Research On Numerical Algorithm Of Valuing Contingent Claims

Contingent claim is a kind of financial contract. If the market is complete and trading strategies allow no arbitrage opportunities, then an equivalent martingale measure exists that allows any contingent claim to be valued by computing an appropriate conditional expectation. If this conditional expectation is difficult to find a closed-form formula, then numerical methods are needed. The financial options and real options are the specific cases of contingent claims. With the development and perfection of financial theory, the research of contingent claim's numerical method becomes more and more important. Numerical algorithm has been played the role of a bridge that links financial theory with financial practical. In this paper, several numerical algorithms of American-type contingent claim's pricing have been presented. Concretely, the main results of this paper are as following:The first method: Expanded binomial model to trinomial model, we have set up the trinomial model of contingent claims which written on a single underlying asset. Expanded standardized symmetric trinomial model to asymmetrical model, we have obtained risk-neutral probabilities of contingent claims pricing written on a single underlying asset.Proposition 1: Let discrete random variableξ^α(t ) be the approximating distribution forξ(t ) over the interval [t , t +Δt], and discrete random variableξ^α(t ) has the following distribution low: Henceδ≥0 is an appropriate constant and p₁ + p₂ + p₃ =1, then the risk-neutral probabilities are as following: Ignore the parts of o(Δt ), we rewrite the risk-neutral probabilities as following: The second method: The valuation of contingent claims whose value depends on multiple sources of uncertain is an important problem in financial economics. By using multinomial model basic on two underlying assets, we draw two parameters into the model and modify standardized symmetric multinomial model to asymmetrical model, and obtained risk-neutral probabilities of contingent claims pricing written on two underlying asset:Proposition 2: Let a pair of discrete random variables {ξ₁^α,ξ₂^α} approximates the joint normal random variable {ξ₁( t ),ξ₂(t )}. The pair of {ξ₁^α,ξ₂^α} satisfies special distribution low as following: Table: The distribution low of pair of discrete random variables Where v_i =λ_iσ_i(Δt)^1/2,i = 1,2,and p₁ + p₂ + p₃ + p₄ + p₅ =1,whenΔt→0, the risk-neutral probabilities written on two underlying assets are as following:Because of the generality of the technology for solving the risk-neutral probabilities written on two underlying assets, the method of solving the risk-neutral probabilities written on multiple underlying assets is the same as above.The third method: By using finite difference method, we set up the valuation of American-type contingent claim (American put) with stochastic volatility of finite Markov chains under the efficiency and convergence of the algorithm. The conclusion improved the results of constant volatility model and binomial probability tree model. It can price other American-type options.Proposition 3: In the world of risk-neutral, the value of American put was obtained at the time 0, which was described by PDE with stochastic volatility of finite Markov chains and accuracy of parting satisfies R (h ,Ï„) = O(Ï„+ h²):At first, solve the equations, get the theory values of American put at the time layer; then compute F_i,j^l, and compare it with EX - i·h:(1) If F_i,j^l≤EX - i·h, then exercise the American put and obtain payment F_i,j^l= EX - i·h (because F_i,j^l, payoff holding the put continue, is smaller than EX - i·h;(2) If F_i,j^l> EX - i·h, then hold the put continuously; That is ; According this method, letj = N-2, N-3…1, 0, respectively. Finally, the claim values at time t = 0 are Here, there must be a F_{i ,0} corresponding present price S = i - h. Such F_{i ,0} is the value of American Put. Here F_{i ,j}^l satisfies special express.The fourth method: On the basic simulation method of Monte Carlo, present the simulation LSM according least-squares Monte Carlo approach. This method can compute the value of American-type contingent claims efficiently. It becomes the basic technology of valuation other American-type contingent claims by combing with stochastic volatility of finite Markov chains.Proposition 4: For any finite choice of M and K , let LSM ( S_t (ω),σ_i; M , K ) denote the discounted cash flow resulting from following the LSM rule of exercising, which basised on the underlying asset with volatilityσi . Then the value of American-type contingent claim, in the risk-neutral world, written on the underlying asset with stochastic volatility of finite Markov chains at time t is as followingThe fifth method: Although the pricing theory of contingent claims basis on the underlying asset dynamics following fractional Brownian motion (FBM) is non-perfection, a great of financial data supports the assumption of FBM. The times return series of financial market have the character of self-similar. Proposition 5: For any finite choice of M and K , let FBM LSM ( S t (ω),σi; M , K ) denote the discounted cash flow resulting from following the LSM rule of exercising, which basis on the underlying asset with volatilityσi. Then the value of American-type contingent claim, in the risk-neutral world, written on the underlying asset with stochastic volatility of finite Markov chains at time t is as followingAccording FBM least-squares Monte Carlo approach, we have some important conclusions about the value of American-type contingent claims. The movement of underlying asset with Hurst index H∈(1, 1/2) is more violent than the movement of underlying asset basis on standard Brownian motion, or the movement of underlying asset with Hurst index H∈(1/2, 1) is more smooth than the movement of underlying asset basis on standard Geometric Brownian motion. The value of contingent claim written on the underlying asset satisfying FBM is more (when H∈(1, 1/2)) than the value of underlying asset basis on standard Brownian motion, or less (when H∈(1/2, 1)) than that one basis on standard Brownian motion.