Choosing Parameters for a Log-Linear Stochastic Volatility Simulation

Summary

The document presents a simulation question about a stock price whose volatility is stochastic. The price follows a geometric Brownian motion with instantaneous volatility given by the exponential of a mean-reverting process. That volatility factor follows an Ornstein–Uhlenbeck-style equation, with its Brownian shock correlated with the price shock. The author describes using Euler discretization and asks whether the specified long-run level, starting volatility state, and mean-reversion regimes explain why simulated prices do not reach the range seen in the source series.

The example supplies a starting price, a one-year horizon, a time step, drift, correlation, and alternative mean-reversion and volatility-of-volatility parameters. It reports that the source prices range from 29 to 50 while the simulated paths remain below 40, and asks whether a negative long-run level is sensible. The document does not include an answer, calibration, or simulation evidence beyond that observation. Its value is in framing parameter interpretation and calibration questions; the stated price range alone does not establish which parameter is responsible.

Key ideas

• The model uses an exponential transformation of a mean-reverting latent process as price volatility.
• The latent volatility shock is correlated with the shock driving the stock price.
• Euler discretization is used to simulate paths under multiple parameter regimes.
• A mismatch between simulated and observed price ranges motivates checking the starting state and parameter calibration.
• The document raises, but does not resolve, whether a negative long-run level is appropriate.

Tags

# Negative theta in Log-linear stochastic volatility model


# Negative theta in Log-linear stochastic volatility model












I was asked to simulate the following geometric Brownian motion to get paths for the SPX stock price. the process follows a Log-Linear stochastic volatility.

$dS_t = \mu S_tdt+e^VS_tdW_1 $

where the volatility follows the process:

$dV_t=\alpha(\theta-V_t)dt+\gamma dW_2 $

where $dW_2=\rho dW_1+\sqrt{1-\rho^2}dZ$

My initial price is 30.2 ($S_0$) and I need 1 year, so T=252. I was asked to use $dt=\dfrac{0.5}{252}$ and was given $\theta = -1$, $\rho = -0.4$ and two regimes for the mean reversion parameters: $\alpha = 1 and 150$ and $\gamma = 0.3 and 8$

I'm using $\mu = 0.02$ right now

In my original time series for SPX, my prices range from 29 to 50, which is a quite big variance

When using Euler discretization, starting with $V_0 = 1$ , a never get prices over 40, so I'm wondering whether the problem here is just in the mean reversion regimes I was asked to use or whether I'm not choosing a reasonable value for $V_0$. Also, does it make any sense to have $\theta = -1$?

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.