European Put Pricing in the Heston Model and Measure Changes

Summary

The document sets out the risk-neutral Heston dynamics for an asset price and its stochastic variance, then asks how to value a European put by discounting its terminal payoff. The author is interested in transforming the put valuation into a form related to a call through put-call symmetry, and points to a separate treatment that uses a Girsanov kernel.

The specific request is for the intermediate measure-change steps: how to construct the kernel and relate the Brownian motions under the original and changed measures. No derivation, option price, numerical example, or answer is included, so the text does not establish a particular symmetry formula or pricing procedure. It is a focused question about connecting risk-neutral payoff valuation, changes of probability measure, and correlated stochastic drivers in the Heston model. Any application would need the assumptions and integrability conditions for the chosen change of measure to be checked.

Key ideas

• The underlying asset and its variance follow risk-neutral Heston dynamics with correlated Brownian drivers.
• A European put is valued as the discounted expectation of its terminal payoff under the risk-neutral measure.
• The author proposes using put-call symmetry to relate the put valuation to a call-like expression.
• The central unresolved issue is deriving the Girsanov kernel and transforming the Brownian motions between measures.
• The post contains a question rather than a worked pricing method or supporting numerical evidence.

Tags

# Pricing a put-option in the Heston Model


# Pricing a put-option in the Heston Model












Assume the Heston Model with dynamics under the martingale measure $Q$ given by

\begin{align} dS_t &= (r-q)S_t dt + \sqrt{v_t}S_tdW_{1,t}^Q\\ dv_t &= \kappa(\theta-v_t)dt + \sigma\sqrt{v_t}dW_{2,t}^Q. \end{align}

How, do I find the price of an European put option, i.e. $$e^{-rT}E^Q[(K-S_T)^+]$$.

My idea is to somehow transform this expression into something that looks like an European call-option (I guess that's the so-call put-call-symmetry). And this post seems to try to tackle the problem. My question is how the last step with the Girsanov Kernel is performed? Some more detailed steps on how to find the kernel and the relation beween the Wiener Processes under the different measures would be helpfull. $$$$

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