TABLE OF CONTENTS In real markets, practitioners are interested in analysis of the sensitivity of the option prices to changes in factors associated with the underlying, such as the price of the underlying, the volatility and the interest rates. These issues are taken up in 6.1, where we use a translation method via shifting paths in the sample space to give a computation method of some sensitivity parameters called Greeks. A simple kind of integration by parts formula (analogous to the one used in standard Malliavin calculus) is used in evaluating one such parameter (the Vega). The results are useful in Monte Carlo simulations for calculating these parameters.
Then in 6.2 we mention the notation of implied volatility and its term structure, whose counterpart is the term structure of interest rates that we will model in 6.3.
We first discuss the various quantities that measure sensitivities. They are called the Greeks because of the notation, but are sometimes also referred to as hedge parameters in risk management for their important roles in constructing hedges against the risk connected to dealing with options and other derivatives.
For convenience, we now refer to the derivative security as an option P written on a stock S. So in the Black-Scholes setting,
as a function of the stock price x, time to maturity t, volatility of the stock ?, riskless interest rate r and the strike price K. From...
