Double integrals simply explained

Double integrals play a significant role in option pricing, especially when we need to understand the volume under a surface and its relation to financial models.

A double integral allows us to integrate over a two-dimensional area. If you have a function f(x, y), the double integral of this function over a region R in the xy-plane gives us the volume under the surface defined by f(x, y) and above the region R.

The double integral of f(x,y) over the region R is written as ∬R f(x,y) dx dy.

The double integral ∬R f(x,y) dx dy represents the volume of the solid that lies under the surface z = f(x, y) and above the region R in the xy-plane.

In option pricing, the price of an option is often viewed as the expected value of its payoff, adjusted for risk, under a probability density function. Double integrals become particularly useful in more complex scenarios:

For complex options, the expected payoff might require integrating over the probability distribution of two or more correlated variables.

For an Asian option, where the payoff depends on the average price of the underlying asset, you might need to integrate over the probability distribution of the average and the terminal price. This involves a double integral over these two variables.

For a basket option, which depends on multiple assets, the price is computed as an expectation over a multidimensional distribution involving each asset’s price. For two assets, this is a double integral over their joint price distribution.

Consider a European call option under the Black-Scholes model. The risk-neutral price of the option is the present value of the expected payoff:

C(S, t) = e^(-r(T-t)) E[(S_T - K)^+]

Here:

- S_T is the price of the underlying at maturity T.
- K is the strike price.
- r is the risk-free rate.
- E denotes the expected value under the risk-neutral measure Q.

The expectation E[(S_T - K)^+] is given by:

- E[(S_T - K)^+] = ∫ from K to ∞ (s - K) f(s) ds

where f(s) is the risk-neutral density of S_T.

If you have two underlying assets with prices S_1,T and S_2,T, and the payoff depends on both, for example, max(S_1,T + S_2,T - K, 0), the price involves a double integral:

- E[max(S_1,T + S_2,T - K, 0)] = ∬ for x+y >= K (x + y - K) f(x, y) dx dy

where f(x, y) is the joint density of (S_1,T, S_2,T).

The bounds of the double integral define the region over which the function is integrated. In option pricing:

- For a European option, the bounds are typically from the strike price K to infinity for the asset price at maturity.

- For two assets, the bounds are defined over the region where the payoff is positive, for example, x + y >= K for a basket call option.

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