Brownian motion and normal distribution have been widely used in the Black--Scholes option-pricing framework to model the return of assets. However, two puzzles emerge from many empirical investigations: 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 an empirical phenomenon called "volatility smile" in option markets. To incorporate both of them and to strike a balance between reality and tractability, this paper proposes, for the purpose of option pricing, a double exponential jump-diffusion model. In particular, the model is simple enough to produce analytical solutions for a variety of option-pricing problems, including call and put options, interest rate derivatives, and path-dependent options. Equilibrium analysis and a psychological interpretation of the model are also presented.

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A Jump-Diffusion Model for Option Pricing

The Theory of Stochastic Processes. By D. R. Cox and H. D. Miller. Pp. x, 398. 70s. (Methuen)

This paper aims to extend the analytical tractability of the Black--Scholes model to alternative models with arbitrary jump size distributions. More precisely, we propose a jump diffusion model for asset prices whose jump sizes have a mixed-exponential distribution, which is a weighted average of exponential distributions but with possibly negative weights. The new model extends existing models, such as hyperexponential and double-exponential jump diffusion models, because the mixed-exponential distribution can approximate any distribution as closely as possible, including the normal distribution and various heavy-tailed distributions. The mixed-exponential jump diffusion model can lead to analytical solutions for Laplace transforms of prices and sensitivity parameters for path-dependent options such as lookback and barrier options. The Laplace transforms can be inverted via the Euler inversion algorithm. Numerical experiments indicate that the formulae are easy to implement and accurate. The analytical solutions are made possible mainly because we solve a high-order integro-differential equation explicitly. A calibration example for SPY options shows that the model can provide a reasonable fit even for options with very short maturity, such as one day.

This paper was accepted by Michael Fu, stochastic models and simulation.

/pdf/option-pricing-under-a-double-exponential-jump-diffusion-4aajyykwc8.pdf

Option Pricing Under a Double Exponential Jump Diffusion Model

Much of the work on path-dependent options assumes that the underlying asset price follows geometric Brownian motion with constant volatility. This paper uses a more general assumption for the asset price process that provides a better fit to the empirical observations. We use the so-called constant elasticity of variance CEV diffusion model where the volatility is a function of the underlying asset price. We derive analytical formulae for the prices of important types of path-dependent options under this assumption. We demonstrate that the prices of options, which depend on extrema, such as barrier and lookback options, can be much more sensitive to the specification of the underlying price process than standard call and put options and show that a financial institution that uses the standard geometric Brownian motion assumption is exposed to significant pricing and hedging errors when dealing in path-dependent options.

/pdf/pricing-and-hedging-path-dependent-options-under-the-cev-4sw015wcjl.pdf

Pricing and Hedging Path-Dependent Options Under the CEV Process

We develop a flexible and analytically tractable framework which unifies the valuation of corporate liabilities, credit derivatives, and equity derivatives. We assume that the stock price follows a diffusion, punctuated by a possible jump to zero (default). To capture the positive link between default and equity volatility, we assume that the hazard rate of default is an increasing affine function of the instantaneous variance of returns on the underlying stock. To capture the negative link between volatility and stock price, we assume a constant elasticity of variance (CEV) specification for the instantaneous stock volatility prior to default. We show that deterministic changes of time and scale reduce our stock price process to a standard Bessel process with killing. This reduction permits the development of completely explicit closed form solutions for risk-neutral survival probabilities, CDS spreads, corporate bond values, and European-style equity options. Furthermore, our valuation model is sufficiently flexible so that it can be calibrated to exactly match arbitrarily given term structures of CDS spreads, interest rates, dividend yields, and at-the-money implied volatilities.

http://users.iems.northwestern.edu/~linetsky/JDCEV.pdf

A jump to default extended CEV model: an application of Bessel processes

This paper develops an eigenfunction expansion approach to pricing options on scalar diffusion processes. All contingent claims are unbundled into portfolios of primitive securities calledeigensecurities. Eigensecurities are eigenvectors (eigenfunctions) of the pricing operator (present value operator). All computational work is at the stage of finding eigenvalues and eigenfunctions of the pricing operator. The pricing is then immediate by the linearity of the pricing operator and the eigenvector property of eigensecurities. To illustrate the computational power of the method, we develop two applications:pricing vanilla, single- and double-barrier options under the constant elasticity of variance (CEV) process and interest rate knock-out options in the Cox-Ingersoll-Ross (CIR) term-structure model.

http://users.iems.northwestern.edu/~linetsky/eigenfunction.pdf

Pricing Options on Scalar Diffusions: An Eigenfunction Expansion Approach

of time the underlying asset price spends outside of a pre-specified price range (occupation time). Proportional and simple double-barrier step options are gradual knockout options with the principal amortized based on the occupation time outside of the range. Delayed double-barrier options are extinguished when the occupation time outside of the range exceeds a prespecified knock-out window (delayed knockout). These contract designs are proposed as alternatives to the currently traded double-barrier options. They alleviate discrete “barrier event” risk and are easier to hedge.

Dmitry Davydov

Papers

Pricing and Hedging Path-Dependent Options Under the CEV Process

Pricing Options on Scalar Diffusions: An Eigenfunction Expansion Approach

Structuring, Pricing and Hedging Double-Barrier Step Options