Notes about Option Pricing

This is a binomial algorithm which I created in June / July 1990. It is unique in that it has a flowchart which uses the European valuations where appropriate. It is also unique in that the price grid for the underlying stock is calculated as an arithmatic array, and then the exponent function is called to convert to price.

This graph demonstrates the dangers of taking derivitaves of binomial option prices.

The problematic graph was calculated from this graph. Even though the prices of the option appear quite reasonable, the derivative is visably inaccurate.

Variable Definitions

1. r = risk free interest rate
2. alpha = return of the underlying. This is
   1. for futures alpha = 0.
   2. for stocks alpha = r - d, where r is the interest rate, and d is the dividend yield.
   3. for currencies alpha = r - r_foreign, where r is the foreign interest rate.
3. sigma = volatility of
4. S = price of underlying
5. K = strike price
6. Tau = amount of time till expiration. Note that the derivative with respect to Tau is not the same as the expected change with respect to time.
7. n = the number of steps in the binomial tree

Page 1

These are recomended values for the first differences. Note the warning about the sinusoidal pattern in the second derivative for some values of delta S.

Page 2

This is the definition of the parameter grid.

Page 3

This describes when to use the European formulae, rather than the American formulae.

Page 4

This defines a grid of option prices to calculate the derivatives. Fortran style loops are used. For i -1, 1, 2, means loop i, starting at -1, ending at 2, incrementing by 1.

Page 5

This is the calculation of the differences, to approximate the derivatives.

Page 6

This copies the derivativs to their output names.

Page 7

This is the definition of the function S, and European call. An alternative valuation of S is also given.

Page 8

This is the definition of the function American Call.

Page 9

This is the definition of the function American Put.

Page 10

This is the first page of a derivation of calculating a stock grid, given the first 2 moments of the distribution of stock price.

1. S0 - current stock price
2. phi - probability of move of size u
3. u - size of up move
4. v - size of down move
5. n - number of steps

Page 11

This is the solution of u

Page 12

This is the solution of v and phi. There is also alternative statement of the price grid.