Binomial trees are frequently used to approximate the movements in the price of a stock or other asset In each small interval of time the stock price is assumed to move up by a proportional amount u or to move down by a proportional amount d Options

Standard approach to Estimating Volatility Define on as the volatility per day between day n-1 and day n, as estimated at end of day Define S: as the value of market variable at end of day i

What loss level is such that we are y%o confident it will not be exceeded in n business days?

Put-call parity p +spe-ql=c+Xer holds regardless of the assumptions made about the stock price distribution It follows that

A bank has sold for $300,000 a European call option on 100,000 shares of a nondividend paying stock S=49,K=50,=5%,=20% T=20 weeks,μ=13%

European Options on Stocks 13.2 Providing a dividend yield We get the same probability distribution for the stock price at time T in each of the following cases 1. The stock starts at price so and provides a dividend yield =q 2. The stock starts at price Soe q l and provides no income

The Stock Price Assumption Consider a stock whose price is s In a short period of time of length 8t, the return on the stock is normally distributed

Categorization of Stochastic Processes Discrete time; discrete variable Discrete time; continuous variable Continuous time; discrete variable Continuous time; continuous variable Options, Futures, and Other Derivatives,5 th edition2002 by John.hull

A Simple Binomial model A stock price is currently $20 In three months it will be either $22 or

Three Alternative strategies Take a position in the option and the underlying Take a position in 2 or more options of the same type(a spread Combination: Take a position in a mixture of calls puts(A combination)