Deriving Option Pricing PDEs with Risk-Neutral Measures and Itô’s Formula

Summary

The document gives a practical outline for deriving a pricing partial differential equation for an option. First, Girsanov’s theorem changes the asset’s stochastic differential equation from the real-world measure to the risk-neutral measure by adjusting the drift for the market price of risk. Under that measure, the discounted option value must be a martingale. Applying Itô’s formula to the discounted value and setting its drift to zero yields the pricing PDE.

The final step is to specify boundary or terminal conditions from the option payoff. The sequence follows the Black–Scholes derivation and can be adapted when the option depends on additional state variables, such as a running maximum in a lookback contract. The document points to a continuous-time stochastic calculus text for further study. It is a concise derivation roadmap rather than a worked numerical example, and the exact PDE and conditions depend on the payoff and model assumptions.

Key ideas

• Girsanov’s theorem changes the asset dynamics to the risk-neutral measure by adjusting the drift.
• The discounted option price must have zero drift under risk-neutral dynamics.
• Applying Itô’s formula and eliminating the drift term produces the pricing PDE.
• Terminal and boundary conditions come from the option’s payoff and features.
• Options with additional state variables require corresponding extensions to the PDE setup.

Tags

# What is the easiest way to learn Option pricing with PDE?


# What is the easiest way to learn Option pricing with PDE?












I was reading about Ito's formula and Girsanov theorem, but I am still struggling to grasp how in reality these are combined to compute the price of an option. What are the main source to understand this topic in a very practical manner?

## Answer by starovoitovs (score 5)

https://quant.stackexchange.com/a/46117

In a practical manner, here is how you get to the PDE of your option:

- Use Girsanov theorem to go from the real-world measure to the risk-neutral measure (basically subtract the market price of risk $\mathrm dW^Q_t = \mathrm d W^P_t - \frac{\mu -r}{\sigma} \mathrm dt$). This will change your SDE.

- Discounted option price $e ^{-rt} v(t, S_t)$ has to be a martingale in the risk-neutral world. Hence use the Ito's formula to calculate the differential $\mathrm d (e ^{-rt} v(t, S_t))$ and set the drift term to zero, which will give you the PDE that your option price must satisfy.

- Set the boundary conditions for the PDE based on the payoffs of the option.

The PDE and boundary conditions are individual to each option, but the derivation is always similar to that of the Black-Scholes PDE (sometimes you will have other differentials involved, for example, running maximum in the look-back options etc).

"Stochastic Calculus for Finance II - Continuous-time models" by Shreve Chapters 4 and 7 are good references for this topic.

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.