Introduction
We propose a very simple approach to price an American put on a worst of multiple, n, underlyings. The idea is based on a moment matching technique. We first establish the prices of the different calls in the worst of basket, pricing different maturities and different strikes. Then we construct a model that is based on an artificial underlying with known implied volatilities and write a partial differential equation (PDE) to solve this product. The idea is to build an approximate one factor model for the worst of put and then price the American option on a one dimensional PDE.
Model & Risk Profile
The payoff at maturity of the American put option on multiple underlyings is given by:
The American feature gives the right to exercise at any time prior to maturity t, where the payoff is:
We suppose that Si (t) are n lognormal processes with known volatilities and known correlations. Any other assumptions on a multi-dimensional process which respects the marginal distribution of each underlying is also possible to consider. The point is to have a model for which you can price any call on a worst of n underlyings. The following table shows the risk profile of a long position:
Considering the risk profile of the product, a lognormal model is not too bad an assumption. Indeed, our exposure with respect to the gamma and the vega remains the same. Also the correlation exposure is the same. Thanks to the robustness of the Black Scholes model we can use conservative values for all the parameters that define our model and be safe in terms of hedging strategies.
Pricing All The Calls
Using a Monte Carlo simulation for a discrete set of maturities. We price in one go, many different options on the worst of several underlyings. Another approach could be to use accurate numerical approximations. Obtaining the prices of these calls is part of pre-processing and has to be done once a day. This creates the implied volatility surface for the artificial underlying which is the worst of n underlyings. This is equivalent to what exotic traders do every day by taking the implied volatilities from the market. Here, the implied volatilities are processed from these European options. The result is surprising, as it shows that the implied volatilities are quite constant especially for long maturities. It is therefore a good assumption to suppose that the distribution of the minimum of n underlyings for a long enough maturity is a lognormal around its forward.
Valuing The American Put
Using a one-dimensional grid, it is possible to derive an algorithm that prices the American put on many underlyings. We also obtain a delta with respect to this fake underlying. Another step is necessary to transform this delta to deltas with respect to each of the underlyings. This is done by computing the different deltas of the worst of with respect to the different underlyings and multiplying each by the delta given by the one-dimensional grid. The other greeks can be computed as well by using chain-rule differentiation.
Results For The Implied Volatility
We obtain a quite astonishing result. The worst of n underlyings behaves almost like a lognormal around its forward. We can see in the following graphs, the fact that the forward of the worst of n underlyings is extremely decreasing and the volatility around this forward a constant. The important result is that the long-term volatility smile for the worst of derivative is flat.
Checking The Results
As there is hardly any benchmark when it comes to pricing American puts on many underlyings, we made the following comparison. We priced a barrier option on multiple underlyings. If one of the underlyings touches the barrier you receive zero otherwise you receive the notional. It is possible to price this product using a Monte Carlo simulation technique. We priced it also using our one factor PDE with the right forward and the right volatility. We obtained a difference that is less than 10 basis points. This benchmarking shows that the method is really fast and accurate.
Taking Into Account The Smile
We have created an artificial underlying and we showed that it was sufficient for pricing and hedging American puts on many underlyings. We can make this method more accurate by having an approach that is more consistent with the trader's view; one can establish the following methodology:
*Price an American put on the fake underlying but keep only the term structure of its forward and use the constant volatility that is taken from the implied volatility for the long maturity.
*Extract from this initial pricing the exercise frontier and for a given time discretization {Tj } the first passage probability distribution {pj }.
*The trader can therefore add to the initial price the cost of thehedge when it comes to price the smile. Computing the probabilities to touch the barrier at any time and summing all the prices of the vanilla put does this. The extra cost of these vanillas when the smile is taken into account weighted by the convexity of the payoff easily does this. The formula is then given by:
Where
Pj is the probability to hit the exercise frontier the first time at Tj ,
And
Conclusion
Combining two different numerical approaches, one based on Monte Carlo simulation and the other one on lattice methods, we are able to develop a model for pricing and hedging multi-dimensional American options. Also the decomposition through time and space gives both an intuitive and accurate account of the smile.
This week's Learning Curve was written by Adil Reghai, an equity derivatives quantitative analyst at Dresdner Kleinwort Wassersteinin London. Robin Gray, structurer at DrKW also contributed to the article.
