The Black-Scholes model [6, 23] has gained wide recognition on financial markets. One of its shortcomings, however, is that it is inconsistent with most observed option prices. Although the model can still be used very efficiently, it has been proposed to relax its assumptions, and, for instance, to consider that the volatility of the underlying asset S is no longer a constant but rather a stochastic process. There are two well-known approaches to achieve this goal. In the first class of models, the volatility is assumed to depend on the variables t (time) and S, giving rise to the so-called local volatility models. The second one, conceptually more ambitious, considers that the volatility has a stochastic component of its own. In the latter, the number of factors is increased by the amount of stochastic factors entering the volatility modeling. Both models are of practical interest. In these contexts, it is relevant to express the resulting prices in terms of implied volatilities. Given a price, the Black-Scholes implied volatility is determined, for each given product (that is for each given strike and expiry date defining, say, the call option) as the unique value of the volatility parameter for which the BlackScholes pricing formula agrees with that given price. Actually, it is common practice on trading floors to quote and to observe prices in this way. A great advantage of having prices expressed in such dimensionless units is to provide easy comparison between products with different characteristics. In principle, the implied volatility can be inferred from computed options prices by inverting the Black-Scholes formula. It is more convenient, however, to directly analyze the implied volatility. Indeed, this approach allows us to shed light on qualitative properties that would otherwise be more difficult to establish. In particular, we derive here several asymptotic formulae that are of practical interest, for example, in the calibration problem. The latter—an inverse problem that consists

Computing the implied volatility in stochastic volatility models

We derive a direct link between local and implied volatilities in the form of a quasilinear degenerate parabolic partial differential equation. Using this equation we establish closed-form asymptotic formulae for the implied volatility near expiry as well as for deep in- and out-of-the-money options. This in turn leads us to propose a new formulation near expiry of the calibration problem for the local volatility model, which we show to be well posed.

http://www.cmap.polytechnique.fr/%7Erama/dea/bbf.pdf

Asymptotics and calibration of local volatility models

1 Derivatives Markets: Introduction and Definitions Management Science has a long tradition of publishing important research in the finance area, including significant contributions to portfolio optimization; asset-liability management; utility theory and stochastic dominance; and empirical finance and derivative securities Within the derivatives area, contributions in the journal have advanced our understanding of the pricing, hedging, and risk management of derivative securities in a wide variety of financial markets, including equity, fixed income, commodity, and credit markets In addition to theoretical advances, several articles have focused on practical applications through the design of efficient numerical procedures for valuing and hedging derivative securities In this paper we survey the option pricing literature over the last four decades, including many articles that have appeared in the pages of Management Science We begin with a description of derivative securities and their properties A derivative security is a financial asset whose payoff depends on the value of some underlying variable The underlying variable can be a traded asset, such as a stock; an index portfolio; a futures price; a currency; or some measurable state variable, such as the temperature at some location or the volatility of an index The payoff can involve various patterns of cash flows Payments can be spread evenly through time, occur at specific dates, or a combination of the two Derivatives are also referred to as contingent claims An option is a derivative security that gives the right to buy or sell the underlying asset, at or before some maturity date T , for a prespecified price K, called the strike or exercise price A call (put) option is a right to buy (sell) Because exercise is a right and not an obligation, the exercise payoff is S−K + ≡max S−K 0 for a call option and K − S + ≡ max K − S 0 for a put option, where S denotes the price of the underlying asset Options can be European style, which can only be exercised at the maturity date, or American style, where exercise is at the discretion of the holder, at any time before or at the maturity date Plain vanilla options, such as those described above, were introduced on organized option exchanges such as the Chicago Board of Options Exchange (CBOE) in 1973 Since then, innovation has led to the creation of numerous products designed to fill the needs of various types of investors Path-dependent options, such as barrier options, Asian options, and lookbacks are examples of contractual forms that have emerged since and are now routinely traded in markets or quoted by financial institutions, or both Even more exotic types of contracts, whose payoffs depend on multiple underlying assets or on occupation times of predetermined regions, have emerged in recent years and have drawn interest This paper surveys past contributions in the field and provides an overview of recent trends We first review the fundamental valuation principles for European-style (§2) and American-style (§3) derivatives Next, we outline extensions of the basic model (§4) and survey numerical methods (§5) An assessment of the future and comments on open problems are formulated in the concluding section (§6) The appendix treats derivatives in the fixed income

/pdf/option-pricing-valuation-models-and-applications-c5v2erj000.pdf

Option Pricing: Valuation Models and Applications

Using an expansion of the transition density function of a one-dimensional time inhomogeneous diffusion, we obtain the first- 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 first- and second-order deviation of the implied volatility from its leading value and obtain approximations which we numerically demonstrate to be highly accurate.

/pdf/asymptotics-of-implied-volatility-in-local-volatility-models-4dsluvxqbs.pdf

Asymptotics of implied volatility in local volatility models

This paper surveys the literature on option pricing from its origins to the present. An extensive review of valuation methods for European- and American-style claims is provided. Applications to complex securities and numerical methods are surveyed. Emphasis is placed on recent trends and developments in methodology and modeling.

50th ANNIVERSARY ARTICLE: Option Pricing: Valuation Models and Applications

We consider European calls and puts on an asset whose forward price F(t) obeys dF(t)=α(t)A(F)dW(t,) under the forward measure. By using singular perturbation techniques, we obtain explicit algebraic formulas for the implied volatility σ B in terms of today's forward price F 0 ≡ F(0), the strike K of the option, and the time to expiry tex . The price of any call or put can then be calculated simply by substituting this implied volatility into Black's formula. For example, for a power law (constant elasticity of variance) model dF(t)=aFβ dW(t) we obtain σ B = a/f aυ 1− β {1 + (1−β)(2+β)/24 (F 0 − K/f aυ)2 + (1 − β)2/24 a 2 tex /f aυ 2−2β +…} where f aυ = ½(F 0 + K). Our formula for the implied volatility is not exact. However, we show that the error is insignificant, rarely approaching 1/1000 of the time value of the option. We also present more accurate (albeit more complicated) formulas which can be used for the implied volatility.

Equivalent Black volatilities

A general procedure for creating Markovian interest rate models is presented. The models created by this procedure automatically fit within the HJM framework and fit the initial term structure exactly. Therefore they are arbitrage free. Because the models created by this procedure have only one state variable per factor, twoand even three-factor models can be computed efficiently, without resorting to Monte Carlo techniques. This computational efficiency makes calibration of the new models to market prices straightforward. Extended Hull- White, extended CIR, Black-Karasinski, Jamshidian's Brownian path independent models, and Flesaker and Hughston's rational log normal models are one-state variable models which fit naturally within this theoretical framework. The ‘separable’ n-factor models of Cheyette and Li, Ritchken, and Sankarasubramanian - which require n(n + 3)/2 state variables - are degenerate members of the new class of models with n(n + 3)/2 factors. The procedure is used to create a new class o...

Diana E. Woodward

Papers

Equivalent Black volatilities

Markov interest rate models