The Moment Estimate Of A Discretizated Stochastic Volatility Model

The Black-Scholes formula has made the important contribution for developmentsof chrematistics and finance. At present theres two kinds of methods to estimate thevolatility.One is using the history variance or standard deviaton of stock returns in actualstock market,and another one is using implied volatility of stock.there are problems ofthese two estimates, both of them have estimation errors in varying degree.Specially, theseerrors will be more obvious when the Black-Sholes models is used to other option pricingbesides stock option.Therefore,how to find a appropriateσwill be the determinant of theBlack-Sholes models is whether or not accurate.In this paper,we suppose that volatilityσ²=exp{X_t}, X_t=ln(σ_t)is an Orustein-Uhienbeck process. Thus,we get a models which there are two noise processes and onlyone observedprocess.Estimate the models like this,especially nonlinear,often uses the knowledge aboutstochastic differential equation,but that is very difficulty.Weconsider using classical mo-ment estimate theory to solve thisproblem.First,under some condition,it can be trans-formed the strong stationary ergodic andαmixing and autoregression quadratic errors-in-variables model. Then we got its moment estimator after discrete it. secondly, use someassumption we prove it is strong Consistent asymptotically normal estimate. Finally, wedo a test of the method through a simulation study.Section 1 introduce option prcing models Black-Scholes formula.dS_t= S_t(μ(S_t,t)dt+σ(S_t,t)dB_t).where B_t is a standard Brownian motion.μ(S_t, t) is returns;σ(S_t, t) is volatility.We considerσ²(t)=exp(X_t), X_t=m(X_t-1)+η_t, {ξ_t} and {η_t} are i.i.d respec-tively.The model which extends as follows.dS_t/S_t=μ(t,S_t,X_t)d_t+σ_tdW_t^S dZ_t=α(t, X_t)dt+b(t, X_t)dW_t^σ,where W_t=(W_t^S, W_t^σ) is a two-dimensional Brownian motion. In particular, definingX_t=ln(σ_t)as an Ornstein- Uhlenbeck process: dX_t = -kZ_tdt+γdW_t^σ, then leadsto a discretization Gaussian process: X_t=e_-kX_t-1+η_t, withi. i. d{η_t} with GaussianN(0, s²), sbeing a known function ofκandγ. It difficult to estimate this model, becausethere are two noise processes and only one observed process. In order to simplify thequestion, assume thatμ(t)≡μis constant. Secondly if X₀ is independentξandη, andifξandηindependent, thenμ= E(h_t) can easily be estimated by the empirical meanof h.This leads to the following modelTaking the loarithm of |h_t|^2λ transforms the first equation of this model:ln(|h_t|^2λ)=X_t+λln(ξ_t²)=λω+X_t+λ(ln(ξ_t²)-ω), whereω=E(ln(ξ₁²))Sinceωis assumed to be known, setting Y_t=ln(|h_t|^2λ)-λωandε_t =λ(ln(ξ_t²))-ω,and if m(·)is quadratic, lead to a stationary ergodic andαmixing and autoregressionquadratic EV model.wherem(X_t-1)=β₀+β₁X_t-1+β₂X_t-1².{η_t}i. i. d. Althoughε_t and Y_t is notindependent, butε_t and Y₁, Y₂,…, Y_t-1is independent. {ε_t}i.i.d,ηandεis independent.E(η₁)=E(ε₁)=0,var(η)=σ_η²ï¼œ+∞, var(ε)=σ_ε²ï¼œ+∞, and for any odd numberk＞1, Eε^k=0.After some necessary condition of estimate and proof are showed, section 2 is mo-ment estimate and proof of strong consistent and asymptotically normality. Take m(X_t-1)=β₀+β₁X_t-1+β₂X_t-1² into the model, we getY_t=β₀+β₁Y_t-1+β₂Y_t-1²-2β₂Y_t-1ε_t-1+β₂ε_t-1²-β₁ε_t-1+ε_t+η_t. (2.2.3) We consider the estimate of m, based on the observations Y₁, Y₂,…, Y_n+3.Setting r (?) (r₁, r₂, r₃)^T, r_s=E(Y_tY_t+s), s=1, 2, 3,β(?) (β₀,β₁,β₂)^T.then r=Wβ.(?)=(1/n sum from t=1 to n Y_tY_t+1, 1/n sum from t=1 to n Y_tY_t+2, 1/n sum from t=1 to n Y_tY_t+3)^T。If W is invertible matrix, andβ=W^-1r, then (?)=(?)^-1(?) is the moment estimateβ。Theorem 2.3.7 (Birkhof fergodictheorem) Let T is the measure-preservingtransformation on(Ω, (?), P), f∈L¹(Ω, (?), P),then (?)f^*∈L¹(Ω, (?), P), s. t(?)1/n sum from t=1 to n f(T^tω)=f^*(ω) (a. e. dP). (2.3.4)and f^*(ω)=E(f(ω)|(?))∈(?),Where (?)={A; A is invariant set} is invariantσ-algebra.Corollary 2.3.8 If T ergodi, then f^*(ω)=E(f(ω)) (a. e. dP).Proposition 2.3.10 (?) and (?) is the strong consistent estimator of W and rrespectively.For stationaryαmixing sequence {Y_t, t≥1}, sufficient conditions of central limittheorems. Let S_n=sum from j=1 to n Y_j andσ_n²=VarS_n.Theorem 2.4.1 (Central Limit Theorems ofαmixing sequence)If EY₁=0, E|Y₁|^2+δ＜∞(someδ＞0) andsum from n=1 to∞α(n)^{δ/(2+δ)}＜∞。(2.4.1) thenσ²:=EY₁²+2sum from j=2 to∞EY₁Y_j＜∞.(2.4.2)and whenσ≠0S_n/σn^1/2(?)N(0,1). (2.4.3)and when the variable is bounded, i.e.δ=∞, condision (2.4.1) change tosum from n=1 to∞α(n)＜∞.(2.4.4)Theorem2.4.2 Let T_n =(T_1n,…,T_kn)^T,θ=(θ₁,…,θ_k)^T,Ifn^1/2(T_n-θ)(?)N(0,∑)where∑= (σ_ij)_k×k, and if the partial derivative of continuous function g(t₁,…,t_k) iscontinuous for every t_i, and when n→∞, thenn^1/2[g(T_1n,…,T_kn)-g(θ₁,…,θ_k)](?)N(0,σ²(θ))whereσ²(θ)=∑∑((?)g/(?)θ_i)(?)g/(?)θ_i)σ_ij)。let T_n=(T_1n,…,T_12n)^T=1/n sum from t=1 to nf(Y_t),θ=(θ₁,…,θ₁₂)^T=Ef(Y_t).Z_t=l^T(f(Y_t)-θ), t≥1. then 1/nS_n=l^T (T_n-θ), whereS_n=sum from t=1 to n Z_t.Proposition 2.4.3 If theαmixing coefieientα(n) of sequence {Z_t, t≥1}, statisfiessum from n=1 to∞α(n)＜∞, then S_n/σn^1/2(?)N(0, 1), and n^1/2(T_n-θ)(?)N(0,∑).Proposition 2.4.4 If g is a continuous function g, g(θ)=β=W^-1r, and g(T_n)=(?)=(?)-¹(?). And if the partial derivative of g(t₁,…,t₁₂)is continuous for every t_i, thewhen n→∞,n^1/2[g(T_1n,…,T_12n)-g(θ₁,…,θ₁₂)](?)N(0,θ²(θ)), andn(1/2)(β-β)(?)N(0,σ²(θ))whereσ²(θ)=∑∑((?)g/(?)θ_i) (?)g/(?)θ_jσ_ij^[5]Proposition(2.4.4) show the moment estimatorβofβis asymptotically normal.