Pricing Exotic Options Under Certain Volatility Models

With the increasing internationalization of the financial market,financial deriva?tives are also constantly updated.Exotic options are getting more and more attention as the form of them can be changeable and there are many different ways of trading.However,many factors can affect the prices of exotic options,which makes the pricing problem extremely complex.On the other hand,several volatility models of option pricing have been widely considered by scholars.We investigate pricing problems of exotic options under different volatility models and obtain a series of option pricing results.In this thesis,we consider three types of options,i.e.Asian options,double volatility barrier options and double barrier Parisian options.Asian option is a path-dependent option and the payoff function depends on the average price of the underlying asset.For Asian option pricing problem,we consider the option under an uncertain volatility model and we obtain an approximation of the worst-case scenario price.At the same time,a method of solving stochastic nonlinear partial differential equation is also found.Double volatility barrier option is a new type of derivative.In order to ensure the interests of traders and reduce investment risk,we put the loarrier on volatility.As a compound option,it depends on both the underlying asset and volatil-ity.For double volatility barrier option,we consider the problem on Heston stochastic volatility model and fractional stochastic volatility model.To obtain the exponential approximation of option price respectively,we combine Green’s function method with eigenfunction expansion method.Parisian option is an extension of barrier option.It is characterized by adding a trigger device on the classical barrier option.That is to say,we need to add a variable to record the time that underlying takes to pass the barrier and return it.The option will only be knocked out or knocked in if the record value reaches the preset time value.For double barrier Parisian option,we obtain an exact expression of the option price by Green’s function,Laplace transform and "moving window" technique.Under uncertain volatility model,underlying asset S1 satisfies the following s-tochastic differential equation:dS1(t)=rS1(t)dt+σ1(t)S1(t)dB1(t)where r is risk-free interest rate,B1(t)is a Brownian motion on the probability space(Ω,(?),P).Suppose that the maturity of option is T1 and the volatility process σ1(t)∈A[σ,σ],(?)t∈[0,T1],which is a family of progressively measurable and[σ,σ]-valued processes.Let σ=σ0,σ=σ0+ε.Then the worst-case scenario Asian option price is approximated by the following theorem.Theorem 1 Suppose that φ1 ∈Cp2(R+)is Lipschitz continuous,the fourth deriva-tive of φ1 exists.Then we have#12 where φ1 is payoff of the option,V1ε is the worst-case scenario Asiaan option price with ε as the length of volatility interval,V10 and V11 satisfy the following equations,respectively:#12#12 where Y1(t)=∫0t S1(u)du.From Theorem 1,we can get Asian option price by computing the approximate expression V10+εV11.V10 is Asian option price under Black-Scholes model and we can compute V11 numerically such as by finite difference method.When it comes to double volatility barrier options,we first price option under Hes-ton stochastic volatility model.Assume that underlying asset S2 satisfies the following stochastic differential equation:#12 where B2s(t)and B2v(t)are Brownian motions with a constant correlation coefficient ρ2.On the other hand,volatility process is a mean-reverting process,v2(t)tends towards a long-term value α2 with rate β2.Suppose that double volatility barrier option with maturity T2 and strike price K2 has upper volatility barrier B2 and lower volatility barrier A2.The expression of option price is shown in the following theorem.Theorem 2 Let the price of double volatility barrier option under Heston stochas-tic volatility model be U(t,S2,V2).The price of volatility risk is A2(t,S2,v2)=A2V2(t).Then U can be expressed as#12 where#12#12#12#12To obtain option price,we first get the partial differential equation of option price by setting up a replicating portfolio.Green’s function and cigenfunction expansion method are applied to obtain the approximate expression of option price.For the case of fractional stochastic volatility model,we suppose that underlying asset S3 satisfies#12where BH(t)=B3H(t)+B3v(t).μ3 is the drift rate of risk asset price process,β3 is the average recurrent rate of the volatility process.We suppose that B3s(t),B3H(t)and B3v(t)are independent of each other.B3S(t)and B3v(t)are two staudard Brownian motions.B3H(t)is fractional Brownian motion with Hurst index H>1/2.Assume that double volatility barrier option with maturity T3 and strike price K3 has upper volatility barrier B3 and lower volatility barrier A3.Then we will show the expression of option price in the following theorem.Theorem 3 Let the price of double volatility barrier option under fractional stochastic volatility model be U(t,S3,v3).We have(?)where#12#12#12From Theorem 2 and Theorem 3,we can see that the option price is the sum of exponential term after applying the eigenfunction expansion method.We consider the model with constant volatility when price double barrier Parisian option.Suppose that underlying asset S4 satisfies following stochastic differential e-quation:dS4(t)=μ4 S4(t)dt+σ4S4(t)dB4(t).Suppose that Parisian option has strike price K4,maturity T4,upper-barrier B4,lower-barrier A4.Define J1 and J2 as the time underlying has spent continually below the lower-barrier or above the upper-barrier.When the time of underlying continually being blow A4 reaches J1 or the one that exceeding B4 at one time gets to J2,option will be knocked out.To obtain option price,we first find a partial differential equation system through analysis and discussion.Next we combine the two coordinates and reduce the three-dimensional partial differential equation system to a two-dimensional one.We obtain the closed-form solution on each domain by Green’s function,Laplace transform and "moving window" technique.Theorem 4 Let V41(S4,t,J1),V42(S4,t)and V43(S4,t,J2)be the option price on domains Ⅰ,Ⅱ and Ⅲ,where#12#12#12 Then Parisian option price is#12(?)#12 where#12#12#12 f4i(z)=VBS(z,Ji),#12 Here,the variables in the expression are#12#12 Note that,actually,Wi,Ji,i=1,2 are Wi’,Ji’,i=1,2 which are shown above.li is the combination of t axis and Ji axis at 45° angle to form a new coordinate axis.The expression of Wi is#12 where n=[τ4i/Ji]+1,i=1,2.Here,Wi(n+1)can be expressed as Wi(n+1)(τ4i)=γi1+γi2+γi3+γi4,(?)n=1,2,…,i=1,2,whereFrom Theorem 4,the price of Parisian option is the form of sums of integrals.By calculating the integral we can get exact price of Parisian option,which means that the price we get is an analytic solution of the partial differential equation.Theorem 1-4 show the approximation method of the worst-case scenario Asian option price with uncertain volatility,the approximate expression of double volatility barrier option price under Heston stochastic volatility model,the approximate expres-sion of double volatility barrier option price under fractional stochastic volatility model and the exact pricing formula of double barrier Parisian option with constant volatility,respectively.We numerically calculate the option prices in these four case.Numerical calculation shows that Asian option price under uncertain volatility model is greater than Black-Scholes price and the price difference between these two models increases with the increase of the ambiguity of the model.In addition,the error of approxima-tion method also increases with the increase of the ambiguity of the model.For double volatility barrier option,we find that the difference in option prices between the two models is not significant.But when change the range of volatility,price increases with the increase of barrier interval.When it comes to double barrier Parisian option,we find that Parisian option price is consistent with the results of previous studies when it degenerates to the single barrier case.On the other hand,no matter how adjust the two barriers,Parisian option price goes up as the difference between barriers goes up.