Derivatives Pricing Problems In Stochastic Volatility Models

In the last30years, the famous Black-Scholes-Merton formula has been proved to be an efficient and important tool for pricing option. However, there are some evidences that the constant volatility assumption in Black-Scholes model is inaccurate, even Black himself noticed that the assumption of constant volatility is inaccurate, as the implied volatility in financial market fluctuates frequently around some mean value. It is more suitable to use stochastic volatility model to describe the complicated behaviors of derivatives. In this thesis, we consider stochastic volatility models and derivatives pricing problem. The main results we obtained are as follows:Part I:Stochastic Volatility Model in Pricing Multi-Asset European Options.In Chapter Two, we consider a stochastic volatility model for pricing multi-asset European options, which are widely used in the real world. Under the assumption that the volatilities are driven by different OU processes, we establish the option pricing equation. Using the singular perturbation method for multi-parameter and multiscale skill, we derive a uniform asymptotic expansion for the option prices, as well as the uniform error estimates.The option pricing equation is Using the singular perturbation and multiscale method, we derive a uniform asymptotic expansion for option prices, as well as the uniform error estimates. Theorem0.1Under the assumptions (A2.1),(A2.2) and (A2.3), we obtain the option price Pεδ in the following form, where the leading term P0(t,x1,x2) called main term of option price is the classical Black-Scholes price when we replacing the stochastic volatility f(y1) and g(y2) by their average <f> and <g>. The second order term P01(t,x1,x2) and P10(t,x1,x2) are called corrected terms of option price due to stochastic volatility and they are combinations of derivatives of P0(t,x1,x2) with respect to stock price x1,x2.Actually, we can derive higher order corrected terms to Black-Scholes price and the full asymptotic expansion of the option price. Moreover we obtain the following uniform error estimates.Theorem0.2Under the assumptions (A2.1),(A2.2) and (A2.3), we derive the uniform error estimate between the real option price Pε,δ and the approximate option price P0+(?)P10+(?)P01where P0(t,x1,x2), P10(t,x1,x2) and P01(t,x1,x2)have been calculated explicitly.Part Ⅱ: Stochastic Model with Stochastic Interest Rate and Derivatives Pricing.In Chapter Three, we consider the stochastic model with stochastic interest rate. It follows from the fast mean reverting stochastic volatility model and CIR model of short rate that we obtain the pricing equation of interest rate derivative. By the singular perturbation method and boundary layer theory, we derive the first order approximate price of derivative and the uniform error estimates.First, we derive the pricing equation of interest rate derivatives, where h(xt) is the payoff function.Second, following the singular perturbation method and multiscale skill, we get the explicit expression of first order approximate price, where the main term Po(t, x, z) is the classical Black-Scholes price with constant volatil-ity(?>) The corrected term due to stochastic volatility is Another corrected term due to stochastic interest rate isAt last, we give the uniform error estimate.Theorem0.3Under the assumptions (A3.1),(A3.2),(A3.3) and (A3.4), we get the error estimate between real option price and approximate price The above error estimate is uniform in time stock price and interest rate.Part III:Leverage Effect of Stochastic Volatility Model.In Chapter Four, we consider the leverage effect of stochastic volatility model in financial market. We describe apparition of leverage effect and define quantitative leverage with time structure. Moreover we explicitly state the conditions for the ex-istence of leverage effect. We respectively study leverage effects for three classes of models:the general stochastic volatility model, the asset pricing model with jumps and the stochastic volatility model with Poisson jumps. The leverage of the first model depends on the correlation parameters. The leverage effect in the second model is actually amplification. The third model has the amplified leverage effect.Considering general stochastic volatility model dSt=μStdt+σ(Yt)StdWt, dYt=α(Yt)dt+β(Yt)dBt, we now state the definition of leverage and condition of leverage effect in this model. Definition0.4We define the leverage quantity over the derivatives'available time [0,t] as the correlation ρRY between asset's return Rt and stochastic volatility σ(Yt). There exits leverage effect in the formulated model for asset price if L(t)<0, for any t∈[0,t].Theorem0.5There exists leverage effect in the general stochastic volatility model if the following condition hold true for any u, s∈[0,t]. where The second model considered is asset pricing model with jump, dR(t)=μdt+σdW(t)+d(Q(t)-βλt). The jump process Q(t) has a special leverage effect on the asset return R(t) in the above model.Proposition0.6In the asset pricing model, the jump process Q(t) have amplified effect on R(t) at any time. Namely,(?)(t)=λt(β2+η)≤0for any t>0,有A>0. We call (?)(t) the magnification of amplified leverage which implied that the jumps' frequency and sizes reinforce the return or loss of asset.The third model is the stochastic model with Poisson jumps. dS(t)=μS(t)dt+σ(Y(t))S(t)dW(t)+S(t-)d(Q(t)-βλt) dY(t)=αa(Yt)dt+β(Yt)dB(t). We find that the leverage of this model consists of two part: one part is the amplified effect (?)(t) of jump process Q(t), the other is related to the leverage effect L(t).Definition0.7We define the leverage quantity L(t) between asset return R{t) and stochastic volatility process σ(Y{t)) at time t as L(t)=(?)(t)Cov(R(t),σ(Y(t))), where (?)(t)=cov(R(t), Q(t)) is called magnification of amplified leverage. There exits amplified leverage effect in model if L(t)<0for any t>0.Theorem0.8(The leverage condition) There exits leverage effect in the stochastic model with Poisson jumps at time t,if the following inequality hold true for any u, s∈[0, t], wherePart IV:Stochastic Volatility Model Under Non-Smoothness ConditionsIn Chapter Five, we consider a stochastic volatility model for pricing European options. By the perturbation analysis and smoothing approximation method, we solve a backward PDE with one small parameter and derive a uniform asymptotic expansion for the option prices, as well as the uniform error estimates, under non-smoothness conditions on the payoff function.Theorem0.9Under the assumptions (A5.1),(A5.2),and (A3.5), we get the error estimate between real option price P(t,x,y) and first order approximate price where P0(t,x) is the Black-Scholes price with content volatility (?), namely Pi(t,x) is the combination of derivatives of P0(t,x), namely The error estimate obtained above is uniform in time t∈[0,T], stock price x∈R and volatility driving process y∈S.