The Binomial Tree Formula

The previous post presented an example binomial tree calculation of the price of an american-style option. The lesson drawn from thta example was that the accurate estimate of volatility is key to price options correctly.

I will look here more closely to the standard formula used in the binomial model, to see how volatility enters into the picture. The up and down price movements in the option underlying asset used to build the binomial tree are calculated as follows:

The up movement U is equal to the exponential of the product of the volatility V and the square root of the time step DT. Since the volatility is annualized, the time step is expressed in years. The down movement is simply the inverse of the up movement.

U = EXP(V*SQRT(DT))

D = 1/U

The actual option price P at the last step is known and it is the intrinsic value. The option price at an intermediate step is calculated as either the binomial value B or the exercise price E, whichever is the largest.

P = MAX(B,E)

The binomial value is

B = (X*U + (1-X)*D)*EXP(-R*DT)

where R is the risk-free interest rate, typically the deposit rate, and

X = (EXP((R-Y)*DT) - D)/(U - D)

where Y is the underlying security yield (the dividend yield if it's a stock) corresponding to the life of the option.

If V is estimated using historical volatility, the formula is

V = STANDARDDEVIATION(LOG(S[n]/S[n-1]))

where S[n] is a series of underlying asset prices over a given period of time (the historical sample).

The standard deviation is, of course the square root of the sample variance:

V = SQR(SUM((L[n]-A)*(L[n]-A))/T)

where L[n] is the logarithm of the return

L[n] = LOG(S[n]/S[n-1]),

A is the sample average

A = SUM(L[n])/T,

and T is the sample size (the total period over which prices are sampled).

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