Variance Risk Premium and Measure Choice in the Heston Model

# On the Variance Risk Premium in the Risk-Neutral Heston Model


# On the Variance Risk Premium in the Risk-Neutral Heston Model












The Heston model assumed the price $S(t)$ of an asset and its variance $v(t)$ follow $$ dS(t)=\mu S(t)dt+\sqrt{v(t)}S(t)dW_{1}(t), \\ dv(t)=\kappa[\theta-v(t)]dt+\sigma\sqrt{v(t)}dW_{2}(t). $$ This model is under the historical measure $\mathbb{P}$. However, the model must be under the risk-neutral measure $\mathbb{Q}$ for options (or any other derivatives) pricing. Since the asset price growth rate is only based on the risk-free interest rate $r$ under $\mathbb{Q}$ (the asset price growth is based on the drift $\mu$ under $\mathbb{P}$), the risk-neutral process of the asset price is given by $$ dS(t)=rS(t)d{t}+\sqrt{v(t)}S(t)d\tilde{W}_{1}(t), $$ where $\tilde{W}_{1}(t)=W_{1}(t)+\frac{\mu-r}{\sqrt{v(t)}}t$ is the Wiener process of $S(t)$ under $\mathbb{Q}$. Motivated by Breeden's CCAPM model, Heston introduced the volatility risk premium $\lambda(t,S(t),v(t))=\lambda v(t)$ to the model such that the risk-neutral process of $v(t)$ is given by $$ dv(t)=\tilde{\kappa}\left[\tilde{\theta}-v(t)\right]dt+\sigma\sqrt{v(t)}d\tilde{W}_{2}(t), $$ where $\lambda$ is the variance risk premium, $\tilde{\kappa}=\kappa+\lambda$ and $\tilde{\theta}=\kappa\theta/(\kappa+\lambda)$ are the risk-neutral parameters, and $\tilde{W}_{2}(t)=W_{2}(t)+\frac{\lambda\sqrt{v(t)}}{\sigma}t$ is the the Wiener process of $\sqrt{v(t)}$ under $\mathbb{Q}$.

When $\lambda=0$, the risk-neutral model admits $\tilde{\kappa}=\kappa$ and $\tilde{\theta}=\theta$, hence these parameters are the same under the physical and risk-neutral measures. From the book The Heston Model and Its Extensions in Matlab and C#, $\lambda$ is zero throughout the book, although it is not necessarily needed.

- What does that mean?

- Why $\lambda=0$ is not necessarily needed?

I would appreciate an explanation, either mathematical or practical. Thank you in advance!

#### Update

I realized that my question was not properly asked, so I would add more context to it.

My confusion came from the idea that the risk-neutral parameters (under $\mathbb{Q}$) would end up the same as the parameters under $\mathbb{P}$ when $λ=0$. Then, why are we discussing the risk-neutral model if, eventually, it appears to be the same because $λ=0$ throughout the book?

## Answer by KaiSqDist (score 2, accepted)

https://quant.stackexchange.com/a/81558

You need to work out the maths.

Question (1.) - If we replace the tilde measures into the stochastic process of $v(t)$, we have:

$$ dv(t)=\tilde{\kappa}\left[\tilde{\theta}-v(t)\right]dt+\sigma\sqrt{v(t)}d\tilde{W}_{2}(t) \\ = (\kappa+\lambda)\left[\frac{\kappa\theta}{(\kappa+\lambda)}-v(t)\right]dt+\sigma\sqrt{v(t)}[dW_{2}(t)+\frac{\lambda\sqrt{v(t)}}{\sigma}dt)]\\ = \kappa[\theta-v(t)]dt+\sigma\sqrt{v(t)}dW_{2}(t) $$

which is the same as the initial stochastic process for the variance. Therefore, regardless of the value for $\lambda$, the risk-neutral and physical stochastic processes are the same.

Question (2.) - answered above.

## Answer by Andrea (score 1)

https://quant.stackexchange.com/a/81559

Disclaimer: I have not read the book.

The theory works like this

- Start with the dynamics under $\mathbb{P}$

- Introduce 2 $\lambda$s

- Solve them by making all tradables martingales under $\mathbb{Q}$ (i.e. matching their market prices)

In our case, at point 3, you can only identify the $S$ market price for risk, since $v$ is not a tradable. So, the model is arbitrage free, but not complete. Now what?

We need a 2nd tradable to resolve the $\lambda$ for $v$. Which one? Pick an option: according to the theory they will all yield the same $\lambda$. So select this option, read its price (today) from the market (in exactly the same way as you read $S(0)$) and solve for $\lambda$ (vol). (remember: all other parameters are known!!!). This is the value to use.

But wait a minute! How do I price an option, if I have not yet calibrated the model???

In reality, we cannot separate $\mathbb{P}$-parameters from the market price of risk, so one has to calibrate them all at the same time in which case the market price of risk of the vol is redundant (just set it to 0 and get rid of it).

Hence, people specify directly the dynamics under $\mathbb{Q}$ and ignore $\mathbb{P}$ and the market price of risk.

If you really wanted to stick to the theory, you would estimate the $\mathbb{P}$-parameters of the vol from time series, and only calibrate $\lambda$(vol) to the smile. But, it won't gain you anything.