The Quantitative Finance Library

Simple Processes and the Pricing and Hedging of Cliquets For data on market prices on 246 cliquets we consider the task of pric- ing these exotic options using a relatively simple path space subsequently stressed to market implied and then predicted stress levels. An additive process transitioning from a Sato process to a Levy process is formulated and estimated on vanilla options. Ask prices constructed from predicted stress levels are observed to have an in sample correlation of 92% with market prices. Interestingly, it is observed that capped cash ?ows have negative stress levels while uncapped products have positive stress levels. We illustrate the e?ect of hedging cliquet liabilities using call options as hedging assets permiting a 10% reduction in ask prices. , D.P.Madan, W.Schoutens (2010)

Dynamics of implied volatility surfaces The prices of index options at a given date are usually represented via the corresponding implied volatility surface, presenting skew/smile features and term structure which several models have attempted to reproduce. However, the implied volatility surface also changes dynamically over time in a way that is not taken into account by current modelling approaches, giving rise to ?Vega? risk in option portfolios. Using time series of option prices on the SP500 and FTSE indices, we study the deformation of this surface and show that it may be represented as a randomly fluctuating surface driven by a small number of orthogonal random factors. We identify and interpret the shape of each of these factors, study their dynamics and their correlation with the underlying index. Our approach is based on a Karhunen?Lo`eve decomposition of the daily variations of implied volatilities obtained from market data. A simple factor model compatible with the empirical observations is proposed. We illustrate how this approach models and improves the well known ?sticky moneyness? rule used by option traders for updating implied volatilities. Our approach gives a justification for use of ?Vega?s for measuring volatility risk and provides a decomposition of volatility risk as a sum of contributions from empirically identifiable factors. , R. Cont, J. da Fons?ca (2009)

Pricing Forward Start Options in Models based on (time-changed) L?vy Processes Options depending on the forward skew are very popular. One such option is the forward starting call option - the basic building block of a cliquet option. Widely applied models to account for the forward skew dynamics to price such options include the Heston model, the Heston-Hull-White model and the Bates model. Within these models solutions for options including forward start features are available using (semi) analytical formulas. Today exponential (subordinated) Levy models being increasingly popular for modelling the asset dynamics. While the simple exponential Levy model imply the same forward volatility surface for all future times the subordinated models do not. Depending on the subordinator the dynamic of the forward volatility surface and therefore stochastic volatility can be modelled. Analytical pricing formulas based on the charcteristic function and Fourier transform methods are available for the class of these models. We extend the applicability of analytical pricing to options including forward start features. To this end we derive the forward characteristic functions which can be used in Fourier transform based methods. As examples we consider the Variance Gamma model and the NIG model subordinated by a Gamma Ornstein Uhlenbeck process and respectively by an Cox-Ingersoll-Ross process. We check our analytical results by applying Monte Carlo methods. These results can for instance be applied to calibration of the forward volatility surface. , P. Beyer, J. Kienitz (2009)

Saddlepoint Methods for Option Pricing Saddle point method , P. Carr, D.P. Madan (2008)

Lecture 5: Static Hedging and Implied Distributions , E.Derman (2008)

Lecture 4: Arbitrage Bounds, Problems with Valuation, Models , E.Derman (2008)

On the qualitative effect of volatility and duration on prices of Asian options We show that under the Black Scholes assumption the price of an arithmetic average Asian call option with fixed strike increases with the level of volatility . This statement is not trivial to prove and for other models in general wrong. In fact we demonstrate that in a simple binomial model no such relationship holds. Under the Black-Scholes assumption however, we give a proof based on the maximum principle for parabolic partial differential equations. Furthermore we show that an increase in the length of duration over which the average is sampled also increases the price of an arithmetic average Asian call option, if the discounting effect is taken out. To show this, we use the result on volatility and the fact that a reparametrization in time corresponds to a change in volatility in the Black-Scholes model. Both results are extremely important for the risk management and risk assessment of portfolios that include Asian options. , P.Carr, C.O.Ewald, Y.Xiao (2008)

Put-Call Symmetry: Extensions and Applications Classic put-call symmetry relates the prices of puts and calls at strikes on opposite sides of the forward price. We extend put-call symmetry in several directions. Relaxing the assumptions, we generalize to unified local/stochastic volatility models and time-changed L?evy processes, under a symmetry condition. Further relaxing the assumptions, we generalize to various asymmetric dynamics. Extending the conclusions, we take an arbitrarily given payoff of European style or single/double/sequential-barrier style, and we construct a conjugate European-style claim of equal value, and thereby a semi-static hedge of the given payoff. , P. Carr, R. Lee (2007)

Markov Functional Modeling of Equity, Commodity and other Assets In this short note we show how to setup a one dimensional single asset model, e.g. equity model, which calibrates to a full (two dimensional) implied volatility surface. We show that the efficient calibration procedure used in LIBOR Markov functional models may be applied here too. In a addition to the calibration to a full volatility surface the model allows the calibration of the joint asset-interest rate movement (i.e. local interest rates) and forward volatility. The latter allows the calibration of compound or Bermudan options. The Markov functional modeling approach consists of a Markovian driver process x and a mapping functional representing the asset states S(t) as a function of x(t). It was originally developed in the context of interest rate models, see [7]. Our approach however is similar to the setup of the hybrid Markov functional model in spot measure, as considered in [5]. For equity models it is common to use a deterministic Num?raire, e.g. the bank account with deterministic interest rates. In our approach we will choose the asset itself as Num?raire. This is a subtle, but crucial difference to other approaches considering Markov functional modeling. Choosing the asset itself as Num?raire will allow for a very efficient numerically calibration procedure. As a consequence interest rates have to be allowed to be stochastic, namely as a functional of x too. The Black-Scholes model with deterministic interest rates is a special case of such a Markov functional model. The most general form of this modeling approach will allow for a simultaneous calibration to a full two dimensional volatility smile, a prescribed joint movement of interest rates and a given forward volatility structure. , C. P. Fries (2006)

Hedging Exotic Options in Stochastic Volatility and Jump Diffusion Models , K. Detlefsen (2005)

Pricing discretely sampled path-dependent exotic options using replication methods A semi-static replication method is introduced for pricing discretely sampled path-dependent options. It depends upon buying and selling options at the reset times of the option but does not involve trading at intervening times. The method is model independent in that it only depends upon the existence of a pricing function for vanilla call options which depends purely on current time, time to expiry, spot and strike. For the special case of a discrete barrier, an alternative method is developed which involves trading only at the initial time and the knockout time or expiry of the option. , M.Joshi (2005)

Cliquet Options: Pricing and Greeks in Deterministic and Stochastic Volatility Models This paper presents a method to determine the price of a cliquet option, as well as its sensitivity to changes in the market, the Greeks, for deterministic (also incorporating skews) and stochastic (Hestonian) volatility and, lognormal and jump-diffusion asset price - processes, with almost machine precision in a fraction of a second. In the pricing algorithms we make use of a new Laplace transform inversion technique, which guarantees fast and numerically stable pricing. The computation of the Greeks is based on a Girsanov type drift adjustment. , P.D.Iseger, E.Oldenkamp (2005)

Numerical Methods and Volatility Models for Valuing Cliquet Options Several numerical issues for valuing cliquet options using PDE methods are investigated. The use of a running sum of returns formulation is compared to an average return formulation. Methods for grid construction, interpolation of jump conditions, and application of boundary conditions are compared. The effect of various volatility modelling assumptions on the value of cliquet options is also studied. Numerical results are reported for jump diffusion models, calibrated volatility surface models, and uncertain volatility models. , H.A.Windcliff, P.A.Forsyth, K.R.vetzal (2005)

Stocks paying discrete dividends: modelling and option pricing In the Black-Scholes model, any dividends on stocks are paid continu- ously, but in reality dividends are always paid discretely, often after some announcement of the amount of the dividend. It is not entirely clear how such discrete dividends are to be handled; simple perturbations of the Black-Scholes model often fall into contradictions. Our approach here is to recognise the stock price as the net present value of all future dividends, and to model the (discrete) dividend process directly. The stock price pro- cess is then deduced, and various option-pricing formulae derived. The Black-Scholes model with continuous dividend payments results as a limit as the time between dividend payments goes to zero. 1 I , R. Korn, L.C.G. Rogers (2004)

Accounting for Biases in Black-Scholes Prices of currency options commonly differ from the Black-Scholes formula along two dimensions: implied volatilities vary by strike price (volatility smiles) and maturity (implied volatility of at-the-money options increases, on average, with maturity). We account for both using Gram-Charlier expansions to approximate the conditional distribution of the logarithm of the price of the underlying security. In this setting, volatility is approximately a quadratic function of moneyness, a result we use to infer skewness and kurtosis from volatility smiles. Evidence suggests that both kurtosis in currency prices and biases in Black-Scholes option prices decline with maturity. , K. Backus, L. Wu, S. Foresi (2004)

On the Pricing of Cliquet Options with Global Floor and Cap In this thesis we present two methods for the pricing and hedging of cliquet options with global floor and/or cap within a Black-Scholes market model with fixed dividends and time dependent volatilities and interest rates. The first is a Fourier transform method giving integral formulas for the price and the greeks. A numerical integration scheme is proposed for the evaluation of these formulas. Using Ito?s Lemma it is proved that the vanilla Black-Scholes PDE is valid. In addition to giving us the gamma for free, it forms the basis for an explicit finite difference method. Both methods outperform Monte Carlo simulation in terms of computational time, with the Fourier method in most cases being the faster one for a given level of accuracy. This tendency is amplified as the number of reset periods increases. Potential future research includes local volatility models and early exercise features for the finite difference method and Levy-process market models for the Fourier method. , M.Kjaer (2004)

Financial Engineering with Reverse Cliquet Options Index-linked securities are offered by banks, financial institutions and building societies to investors looking for downside risk protection whilst still providing upside equity index participation. This article explores how reverse cliquet options can be integrated into the structure of a guaranteed principal bond. Pricing problems are discussed under the standard Black-Scholes model and under the constant-elasticity-of-variance model. Forward start options are the main element of this structure and new closed formulae are obtained for these options under the latter model. Risk management issues are also discussed. An example is described showing how this structure can be implemented and how the financial engineer may forecast the coupon payment that will be made to investors buying this product without exposing the issuing institution to risk of loss. , B.A.Eales, R.Tunaru (2004)

Bessel processes, the integral of geometric Brownian motion, and Asian options This paper is motivated by questions about averages of stochastic processes which originate in mathematical finance, originally in connection with valuing the so-called Asian options. Starting with [Y], these questions about exponential functionals of Brownian motion have been studied in terms of Bessel processes using the Hartman-Watson theory of [Y80]. Consequences of this approach for valuing Asian options proper have been spelled out in [GY] whose Laplace transform results were in fact regarded as a noted advance. Unfortunately, a number of difficulties with the key results of this last paper have surfaced which are now addressed in this paper. One of them in particular is of a principal nature and originates with the Hartman-Watson approach itself: this approach is in general applicable without modifications only if it does not involve Bessel processes of negative indices. The main mathematical contribution of this paper is the developement of three principal ways to overcome these restrictions, in particular by merging stochastics and complex analysis in what seems a novel way, and the discussion of their consequences for the valuation of Asian options proper. , P. Carr, M. Schroder (2003)

Pricing Forward Start Options under the CEV Model With Applications in Financial Engineering Index-linked securities are offered by banks, financial institutions and building societies to investors looking for downside risk protection whilst still providing upside equity index participation. This article explores how reverse cliquet options can be integrated into the structure of a guaranteed principal bond. Pricing problems are discussed under the constant-elasticity-of-variance model. Forward start options are the main element of this structure and new closed formulae are obtained for these options under the square-root process model. Risk management issues are also discussed. An example is described showing how this structure can be implemented and how the financial engineer may forecast the coupon payment that will be made to investors buying this product. , Unknown (2002)

Unified Pricing of Asian Options A simple and numerically stable 2-term partial differential equation characterizing the price of any type of arithmetically averaged Asian option is given. The approach includes both continuously and discretely sampled options and it is easily extended to handle continuous or discrete dividend yields. In contrast to present methods, this approach does not require to implement jump conditions for sampling or dividend days. , J.Vecer (2002)

Robust Numerical Methods for PDE Models of Asian Options We explore the pricing of Asian options by numerically solving the the associated partial differential equations We demonstrate that numerical PDE techniques commonly used in finance for standard options are inaccurate in the case of Asian options and illustrate mod indications which alleviate this problem In particular the usual methods generally produce solutions containing spurious oscillations We adapt flux limiting techniques originally de veloped in the field of computational fluid dynamics in order to rapidly obtain accurate solutions We show that flux limiting methods are total variation diminishing and hence free of spurious oscillations for non conservative PDEs such as those typically encountered in finance for fully explicit and fully and partially implicit schemes We also modify the van Leer flux limiter so that the second order total variation diminishing property is preserved for non uniform grid spacing , R. Zvan, P.A. Forsyth, K.R. Vetzal (2000)

Pricing Exotics under the Smile The volatility implied from the market prices of vanilla options, using the Black Scholes formula, is seen to vary with both maturity and strike price. This surface is known as the volatility smile. It can be considered as a correction for second order effects where the market departs in practice from the assumptions underlying the Black Scholes model. Recent years have seen a surge in the market for exotic path dependent options. Both the liquidity and the volumes of trades of products such as barrier options, compound options and range notes have increased dramatically. These products can have large second order exposures and their traded prices can be significantly offset from the theoretical values calculated under the Black Scholes assumptions. Considerable research effort has been focused on the search for a consistent framework to value both european and exotic options. The objective being to find a methodology which can be practically implemented in a risk management system. This paper details the Exotic Smile model which has been developed and implemented within J. P. Morgan. The paper begins with discussion of the market conditions that are behind the volatility smile. The theoretical framework for the model is then presented, leading to a description of the practical implementation. The results from the model are then compared with market exotic prices. Finally, we discuss the implications of the model to the risk management of exotic products. , M. Jex, R. Henderson, D. Wang (1999)

Option Valuation Using Fast Fourier Transforms , P.Carr, D.Madan (1999)

Pricing Parisian-Style Options with a Lattice Method , M. Avellaneda, L. Wu (1998)

Static Options Replication This paper presents a method for replicating or hedging a target stock option with a portfolio of other options. It shows how to construct a replicating portfolio of standard options with varying strikes and maturities and fixed portfolio weights. Once constructed, this portfolio will replicate the value of the target option for a wide range of stock prices and times before expiration, without requiring further weight adjustments. We call this method static replication. It makes no assumptions beyond those of standard options theory. You can use the technique to construct static hedges for exotic options, thereby minimizing dynamic hedging risk and costs. You can use it to structure exotic payoffs from standard options. Finally, you can use it as an aid in valuing exotic options, since it lets you approximately decompose the exotic option into a portfolio of standard options whose market prices and bid-ask spreads may be better known. , E.Derman, I.Kani (1994)