三因子期权定价Monte Carlo Euler离散化偏差问题排查求助

Here are the key suspicious points in your implementation that likely cause the underpricing:

• Static State Variable Calculation: Your code computes r, rf, v, and Q using only initial values (r_0, v_0, etc.) for all time steps, instead of iterating through each step to update variables dynamically. Euler-Maruyama requires updating each state variable at every time step using the previous step's value. For example:

  • The r process should be built incrementally: r(:,t) = r(:,t-1) + p.alpha1*(p.beta1 - r(:,t-1))*dt(t) + p.eta1*sqrt(dt(t))*... (using the prior r value, not r_0 for all steps).
  • Same for v: you’re using sqrt(v_0) in every shock term, but it should be sqrt(max(v(:,t-1), 0)) to update volatility path-by-path.

• Incorrect Delta Process Dynamics: The delta process is defined as delta = delta0_vals(i) + delta0_vals(i)*Q, but Q uses static initial values instead of the dynamic paths of r, rf, and v. Typically, an exchange rate-like delta process follows an SDE like dδ/δ = (r - rf)dt + sqrt(v)dW1 + ξ1 r dW2 + ξ2 rf dW3—your current drift and shock terms don’t incorporate time-varying r(t) and rf(t) from each path.

• Discounting Error: You’re using a single zero-coupon bond (ZCB) value computed from r_0 to discount all payoffs. In Monte Carlo, each path’s payoff should be discounted using the path-specific integral of the short rate: exp(-sum(r(:,1:n).*dt)) for each simulation. Using a fixed ZCB ignores the stochastic nature of the domestic rate, leading to incorrect discounting.

• Low Simulation Count: M=1000 is a small number of Monte Carlo paths, which can introduce significant sampling noise. While this might not explain systematic underpricing, increasing M (e.g., to 10,000 or 100,000) would help confirm if the bias stems from discretization or sampling error.

These issues are not just expected discretization bias—they’re fundamental mistakes in implementing the Euler-Maruyama scheme for multi-factor stochastic processes. Fixing the dynamic iteration of state variables and path-dependent discounting should align your results with the paper.

内容的提问来源于stack exchange，提问作者Umberto Mannello