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

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.

## 1 Answer

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.

• Great answer. A question that has puzzled me for a long time too (in the general context of hedging derivs under model misspecification). You’re so right, nobody actually cares about the mean reversion in the drift, the problem is the vol estimation ! Nice. – Ivan Mar 24 '18 at 20:05
• @Ivan: Thank you. I think also that persistent mean reversion or trending behavior could give different results and be exploited in a realistic delta-hedging strategy where rebalancing is not continuous (but over discrete intervals) and with transaction costs. – RRL Mar 24 '18 at 20:25
• @RRL thanks! Quite interesting. The way I see it, the pricing function would look just like as if we were using a Bachelier model instead of mean reverting model, right? – Lisa Mar 25 '18 at 0:28
• @Lisa: You're welcome. You are correct about Bachelier model. – RRL Mar 25 '18 at 1:25