Lambda as the Market Price of Volatility Risk in Heston

Summary

The document identifies lambda in the Heston stochastic volatility model as the market price of volatility risk. The question concerns equations in which lambda appears alongside the mean-reversion rate, correlation between the price and volatility shocks, and volatility of volatility. The answers point to the original paper’s discussion of the parameter and explain that it represents compensation associated with exposure to changing volatility.

One answer gives an economic intuition: investors may have different attitudes toward volatility changes, and that preference affects option pricing. It further describes volatility increases as tending to coincide with adverse economic conditions and falling consumption, which motivates a negative sign for lambda in the explanation offered. The material is brief and conceptual; it does not derive the pricing equations, define a calibration procedure, or establish that lambda must be negative in every model or market. Readers should treat the sign discussion as contextual intuition rather than a universal parameter constraint.

Key ideas

• In the cited Heston formulation, lambda denotes the market price of volatility risk.
• The parameter captures compensation for exposure to changes in volatility in option pricing.
• The answer links volatility risk pricing to investors’ attitudes toward volatility shocks.
• The negative sign discussed is an economic intuition in the answer, not a universal rule established by the text.

Tags

# What is "Lambda" in Heston's original paper on stochastic volatility models?


# What is "Lambda" in Heston's original paper on stochastic volatility models?












In his paper (link), he has the equations:

> b1 = k + ƛ - (ρ * σ) b2 = k + ƛ

k is the rate of mean reversion, ρ is the correlation between the two Wiener processes, σ is vol of vol, what is ƛ?

I have yet to figure out what ƛ is.

Thanks!

## Answer by Magic is in the chain (score 4, accepted)

https://quant.stackexchange.com/a/45235

It is on page 329 (which is the third page of the article) and represents the market price of volatility risk. I have copied below from the original article:

## Answer by Alex C (score 1)

https://quant.stackexchange.com/a/45234

That is the "price of volatility risk" (see Page 329)

When volatility can change the "attitude" of investors to these changes becomes important for pricing options. This like/dislike for vol increases is captured in the parameter $\lambda$. In practice vol goes up i.e. $dv$ is positive, in bad economic times, such as recessions, when $dC$ is negative. So $\lambda$ is negative.

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.