European Call price for an asset with mean reverting (Vasicek model) dynamics

Question

Let's look at a stock with a mean reverting price dynamics: $$dS_t = a(S-S_0)dt + \sigma dW_t$$

If we let $\sigma=0.25$ and $a=-0.5$ then the variance of this process is: $$Var(S_t) = 0.199\sim0.2$$ see the Wiki article about for this kind of proces: https://en.wikipedia.org/wiki/Vasicek_model

How do I derive the Arbitrage free pricing function for a Call option with strike K and underlying being the stock with MR as described above.

Answer 1

The whole point of no-arbitrage pricing in a complete market is that a general underlying model of the form

$$d S_t = \mu(S_t,t)\, dt + \sigma(S_t,t) \, dW_t$$

can be replaced with the risk-neutral process.

$$d S_t = (r - \sigma^2/2)\, dt + \sigma(S_t,t) \, dW_t$$

for the purpose of finding the theoretical fair option price. This, of course, follows from the possibility of continuous hedging and, mathematically, through a change of measure.

You introduce two twists in that the drift imposes mean reversion and you set $\sigma(S_t,t) = \sigma = \text{constant}$. Had you chosen $\sigma(S_t,t) = \sigma S_t$, this would revert to the Black-Scholes model as far as the option price is concerned. The form of the drift is irrelevant.

Assuming $\sigma(S_t,t) = \sigma$ will then give the closed-form option price for arithmetic Brownian motion.

There is, however, one issue that needs to be addressed -- the estimation of $\sigma$. Without mean reversion, and autocorrelation of returns, the volatility can be estimated using price data observed at discrete time intervals and independence would imply $\sqrt{t}$ scaling. The parameter $\sigma$ used in the option pricing formula would, for example, be obtained by estimating volatility $\hat{\sigma}$ over intervals of length $\delta t$ and assigning $\sigma = \hat{\sigma}/\sqrt{\delta t}$.

This would not be the case if the real price dynamics were mean reverting.

See the paper by Lo and Wang.