, L [v](z) is the bilateral Laplace transform of v for z ∈ C Z ∞ L [v](z) = e−zxv(x) dx, (3.4) −∞

φ(ω) is the characteristic function of XT

iωXT φ(ω) = EQ(e ), (3.5) where µ ¶ ST XT = ln − rT, (3.6) S0 Q is the risk-neutral measure, R is a real constant such that x 7→ e−Rx|v(x)| is bounded and integrable and the generating function mgf(u) of XT satisfies mgf(−R) < ∞. Applying the approach of Raible(2000) to the valuation of European call options re- produces the formulas of Attari(2004) and Carr & Madan(1999), which are discussed in Chapter 4. The purpose of the calibration procedure is to find model parameters that reproduce the prices of traded options. This can be formally stated as an optimization problem. The objective function in this problem is some distance between the market and model prices. The objective function is minimized with respect to model parameters. Typical possibilities to specify the distance between the market and the model are the root mean squared error of absolute or relative weighted differences of prices or implied volatilities. Detlefsen & Hardle(2006) point out that these different measures of the calibration error give rise to different sets of model parameters and that the resulting

15 3. PRICING EXOTIC OPTIONS IN STOCHASTIC VOLATILITY MODELS values of exotic contracts vary significantly. Their experiments have shown that the calibration error measures can be separated in two groups: Calibrations on relative prices, absolute implied volatilities or relative implied volatilities result in similar val- ues of exotics. Calibrations on absolute prices lead to exotic option prices that are quite different from the prices of the first group. The authors of this test have chosen the same weight to all points of the same maturity. Moreover, the weights have been assigned in such a way that all maturities have the same influence on the objective function. Possible choices of the optimization methods for the calibration of stochastic volati- lity models are described in Maruhn(2009), Mikhailov & Nogel(2003), Ben Hamida & Cont(2005), Madsen et al.(2004) and Feoktistov(2006). We use a combination of the Differential Evolution and the Levenberg-Marquardt optimizer in all numerical experi- ments that are described in this thesis and require model calibration. The Differential Evolution algorithm is used to find a good initial guess for the Levenberg-Marquardt optimization routine.1 Several runs of the Differential Evolution2 result in a small set of different initial values for the Levenberg-Marquardt algorithm. For each of these initial values a local optimum is found using the Levenberg-Marquardt optimizer. Thus a set of guesses for the global optimum is obtained. We apply an objective function to this set and find the global minimum. The main contribution of the Differential Evolution step to the described two-step optimization algorithm3 is to provide the features of global optimization. If we used the Levenberg-Marquardt step only, it would be a local optimization algorithm that does not suit our purpose of calibrating the stochastic vo- latility model. The main contribution of the Levenberg-Marquardt step is to increase the accuracy and the speed of the optimization algorithm. Numerical experiments show that if we use the Differential Evolution step only, a lot of computation time is spent when the optimum is already reached. At this point of the algorithm the Differential Evolution produces unnecessary “jumps near the optimum” because of the heuristic na- ture of this optimizer. If at this point we turn on the Levenberg-Marquardt algorithm, the optimum is found in one or two iterations because of the deterministic nature of

1Maruhn(2009) proposes a technique that allows to avoid the Differential Evolution step. 2Typically three to ten runs. 3The first step is the Differential Evolution, the second step is the Levenberg-Marquardt optimizer.

16 3.3 Monte-Carlo pricing of exotic options this optimizer. The Differential Evolution algorithm is described in Price et al.(2005).1 The Levenberg-Marquardt algorithm is described in Gill & Murray(1978).2

3.3 Monte-Carlo pricing of exotic options

Recently a few papers on efficient discretization of the continuous-time stochastic vo- latility processes have emerged. Most of them deal with the Heston model. Kahl & Jackel(2005a) apply an implicit Milstein scheme for the Heston variance process V (t), coupled with a particular discretization Sˆ(t) for the stock process S(t). Their scheme is

q ∆ √ ln Sˆ(t + ∆) = ln Sˆ(t) − (Vˆ (t + ∆) + Vˆ (t)) + ρ Vˆ (t)Z ∆ 4 V µq q ¶ 1 √ √ 1 + Vˆ (t + ∆) − Vˆ (t) (Z ∆ − ρZ ∆) + θρ∆(Z 2 − 1), 2 X V 4 V (3.7)

q √ 1 2 2 Vˆ (t) + κη∆ + θ Vˆ (t)ZV ∆ + θ ∆(ZV − 1) Vˆ (t + ∆) = 4 , (3.8) 1 + κ∆ where ZS and ZV are standardized Gaussian variables with correlation ρ. Ander- sen(2007) points out that this scheme can lead to negative values of V (t) if 4κη ≤ θ2. A possible solution to this problem is to use Euler discretization whenever Vˆ (t) drops below zero, and simultaneously using Vˆ (t + ∆)+ and Vˆ (t)+, rather than Vˆ (t + ∆) and Vˆ (t), in (3.7). Andersen(2007) proposes, however, using a quadratic exponential approximation of the density near zero, rather than just truncating. Broadie & Kaya(2006) propose a simulation scheme of the Heston model that consists of the following three steps: (1) Sample Vˆ (t + ∆) from the non-central chi-square dis- R ˆ ˆ t+∆ tribution; (2) Conditional on V (t + ∆) and V (t) draw a sample t V (u) du using

Z µ Z ¶ t+∆ p 1 t+∆ V (u) dWV (u) = V (t + ∆) − V (t) − κη∆ + κ V (u) du ; (3.9) t θ t

1An open source code is available from http://www.icsi.berkeley.edu/∼storn/code.html 2 An open source code is available from http://quantlib.org.

17 3. PRICING EXOTIC OPTIONS IN STOCHASTIC VOLATILITY MODELS

R t+∆ (3) Draw a sample of log-spot conditional on V (t + ∆) and t V (u) du: ρ ln S(t + ∆) = ln S(t) + (V (t + ∆) − V (t) − κη∆) (3.10) θ µ ¶ Z t+∆ p Z t+∆ p κρ 1 2 + − V (u) du + 1 − ρ V (u) dWV (u). (3.11) θ 2 t t This algorithm is bias-free by construction. Modifications of this algorithm that are necessary to calculate accurate greeks are described in Broadie & Kaya(2004). Recent papers of Zhu(2008), van Haastrecht & Pelsser(2008) and Smith(2007) also propose Monte-Carlo simulation schemes for the Heston model and report computing times and accuracy comparable with the schemes of Andersen(2007). In Chapter 6 we use a simulation scheme of Andersen & Brotherton-Ratcliffe(2005) which is based on a moment-matched log-normal approximation ³ ´ −κ∆ − 1 Γ(t)2+Γ(t)Z Vˆ (t + ∆) = η + (Vˆ (t) − η)e e 2 V , (3.12) where à ! 1 θ2Vˆ (t)2κ−1(1 − e−2κ∆) Γ(t)2 = ln 1 + 2 . (3.13) (η + (Vˆ (t) − η)e−κ∆)2

3.4 Analytical pricing of exotic options

Some types of exotic options can be priced in a model-independent way under certain assumptions. If a particular stochastic volatility model fulfills these assumptions, this is a very convenient way to price the option in this model. An example of such a situation is pricing variance swap under diffusion assumptions. Since the underlying process in the Heston model is a diffusion, variance swaps can be priced in this model using the replication formula even without calibration. A variance swap is a on the realized annualized variance. Its payoff is some notional amount times 2 2 R0,T − Kvol, (3.14) where   µ ¶ X S 2 2 2  tn  Rti,tj =u ˜ ln − Mti,tj , (3.15) Stn−1 ti+1

µ ¶2 1 Stj Mti,tj = ln , (3.16) j − i Sti

18 3.4 Analytical pricing of exotic options

2 1 u˜ is an annualization and rescaling factor and Kvol is a volatility strike. Since this is a forward contract, this payoff can be negative. There exist two versions of the variance 2 swap payoff. In the second version the term Mti,tj is not substracted . The properties of the second version of the variance swap are used in the definition of the Bergomi model. The floating leg of a variance swap can be approximated as a total variance of the underlying. The total variance, in turn, can be replicated using an infinite strip of

European options under assumption of no jumps. The fair value of total variance WT in this case is3 4 ·Z T ¸ µZ 0 Z ∞ ¶ 2 EQ σSt dt = 2 p(k) dk + c(k) dk , (3.17) 0 −∞ 0 where y C˜(S0e ) c(y) = y , (3.18) S0e

y P˜(S0e ) p(y) = y , (3.19) S0e C˜(K) and P˜(K) denoting undiscounted call and put prices. However, if the calibrated parameters of the Heston model are already available, a simpler formula can be used to calculate fair value of total variance, namely

·Z T ¸ −κT 2 1 − e 2 EQ σSt dt = (σt − η) + ηT. (3.20) 0 T The fair strike of a can be also approximated analytically. The volatility swap is a forward contract on the realized annualized volatility. Its payoff is

R0,T − Kvol. (3.21)

Brenner & Subrahmanyam(1988) show that the Black-Scholes formula for the at-the- money-forward call with volatility σ can be approximated by √ S σ T CBS(σ) ≈ 0√ . (3.22) 2π

1 2 252 Typical value of the annualization and rescaling factor isu ˜ = 10000 × j−i . 2 The impact of the term Mti,tj on the price is negligible. Its omission makes the payoff additive. 3The derivation of this formula is given in Gatheral(2005). 4This formula assumes zero interest rates and the absence of dividends. Thorough analysis of the effect of dividends on the valuation of variance swaps is given in Buhler(2009)

19 3. PRICING EXOTIC OPTIONS IN STOCHASTIC VOLATILITY MODELS

Feinstein(1989) applied this result in order to approximate the fair strike of the volatility swap. Indeed √ S0σimpl(F0) T BS BS √ ≈ C (σimpl(F0)) = EQ(C (R0,T )) 2π √ √ S0R0,T T S0EQ(R0,T ) T ≈ EQ( √ ) = √ , (3.23) 2π 2π where F0 is the forward price and Equation (3.22) is used in both cases when the ap- proximation sign occurs. Therefore the fair strike of the volatility swap is approximately

EQ(R0,T ) ≈ σimpl(F0). (3.24)

This approximation is valid only at the inception time of the volatility swap. Carr & Lee(2007) propose a synthetic portfolio that results in a robust replication strategy and gives the possibility to approximate values of volatility swap at all times between inception and maturity. An efficient method of calculating prices of forward-start options in the Heston model is described in Lucic(2003). We use it in Chapter 6 when calculating prices of call- spread cliquets. A number of analytical formulae for option pricing under assumption of continuity of the payoff function and/or the underlying process is available (e.g. Raible(2000), Borovkov & Novikov(2002)). Eberlein et al.(2008) analyze conditions under which the valuation formulae hold in a general setting, i.e. for discontinuous payoffs and for variables that might not possess a Lebesgue density.

20 4

Accelerating the calibration of stochastic volatility models

When implementing a calibration algorithm for an option pricing model with known characteristic function of the asset’s return1, one has to choose a method for pricing vanilla options. In this chapter we compare the following methods: (1) Direct inte- gration, (2) Fast Fourier Transform (FFT), (3) Fractional FFT. Before choosing one of these techniques, it is important to consider all possible ways of improving accu- racy and calculation speed of each of these methods. These improvements can include mathematical modifications as well as implementation techniques. In this chapter we compare optimized implementations of the calibration algorithm based on each of the above mentioned valuation methods. It helps to identify the factors which are most im- portant for accuracy and speed of calibration. We show that using an additional cache technique makes the calibration with the direct integration method at least seven times faster than the calibration with the fractional FFT method. In the existing literature unoptimized versions of direct integration are criticized. Carr & Madan(1999) point out the inability of the direct integration method to harness the computational power of FFT. Lee(2004) and Carr & Madan(1999) point out the numerical instability of the direct integration method in case of using a decomposition of an option price into probability elements. However the modification of the direct integration method described in Attari(2004) is free from this instability. In the present

1E.g., Heston, Bates, Barndorff-Nielsen&Shephard models or Levy models with stochastic time.

21 4. ACCELERATING THE CALIBRATION OF STOCHASTIC VOLATILITY MODELS chapter we compare systematically all advantages and disadvantages of the three valua- tion methods. A special attention is paid to the possibility of a simultaneous valuation of a set of options and to the efficiency of the applied numerical integration methods. The outline of this chapter is as follows. Section 4.1 lists the characteristic functions of stochastic volatility models that are used in our numerical experiments. Section 4.2 describes the pricing methods compared in this chapter. In Section 4.3, we describe the caching technique that accelerates the calibration with the direct integration method. Section 4.4 elaborates the details of a numerical experiment that compares the speed of calibration for the compared methods.

4.1 Characteristic functions

The methods discussed in this chapter can be applied to calibrate a bundle of models. The only model-specific element of the calibration algorithm is a calculation of the characteristic functions φ(ω) defined by (3.5). The derivation of exact expressions for the function φ(ω) in stochastic volatility models can be found in Gatheral(2005), Cont & Tankov(2003) and Zhu(2000). This subsection lists the characteristic functions of 1 the variable XT for the models used in our calibration tests.

Heston model. The characteristic function of XT in the Heston model is given by (4.33).

Bates model. The characteristic function of XT in the Bates model is

φ(ω) = exp{Υ1 + Υ2 − Υ3}, (4.1) where 1 − ge−dT Υ = ηκθ−2((κ − ρθωi − d)T − 2 ln( )), (4.2) 1 1 − g

1 − e−dT Υ = σ2θ−2(κ − ρθωi − d) , (4.3) 2 0 1 − ge−dT

iω 2 Υ3 = λµJ iωT + λT ((1 + µJ ) exp(σJ (iω/2)(iω − 1)) − 1), (4.4)

1 The variable XT is defined by (3.6).

22 4.1 Characteristic functions d and g are given by (4.34) and (4.35). Barndorff-Nielsen&Shephard model with the Gamma Ornstein-Uhlenbeck latent state. The characteristic function of XT in this model is given by

φ(ω) = exp{−Θ1 + Θ2}, (4.5) where −1 −1 2 −λT 2 Θ1 = ρ(b − ρ) T − λ (u + iu)(1 − e )σ0/2, (4.6)

b − f Θ = a(b − f )−1(b ln( 1 ) + f λT ), (4.7) 2 2 b − iuρ 2

−1 2 −λT f1 = iuρ − λ (u + iu)(1 − e )/2, (4.8)

−1 2 f2 = iuρ − λ (u + iu)/2. (4.9)

Levy models with stochastic time. The characteristic functions of XT in the Levy models with stochastic time can be composed from the characteristic exponent of the Levy component and the characteristic function of the rate of time change according to the formula ϕ(−ic(ω)) φ(ω) = , (4.10) ϕ(−ic(−i))iω where the characteristic exponent of the Levy process Xt is the logarithm of the char- acteristic function of the value of the process at time t = 1

c(ω) = ln E [exp(iωX1)] . (4.11)

The characteristic function of the CIR stochastic clock is exp(κ2ηt/λ2) exp(2y iζ/(κ + γ coth(γt/2))) ϕ(ζ) = 0 , (4.12) (cosh(γt/2) + κ sinh(γt/2)/γ)2κη/λ2 where p γ = κ2 − 2λ2iζ. (4.13)

The characteristic function of the GOU stochastic clock is · µ µ ¶ ¶¸ λa b ϕ(ζ) = exp iζy λ−1(1 − e−λt) + b ln − iζt . 0 iζ − λb b − iζλ−1(1 − e−λt) (4.14)

23 4. ACCELERATING THE CALIBRATION OF STOCHASTIC VOLATILITY MODELS

The characteristic exponent of the Variance Gamma process is

− ln(1 + 1 ω2σ2ν − iθνω) c(ω) = 2 , (4.15) ν where θ and σ are drift and volatility of the subordinated arithmetic Brownian motion, ν is variance of the Gamma subordinator. The characteristic exponent of the Normal Inverse Gaussian process is √ 1 − 1 + ω2σ2ν − 2iθων c(ω) = , (4.16) ν where θ and σ are the drift and volatility of the subordinated arithmetic Brownian motion and ν is variance of the Inverse Gaussian subordinator.

4.2 Pricing methods

In this section we describe the pricing methods and point out their limitations. Direct integration. The direct integration method implies computing vanilla call op- tion values using one-dimensional numerical quadrature, for example Gaussian quadra- ture.1 For this method Attari(2004) obtains an efficient formula2

1 C(S ,T,K) = S − e−rT K(I + ), (4.17) 0 0 2 where

Z +∞ Im(φ(ω)) Re(φ(ω)) 1 (Re(φ(ω)) + ω ) cos(ωl(K)) + (Im(φ(ω)) − ω ) sin(ωl(K)) I = 2 dω, π 0 1 + ω (4.18) µ ¶ Ke−rT l(K) = ln , (4.19) S0 and K denotes the . The advantages of this formula in comparison with the formula of Heston(1993) are: 1.) Formula (4.17) contains only one integral instead of two.

1The first analytical formula for pricing vanilla options in stochastic volatility models with known characteristic function was obtained in Heston(1993). Bakshi & Madan(2000) extend this approach and point out its theoretical advantages. 2An equivalent formula has been obtained by Lewis(2001). This formula can also be used with the direct integration method.

24 4.2 Pricing methods

2.) The integrand in (4.17) has a quadratic term in the denominator. This gives a faster rate of decay. The implementation of this method should control the branches of the complex loga- rithm that appears in the characteristic function. It slightly complicates the implemen- tation of this method, but does not affect the accuracy and the speed of the calculations. One possible solution of this problem is a reimplementation of the complex logarithm routine with storing the returned value and the branch number at the previous step of the algorithm. An alternative solution is described in Kahl & Jackel(2005b). Lord & Kahl(2006) and Albrecher et al.(2007) show that for some models this problem can be solved by using an appropriate representation of the characteristic function. Fast Fourier Transform. Carr & Madan(1999) suggest a transformation of the vanilla pricing formula that allows to use the FFT technique. The value of a call op- tion can be expressed as

−γk Z +∞ e −iku C(S0, T, k) = e ψ(u) du (4.20) π 0 where k denotes the log of the strike price, γ is a damping parameter and

e−rT φb(u − (γ + 1)i) ψ(u) = , (4.21) γ2 + γ − u2 + (2γ + 1)ui where

iωxb φb(ω) = EQ(e ) (4.22)

is the characteristic function of the log price xb = ln(ST ). The integral in (4.20) is approximated using an integration rule

Z +∞ NFFTX−1 −iku −ikuj e ψ(u) du ≈ e ψ(uj)wjδ, (4.23) 0 j=0

uj = jδ, (4.24)

25 4. ACCELERATING THE CALIBRATION OF STOCHASTIC VOLATILITY MODELS

where NFFT is the number of grid points and the weights wj implement the integra- tion rule. The crucial limitation of the FFT method is that the grid points uj must be chosen equidistantly. This limitation prohibits the use of the most effective integration rules such as the Gaussian quadrature. The FFT pricing method simultaneously computes the values of the integral approxi-

NFFT λ mations (4.23) for the set of log-strikes {km = −( 2 ) + mλ, m = 0,...,NFFT − 1}. The simultaneous calculation for all strikes is not an exclusive advantage of the FFT- based methods, because a slightly modified direct integration method also has this advantage. This simple modification is described in the next section. The second important restriction is that the grid spacings must satisfy the condition

2π λδ = . (4.25) NFFT

If this condition is satisfied, the sums in (4.23) can be expressed in the form

NFFT −1 NFFT −1 NFFT −1 X X X 2π −ikuj −iλδjm −i( N )jm e ψ(uj)wjδ = e hj = e FFT hj, (4.26) j=0 j=0 j=0 which allows the application of the FFT procedure invoked on the vector h = {hj = i( Nλ jδ) e 2 ψ(uj)wjδ, j = 0,...,NFFT − 1}. Fractional Fast Fourier Transform. Chourdakis(2005) has shown how the method of Carr & Madan(1999) can be accelerated using the fractional FFT algorithm. This algorithm rapidly computes sums of the form

NX−1 −i2πkjα Dk(h, α) = e hj (4.27) j=0 for any value of α. The fractional FFT method can be applied without the need to impose the restriction (4.25). However, the fractional FFT method does not overcome the crucial limitation of the FFT method because the grid points ui still must be chosen equidistantly. Fractional FFT is implemented by invoking three FFT procedures, i.e.,

−iπk2α N−1 −1 Dk(h, α) = (e )k=0 ¯ Dk (Dj(y) ¯ Dj(z)), (4.28)

26 4.2 Pricing methods where

−iπj2α N−1 N−1 y = ((hje )j=0 , (0)j=0 ), (4.29)

iπj2α N−1 iπ(N−j)2α N−1 z = ((e )j=0 , (e )j=0 ), (4.30)

Dk(h) denotes the FFT sum

NX−1 −i 2π kj Dk(h) = e N hj, (4.31) j=0

−1 Dk (h) is the inverse FFT sum

NX−1 −1 1 i 2π kj D (H) = e N H , (4.32) k N j j=0 and ¯ denotes element-by-element vector multiplication. The fractional FFT pricing method is faster than the FFT pricing method, because the absence of the restriction (4.25) allows the use of sparser grids. This effect is more important in terms of computing time than the disadvantage of using three FFT routines instead of one.1 The accuracy of the prices calculated with the FFT or the fractional FFT methods strongly depends on the choice of the damping parameter γ. Using the same value of the damping parameter in all pricing situations would be fatal for the calibration procedure. In particular, if we use a reasonable grid size NFFT < 4096 there is no value of γ that leads to an acceptable pricing error for all possible parameter values. Lord & Kahl(2007) provide an example of two pricing inputs with non-overlapping sets of acceptable damping parameters. Only an extremely fine FFT grid will result in overlapping sets of acceptable damping parameters. But fine FFT grids are impractical because they slow down the calibration. Therefore the recommendations of Lee(2004) and Lord & Kahl(2007) for the choice of γ are not just an additional improvement of the FFT-based methods but a necessary requirement for the implementation of these methods.

1See Chourdakis(2005)

27 4. ACCELERATING THE CALIBRATION OF STOCHASTIC VOLATILITY MODELS

4.3 Caching technique

The most time-consuming part of the computation is the evaluation of the characteristic 1 function φ(ω). For example the characteristic function of XT in the Heston model

1 − ge−dT φ(ω) = exp{ηκθ−2((κ − ρθωi − d)T − 2 ln( )) 1 − g 1 − e−dT + σ2θ−2(κ − ρθωi − d) }, (4.33) 0 1 − ge−dT d = ((ρθωi − κ)2 − θ2(−iω − ω2))1/2, (4.34) κ − ρθωi − d g = , (4.35) κ − ρθωi + d contains two complex exponents,2 one complex logarithm and one complex square root. Therefore an extremely important requirement for an effective implementation of the calibration algorithm is the following: The number of evaluations of the characteristic function should be as low as possible. If the calibration algorithm uses the direct in- tegration method to compute the values of vanilla options, a caching technique should be used to avoid unnecessary recalculations of the characteristic function. If the caching technique is not used, the calculation of the values of vanilla options at each iteration of the optimization algorithm includes the following steps: 1. Loop over expiries of the vanilla options. 2. Loop over strikes of the vanilla options.

3. Loop over the points ωi, i = 1,...,U that are used to evaluate the integral in (4.17) numerically.

4. Evaluate the characteristic function in ωi.

5. Evaluate the integrand in ωi. 6. Calculate the value of the vanilla option.

However, the value of the characteristic function does not depend on the strike. If we use the same grid ωi, i = 1,...,U for all options and run the described algorithm,

1 The literature provides two specifications for the characteristic function of XT in the Heston model. The first one is used in Heston(1993). The second one can be found in Schoutens et al.(2004) or in Gatheral(2005). We use the second specification. For justification of this choice see Albrecher et al.(2007) 2We do not count identical repeated terms.

28 4.3 Caching technique we recalculate the same values of the characteristic function at each step of the strike- loop. We can use the following modification of the algorithm in order to avoid these recalculations: 1. Loop over expiries of the vanilla options. 2. Loop over strikes of the vanilla options.

3. Loop over the points ωi, i = 1,...,U that are used to evaluate the integral in (4.17) numerically. 4. If we are at the first step of the strike-loop, evaluate the characteristic function in

ωi and save this value in the cache. 5. If we are not at the first step of the strike-loop, read the value of the characteristic function in ωi from the cache.

6. Evaluate the integrand in ωi. 7. Calculate the price of the vanilla option.

The numerical evaluation of the integral in (4.17) requires a choice of the numeri- cal upper integration limit. Suppose a maximum tolerable truncation error is given. Then the numerical upper integration limit ω depends on the maturity T and the strike K of the vanilla option: ω = ω(T,K). We can still use the same ω-grid for all T and K - we just define the index of the last integration point as an integer-valued function U(T,K) that satisfies the condition

ωU(T,K) ≤ ω(T,K) < ωU(T,K)+1. (4.36)

The grid at step 3 of the algorithm can now be defined as ωi, i = 1,...,U(T,K). In most cases the function U(T,K) is an increasing function of K. It leads to a different number of loop iteration at step 3 for different K. Therefore, we have to modify the described algorithm once more in order to take this fact into account. We can use a reverse order of strikes or we can control at each point ωi whether the characteristic function has been already evaluated at this point. We can also combine these two solutions. In this case the algorithm is: 1. Loop over expiries of the vanilla options. 2. Loop over strikes of the vanilla options. Use a reverse order of strikes.

3. Loop over the points ωi, i = 1,...,U(T,K) that are used to evaluate the integral in (4.17) numerically.

29 4. ACCELERATING THE CALIBRATION OF STOCHASTIC VOLATILITY MODELS

4. If the value of the characteristic function in ωi is still not in the cache, evaluate it and save this value in the cache.

5. If the value of the characteristic function in ωi is already in the cache, use this precomputed value.

6. Evaluate the integrand in ωi. 7. Calculate the value of the vanilla option.

There is a further possibility to accelerate this algorithm. Some terms of the character- istic function do not depend on T . These terms can be precomputed before starting the loop over expiries of the vanilla options. For example, we recommend to compute the term (4.34) only once and store it, because it contains a time-consuming square root operator. Dobranszky(2009) shows how to achieve better convergence of the calibration algorithm using control variates for the Fourier inversion.

4.4 Numerical experiment

The FFT-based methods of pricing vanilla options are very popular because they si- multaneously give option values for a range of strikes. This simultaneous calculation saves computing time because the characteristic function need not to be recomputed for different strikes. However, the direct integration method also has this useful feature - we just have to use the caching technique described in the previous section. Therefore the possibility of simultaneous pricing for different strikes cannot be considered as a criterion for comparison of pricing methods.1 We have to define other criteria for the comparison. The first of these criteria is the speed of the numerical integration method. Obviously, there are a lot of techniques of simple numerical integration in the general case that are both faster and more accurate than integration using FFT. They are designed to minimize the number of integrand evaluations. One of these techniques is the Gaussian quadrature formula. This section shows that the grid for the numerical integration

1 The FFT algorithm reduces the number of multiplications in the required NFFT summations from 2 an order of NFFT to that of NFFT ln2 NFFT (Carr & Madan(1999)). However the computing time required for these multiplications is negligible in comparison with the time required for the evaluations of the characteristic function. Therefore we concentrate on the number of the calculations of the characteristic function only.

30 4.4 Numerical experiment

(4.17) with six-point Gaussian quadrature is at least seven times more economical than the FFT-grid in (4.20). The second criterion is the rate of decay of the integrand. The integrand in (4.17) decays at a quadratic rate. This is the main reason why we use the pricing formula from Attari(2004) rather than the formula from Heston(1993). The rate of decay of the integrand in (4.20) is also quadratic. Therefore this criterion does not indicate any advantages of Formula (4.20) relative to Formula (4.17). As we have already pointed out, the number of the evaluations of the characteristic function is the main factor driving the calibration time. We carry out a numerical experiment to compare the influence of this factor in each pricing method. Then we conduct a second numerical experiment where we compare the calibration time directly. First, we notice that the number of evaluations of the characteristic function during the calibration procedure is equal to Ngrid · NM , where NM denotes the number of maturities in the set of vanilla options which are used to calibrate a model, Ngrid is 1 equal to NFFT for the FFT-based pricing methods, and P NM t=1 maxj U(Tt,Kj) Ngrid = (4.37) NM for the direct integration method. Note that Ngrid in (4.37) is approximately equal to the average size of the numerical integration grid at the last strike of each calibration maturity. In order to compare the performance of the above-mentioned pricing meth- ods, we define some benchmark accuracy levels. We then estimate grid sizes Ngrid that lead to the desired accuracy. The results are summarized in Table 1.2 These results are based on the following numerical experiment. Values of 100 vanilla options (10 maturities from 0.1 to 5.0 years, 10 strikes for each maturity) are calculated with 100 random (but reasonable3) sets of parameters of the Heston model. These calculations are performed with different grid sizes. The results of the calculations with extremely

1Application of the fractional FFT method results in a smaller grid size and in fewer evaluations of the characteristic function. 2 If the direct integration method is used, Ngrid is not necessary an integer value (see (4.37)).

However we report only an integer part of Ngrid for a more natural interpretation. 3The parameters are drawn from the following ranges: long-run variance η ∈ [0.01, 1.0], mean- reversion rate κ ∈ [1.0, 4.0], volatility of variance θ ∈ [0.01, 10.0], short-term volatility σ0 ∈ [0.1, 1.0], correlation ρ ∈ [−1.0, 1.0].

31 4. ACCELERATING THE CALIBRATION OF STOCHASTIC VOLATILITY MODELS

Numerical integration Accuracy Grid size for the Grid size for the grid size for the (implied volatility fractional FFT FFT method direct basis points) method integration method

2.0 4096 1024 96 1.0 4096 2048 126 0.2 8192 2048 162 0.02 16384 4096 582

Table 4.1: Grid sizes that are needed to obtain some benchmark accuracy levels

fine grids are used as the benchmark for accuracy estimations.1 Suppose we obtaine a value p1 with a reasonable grid size and a price p2 with extremely fine grid size.

To estimate the accuracy of the price p1 we compute Black-Scholes implied volatilities

σ(p1) and σ(p2) for both prices p1 and p2. The accuracy is defined as the absolute difference |σ(p1) − σ(p2)| between these implied volatilities. Table 4.1 reports the min- imum grid sizes that lead to the desired accuracy in all 10000 pricing situations.2 The same experiment is carried out for the Bates model, the Barndorff-Nielsen&Shephard model and four models based on time-changed Levy processes (NIG-CIR, NIG-GOU, VG-CIR, VG-GOU).3 The results are very similar to the results reported in the Table 4.1. For all these models the grid for the fractional FFT method must be at least seven times finer than the grid for the direct integration method to obtain the same accuracy in both methods. Our second numerical experiment compares the speed of the calibration directly. We select 100 random business days from January, 2000 to November, 2006. For each of these days we use historical market data on DAX vanilla option prices as an input to the calibration routine. Each calibration input contains 80 to 155 options with 8 to 12

1It is also checked that these benchmark values are identical for all three pricing methods. We also test the benchmark values against Monte-Carlo simulations. Occasionally there are some reference prices in the literature. In these cases we also compare our prices with these references (Table 3 in Lord, Koekkoek and van Dijk (2006), Figure 6 in Kahl & Jackel(2005b)). 210000 pricing situations correspond to 100 options and 100 parameter sets. 3The description of all these models can be found in Schoutens et al.(2004).

32 4.4 Numerical experiment

Fractional Direct Model FFT FFT integration

Heston 466 239 15 Bates 620 316 20 BNS 405 208 13 VG-CIR 540 281 17 VG-GOU 522 269 17 NIG-CIR 546 280 18 NIG-GOU 521 273 17

Table 4.2: Average calibration time (in seconds). different maturities. We run the calibration procedure with the three different vanilla pricing methods applied to seven different models. Each calibration consists of three runs of the Differential Evolution algorithm and three runs of the Levenberg-Marquardt algorithm. The grid sizes are set as in the second line of Table 1 (accuracy = 1.0 ba- sis points). The average calibration time1 is compared in Table 4.2. This table shows that the calibration with the direct integration method is approximately 16 times faster than the calibration with the fractional FFT method. It corresponds to the ratio of grid sizes: 126 points for the direct integration and 2048 for the fractional FFT. However, we should take into account that we are extremely inflexible in the choice of the FFT grid - the number of the FFT grid points must be a power of two. Therefore the differ- ence in calibration speed between the fractional FFT method and the direct integration method highly depends on the desired accuracy. For example, if the desired accuracy is 0.02 basis points, the direct integration grid size is approximately 7 times smaller than the fractional FFT grid (the last line of Table 4.1). It results in a corresponding ratio of calibration times for these methods.

1The computations were done on a Centrino Pentium M, 1.5GHz CPU

33 4. ACCELERATING THE CALIBRATION OF STOCHASTIC VOLATILITY MODELS

34 5

Forward-start options in the Barndorff-Nielsen&Shephard model

In this chapter we derive a semi-analytical formula for pricing forward-start options in the Barndorff-Nielsen&Shephard model. In terms of computational time, this formula is equivalent to one-dimensional integration. The pricing of forward-start options is one of the problems where realistic modeling of volatility dynamics is of particular importance. The Barndorff-Nielsen&Shephard model tries to meet this requirement. The general case of this model is introduced in Barndorff-Nielsen & Shephard(2001). An empirical performance of the Barndorff- Nielsen&Shephard model is analyzed in Tompkins(2001). Possible patterns of implied volatility surfaces generated by this model are described in Cont & Tankov(2003). A possible modification of this model and parameter estimation technique is described in Hubalek & Posedel(2008). We use the affine1 property of the Barndorff-Nielsen&Shephard model in order to derive a formula for pricing forward-start options. Applications of the affine processes in pricing have been studied in Duffie et al.(2003) and Keller-Ressel(2008). The forward-start option pricing in other stochastic volatility models has been studied in Lucic(2003), Kruse & Noegel(2005) and Bloch(2008). In Section 5.1 we derive a semi-analytical formula for pricing forward-start options

1The definition of affine stochastic volatility models is given in Keller-Ressel(2008)

35 5. FORWARD-START OPTIONS IN THE BARNDORFF-NIELSEN&SHEPHARD MODEL in the Barndorff-Nielsen&Shephard model. Section 5.2 provides numerical examples based on this formula.

5.1 Derivation

The payoff of a forward-start option is µ ¶ S max T − k, 0 , (5.1) ST0 where T is a maturity, T0 is a forward-start time and k is a relative strike. Some techniques of pricing plain vanilla options require only the knowledge of the characteristic function of the logarithm of the underlying1. Since forward-start options

ST can be seen as options on the underlying UT = , these techniques can be extended ST0 to forward-start options. The only modification that must be made is to replace the characteristic function of the logarithm of the underlying

iu ln ST Ψ(u) = EQ(e ) (5.2) by the forward characteristic function

iu ln ST ST Φfwd(u) = EQ[e 0 ]. (5.3)

Keller-Ressel(2008) has derived a general expression for the forward characteristic func- tion of the affine models in terms of the solutions of the generalized Riccati equations. In this chapter we use this result to derive an implementable formula for the forward characteristic function of the Barndorff-Nielsen&Shephard model. The risk-neutral dynamics in the general case of the Barndorff-Nielsen&Shephard model is 2 p d(ln St) = δdt + VtdWt + ρdJλt, (5.4)

dVt = −λVtdt + dJλt, (5.5) where Jt is the Levy subordinator that drives the model, δ is the drift that is determined by the martingale condition for St, λ is the mean-reversion rate, ρ is the comovement

1Examples of such techniques are described in Chapter 4. 2Barndorff-Nielsen&Shephard model with the Gamma-Ornstein-Uhlenbeck latent state (2.6)-(2.7) is a special case of the general Barndorff-Nielsen&Shephard model.

36 5.1 Derivation parameter. In general, the cumulant generating function of affine models is defined by some func- tions φ(t, u, w) and ψ(t, u, w) and has the form

u ln St+wVt K(u, w) = ln EQ[e ] = φ(t, u, w) + V0ψ(t, u, w) + u ln S0, (5.6)

u ln St+wVt for all u, w ∈ C, where EQ[e ] < ∞. The forward characteristic function of affine models has the form

ST iu ln S T0 −iu ln ST0 iu ln ST Φfwd(u) = EQ[e ] = EQ[e EQ[e |FT0 ]] =

EQ[exp(φ(T − T0, iu, 0) + VT0 ψ(T − T0, iu, 0))] =

exp(φ(T − T0, iu, 0))EQ[exp(VT0 ψ(T − T0, iu, 0))] =

exp[φ(T − T0, iu, 0) + φ(T0, 0, ψ(T − T0, iu, 0))

+ V0ψ(T0, 0, ψ(T − T0, iu, 0))]. (5.7)

In the Barndorff-Nielsen&Shephard model 1 ψ(t, u, w) = (u2 − u)(1 − e−λt) + e−λtw, (5.8) 2λ

Z t φ(t, u, w) = F (u, ψ(s, u, w)) ds, (5.9) 0 where F (u, w) = λκ(w + ρu) − uλκ(ρ), (5.10) and κ(u) is the cumulant-generating function of the Levy subordinator Jt that drives the model. In this chapter we consider an example of the Barndorff-Nielsen&Shephard model where the latent state follows the Gamma-Ornstein-Uhlenbeck process. In this case the Levy subordinator Jt is a

XNt Jt = xn, (5.11) n=1 where Nt is a Poisson process with intensity a and each xn follows an exponential law 1 with mean b . The cumulant-generating function of Jt is au κ(u) = . (5.12) b − u

37 5. FORWARD-START OPTIONS IN THE BARNDORFF-NIELSEN&SHEPHARD MODEL

We calculate the integral in (5.9) analytically in order to accelerate the pricing algo- rithm. Equations (5.10) and (5.12) yield

λa(Ae−λs + B) F (u, ψ(s, u, w)) = , (5.13) (b − ρ)(Ce−λs + D) where

· ¸ 1 A = − (u2 − u) − w (b − ρ + uρ), 2λ 1 B = −uρ2 + u2ρ2 + (u2 − u)(b − ρ + uρ), 2λ 1 C = (u2 − u) − w, 2λ 1 D = b − ρu − (u2 − u). (5.14) 2λ Separating real and complex parts in (5.13) and applying the complex numbers division rule yields

£ ¤ λa(e−λsRe(A) + Re(B) + i e−λsIm(A) + Im(B) ) F (v, ψ(s, v, w)) = (b − ρ)(e−λsRe(C) + Re(D) + i [e−λsIm(C) + Im(D)]) λa = × (b − ρ)(e−λsRe(C) + Re(D))2 + (e−λsIm(C) + Im(D))2 [(e−λsRe(A) + Re(B))(e−λsRe(C) + Re(D))+ (e−λsIm(A) + Im(B))(e−λsIm(C) + Im(D))+ i{(e−λsIm(A) + Im(B))(e−λsIm(C) + Im(D))− (e−λsRe(A) + Re(B))(e−λsIm(C) + Im(D))}]. (5.15)

Let us introduce an auxiliary integral Z αe−2px + βe−px + γ G(α, β, γ, ζ, η, ν, p, x) := dx, α, β, γ, ζ, η, ν, p, x ∈ R (5.16) ζe−2px + ηe−px + ν that can be evaluated analytically, in fact,

px µ ¶ arctan(√η+2e ν ) αx αη γη 4ζν−η2 G(α,β, γ, ζ, η, ν, p, x) = + 2β − − p ζ ζ ν p 4ζν − η2 µ ¶ 1 γ α ln(ζ + ηepx + νe2px) + − + C,¯ C¯ = const. (5.17) 2 ν ζ p

38 5.2 Numerical examples

Combining (5.9) and (5.15) yields

λa φ(t, v, w) = [G (t) − G (0) + i{G (t) − G (0)}] , (5.18) (b − ρ) 1 1 2 2 where

G1(x) = G(Re(A)Re(C) + Im(A)Im(C), Re(A)Re(D) + Re(B)Re(C) + Im(A)Im(D) + Im(B)Im(C), Re(B)Re(D) + Im(B)Im(D), (Re(C))2 + (Im(C))2, 2Re(C)Re(D) + 2Im(C)Im(D), (Re(D))2 + (Im(D))2, λ, x), (5.19)

G2(x) = G(Im(A)Re(C) − Re(A)Im(C), Im(A)Re(D) + Im(B)Re(C) − Re(A)Im(D) − Re(B)Im(C), Im(B)Re(D) − Re(B)Im(D), (Re(C))2 + (Im(C))2, 2Re(C)Re(D) + 2Im(C)Im(D), (Re(D))2 + (Im(D))2, λ, x). (5.20)

Substituting (5.8) and (5.18) into (5.7) yields an analytic formula for the forward char- acteristic function of the Barndorff-Nielsen&Shephard model.

5.2 Numerical examples

In this subsection we illustrate how the price and greeks of a forward-start call depend on each of the Barndorff-Nielsen&Shephard parameters. The solid lines in Figures 5.1- 5.5 show the prices of a forward-start call for a range of values of one parameter when all the other parameters are kept constant. The five figures correspond to five Barndorff- Nielsen&Shephard parameters. The strike of the option is 1.0, the maturity is 3 years,

39 5. FORWARD-START OPTIONS IN THE BARNDORFF-NIELSEN&SHEPHARD MODEL

√ Figure 5.1: Comovement scenario. λ = 1.0, b = 50.0, a = 0.5, V0 = 0.2

√ Figure 5.2: Mean-reversion rate scenario. ρ = −5.0, b = 50.0, a = 0.5, V0 = 0.2

40 5.2 Numerical examples

√ Figure 5.3: Exponential law parameter scenario. ρ = −5.0, λ = 1.0, a = 0.5, V0 = 0.2

√ Figure 5.4: Poisson intensity scenario. ρ = −5.0, λ = 1.0, b = 50.0, V0 = 0.2

41 5. FORWARD-START OPTIONS IN THE BARNDORFF-NIELSEN&SHEPHARD MODEL

Figure 5.5: Initial latent state scenario. ρ = −5.0, λ = 1.0, b = 50.0, a = 0.5

√ Figure 5.6: Forward start time scenario. ρ = −5.0, λ = 1.0, b = 50.0, a = 0.5, V0 = 0.2, τ = 2.0

42 5.2 Numerical examples the forward-start time is 1 year, the interest rate is set to 0.0 and there are no divi- dends. Dashed lines show the values of greeks corresponding to particular parameters. Corresponding numerical values are reported in Tables 5.1-5.5. Figure 5.6 and Table 5.6 show price and theta for different forward start periods. To calculate theta we set T = τ + T0, keep τ constant and calculate the derivative with respect to T0. Reported values illustrate an important feature of theta in the Barndorff- Nielsen&Shephard model. Zero values of delta, gamma1 and absence of interest rates does not necessarily lead to zero value of theta. This feature can be explained by the PIDE of the Barndorff-Nielsen&Shephard model2. This PIDE contains not only delta, gamma and theta, but also first-order and second-order sensitivities to latent state.

1Delta and gamma of forward-start options are approximately equal to zero. 2This PIDE can be found in Hilber(2005) and Schwab et al.(2007).

43 5. FORWARD-START OPTIONS IN THE BARNDORFF-NIELSEN&SHEPHARD MODEL

Sensitivity to Comovement Price comovement

-10 0.10775 -0.00612 -9.5 0.10470 -0.00608 -9 0.10166 -0.00604 -8.5 0.09866 -0.00598 -8 0.09568 -0.00590 -7.5 0.09276 -0.00580 -7 0.08988 -0.00568 -6.5 0.08707 -0.00554 -6 0.08434 -0.00537 -5.5 0.08171 -0.00516 -5 0.07918 -0.00492 -4.5 0.07679 -0.00464 -4 0.07454 -0.00432 -3.5 0.07247 -0.00395 -3 0.07060 -0.00352 -2.5 0.06895 -0.00304 -2 0.06756 -0.00250 -1.5 0.06646 -0.00190 -1 0.06567 -0.00124 -0.5 0.06523 -0.00052 √ Table 5.1: Comovement scenario. λ = 1.0, b = 50.0, a = 0.5, V0 = 0.2

44 5.2 Numerical examples

Sensitivity to Mean-reversion rate Price mean-reversion rate

0.05 0.10863 -0.07332 0.15 0.10189 -0.06171 0.25 0.09624 -0.05132 0.35 0.09159 -0.04200 0.45 0.08781 -0.03364 0.55 0.08483 -0.02613 0.65 0.08256 -0.01940 0.75 0.08092 -0.01338 0.85 0.07986 -0.00804 0.95 0.07929 -0.00333 1.05 0.07917 0.00075 1.15 0.07942 0.00420 1.25 0.07999 0.00705 1.35 0.08082 0.00931 1.45 0.08184 0.01101 1.55 0.08301 0.01224 1.65 0.08427 0.01304 1.75 0.08560 0.01355 1.85 0.08697 0.01385 1.95 0.08837 0.01400 √ Table 5.2: Mean-reversion rate scenario. ρ = −5.0, b = 50.0, a = 0.5, V0 = 0.2

45 5. FORWARD-START OPTIONS IN THE BARNDORFF-NIELSEN&SHEPHARD MODEL

Sensitivity to Exponential law Price exponential law parameter parameter

5 0.30915 -0.03135 15 0.16042 -0.00696 25 0.11562 -0.00285 35 0.09491 -0.00149 45 0.08325 -0.00090 55 0.07587 -0.00060 65 0.07081 -0.00042 75 0.06713 -0.00031 85 0.06436 -0.00024 95 0.06219 -0.00019 105 0.06044 -0.00015 115 0.05902 -0.00012 125 0.05783 -0.00010 135 0.05682 -0.00009 145 0.05596 -0.00008 155 0.05521 -0.00006 165 0.05456 -0.00006 175 0.05399 -0.00005 185 0.05348 -0.00004 195 0.05302 -0.00004 √ Table 5.3: Exponential law parameter scenario. ρ = −5.0, λ = 1.0, a = 0.5, V0 = 0.2

46 5.2 Numerical examples

Sensitivity to Poisson intensity Price Poisson intensity

0.01 0.04570 0.07381 0.26 0.06357 0.06851 0.51 0.07980 0.06119 0.76 0.09420 0.05417 1.01 0.10698 0.04834 1.26 0.11847 0.04374 1.51 0.12894 0.04012 1.76 0.13859 0.03722 2.01 0.14759 0.03486 2.26 0.15606 0.03287 2.51 0.16406 0.03118 2.76 0.17167 0.02972 3.01 0.17893 0.02843 3.26 0.18590 0.02729 3.51 0.19259 0.02627 3.76 0.19904 0.02535 4.01 0.20527 0.02451 4.26 0.21131 0.02374 4.51 0.21715 0.02304 4.76 0.22283 0.02239 √ Table 5.4: Poisson intensity scenario. ρ = −5.0, λ = 1.0, b = 50.0, V0 = 0.2

47 5. FORWARD-START OPTIONS IN THE BARNDORFF-NIELSEN&SHEPHARD MODEL

Sensitivity to Initial latent state Price initial latent state

-10.0 1.18497 -0.06705 -9.5 1.14325 -0.09739 -9.0 1.09053 -0.11020 -8.5 1.03630 -0.10368 -8.0 0.98930 -0.08257 -7.5 0.95457 -0.05642 -7.0 0.93191 -0.03597 -6.5 0.91630 -0.02908 -6.0 0.90020 -0.03765 -5.5 0.87681 -0.05711 -5.0 0.84257 -0.07979 -4.5 0.79737 -0.10049 -4.0 0.74245 -0.11899 -3.5 0.67840 -0.13728 -3.0 0.60512 -0.15578 -2.5 0.52277 -0.17336 -2.0 0.43212 -0.18878 -1.5 0.33458 -0.20059 -1.0 0.23256 -0.20601 -0.5 0.13130 -0.19345

Table 5.5: Initial latent state scenario. ρ = −5.0, λ = 1.0, b = 50.0, a = 0.5

48 5.2 Numerical examples

Forward-start time Price Theta

0.05 0.09442 -0.02244 0.1 0.09331 -0.02164 0.15 0.09224 -0.02087 0.2 0.09121 -0.02011 0.25 0.09022 -0.01937 0.3 0.08927 -0.01866 0.35 0.08835 -0.01796 0.4 0.08746 -0.01729 0.45 0.08661 -0.01664 0.5 0.08579 -0.01600 0.55 0.08501 -0.01538 0.6 0.08425 -0.01478 0.65 0.08352 -0.01420 0.7 0.08282 -0.01364 0.75 0.08215 -0.01309 0.8 0.08151 -0.01257 0.85 0.08089 -0.01205 0.9 0.08029 -0.01156 0.95 0.07973 -0.01108 1.0 0.07918 -0.01061 √ Table 5.6: Forward start time scenario. ρ = −5.0, λ = 1.0, b = 50.0, a = 0.5, V0 = 0.2, τ = 2.0

49 5. FORWARD-START OPTIONS IN THE BARNDORFF-NIELSEN&SHEPHARD MODEL

50 6

On the cost of poor volatility modeling – The case of cliquets

In this chapter we conduct pricing and hedging experiments in order to check whether simple stochastic volatility models are capable of capturing the forward volatility and forward skew risks correctly. As a reference we use the Bergomi model that treats these risks accurately per definition. Results of our experiments show that the cost of poor volatility modeling in the Heston model, the Barndorff-Nielsen&Shephard model and a Variance-Gamma model with stochastic arrival is too high when pricing and hedging cliquet options. This chapter is structured as follows: Definitions of forward volatility and forward skew are introduced in Section 6.1. Examples and properties of cliquet options are discussed in Section 6.2. We conduct a price comparison in Section 6.3. The performance of hedge strategies based on simpler popular models is analyzed in Section 6.4.

6.1 Forward volatility and forward skew

Analysis of influence of implied volatility dynamics on pricing and hedging exotic op- tions should take into account the sensitivity of a particular product to forward vola- tility and forward skew. However, we should be careful when using the terms forward volatility and forward skew. Sometimes these terms are used without defining them. It can be confusing since there are alternative non-equivalent definitions of forward volatility. We should distinguish between these definitions. We should also distinguish

51 6. ON THE COST OF POOR VOLATILITY MODELING – THE CASE OF CLIQUETS between forward-vol-sensitive options and forward-skew-sensitive options. We may give the following non-equivalent definitions of forward volatility (forward im- plied volatility): Definition 1. Forward volatility is the implied volatility for a relative strike k and maturity T2 that is observed at a future time-point T1. This value is unknown today. It is also model-independent: one needs no model to observe this value - one needs to wait only until time T1. We call this forward volatility the future volatility. Definition 2. Forward volatility is the Black-Scholes volatility implied from a price of a forward-start call computed in another model. In other words: At time t compute the price CF of a forward-start call maturing at T2 and relative strike k, which will be

T1,T2 fixed at T1, in some model. Find the volatility σf (t, k), such that the Black-Scholes T1,T2 price of this forward-start call with the volatility σf (t, k) is equal to CF . The value T1,T2 σf (t, k) is forward volatility. This value is known today and it is model-dependent. We will refer to this forward volatility as the forward-start-vanilla-implied forward vo- latility. A similar definition of forward volatility is given in Keller-Ressel(2008). Definition 3. Forward volatility is the forward variance swap volatility process s q T2 T1 (T2 − t)V − (T1 − t)V T1,T2 T1,T2 t t σf (t) = ξ (t) = , T2 − T1

T1 T2 where Vt and Vt are the implied variance swap variances, t < T1 < T2. The starting T1,T2 value σf (0) of this process is known today. This definition is convenient for modeling and is used in the Bergomi model described below. We will refer to this forward volatility as the variance-swap-implied forward volatility. Although these definitions are not equivalent, all these definitions are suited for the definition of the term forward-volatility-sensitive option and reflect the same qualitative aspect of the implied volatility dynamics. Definitions 1 and 2 can be adapted for the definition of forward skew. We can define intuitively: T1,T2 T1,T2 σ (t, k2) − σ (t, k1) forward skew = f f , k2 − k1 or exactly: ¯ T1,T2 ¯ dσf (t, k)¯ forward skew = ¯ , d ln k ¯ k= F (t,T2) F (t,T1)

52 6.2 Cliquet options

where σf is the forward volatility and F (t, τ) is the forward price for maturity τ cal- culated at time point t. This reflects the slope of the smile curve as a function of the strike and is related to the risk reversal quotes in the market. A simple example of a forward-volatility-sensitive option is a forward-start call. A sim- ple example of a forward-skew-sensitive option is a forward-start call spread. Further examples of forward-volatility- and forward-skew-sensitive options are introduced in the next section.

6.2 Cliquet options

A is a derivative that pays off some function of a set of relative returns of an underlying. Typically this function incorporates local or global caps and floors, minimum or maximum functions, sums and fixed coupons. The relative returns are typically calculated on a monthly, semi-annual or annual basis. Wilmott(2002) shows an example where the sensitivity of a cliquet option to a deter- ministic volatility is negligible in comparison with the real sensitivity of this option to volatility dynamics. To show this fact, he considers a globally floored, locally capped cliquet and analyzes a model where the actual volatility is chosen to vary in such a way as to give the option its highest or lowest possible value. This model exploits the property that an increase in volatility leads to an increase of the option value when gamma is positive and to a decrease of the option value when gamma is negative. Schoutens et al.(2004) compare prices of cliquet options in seven stochastic volatility models calibrated to the implied volatility surface of the Eurostoxx 50 index. They observe a price range of more than 40 percent amongst these models. They show that different models can produce almost the same marginal distribution of the underlying, but at the same time totally different cliquet prices. They demonstrate how the fine- grain structure of the underlying process influences the exotic option values. Here we introduce definitions of particular cliquet options that are referred to in this chapter. Reverse Cliquet. The payoff of a reverse cliquet is à ! NX−1 − max 0,C + ri , (6.1) i=0

53 6. ON THE COST OF POOR VOLATILITY MODELING – THE CASE OF CLIQUETS where

STi+1 − STi ri = , STi − ri = min(ri, 0),

0 = T0 < T1 < T2 < ... < TN , C > 0.

In this chapter, all numerical examples for the reverse cliquet are calculated using the same contract specification as in Bergomi(2005). The length of the reset period Ti+1−Ti is one month. The number of reset periods is N = 36, therefore the maturity is three years. The maximum possible payoff is equal to the coupon C = 50%. This option is called “reverse cliquet” because the final payoff depends on negative returns only. This option is both forward-volatility- and forward-skew-sensitive. The forward-skew- sensitivity of the reverse cliquet can be explained intuitively. If NX−2 − C + ri > 0, (6.2) i=0 the value of the reverse cliquet in last period is equal to the value of a call-spread option. Therefore the value of the corresponding forward-start call spread has an effect on the value of the whole structure. Consequently, the price of the reverse cliquet depends on the difference of two forward-start-vanilla-implied forward volatilities with two strikes corresponding to strikes of this forward-start call spread. This difference is the forward skew according to Definition 2 in Section 6.1. Napoleon. This contract consists of several building blocks. The payoff of each building block, which is settled individually, is

max(0,C + min ri), (6.3) i=0,N−1 where

STi+1 − STi ri = , STi 0 ≤ T0 < T1 < T2 < ... < TN , C > 0.

In this chapter we consider an example of a Napoleon option that consists of three building blocks. Each building block has N = 12 reset periods. The length of each

54 6.2 Cliquet options

reset period Ti+1 − Ti is one month. Therefore the maturity of this contract is three years and the possible payments occur at the end of each year. The maximum possible payment at the end of each year is equal to the coupon C = 8%. This contract type is analyzed in Bergomi(2004), Bergomi(2005) and Gatheral(2005). Numerical experiments in Bergomi(2005) show that this option is extremely forward- volatility-sensitive but almost forward-skew-insensitive. Accumulator. The payoff of an accumulator is

à ! NX−1 max 0, max(min(ri, cap), floor) , (6.4) i=0 where

STi+1 − STi ri = , STi 0 = T0 < T1 < T2 < ... < TN .

In this chapter we consider an example of an accumulator where the floor is set to

-1%, the cap is 1%, each reset period Ti+1 − Ti is one month and the maturity of the option is three years(N = 36). This option is forward-skew-sensitive. This contract can also be forward-volatility-sensitive but only in cases of strong forward-skew. The intuition for this behavior is similar to the intuition for skew- and volatility- sensitivity of a standard one month call-spread. In the Black-Scholes model this call-spread has negligible vega. However, the absolute value of vega of the call-spread increases in the models that takes skew into account and in the presence of skew.

Call spread cliquet. This option pays at the end of each reset period [Ti,Ti+1] the amount

µ ¶ µ ¶ STi+1 STi+1 max − k1, 0 − max − k2, 0 . (6.5) STi STi A call spread cliquet can be seen as a portfolio of forward start call spreads. As in the previous examples we consider N = 36 monthly reset periods Ti+1 − Ti. The strikes are set to k1 = 0.95 and k2 = 1.05. This option inherits forward-skew-sensitivity from its forward-start call-spread building blocks.

55 6. ON THE COST OF POOR VOLATILITY MODELING – THE CASE OF CLIQUETS

6.3 Price comparison

In this section we compare the theoretical prices of cliquet options in the Bergomi model and in calibrated versions of the discussed simpler models. We take the Bergomi model to be the true data-generating process. This assumption allows us to gauge the differences between this recent model and simpler popular models. The parameter values we use are reported in Tables 6.1-6.2. Initial implied variance swap variances are 2 generated in the Heston model with the parameters σ0Heston, ηHeston and κHeston. The meaning of these parameters is described in Subsection 2.1 about the Heston model.

Scenario ω θ k1 k2 ρ ν 1 1.0 0.1 5.0 0.4 0.2 -0.2 2 1.0 0.5 5.0 0.4 0.2 0.2 3 1.0 0.2 4.8 0.4 0.2 0.2 4 1.4 0.3 6.0 0.25 0.0 0.1 5 1.5 0.3 6.0 0.25 0.0 0.2

Table 6.1: Values of parameters ω, θ, k1, k2, ρ and ν.

2 Scenario χ ρSX ρSY ηHeston σ0Heston κHeston 1 -0.06 -0.6 -0.3 0.04 0.04 1.0 2 -0.06 -0.6 -0.3 0.09 0.04 1.0 3 -0.06 -0.6 -0.3 0.09 0.04 1.0 4 -0.07 -0.7 -0.35 0.05 0.04 1.0 5 -0.0625 -0.7 -0.35 0.09 0.04 1.0

2 Table 6.2: Values of parameters χ, ρSX , ρSY , ηHeston, σ0Heston.

We use parameters reported in Tables 6.1-6.2 for the Bergomi model to generate five implied volatility surfaces. Then we calibrate the simpler models to these implied volatility surfaces and compare price differences. The calibrated parameters for the simpler models are reported in Tables 6.3-6.5. We use the calibration algorithm described in Chapter 4. Of course, the calibrated values of mean-reversion rate, long-run variance and short- term volatility in the Heston model1 should not coincide with the corresponding param-

1See Table 6.3.

56 6.3 Price comparison eters that we use to generate initial implied variance swap variances. We cannot expect such a correspondence, because we use the set of vanilla options as a calibration input. In the absence of jumps1 a set of prices of vanilla options with all possible maturities and strikes contains more information than a set of prices of variance swaps with all possible maturities.

Scenario η κ θ σ0 ρ 1 0.308888 1.524 2.57848 0.251838 -0.659371 2 0.0901768 1.0 0.231635 0.194986 -0.514877 3 0.0909267 1.0 0.149329 0.193474 -0.609845 4 0.053176 3.53349 0.521236 0.201211 -0.67917 5 0.091009 1.12328 0.277384 0.194058 -0.605952

Table 6.3: Calibrated Heston parameters.

Scenario ρ λ b a σ0 1 -5.17388 0.0403766 11.3553 9.98816 0.164977 2 -2.04324 0.83257 49.3421 4.00691 0.174382 3 -2.86624 0.804772 93.3514 7.78661 0.169629 4 -2.7184 2.0589 47.182 1.53604 0.159252 5 -1.99725 0.905518 36.6755 2.85488 0.168391

Table 6.4: Calibrated Barndorff-Nielsen&Shephard parameters.

Scenario θ σ ν κ η λ y0 1 -0.19297 0.256868 0.273051 1.47153 1.56324 3.98489 1.0 2 -0.526388 0.19289 0.0210463 1.62598 0.941867 0.726751 1.0 3 -0.76172 0.184905 0.0153841 1.58304 0.912081 0.500934 1.0 4 -0.348248 0.197137 0.0573394 3.27292 0.712477 1.76894 1.0 5 -0.413335 0.193394 0.033752 1.7588 0.964997 1.01945 1.0

Table 6.5: Calibrated VGSA parameters.

Calibrated values of the mean-reversion rate in the Heston model for scenarios 2 and 3 are exactly equal to 1.0. That is explained by the parameter bounds that are used to calibrate the Heston model. For the mean-reversion rate we use the bounds 1Both Heston and Bergomi processes do not have jumps.

57 6. ON THE COST OF POOR VOLATILITY MODELING – THE CASE OF CLIQUETS

[1.0; 4.0] which correspond to the time interval between three months and one year. Values of cliquet options in all described models are reported in Tables 6.6-6.10. These values are calculated using a Monte-Carlo simulation with one million paths using anti- thetics as a variance reduction technique and a path generation of the variance process based on Andersen & Brotherton-Ratcliffe(2005) in order to prevent the variance pro- cess to take negative values1. Using these values we generate 20 figures (5 scenarios x 4 instruments) that allow us to compare between different models. In 16 of these 20 figures the best approximation of the reference (Bergomi) price is the value calcu- lated in the Heston model. However, in more than half of the pricing experiments the relative difference between reference price and value calculated in the Heston model is more than 3.5%, which is unacceptable for practical applications.

Bergomi Heston BNS VGSA Napoleon 0.0595 0.0458 0.0331 0.0727 Reverse Cliquet 0.0258 0.0401 0.0203 0.0880 Accumulator 0.0666 0.0796 0.0424 0.1821 Call Spread Cliquet 2.0191 1.9389 1.9277 2.0165

Table 6.6: Comparison of theoretical cliquet values. Scenario 1.

Bergomi Heston BNS VGSA Napoleon 0.0153 0.0148 0.0176 0.0274 Reverse Cliquet 0.0031 0.0025 0.0048 0.0144 Accumulator 0.0228 0.0231 0.0252 0.0446 Call Spread Cliquet 1.7776 1.7929 1.8087 1.8716

Table 6.7: Comparison of theoretical cliquet values. Scenario 2.

6.4 Hedge performance

Practical implementations of the hedge strategies corresponding to the theoretical val- ues analyzed in Section 6.3 necessarily involve a number of approximations. The two main sources of discrepancies between theoretical and realized hedge performance are

1In our experiments the highest standard error for reverse cliquets was 0.00002, for Napoleons 0.00004, for accumulators 0.00004, for call spread cliquets 0.0002.

58 6.4 Hedge performance

Bergomi Heston BNS VGSA Napoleon 0.0117 0.0113 0.0132 0.0254 Reverse Cliquet 0.0011 0.0016 0.0020 0.0115 Accumulator 0.0217 0.0216 0.0239 0.0410 Call Spread Cliquet 1.7763 1.7830 1.8004 1.8619

Table 6.8: Comparison of theoretical cliquet values. Scenario 3.

Bergomi Heston BNS VGSA Napoleon 0.0282 0.0279 0.0374 0.0433 Reverse Cliquet 0.0137 0.0110 0.0245 0.0422 Accumulator 0.0422 0.0382 0.0480 0.0887 Call Spread Cliquet 1.8994 1.8658 1.9050 1.9323

Table 6.9: Comparison of theoretical cliquet values. Scenario 4.

Bergomi Heston BNS VGSA Napoleon 0.0170 0.0163 0.0216 0.0305 Reverse Cliquet 0.0042 0.0034 0.0084 0.0199 Accumulator 0.0241 0.0254 0.0286 0.0555 Call Spread Cliquet 1.7829 1.8068 1.8294 1.9003

Table 6.10: Comparison of theoretical cliquet values. Scenario 5.

59 6. ON THE COST OF POOR VOLATILITY MODELING – THE CASE OF CLIQUETS discrete rehedging and the practice of recalibrating the model used to compute the hedge ratios. In this section we analyze the magnitude of the discrepancies caused by these two sources. We concentrate on the hedge performance of the Heston model only. There are two reasons for this choice. First of all, the results of the previous section have shown that the differences between cliquet values in the Heston model and in the reference (Bergomi) model are less than the corresponding differences for other models. Sec- ondly, our analysis of the hedge performance requires a huge number of recalculations of cliquet values. It would be extremely time-consuming, if each value recalculation was done using the Monte-Carlo method. Fortunately, there is a much faster method to calculate the values of call spread cliquets in the Heston model. Lucic(2003) shows how to calculate the value of a forward-start option in the Heston model in a few millisec- onds. Since a call spread cliquet can be seen as a portfolio of forward start call spreads, the same pricing method can be applied to calculate values of call spread cliquets. The most natural way to evaluate the performance of a hedge strategy is to consider the realized profit-and-loss distribution. We construct such distributions by simulating the following implementations of the hedge strategies prescribed by the Heston model:

1. Hedging with constant parameters (HCP): The Heston model is calibrated to the initial implied volatility surface and the subsequent hedge adjustments are done holding the calibrated parameters constant through time. This is in line with the assumptions underlying the Heston model.

2. Hedging with recalibration (HR): The Heston model is recalibrated after each increment in the Bergomi model. This is in line with standard practice.

To construct such distributions, we conduct simulation experiments where each itera- tion has the following structure:

1. Simulate an increment in a realization of the Bergomi model.

2. If relevant, calibrate the simpler models to the resulting implied volatility surface.

3. Adjust hedges according to the, possibly recalibrated, Heston model. The hedges are adjusted on a weekly basis.

4. At expiry, record hedge errors.

60 6.4 Hedge performance

We investigate the performance of two dynamic hedging strategies

1. Delta and short-term vega (DSV),

2. Delta and parallel shift vega (DPV), where short-term vega is the sensitivity of the option price to the parameter σ0 of the Heston model, parallel shift vega is the sensitivity of the option price to a simultaneous shift of the parameters σ0 and η so that p √ ∆σ0 = η + ∆η − η. (6.6)

The comparison between HCP and HR implementations is done on the basis of the DSV strategy. The DPV strategy is analyzed using the HCP implementation. Histograms of absolute cumulative hedging errors in the resulting three experiments (HR-DSV, HCP- DSV, HCP-DPV) are shown in Figures 6.1-6.3. The total number of hedge scenarios is 100 in the experiment HR-DSV, 299 in HCP-DSV, 2782 in HCP-DPV. The underlying path is generated using parameters of scenario 5 (see Tables 6.1-6.2). The option that is hedged is a 36-periods 95-105% call spread cliquet with monthly resets. The value of the call spread cliquet in the Heston model at the issue date was 1.8068 in all experiments. Relative hedging errors are shown in percent of the cliquet value. Relative frequencies are shown in percent of the total number of experiments. The experiment HCP-DPV shows better performance than HR-DSV and HCP-DSV.1 The strategy HCP-DSV performs the worst. However, all experiments show that the hedging error can be unacceptably high. This observation confirms the assertion that the risk of using Heston’s model for hedging cliquet options is too high.

1We use the variance of the realized relative hedging errors to compare the performance of the strategies.

61 6. ON THE COST OF POOR VOLATILITY MODELING – THE CASE OF CLIQUETS

Figure 6.1: Experiment HR-DSV. Histogram of relative cumulative hedging errors of a call spread cliquet.

62 6.4 Hedge performance

Figure 6.2: Experiment HCP-DSV. Histogram of relative cumulative hedging errors of a call spread cliquet.

63 6. ON THE COST OF POOR VOLATILITY MODELING – THE CASE OF CLIQUETS

Figure 6.3: Experiment HCP-DPV. Histogram of relative cumulative hedging errors of a call spread cliquet.

64 7

Conclusion

We have shown that an efficient implementation of the direct integration method results in a sizable speed up of the calibration of stochastic volatility models. This method even outperforms the calibration with the fractional FFT. The simultaneous pricing of options with different strikes is not an exclusive advantage of the FFT methods com- pared to the direct integration method, because an application of a cache technique leads to simultaneous pricing of options with different strikes in the framework of di- rect integration. Taking this into account we argued that the pricing methods differ in two aspects only: the numerical integration technique and the pricing formula. The combination of these factors results in higher calculation speed of the direct integration method in comparison to the FFT and fractional FFT methods. Specifically: (1) Gaus- sian quadrature is a much faster numerical integration technique than the FFT, (2) the transformed pricing formula of Attari (2004) provides approximately the same rate of decay of the integrand in comparison with the main formula of the FFT method. As we have pointed out in Chapter 4, the direct integration method is frequently criticized in the literature. However this critique is valid only if we consider an unoptimized implementation of the general formula. The use of the modified pricing formula and the caching technique makes the direct integration method the best choice for practical applications. The second innovative result which is obtained in this thesis is the pricing formula for forward-start options in the Barndorff-Nielsen&Shephard model. The pricing of this type of options in the general form of the Barndorff-Nielsen&Shephard model re-

65 7. CONCLUSION quires numerical integration1 for each evaluation of the forward characteristic function. Together with the integration in the Fourier space this yields computational times equivalent to two-dimensional integration. However, if we use a particular form of the Barndorff-Nielsen&Shephard model with the latent state following the Gamma- Ornstein-Uhlenbeck process, an analytical expression for the forward characteristic function is available. It reduces the computational time for pricing forward-start op- tions to a time equivalent to one-dimensional integration, i.e. just a few milliseconds on a standard PC. This particular form of the Barndorff-Nielsen&Shephard model is not the only one where such an analytical simplification for the forward-start options is available. Each choice of the latent state dynamics however requires its own formal analysis similar to the one described in Chapter 5. This is a possible direction of further research in the area of efficient pricing in the Barndorff-Nielsen&Shephard model. Hedging experiments described in this thesis show that the cost of poor volatility model- ing in popular stochastic volatility models (Heston, Barndorff-Nielsen&Shephard, Levy with stochastic clock) is too high. A possible cause of relatively high cliquet hedg- ing errors produced by these models is that they are developed to fit vanilla prices and to control forward volatility and forward skew simultaneously. A lot of modeling and numerical effort is spent to reach all these targets in one model. However, to our knowledge, today there exists no model that attains both these targets simultaneously. Therefore, we would recommend to change the requirements for the model that should be used for pricing and hedging cliquet options. Specifically, we recommend to abandon the requirement to fit vanilla prices. This would facilitate direct modeling of forward volatility and forward skew. When pricing and hedging cliquets, accurate modeling of forward volatility, forward skew and vega-hedging are incomparably more impor- tant than fitting vanilla prices. Therefore we expect better hedging performance if we do not complicate models of smile dynamics by the requirement to fit vanilla prices. Bergomi’s model is a good example of this approach. In our further research we plan to investigate whether it is possible to obtain acceptable cliquet hedging performance using direct forward smile modeling without fitting vanilla prices.

1See formula (5.9).

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72 Curriculum vitae

Fiodar Kilin holds a diploma in applied mathematics from the Belarus State University and a Ph.D. in quantitative finance from the Frankfurt School of Finance and Man- agement. He worked five years at Quanteam AG as a senior consultant. He currently consults financial institutions in the area of quantitative finance. His main research interests are stochastic volatility models and exotic equity derivatives.

73 Declaration

I herewith declare that I have produced this paper without the prohibited assistance of third parties and without making use of aids other than those specified; notions taken over directly or indirectly from other sources have been identified as such. This paper has not previously been presented in identical or similar form to any other German or foreign examination board.

The thesis work was conducted from March 1, 2006 to June 14, 2009 under the supervision of Professor Uwe Wystup at the Frankfurt School of Finance & Management.

Frankfurt am Main, June 14, 2009