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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.