Thomas Rådberg and Oskar Rundlöf

Present their master thesis

Pricing of Emerald Options Using Local Volatility
   
Abstract

This paper concerns the pricing of a structured product known as an
Emerald option. The Emerald is a path-dependent derivative written on
a basket of underlying assets. We will emphasise the problems
associated with a closed form valuation, and hence resort to a
numerical solution with Monte Carlo simulation. Antithetic variates
are implemented to reduce variance in the estimates. With available
market prices of call options written on the equity indices in the
basket, we estimate volatilities using the Dupire s Local Volatility
model. This model assumes that future volatilities will be a
deterministic function of the underlying assets  value and time. The
local volatility surface is derived through an analytical formula,
where market prices are differentiated with respect to strike and
time. Since option prices are only available in a finite number, we
use smoothing splines to interpolate additional values. We deduce that
the interpolation is a significant source of pricing error, since
accuracy contradicts necessary smoothness. Alternative methods of
volatility estimation are briefly discussed. Model validation is done
by comparing estimated call prices with market prices. Our model
generates accurate prices for long-term options, while short-term
options are priced with a slightly higher error. Finally, we price the
Emerald option with our local volatility model. This part concerns the
issue of asset correlation. We apply Cholesky decomposition to the
correlation matrix, and generate correlated random walks.