Extending Black–Scholes PDEs to Path-Dependent Options

Summary

Black–Scholes pricing can apply to some path-dependent options by adding the relevant history measure to the model’s state and adjusting the PDE boundary conditions. The document uses barrier, Asian, and lookback options to show how this works: a barrier option can use a related PDE with a boundary at the barrier, while Asian and lookback options require tracking a running average or an extreme price. For a knock-in barrier call, its value can also be related to the corresponding knock-out and vanilla call values.

The central point is that path dependence does not automatically rule out PDE pricing; the treatment depends on the contract’s payoff and what information must be tracked. The discussion is conceptual and gives no derivation, numerical example, or assessment of computational difficulty. It also does not cover every path-dependent contract or explain when alternative methods may be preferable. A recommended reference is a chapter on stochastic calculus and finance.

Key ideas

• Path dependence may be handled by expanding the pricing model’s state to track relevant history.
• Barrier options can use a Black–Scholes PDE with a boundary condition at the barrier.
• Asian option pricing requires the running average as an additional state variable.
• Lookback option pricing requires tracking a running minimum or maximum.
• A knock-in barrier call can be related to a knock-out call and a vanilla call.

Tags

# Can we use Black-Scholes to price path dependent options?


# Can we use Black-Scholes to price path dependent options?












I know that we can use the Black-Scholes framework to price vanilla products like a European call or put, where the payoff only depends on the share price at maturity.

But can we use it to price path dependent options - those options where the payoff depends not only on the price of the underlying at maturity, but on the entire price history over the life of the contract?

Specifically, what part of the model/derivation allows or disallows us to price such path dependent options?

## Answer by starovoitovs (score 5)

https://quant.stackexchange.com/a/47157

It depends very much on the individual option you are pricing.

Sometimes you can get a Black-Scholes PDE with some extended state and boundary conditions.

- up-and-out barrier option will have virtually the same pricing PDE, and zero boundary condition at $S=0$ and $S=B$ (barrier level). The up-and-in barrier option can be priced by $$C_\text{up-and-in} + C_\text{up-and-out} = C_\text{call}$$

- PDE for Asian options will include running average

- lookback options will include running minimum / maximum

Chapter 7 in "Stochastic Calculus for Finance 2" by Shreve can give you a good insight.

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.