Modelling FX smile : from stochastic volatility to skewness

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A Jump Diffusion Model for Option Pricing with Three Properties: Leptokurtic Feature, Volatility Smile, and Analytical Tractability

Brownian motion and normal distribution have been widely used, for example, in the Black-Scholes-Merton option pricing framework, to study the return of assets. However, two puzzles, emerged from many empirical investigations, have got much attention recently, namely (a) the leptokurtic feature that the return distribution of assets may have a higher peak and two (asymmetric) heavier tails than those of the normal distribution, and (b) an empirical abnormity called ``volatility smile'' in option pricing. To incorporate both the leptokurtic feature and ``volatility smile'', this paper proposes, for the purpose of studying option pricing, a jump diffusion model, in which the price of the underlying asset is modeled by two parts, a continuous part driven by Brownian motion, and a jump part with the logarithm of the jump sizes having a double exponential distribution. In addition to the above two desirable properties, leptokurtic feature and ``volatility smile'', the model is simple enough to produce analytical solutions for a variety of option pricing problems, including options, future options, and interest rate derivatives, such as caps and floors, in terms of the $Hh$ function. Although there are many models can incorporate some of the three properties (the leptokurtic feature, ``volatility smile'', and analytical tractability), the current model can incorporate all three under a unified framework.

What Type of Process Underlies Options? A Simple Robust Test

We develop a simple robust test for the presence of continuous and discontinuous (jump) com­ponents in the price of an asset underlying an option. Our test examines the prices of at­the­money and out­of­the­money options as the option maturity approaches zero. We show that these prices converge to zero at speeds which depend upon whether the sample path of the underlying asset price process is purely continuous, purely discontinuous, or a mixture of both. By applying the test to S&P 500 index options data, we conclude that the sample path behavior of this index contains both a continuous component and a jump component. In particular, we find that while the pres­ence of the jump component varies strongly over time, the presence of the continuous component is constantly felt. We investigate the implications of the evidence for parametric model specifications.Jumps; continuous martingale; option pricing; Levy density; double tails; local time.

WEATHER DERIVATIVES: MANAGING RISK WITH MARKET-BASED INSTRUMENTS

Accurate pricing of weather derivatives is critically dependent upon correct specification of the underlying weather process. We test among six likely alternative processes using maximum likelihood methods and data from the Fresno, CA weather station. Using these data, we find that the best process is a mean-reverting geometric Brownian process with discrete jumps and ARCH errors. We describe a pricing model for weather derivatives based on such a process.Risk and Uncertainty,

Moment Methods for Exotic Volatility Derivatives

The latest generation of volatility derivatives goes beyond variance and volatility swaps and probes our ability to price realized variance and sojourn times along bridges for the underlying stock price process. In this paper, we give an operator algebraic treatment of this problem based on Dyson expansions and moment methods and discuss applications to exotic volatility derivatives. The methods are quite flexible and allow for a specification of the underlying process which is semi-parametric or even non-parametric, including state-dependent local volatility, jumps, stochastic volatility and regime switching. We find that volatility derivatives are particularly well suited to be treated with moment methods, whereby one extrapolates the distribution of the relevant path functionals on the basis of a few moments. We consider a number of exotics such as variance knockouts, conditional corridor variance swaps, gamma swaps and variance swaptions and give valuation formulas in detail

Dampened Power Law: Reconciling the Tail Behavior of Financial Security Returns

This paper proposes a stylized model that reconciles several seemingly conflicting findings on financial security returns and option prices. The model is based on a pure jump Levy process, wherein the jump arrival rate obeys a power law dampened by an exponential function. The model allows for different degrees of dampening for positive and negative jumps, and also different pricing for upside and downside market risks. Calibration of the model to the S&P 500 index shows that the market charges only a moderate premium on upward index movements, but the maximally allowable premium on downward index movements.dampened power law; alpha-stable distribution; central limit theorem; upside movement; downside movement

Moment Methods for Exotic Volatility Derivatives

The latest generation of volatility derivatives goes beyond variance and volatility swaps and probes our ability to price realized variance and sojourn times along bridges for the underlying stock price process. In this paper, we give an operator algebraic treatment of this problem based on Dyson expansions and moment methods and discuss applications to exotic volatility derivatives. The methods are quite flexible and allow for a specification of the underlying process which is semi-parametric or even non-parametric, including state-dependent local volatility, jumps, stochastic volatility and regime switching. We find that volatility derivatives are particularly well suited to be treated with moment methods, whereby one extrapolates the distribution of the relevant path functionals on the basis of a few moments. We consider a number of exotics such as variance knockouts, conditional corridor variance swaps, gamma swaps and variance swaptions and give valuation formulas in detail.volatility derivatives; operator methods; moment methods; conditional corridor variance swaps; variance knockout options