MCMC methods in Stochastic Volatility Model

Dingxian Cao, Peiyang Yu

Outline

  1. Introduction:Price and Volatility
  2. Diffusion models for Price and Volatility
  3. MCMC for sampling process sample path
  4. Real Data example
  5. Project

Denotations

  • \( \{s_t, t ∈ [0,T]\} \): Price series
  • R = \( log(s_T /s_0) \): Return rate over[0,T]
  • \( σ_t \): volatility at time t
  • \( v_t = σ_t^2 \): variance
  • \( \bar v = (1/T)\int_0^T v_τdτ \): mean variance
  • \( B_t \) or \( W_t \): standard Wiener Process/Brownian Motion
  • \( \phi() \): standard normal p.d.f.

1.1 Introduction: Price and Volatility

appl

Geometric Brownion Motion\[ dS_{t}=\mu S_{t}\,dt+\sigma S_{t}\,dW_{t}\\dlog(S_t)=(\mu -\frac{1}{2}\sigma^2)dt + \sigma dW_t \] In our problem: \( \sigma \rightarrow \sigma_t \)

1.2 Introduction- SDE and diffusion process

\[ \begin{array}{aligned} d log(s_t) = (θ_1 + θ_2σ_t^2)dt + σ_tdB_t \\ dv_t = μv_tdt + ξv_tdB_t \\ R = log(s_T /s_0) \\ \bar v = (1/T)\int_0^T v_τdτ \end{array} \]

a. Diffusion process is a solution to a stochastic differential equation.
b. Diffusion Process is a continuous Markov Process.
c. Use a SDE to infer a underlying diffusion process.
d. The mean variance is an integral of square volatility over the path.

1.2 Introduction - Problem

A result:

When ρ = 0, f(R|\( \bar v \)) = \( N(η,λ^2) \),
\( η = (θ_1 + θ_2 \bar v)T~~ and~~ λ^2 = T\bar v \),

Problem:

How to derive f(\( \bar v \) | R=r).

2.1 Diffusion models for price and volatility

Price \( \{s_t\} \) model:

\[ d log(s_t) = (θ_1 + θ_2σ_t^2)dt + σ_tdB_t \]

Volatility \( \{v_t\} \) model:

\[ dv_t = μv_tdt + ξv_tdB_t \]

mean variance: \( \bar v = (1/T)\int_0^T v_τdτ \)

Goal: \( f(\bar v|R=r) \)

3. Bayesian idea and MCMC

Data: R=r,
likelihood function: \( \phi(r,η(\bar v),λ^2(\bar v)) \)

  1. Generate an initial sample path \( \{v_t^{(0)}, t ∈ [0,T]\} \) using Milstein scheme, then calculate \( \bar v_0=(1/T)\int_0^T v^{(0)}_τdτ \)
  2. Propose a new sample paths by simulating \( \{v_t^{(l)}, t ∈ [0,T]\} \) and as well calculate \( \bar v_l \).
  3. Accept the proposed diffusion with probability: \[ min(1,\frac{\phi(r,η(\bar v_l),λ^2(\bar v_l))}{\phi(r,η(\bar v_{l-1}),λ^2(\bar v_{l-1}))}) \]
  4. Set l=l+1 and go to 2.

p.s. model fitting issues.

4. Real Data

Standard and Poor’s 500 index (S&P 500) from 5/5/1995 to 14/4/2003.

4. Real Data

\[ \begin{array}{aligned} d log(s_t) = (θ_1 + θ_2σ_t^2)dt + σ_tdB_t \\ dv_t = θ_4(θ_5 − v_t)dt + θ_3v_tdw_t. \end{array} \]

  • (θ2,θ4)=0 and \( (θ_1 , θ_3^2 , θ_5 , ρ) \) = (0.022, 0.029, 3.09, -0.84).

  • \( E(\bar v|r) \) are (0.012, 0.0139, 0.0144) for r = (0.039, 0.05,0.055)

5. Our Project

  1. Fit the SDE
  2. Simulate sample path from \( \{v_t^{(l)}, t ∈ [0,T]\} \)

\[ Thank~~You \]