Risk-Neutral Volatility Dynamics and the Market Price of Risk
Summary
The document asks why stochastic-volatility modeling often introduces a market price of risk when option pricing can begin directly under a risk-neutral measure. It describes a traditional setup in which variance has historical dynamics, its drift changes under a risk-neutral measure through an unspecified risk premium, and practitioners then replace that drift with a similar functional form whose parameters are calibrated to vanilla option smiles.
The cited discussion argues that the variance process is a modeling construct tied to forward variance, so its drift reflects how the forward-variance term structure evolves. This makes an arbitrary market-price-of-risk specification less directly meaningful for pricing than the calibrated risk-neutral dynamics. The text raises conceptual questions rather than deriving a resolution, and it does not provide empirical tests or a full derivation. Its focus is the distinction between describing real-world dynamics and specifying dynamics used to value derivatives.
Key ideas
- Historical and risk-neutral measures can assign different drifts to stochastic variance.
- A market price of risk parameterizes the change in drift when moving between measures.
- Risk-neutral volatility parameters are often calibrated to observed vanilla option smiles.
- The document presents the view that variance drift reflects the evolution of forward variance and term structure.
- It distinguishes real-world modeling from the dynamics needed for risk-neutral derivative pricing.
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# Why is the market price of risk a non-entity according to Bergomi?
# Why is the market price of risk a non-entity according to Bergomi?
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 :
- 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 $\mathbf{P}$)
- 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 $\mathbf{Q}$)
- (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 : $$dV_t = \mu(t,S_t,V_t,p^{\star})dt + \alpha dZ_t^{\mathbf{Q}}$$
- 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.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.