Deriving Option Pricing PDEs Beyond Black–Scholes
Summary
The document outlines the Black–Scholes derivation: apply Itô’s formula to an option whose value depends on the underlying, combine it with the underlying in a portfolio, hedge away the source of randomness, and use no-arbitrage to obtain the pricing PDE. It then asks how this procedure changes when volatility is stochastic and the option value depends on both price and volatility.
The central issue is that one option and the underlying may not hedge two independent sources of risk. The discussion raises the need for additional traded instruments or assumptions about risk premia to price volatility risk, then uses Feynman–Kac to connect a suitable PDE with an expectation under a risk-neutral measure. The document poses these questions rather than resolving them, so it offers no derivation or empirical evidence for a specific stochastic-volatility model.
Key ideas
- The Black–Scholes PDE follows by applying Itô’s formula and hedging the underlying’s random component.
- With stochastic volatility, the option value may depend on both the underlying price and volatility.
- A single option and the underlying may not eliminate two independent sources of risk.
- A risk-neutral expectation representation requires a pricing measure consistent with the model.
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Full text
# What is the recipe for deriving a PDE for the price of an option? # What is the recipe for deriving a PDE for the price of an option? In the Black Scholes setting, here is how my understanding is of how we derive the PDE for the value of an option. - We assume that the price of the option is Markovian in our state variable $S_t$. Then we can use Ito's formula to get the SDE of our option process. - We form a portfolio of the underlying and the option. Using Ito, we can know get a SDE for our portfolio process. - Using the SDE of the portfolio we make it independent of $dS_t$ (and thus also $dW_t$, i.e. it is locally riskless) by choosing our portfolio weights appropiately. - Using arbitrage arguments, we know then that the remaining $dt$-term in the portfolio process must equal $r(t)$. - This condition is the Black Scholes PDE. - Using a Feynman-Kac representation, we can convert this PDE representation to a discounted expectation but under a "different" (risk-neutral) measure. Now, for a non-Black Scholes setting (e.g. stochastic volatility), how must this procedure be modified to produce the PDE for the option AND the risk-neutral expectation? In particular, I have 2 questions: - Do we still assume that the price of the option is Markovian in $S_t$ but now also in $\sigma_t$, the volatility? - When forming the portfolio, I need to ensure that the $dS_t$ and the $d\sigma_t$ terms in the SDE of the portfolio process disappear, so that we have a locally riskless portfolio. But how can I do this? If I am investing in the option + the underlying, I have 2 variables but 3 equations.
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