Rescaling a Volatility-Scaled Heston Model for Standard Pricing Inputs

Summary

The document explains how to translate a Heston stochastic volatility model whose spot diffusion includes a separate volatility scale into the more common form used by standard pricing tools. The original setup has a variance state that mean-reverts toward one, while the spot process is multiplied by a constant volatility parameter. A change of variable absorbs the square of that parameter into a new variance state.

Under this transformation, the spot process has the conventional square-root variance form. The new variance mean reverts toward the squared scale parameter, its initial value is also the squared parameter, and its volatility-of-variance coefficient is the original coefficient multiplied by the scale. This provides a way to map parameters into a pricer that omits the separate scale parameter. The response gives the transformation and resulting parameter relationships, but no computed option price or numerical validation. It also distinguishes this parameterization from the original Heston model, where the initial variance need not equal one.

Key ideas

• Absorb the squared spot volatility scale into a rescaled variance state.
• The transformed variance mean-reversion level equals the squared scale parameter.
• The transformed initial variance is the squared scale parameter when the original state starts at one.
• The transformed volatility-of-variance coefficient is the original coefficient times the volatility scale.
• Parameter conversion makes the model compatible with a standard Heston pricer.

Tags

# Pricing in the Heston Model


# Pricing in the Heston Model












The dynamics of the Heston Model is

\begin{align*} \frac{dS}{S} & = \lambda \sqrt{\nu} d W^S \\[0.5em] d \nu & = k (1- \nu )dt + \epsilon \sqrt{\nu} dW^\sigma \end{align*}

where $\lambda$ is the instantaneous volatility. Let $\nu_0 = 1$. The Brownian motions are correlated with $\rho dt$.

Now I want to use this online pricer: https://kluge.in-chemnitz.de/tools/pricer/heston_price.php to determine the prices.

How would I go about this without $\lambda$ being specified in the model of the pricer?

Say I want to find the price for $S_0=100$, $K=90$, $\epsilon = 0.3$, $\kappa = 0.05$, $\rho = 0.5$ and $\lambda = 0.2$.

I know that by using Ito on the spot I get

$$ d \log S_t = \lambda \sqrt{\nu_t} d W_t^S - \frac{1}{2} \lambda^2 \nu_t dt $$

Do I somehow need to use the relationship between $\lambda^2 \nu_t$? If so, how?

## Answer by jherek (score 1)

https://quant.stackexchange.com/a/45020

This is not the typical Heston stochastic differential equation (SDE). In the original Heston paper, the SDE is defined without $\lambda$, that is $\lambda=1$ and $v(0)=v_0$ not necessarily 1.

In your case you have to do the change of variable $y= \lambda^2 v$ which leads to $$dS/S = \sqrt{y}dW_S$$ $$dy = k(\lambda^2 - y) + \epsilon\lambda\sqrt{y} dW_y$$ and $y(0)=\lambda^2$.

the initial vol is $\sqrt{y(0)}$ and the vol of vol (actually really a vol of var) is $\xi=\epsilon\lambda$.

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.