1 / 14

Binomial Method for Options Pricing

Binomial Method for Options Pricing. Amit Kumar. 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.

kristopher-tannis
Download Presentation

Binomial Method for Options Pricing

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Binomial Method for Options Pricing Amit Kumar

  2. 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.

  3. 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.

  4. 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.

  5. u2S uS duS S dS d2S S – Tree

  6. The Tree for C • 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:

  7. Cuu = max [0, u2S-K] Cu Cdu = max [0, duS-K] C Cd Cdd = max [0, d2S-K] C-Tree

  8. 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

  9. 270(0.216) 180(0.36) 90(0.432) 120(0.6) 60(0.48) 80 40(0.4) 30(0.288) 20(0.16) 10(0.064) Asset values and probabilitieswhen, n=3, S=80

  10. 270(0.36) 180(0.6) 90(0.48) 120 60(0.4) 30(0.16) Asset values and probabilities …when n=2, if S = 120

  11. 90(0.36) 60(0.6) 30(0.48) 40 20(0.4) 10(0.16) Asset values and probabilities …when n=2, if S = 40

  12. 190 107.272 10 60.463 5.454 34.065 2.974 0 0 0 Call value tree…

  13. 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]; }

  14. 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.

More Related