Abstract
Problem statement: We presented option pricing when the stock prices follows a jump- diffusion model and their stochastic volatility fol lows a fractional stochastic volatility model. This proposed model exhibits the a memory of a stochasti c volatility model that is not expressed in the classical stochastic volatility model. Approach: We introduce an approximated method to fractional stochastic volatility model perturbed by the fracti onal Brownian motion. A relationship between stochastic differential equations and partial diffe rential equations for a bivariate model is presente d. Results: By using an approximate method, we provide the approximate solution of the fractional stochastic volatility model. And European options a re priced by using the risk-neutral valuation. Conclusion/Recommendations: The formula of European option is calculated by usi ng the technique base on the characteristic function of an underlyin g asset which can be expressed in an explicit formula. A numerical integration technique to simul ation fractional stochastic volatility are presente d in this study.
Highlights
F= thist0i≤o(tΩn≤T,wF..,ilAPl)lbl.bepedreoafciepnsresodebsainbtihtlhaititys space with filtration we shall consider in space
T < ∞ a geometric Brownian motion model with jumps and with fractional stochastic volatility is a model of the form: ( ) dSt = St μdt + vt dWt + St−YdNt where, ω > 0 is the mean long-term volatility, θ∈R is the rate at which the volatility reverts toward its longterm mean, ξ>0 is the volatility of the volatility process and (Bt )t∈[0,T] is a fractional Brownian motion
According to the formula as given in Theorem 3, we firstly choose a real number ε > 0, the solution that we get is the value of a European call option of the approximation model (10) with (11) and this value can be used as an approximating value of a call option of the fractional model (7) including model (8) as ε approaches zero
Summary
F= this (sLFetect)t0i≤o(tΩn≤T,wF..,ilAPl)lbl.bepedreoafciepnsresodebsainbtihtlhaititys space with filtration we shall consider in space. T < ∞ a geometric Brownian motion (gBm) model with jumps and with fractional stochastic volatility is a model of the form:. Where, ω > 0 is the mean long-term volatility, θ∈R is the rate at which the volatility reverts toward its longterm mean, ξ>0 is the volatility of the volatility process and (Bt )t∈[0,T] is a fractional Brownian motion. Assume that the processes (St) and (vt) are Ftmeasurable. The fractional version of Eq 1 is given by: where μ ∈R, S = (St )t∈[0,T] is a process representing a price of the underlying risky assets, W = (Wt )t∈[0,T] is the standard Brownian motion, N = (Nt )t∈[0,T] is a Poisson process with intensity λ and St-Yt represents the amplitude of the jump which occurs at time t. The volatility process vt := σt in (1) is modeled by:
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