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?


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

  1. 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.

  2. 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.

  3. 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.

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