Replicating Claims in Markovian Stochastic Volatility Models

Summary

The document sets up a claim replication problem in a Markovian stochastic volatility model. An asset price and its variance evolve with correlated Brownian shocks, and two additional traded assets are represented as smooth functions of price and variance. The question asks whether a claim on one traded asset at maturity can be replicated using the underlying and the other traded asset. It proposes choosing hedge weights to cancel exposure to both state variables, then asks how the residual drift and martingale assumptions enter the argument.

No answer or derivation is included, so the proposed hedge is not validated and no replication conclusion is established. In particular, the text does not show the conditions under which the hedge is self-financing, whether the assets span the relevant risks, or how the dynamics imply the needed pricing relation. It is useful as a mathematical setup and a prompt about delta and volatility-risk hedging, but not as a complete replication method.

Key ideas

• The model has an asset price and a stochastic variance driven by correlated Brownian motions.
• Two traded assets are specified as functions of price, variance, and time.
• The proposed hedge selects weights to cancel instantaneous exposure to the price and variance risks.
• The document leaves the residual drift and martingale argument unresolved.

Tags

# How to replicate a claim in a stochastic volatility model?


# How to replicate a claim in a stochastic volatility model?












Given a Markovian stochastic volatility model with an asset $S$ and a variance process $V$ given by $$ dS_t = \mu_t S_tdt + \sqrt{V_t}S_tdW_t, \\ dV_t = \alpha(S_t,V_t,t)dt + \eta \beta(S_t,V_t,t)\sqrt{V_t}dZ_t, \\ dW_tdZ_t = \rho dt, $$ where $W$ and $Z$ are correlated Brownian motions. Now suppose that we have two traded assets $F$ and $G$, which can be written as $F_t = f(t,S_t,V_t)$ and $G_t = g(t,S_t,V_t)$, where $f$ and $g$ are deterministic smooth functions. Additionally I can assume that $S,F,G$ are martingales.

Question: How can the claim $F_T$ be replicated by trading in the options $S$ and $G$?

My idea was the following: Define a portfolio $\Pi = F - \phi S - \psi G$. Using the self financing property and Ito's formula, I can calculate $d\Pi$ and choose $\phi$ and $\psi$ such that the portfolio is instantaneously risk free, i.e. I choose $$ \phi = F_S - G_S \frac{F_V}{G_V}, \\ \psi = \frac{F_V}{G_V}. $$ This leaves me with $d\Pi = (...)dt$. However I am not sure how to go on from here and how this does help me. Additionally, I am unsure on where I can/should use that the processes are martingales.

Was my initial Ansatz with the portfolio $\Pi$ correct? If yes, how should I proceed from here? If no, what would be the right way to start?

Thanks in advance!

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.