A Finite Volume Element Method For The Valuation Of Options On Assets With Stochastic Volatilities

In this paper,a new kind of finite volume element method for the Valuation of Options on Assets with Stochastic Volatilities is presented .The Finite Volume Method,also called as Generalized Difference Method, was firstly put up by professor Ronghua Li in 1982 in [16] .In general, trial and test spaces are chosen as the Lagrange-type finite element space and the piecewise constant function space respectively.Numerical experiments, performed to demonstrate the usefulness of the method, will be presented. Compared with the original problem, the finite volume method presented in this paper come out a better result. The finite volume method constructed here keep the local conservation law of the physical variables, and it simplifies computation . It neither change the limited function of the original problem, so we can take advantage of the finite volume method theory to make the theorem of predictive esti-mate and error estimate and so on more direct in the finite volume method approximation.Option is a kind of financial derivatives which gives holders the right to buy or sell certain underlying assets by certain price. American put option let holders sell certain underlying assets by the contracted price E at any time during the valid period T of the contract. In this paper, we choose a stock without dividend as underlying asset and denote the stock price as f at time T .The value of the American put option on the above stock is represented by f which satisfies the following a two-dimensional Black-Scholes equation : (0-0-1)for (x, y, t)∈(0,X)×(ζ, Y)×(0, T ) with appropriate final (pay-off) and boundary conditions, where x denotes the price of the underlying stock, y =σ²,εandμare constant from a stochastic process governing the varianceσ²,ρis the instantaneous correlation between x and y, and X,ζ, Y and T are positive constants defining the solution domain.To assist the formulation of the finite element volume method, it is con-venient to write (0-0-2) in the following divergence form:(?) (0-0-2)where (?), (?), A =(?) . In this form the coefficientsA, b and c are yet to be determined.Let us now consider final (or pay-off) and boundary conditions for (0-0-1) or(0-0-2).The first final condition is the ramp payoff given byf(x, y,T) = max(0, x - E1), (x,y)∈(?), (0-0-3)where E1 < X denotes the exercise price of the option. A second choice is the cashor- nothing payoff given byf(x,y,T) = BH(0,x- E1),(x,y)∈(?), (0-0-4)where B > 0is a constant and H denotes the Heaviside function. Obviously, this final condition is a step function which is zero if x < E1 and X-E1 if x≥E. Another choice is the bullish vertical spread payoff defined byf(x,y,T) = max(0,x- E1) - max(0,x - E2), (x,y)∈(?) , (0-0-5)where E1 and E2 are two exercise prices satisfying E1< E2. This represents a portfolio of buying one call option with the exercise price E1 and writing one call option with the same expire date but a larger exercise price (i.e., E2). The solution domain of the above problem contains four boundary surfaces defined by x = 0, x = X, y =ζand y = Y. The boundary conditions at x = 0 and x = X are simply taken to be the extension of the final conditions at the points, f(0,y, t) = d(0, y, T) = 0, f(X, y, t) = f (X, y, T). (0-0-6)To determine the boundary conditions at y =ζand y = Y we need to solve the standard one-dimensional Black-Scholes equation obtained by takingξ=μ= 0 in (0-0-1) for two particular values with the boundary and final conditions defined above. In our numerical experiments given in this paper we use the algorithm in [12] to derive the numerical calues of these two face boundary conditions.A new kind of finite volume element method is following. The original subdivision see in (picture 0.25), the dual subdivision see in(picture0.25),shaded area. Trial space is based on a linear function of the triangle network space and test function space is based on a dual partition of the piecewise constant function space.wo get the discrete format using the Crank-Nicolson on time.Integrating (0-0-2) over (?).we have (0-0-7) We now consider the approximation of the middle term in (0-0-7),Using Green's formula,on the dual subdivision,we have(?), (0-0-8)where (0-0-9)Further dealt with the third term in (0-0-7),we have(?)cfdxdy,?)cfdxdy, (0-0-10)where(?),n = 1,2,3,…,6. The six regional are posed by the intersection of the dual subdivision and the original subdivision see in (picture 0.25).f_0,j = f(x₀,y_j),f_Nx,j = f(X,y_j),f_i,0 = f(x_i,y₀),f_i,Ny = f(x_i,Y), In order to calculate the (0-0-9),Trial space is based on a linear function of the triangle network space seeing in 0.25. Take the picture (0.26) for example. Taylor started In the right angle vertex ,we have the Taylor expansion.we have the partial derivatives: where h_x = x_i- x_i-1, h_y = y_j - y_j-1, withα₂₁,α₂₂,b₂ the second and the thirdterms in (0-0-7) can be solved on the dual unit. we have: (?) (0-0-11)The following is the fully discrete format:The same to the discretization of (0-0-7), for (0-0-11) we havewhereh_x = x_i- x_i-1, h_y = y_j-y_j-1, k₁^*.set (N_x -1)²Ã—(N_y -1)² matrix G,the element G_i,j,we have(0-0-13) Except (?),others are zero.Then, we apply the two-level implicit time-stepping method with a splitting parameterθ∈[0.5, 1],Then, the above system can be rewritten as (?), (0-0-14)Whenθ= 0.5,the time-stepping scheme becomes the Crank-Nicolson scheme and whenθ= 1 it is the backward Euler scheme. Both of these schemes are unconditionally stable, and they are of second- and first-order accuracy,respectively.