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Pricing and hedging long-term options. (English) Zbl 0989.91041

The authors study the difference between the short-term and long-term options. The following model that incorporates stochastic volatility, stochastic interest rates R(t) and random jumps is considered. Let the spot interest rate follow a square-root diffusion

dR(t)=[θ R -k R R(t)]dt+σ R R(t)dw R (t)

where k R ,θ R /k R and σ R are respectively the speed of adjustment, the long-run mean, the volatility coefficient of process R(t); w R (t) is a standard Brownian motion. The underlying stock is assumed to pay a constant dividend yield, denoted by δ, and its price S(t) changes, under risk-neutral measure, according to the jump-diffusion stochastic differential equation

dS(t)/S(t)=[R(t)-δ-λμ J ]dt+V(t)dw S (t)+J(t)dq(t),

where V(t) also follow a square-root equation

dV(t)=[θ V -k V V(t)]dt+σ V V(t)dw V (t)·

The intensity of the jump component is measured by λ, the size of percentage price jumps at time t is represented by J(t), which is lognormal, identically and independently distributed over time with unconditional mean μ J ; q(t) is a Poisson counter with

P{dq(t)=1}=λdt,P{dq(t)=0}=1-λdt·

Finally, let Cov t [dw S (t),dw V (t)]ρdt, q(t) and J(t) be uncorrelated with each other or with w S (t) and w V (t). The option pricing formula for a European put option is derived for the considered model. The authors study the option deltas and state-price densities under alternative models for short-term and long-term options. A description of the regular and LEAPS S&P 500 option data is provided and the estimation of structural parameters by using the method of simulated moments is presented. The difference between information in short-term and long-term options is studied. The hedging of the underlying stock portfolio and evaluation of the relative effectiveness of the underlying asset, short-term and medium-term options in hedging LEAPS are presented.

MSC:
91B28Finance etc. (MSC2000)