CAM colloquium - February 27

John Walsh
University of British Columbia

"Skorokhod Embedding and the Convergence of the Tree Scheme for European Options"

The binomial and trinomial tree schemes are commonly used as discrete approximations to the Black-Scholes model. In fact, one can consider them as numerical schemes to solve an initial value problem for the Black-Scholes PDE. The key fact is that for financial applications, the initial data is the payoff of the option, which is commonly not a smooth function. The question arises: what is the rate of convergence of these schemes? The fact that the data is non-smooth leads to some interesting behavior. One can say generally that the tree schemes are first-order schemes for the usual calls and puts (though only of order 1/2 for digital options.) However, if one actually computes these, one finds that there are some strange bumps and grinds in the convergence.

We use the old probabilists' trick of Skorokhod embedding to embed the tree scheme in a logarithmic Brownian motion. A comparison of the
embedded tree with the logarighmic Brownian motion allows us to see why these bumps and grinds appear.