The Quantitative Finance Library

Asymptotics of Implied Volatility in Local Volatility Models Using an expansion of the transition density function of a 1-dimensional time inhomogeneous diffusion, we obtain the ?rst and second order terms in the short time asymptotics of European call option prices. The method described can be generalized to any order. We then use these option prices approximations to calculate the ?rst order and second order deviation of the implied volatility from its leading value and obtain approximations which we numerically demonstrate to be highly accurate. The analysis is extended to degenerate diffusion's using probabilistic methods, i.e. the so called principle of not feeling the boundary. , J.Gatheral, E.P.Hsu, P.M.Laurence, C.Ouyang, T-H.Wang (2010)

A Practical Guide to Implied and Local Volatility We consider a stochastic local volatility model with domestic and foreign stochastic interest rates such that the volatilitydecomposes into a deterministic local volatility plus some bias terms. Assuming a collapse process for the variance with the same random variable for all time and deterministic zero-coupon bond volatility functions, we are going to describe in detail the implementation of that model, focusing on the computation of a proper deterministic local volatility. To do so, we choose to generate an implied volatility surface without arbitrage in space and in time by parametrising a mixture of shifted lognormal densities under constraints and we use a Differential Evolution algorithm to calibrate the model's parameters to a finite set of option prices. We will therefore need to devise an evolutionary algorithm that handle constraints in a simple and efficient way. Using some of the improvements made to the DE algorithm combined with simple and robust constraints handling mechanisms we will propose a modified algorithm for solving our optimisation problem under constraints which greatly improves its performances. , D.A.Bloch (2010)

Local Volatility Enhanced by a Jump to Default A local volatility model is enhanced by the possibility of a single jump to default. The jump has a hazard rate that is the product of the stock price raised to a prespecified negative power and a deterministic function of time. The empirical work uses a power of -1.5. It is shown how one may simultaneously recover from the prices of credit default swap contracts and equity option prices both the deterministic component of the hazard rate function and revised local volatility. The procedure is implemented on prices of credit default swaps and equity options for GM and FORD over the period October 2004 to September 2007. , P.Carr, D.P.Madan (2010)

New Approximations in Local Volatility Models For general time-dependent local volatility models, we propose new ap- proximation formulas for the price of call options. This extends previous results of [BGM10b] where stochastic expansions combined with Malliavin calculus were performed to obtain approximation formulas based on the local volatility At The Money. Here, we derive alternative expansions involving the local volatility at strike. Averaging both expansions give even more accurate results. Approximations of the implied volatility are provided as well. , E.Gobet, A.Suleiman (2010)

Analytical Formulas for Local Volatility Model with Stochastic Rates This paper presents new approximation formulae of European options in a local volatility model with stochastic interest rates. This is a companion paper to our work on perturbation methods for local volatility models http://papers.ssrn.com/sol3/papers.cfm?abstract_id=1275872 for the case of stochastic interest rates. The originality of this approach is to model the local volatility of the discounted spot and to obtain accurate approximations with tight estimates of the error terms. This approach can also be used in the case of stochastic dividends or stochastic convenience yields. We finally provide numerical results to illustrate the accuracy with real market data. , E.Benhamou, E.Gobet, M.Miri (2009)

Volatility and Dividends :Volatility Modelling with Cash Dividends and simple Credit Risk This article shows how to incorporate cash dividends and credit risk into equity derivatives pricing and risk management. In essence, we show that in an arbitrage-free model the stock price process upon default must have the form St = (F *t - D t )Xt + Dt where X is a (local) martingale with X0 = 1, the curve F * is the risky forward and D is the floor imposed on the stock price process in the form of appropriately discounted future dividends. We show that the method presented is the only such method which is consistent with the assumption of cash dividends and simple credit risk. We discuss the implications for implied volatility, no-arbitrage conditions and we derive a version of Dupire?s formula which handles cash dividend and credit risk properly. We discuss pricing and risk management of European options, PDE methods and in quite some detail variance swaps and related derivatives such as gamma swaps, conditional variance swaps and corridor variance swaps. Indeed, to the our best if our knowledge, this is the first article which shows the correct handling of cash dividends when pricing variance swaps. , H.Buelher (2009)

Local Volatility Enhanced by a Jump to Default A local volatility model is enhanced by the possibility of a single jump to default. The jump has a hazard rate that is the product of the stock price raised to a prespeci?ed negative power and a deterministic function of time. The empirical work uses a power of It is shown how one may simultaneously recover from the prices of credit default swap contracts and equity option prices both the deterministic component of the hazard rate function and revised local volatility. The procedure is implemented on prices of credit default swaps and equity options for GM and F ORD over the period October 2004 to S eptember 2007: , P.Carr (2009)

Closed Forms for European Options in a Local Volatility Model Because of its very general formulation, the local volatility model does not have an analytical solution for European options. In this article, we present a new methodology to derive closed form solutions for the price of any European options. The formula results from an asymptotic expansion, terms of which are Black-Scholes price and related Greeks. The accuracy of the formula depends on the payoff smoothness and it converges with very few terms. , E. Benhamou, E. Gobet, M. Miri (2008)

Incorporating an Interest Rate Smile in an Equity Local Volatility Model The focus of this paper is on finding a connection between the interest rate and equity asset classes. We propose an equity interest rate hybrid model which preserves market observable smiles: the equity from plain vanilla products via a local volatility framework and the interest rate from caps and swaptions via the Stochastic Volatility Libor Market Model. We define a multi-factor short-rate process implied from the Libor Market Model via an arbitrage-free interpolation and combine it with the local volatility equity model for stochastic interest rates. We show that the interest rate smile has a significant impact on the equity local volatility. The model developed is intuitive and straightforward, enabling consistent pricing of related hybrid products. Moreover, it preserves the non-arbitrage Heath, Jarrow, Morton conditions. , L. A. Grzelak, N. Borovykh, S. V. Weeren, K. Oosterlee (2008)

Implied Levy Volatility , J.M.Corcuera, F.Guillaume, P.Leoni, W.Schoutens (2008)

Stochastic Interest Rates for Local Volatility Hybrids Models , E.Benhamou, A.Rivoira, A.Gruz (2008)

Lecture 6: Extending Black-Scholes ; Local Volatility Models , E.Derman (2008)

Lecture 7: Local Volatility Continued , E.Derman (2008)

Lecture 8: Local Volatility Models ; Implication , E.Derman (2008)

Lecture 9: Patterns of Volatility Change , E.Derman (2008)

Option Pricing with Quadratic Volatility: A Revisit , L.Andersen (2008)

Hyp Hyp Hooray , P.Jaeckel (2008)

Hyperbolic local volatility , P.Jaeckel (2007)

Smile interpolation and calibration of the local volatility model , N.Kahale (2005)

Local and Stochastic Volatility Models: An Investigation into the Pricing of Exotic Equity Options , L.Majmin (2005)

Arbitrage-Free Smoothing of the Implied Volatility Surface , M.R. Fengler (2005)

From Local Volatility to Local Levy Models , P.Carr, D.Madan, M.Yor, H.Geman (2004)

Pricing with a Smile , B. Dupire (2004)

Lecture 1: Stochastic Volatility and Local Volatility , J.Gatheral (2002)

Implied Volatility: Statics, Dynamics, and Probabilistic Interpretation Given the price of a call or put option, the Black-Scholes implied volatility is the unique volatility parameter for which the Bulack-Scholes formula recovers the option price. This article surveys research activity relating to three theoretical questions: First, does implied volatility ad- mit a probabilistic interpretation? Second, how does implied volatility behave as a function of strike and expiry? Here one seeks to characterize the shapes of the implied volatility skew (or smile) and term structure, which together constitute what can be termed the statics of the implied volatility surface. Third, how does implied volatility evolve as time rolls forward? Here one seeks to characterize the dynamics of implied volatility. , R. Lee (2002)

Black-Scholes Goes Geometric , C.Albanese, G.Campolieti, P.Carr, A.Lipton (2001)

Equivalent Black Volatilities , P.S.Hagan, D.E.Woodward (1998)

Calibrating Volatility Surfaces via Relative-Entropy Minimization , M. Avellaneda, C. Friedman, R. Holmes, D. Samperi (1997)

Trading and Hedging Local Volatility , E.Derman, I.Kani, M.Kamal (1996)

The Local Volatility Surface Unlocking the Information in Index Option Prices , E.Derman, I.Kani, J.Z.Zou (1995)