Forward Implied Volatility Farshad Behvand Christ Church Oxford University January 4, 2010 A thesis submitted in partial fulfillment of the requirements for the MSc in Mathematical Finance Abstract We look at the pricing of structures that depend on forward volatility, such as globally floored cliquets. We start off by analysing market standard models, which prescribe the dynamics of volatility as either deterministic or instantaneous. We then move onto to analysing the Discrete Stochastic Implied Volatility (DSIV) model. This model specifies the dynamics of forward implied volatility directly at discrete times and as a result has more realism of and control over the dynamics. For each of these models we explore in detail the implied forward volatility dynamics and how to price exotic cliquet options. We then compare these prices against Totem marks, which represent actual market price information. Acknowledgements I would like to thank Jochen Theis for all his recommendations, inputs, support and patience. He has really helped me in developing a greater understanding of this topic. I would also like to thank my lecturers at Oxford university and colleagues at Markit, in particular Lukasz Dobrek for their support and input on various issues and topics in mathematical finance. Finally, my family and fianc´eefor standing by me. 1 Contents 1 Introduction 4 2 Cliquet Options Market 6 3 Local Volatility 8 3.1 Model Outline and Dynamics . .8 3.2 Model Setup and Pricing . .9 4 Stochastic Volatility 15 4.1 Heston Stochastic Volatility Model . 15 4.2 Time Dependent Heston Stochastic Volatility Model . 21 4.2.1 Characteristic Function for the Time Dependent Heston Model 21 4.2.2 European & Forward Starting Option Prices . 23 4.2.3 Calibration ............................ 26 4.2.4 Monte Carlo Numerical Implementation . 30 4.3 Exotic Cliquet Pricing . 32 5 Discrete Stochastic Implied Volatility Model 35 5.1 ModelOutline............................... 36 5.2 Model Restrictions . 39 5.3 Arbitrage Freeness . 40 5.3.1 Volatility Conditions . 40 5.3.2 Existence of Risk-Neutral Measure . 42 2 5.3.3 Model Completeness . 44 5.4 Pricing................................... 45 5.5 Calibration ................................ 47 5.6 Exotic Cliquet Pricing . 49 6 Summary 51 A Appendix 54 A.1 SABR Model . 54 A.2 Implied Forward and Cliquet Volatility . 54 A.3 Totem Consensus . 56 A.4 Exotic Cliquet Totem Results for Heston Model . 57 A.5 Monte Carlo Results for Time Dependent Heston Model . 60 A.6 Matlab Code . 61 3 1 Introduction In this thesis we address the issue of volatility modelling for derivatives that depend on \forward volatility", such as reverse cliquets, Napoleons, accumulators and struc- tured notes with embedded forward starting features. Traditionally, smile models have been assessed according to how well they fit the vanilla/spot implied volatility surface (e.g. the Local Volatility model). However, the pricing of exotic structures which have forward starting features embedded within them are more dependent on the assumptions made on the forward implied volatility dynamics than on the price of today's vanilla options prices. A good volatility model not only has to be able to calibrate to the forward starting options but also to the spot implied volatilities. For example, on the reset date t1 a forward starting option turns into a European option. Since such options are liquidly traded, the model must be able to internally produce future implied volatility surfaces that are consistent with historic behaviour. Many researchers and practitioners have tried to address this issue with stochastic volatility models. In the literature it is frequently stated that models of this type are well suited to price forward starting contracts, such as cliquets. However, this is not easily verified in the markets: figure 1, shows the cliquet implied volatility error (Ap- pendix A.2 has further details on calculation and table descriptions) when calibrating the Heston Model (4.1) to the vanilla options market and then pricing quarterly and annually resetting vanilla resetting cliquets of different maturities (see equation (1) for payoff definition). Clearly, the model does not match the market prices of the cliquets. Furthermore, this model with this calibration is clearly not suitable to price exotic cliquet options as it does not fit the forward implied volatility skew observed in the market. We investigate this issue by looking at the dynamics of several different volatility models in this thesis. We assess the quality of the models on the following criteria [8]: • realism of and control over volatility dynamics • ease of calibration to vanilla and forward starting options • continuity in time - ability to price all structures for all times 4 • speed and suitability for practical use In practice, to the author's understanding no volatility model has all of these prop- erties. Generally, something needs to give. This thesis is organised as follows. In sections 3 and 4 we look at three continu- ous time models: the Local Volatility Model [10], the Heston Model [19] and the Time dependent Heston Model [11]. In section 5 we construct the Discrete Stochastic Implied Volatility Model (DSIV Model) [1], a risk-neutral model for the stock price such that all available information on current and future smiles can be incorporated. For each of these models we: • describe the forward volatility dynamics implied by the model • discuss in detail how to calibrate and price with the model • compare the model prices of exotic cliquets against Totem market consensus Finally, we end with a summary of our findings and outline possible future research. (a) Annual resetting cliquet (b) Quarterly resetting cliquet Figure 1: S&P500 Implied cliquet volatility error (σ ^SV − σ^Mkt) 5 2 Cliquet Options Market Cliquets are a form of \volatility product" within the equity markets. They are con- tracts whose payoff depends in some way on the performance of an asset over future time periods. Cliquets are often referred to as a forward volatility product and for- ward implied volatility surfaces can be backed out of vanilla cliquet prices (Appendix A.2). Here are the payoffs of the cliquet options we are going to price in this thesis [24], [2]: • Vanilla call cliquet: n + X Sti+1 − α (1) S i=1 ti • Napoleon: M + X Sti+1 Cj + min − 1; 0 (2) 8ti2[tj−1;tj ] S j=1 ti • Reverse cliquets: ( n )+ X Sti+1 C + min − 1; 0 (3) S i=1 ti • Globally floored cliquet: ( n )+ X Sti+1 r _ − 1 ^ r (4) − S + i=1 ti for C > 0 and α > 0 where S is the stock price, α a constant strike (moneyness), C and Cj are constant coupons or fixed returns, r+ and r− are a cap and floor respectively, t0; :::; tn are the reset dates and the final payment is made at tn (it is also possible for the structure to pay at every reset date). The pricing and risk management of such products is not straightforward and the market has not yet settled on an agreed reference model. 6 Resetting cliquets on equity indices are liquidly traded as products in their own right; they are not traded on model marks. It is not common to have cliquets on single stock equity, however they do exist and pose further problems on modelling the discrete dividends1 within the exotic models. In this thesis we work with major equity indices and as a result have market data on both forward starting and spot implied volatil- ity surfaces. Figure 2 shows the complete range of parameters for which we have calibration data available for our examples, which are typical of the market. Here SpotVol refers to the vanilla option data and 12MFwdVol, 6MFwdVol, etc, refers to the cliquet option data with resetting frequency of twelve months, six months, etc. Note that moneyness is K and for vanilla options this is based on the spot level. Sti Figure 2: Instruments for which we have calibration data available To hedge exotic cliquets we can use a combination of forward starting options, reset- ting cliquets, vanilla options and the underlying option. Also, to hedge our forward starting options we use a combination of vanilla options and the underlying. There- fore, it is vital that our model is able to reproduce the prices of the calibration instruments to prevent slippage. 1There is no market standard model to factor in discrete dividends into exotic option pricing; one way is to model the stock process as the pure stock process (Hans Buehler [6]) 7 3 Local Volatility The first model we review is the Local Volatility Model. It was one of the first volatility models used in practice to price resetting cliquets. Most financial institutions moved away from using this model with cliquets once they understood the dynamics of forward volatilities it produces. We start by outlining the model dynamics and the associated problems with Local Volatility Models. We then show to price with such models and use the model to price vanilla cliquets and compare these prices against the market as represented by the Totem consensus. 3.1 Model Outline and Dynamics The local volatility model describes the stochastic evolution of the underlying stock price St by means of a volatility term that is a deterministic function of the stock price: dSt = (rd(t) − q(t))Stdt + σ(St; t)Stdz (5) where rd is the instantaneous domestic interest rate and q the continuous dividend yield. The local volatility function σ(S; t) depends locally on both the time and the value of the underlying asset. The distribution of the asset price at some time t is thus no longer log-normal; rather, the goal is to determine the underlying process from the market-observed smile by determining the local volatility function.