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See http://kalx.net/fms/rough/pp.html for probability preliminaries.

This document gives a careful derivation of the classic Black-Scholes/Merton partial differential equation. Instead of stopping at the necessary conditions for a self-financing hedge it also shows these conditions are sufficient.

This document describes the discrete time multi-period model and proves the Fundamental Theorem of Asset Pricing for this model.

## Initial Hedge

Suppose you have to hedge a call option. You are paid upfront and have to reproduce the call option payoff at expiration using only the underlying and cash that you can borrow or invest at 0 interest rate.

Take the money you get and borrow what you need to set up your initial delta hedge in the underlying.

What value of the underlying at expriation would result in a perfect hedge?

You need to take an active role and adjust your hedge over time, but you can't really change it continuously like the B-S/M theory assumes.

## First Rehedge

One approach is to place limit orders above and below the the current stock levels. Limit orders have a price and quantity. Fix the above and below prices. The quantity should be for the amount you need to change your initial delta hedge. Once you set the prices,
all you need to know is when the limit order will be executed to compute the change in your hedge.

As a first guess, calculate the delta at the above and below levels using the same time to expiration in order to determine the limit order quantities. Calculate C(t) - V(t)/t = C(t) - (M(t) + N(t) S(t))/t when the underlying hits the above and below limit. What is the probablity the stock never hits either limit?

This document gives a careful derivation of the classic Black-Scholes/Merton partial differential equation. Instead of stopping at the necessary conditions for a self-financing hedge it also shows these conditions are sufficient.

This document describes the discrete time multi-period model and proves the Fundamental Theorem of Asset Pricing for this model.

Take the money you get and borrow what you need to set up your initial delta hedge in the underlying.

What value of the underlying at expriation would result in a perfect hedge?

You need to take an active role and adjust your hedge over time, but you can't really change it continuously like the B-S/M theory assumes.

As a first guess, calculate the delta at the above and below levels using the same time to expiration in order to determine the limit order quantities. Calculate C(t) - V(t)/t = C(t) - (M(t) + N(t) S(t))/t when the underlying hits the above and below limit. What is the probablity the stock never hits either limit?

Last edited May 14, 2013 at 2:39 AM by keithalewis, version 14