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Binomial Method for Options Pricing

Amit Kumar

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References: Book: Computational Methods for Option Pricing, by authors: Yves

Achdou , and Olivier Pironneau.

J. Cox, S. Ross, and M. Rubinstein. Option pricing: A Simplified

Approach. Journal of Financial Economics, 7:229264, 1979.

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Basic Equation for call value: C(ST, T) = (ST – K)+

Binomial Method for options pricing uses a tree of possible events. The Tree for S – determines the intermediate value of the

underlying asset, along the time to maturity. The Tree for C – determines the intermediate values of the

Call option, along the time to maturity.

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The Tree for S Consider a simple situation where the underlying

asset(i.e., Sn = St, t = ndt) can evolve only in two ways:

either it goes up by a factor u >= 1: Sn+1 =uSn with probability p or,

it goes down by a factor d <= 1: Sn+1 = dSn with probability 1-p.

At n= 0, S0 is known, then at n = 1, S1 € {u S0,d S0}, at n= 2, S2 € {u2S0 ,udS0, d2S0} with probability p2, 2p(1-p), (1-p)2, and so forth.

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S – Tree

S

uS

dS

d2S

duS

u2S

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By definition the price of the European vanilla call option can be approximated by the following equation

It is seen that Cn = (Sn – K)+ at t = ndt, also has only two possible changes, a growth with a probability p and a decrease with a probability 1-p.

Let , m = 0, …,n be the possible values of Cn. Then the expected value of Cn+1 knowing Cn = is , so the analogue equation to Black-Scholes is:

The Tree for C

mnC

)()1()())((* 0

0

KSduppNj

KSEeC jjNjjNN

jN

trNN

mn

mn CppC 1

11 )1(

mn

mn

mn

tr CCppCe

))1(( 1

11

mnC

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C-Tree

C

Cu

Cd

Cuu = max [0, u2S-K]

Cdu = max [0, duS-K]

Cdd = max [0, d2S-K]

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Example:S = 80, n=3, K=80, u=1.5, d=0.5 r=1.1

Suppose the probability function is defined as: p = (r-d)/(u-d) = 0.6 in this case.

Relevant values of r: r-1 = 0.909, r-2 = 0.826, r-3 = 0.751

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Asset values and probabilitieswhen, n=3, S=80

80

120(0.6)

40(0.4)

20(0.16)

60(0.48)

180(0.36)

270(0.216)

90(0.432)

30(0.288)

10(0.064)

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Asset values and probabilities …when n=2, if S = 120

120

180(0.6)

60(0.4)

30(0.16)

90(0.48)

270(0.36)

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Asset values and probabilities …when n=2, if S = 40

40

60(0.6)

20(0.4)

10(0.16)

30(0.48)

90(0.36)

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Call value tree…

34.065

60.463

2.974

0

5.454

107.272

190

10

0

0

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Sample Code … that has O(M2) operations// Source: Computational Methods for Option Pricing, by authors: Yves Achdou , and Olivier Pironneau.double binomial(const double S0){ double disc = exp(-r*dt); double u = ( 1+sqrt(exp(sigmap*sigmap*dt)-1))/disc; double d = (1-sqrt(exp(sigmap*sigmap*dt)-1))/disc, p = 0.5; double S[M],C[M]; S[0] = S0; for(int m=1; m<M; m++) { for(int n=m; n>0; n--) S[n] = u*S[n-1]; S[0] = d*S[0]; } for(int n=0; n<M; n++) C[n] = S[n]>K?S[n]-K:0; for(int m=M-1; m>0; m--) for(int n=0; n<m; n++) C[n] = (p*C[n+1]+(1-p)*C[n])*disc; return C[0];}

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What can be done … Adapting binomial method to a time dependent volatility

function? Study the influence of the choice of p, u, and d on the

results.