I am reading Bergomi's book Stochastic Volatility Modelling. In the chapter 6 dedicated to the Heston model, page 202, he describes the traditional approach to first generation stochastic volatility models :

  1. You start with historical dynamics for the instantaneous variance : $$dV_t = \mu(t,S_t,V_t,p)dt + \alpha dZ_t^{\mathbf{P}}$$ where $p$ are model parameters ($Z^{\mathbf{P}}$ is hence a Brownian motion in the "historical" probability $\mathbf{P}$)

  2. In risk-neutral dynamics the drift of $V$ is altered by the "market price of risk" $\lambda$ which is an arbitrary function of $t$, $S_t$ and $V_t$ : $$dV_t = (\mu(t,S_t,V_t,p) + \lambda(t,S_t,V_t,p))dt + \alpha dZ_t^{\mathbf{Q}}$$ ($Z^{\mathbf{Q}}$ is a Brownian motion in a risk-neutral probability $\mathbf{Q}$)

  3. (Quoting Bergomi) A few lines down the road, jettison "mark price of risk" and conveniently decide that risk-neutral drift has same functional form a historical drift except parameters now have stars : $$dV_t = \mu(t,S_t,V_t,p^{\star})dt + \alpha dZ_t^{\mathbf{Q}}$$

  4. Calibrate starred parameters $p^{\star}$ on vanilla smile.

Bergomi writes : "discussions surrounding the market price of risk and the uneasiness generated by its a priori arbitrary form and hasty disposal are pointless -- the market price of risk is a non-entity." He points out right after that $V_t$ is an artifical object as for different $t$ the quantity $V_t$ represents a different forward variance and that its drift is then only a reflection of the term-structure of forward variances.

What does he mean by all of that ?

In fact, in never understood the market price of risk even in the context of the Black-Scholes model : we aim at risk-neutral pricing (we care about market models for which a risk-neutral measure exist, no always necessarily for models where a unique risk-neutral measure exist) so why even bother to give model dynamics in the historical measure and not give them immediately in a risk neutral measure ?

For instance : would you try to derive Heston PDE with delta and option hedging, at some point you would come across the market price of risk while if you would start in a risk neutral measure and use Feynman-Kac you wouldn't bother with market price of risk at all.



Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge that you have read and understand our privacy policy and code of conduct.

Browse other questions tagged or ask your own question.