| Option pricing is a very important problem in financial mathematics. The price ofoption, i.e. premium, is the amount of money the buyer paid to buy the option in aoption trading. The price of option depends on the changes of underlying asset price.Due to the underlying asset is a risk asset, therefore, its price changes are random. Thus,the price of the option is random. In mathematics, the price of option is the solution ofa backward parabolic partial diferential equation.1973, Fischer Black and Myron Scholes established the classical Black-Scholes modelbased on the assumption that the price of the underlying asset follows the geometricBrownian motion. They obtained the Black-Scholes (B-S in short) equation, and gavethe pricing formula for the European call option, i.e. the classical Black-Scholes formula.Then, Cox et al. proposed a discrete time option pricing model, i.e. binomial treemethods. The two models are still popular in the world now.However, in the recent decades, a large number of empirical studies show that theprice of the underlying asset is not necessarily follow the geometric Brownian motion. So, researchers focus their research on improving the Black-Scholes model. For example, fractional Black-Scholes model, subdiffisive (time changed) Black-Scholes model.This thesis is devoted to studies the option pricing problem under different time-changed Black-Scholes model. To be specific, we consider the following four models.In Chapter3, we proposed a time changed Brownian-fractional Brownian Black-Scholes model. In this model, we assume that the price of the underlying stock satisfies here S0,μ,σ are constants, Tα(t) is α-stable subordinator, α∈(2/3,1); Mα,H(t) αB(Tα(t))+bBH(Tα(t)), H∈(1/2,1). We assume that B(τ),BH(τ)和Tα(t) are all inde-pendent.We first obtained the partial differential equation which the European call option satisfied in a discrete time setting. Then we obtained the pricing formula for European option and call-put parity equation. The theorems are:Theorem0.4.1. When the underlying stock price satisfies(3.4), then the value of European call option V=V(t, St) satisfies where hereTheorem0.4.2. call-put parityLet V(t, S) and P(t, S) are European call and put option respectively, and have the same expiry date T, strike price K, risk-free rate r. Then when the price of the underlying stock satisfies (3.4), we haveWe also gave the corresponding numerical results. In the last, we discussed the implied volatility, and obtained the volatility skrew.In Chapter4, we discussed the time changed Merton model with transaction costs. The model assumed that the price of the underlying stock St satisfies S0,μ and a are constants. B(Tα(t)) is a subdiffusive process, B(τ) is the standard Brownian motion. α∈(2/3,1), Tα(t) is the inverse α-table subordinator. Nt is a Poisson process with intensity λ>0, and J is a positive random variable. Assume that Ta(t), B(τ), Nt and J are independent.We obtained the partial differential equation which the European call option satisfied, and gave the option pricing formula. The theorem is: Theorem0.4.3. The value of the European call option V=V(t, St) satisfies with boundary condition Furthermore, Vt, St is given by hereIn Chapter5, we proposed a time-changed CEV model. This model assumes that the underlying stock priceZt satisfies where μ,σ,β, Zo are all constants. dbH(τ)=ω(τ)(dτ)H is a modified fractional Brownian motion, H∈[1/2,1),ω(τ) is the companion Gaussian white noise with zero mean and unit variance. In particular, it is assumed that the Sa(t) is independent of bH(t).Under some assumptions, we obtained the partial differential equation which the European call option satisfied, and gave the pricing formula. Then, we get an asymptotic representation of the European call option price. The theorems are:Theorem0.4.4. Assume that the price of option C(t,Zt) belongs to C1,2([0,T)× [0,+∞)), then C(t, Zt) satisfies the following partial differential equation with boundary conditionTheorem0.4.5. Suppose that αH>1/2, then the solution of problem (2.12)-(2.2),i.e. the value of a European call option C(t, Z(t)) is given by where Here Γ(ε) is the Gamma function.Theorem0.4.6. Suppose the European call option price C(t, Z) which is the solu-tion of (2.1)-(2.2) has an asymptotic expansion price such as Then Co(t, Z) with the final condition C0(T, Z)=[Z-K)+is given by where and each Cn(t, Z), n=1,2,..., with the final condition Cn(T, Z)=0is recursively given by In Chapter6, we discussed the European option pricing under the Merton model of the short rate which based on the time-changed process. We first obtained the pricing formula for a zero-coupon bond, then based on it the pricing formula for European options and the call-put parity were obtained. The results are: Theorem0.4.7. Under the time-changed Merton model of the short rate, the price of the European call option V(S,r,t) satisfies the following partial differential equation and the boundary condition is here Furthermore, the pricing formula for V(S,r,t) is whereThe call-put parity is Theorem0.4.8. Let c(S,r,t), p(S,r,t) are the price of the European call and put option with the same strike price K and expiry date T. P(S, r, t) is the price of a zero-coupon bond. The the call-put parity is... |
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