Prof Avner Kalay Options and Futures The Binomial
Prof. Avner Kalay - Options and Futures The Binomial Model $120 $20 C 100 =? $100 $90 Strategy: Buy 1 stock sell 1. 5 calls $0
Prof. Avner Kalay - Options and Futures The Binomial Model CF today CF at T (S = 90) CF at T (S=120) Buy Stock -$100 $90 $120 Sell 1. 5 calls $1. 5 C $0 ______ -$30 _______ 1. 5 C - 100 $90
Prof. Avner Kalay - Options and Futures The Binomial Model - Investment today of $100 -1. 5 C yields $90 for sure. Hence, - [100 -1. 5 C](1+r) = 90 If r=10% C = (1/1. 5)[100 -90/1. 1] = 12. 12
Prof. Avner Kalay - Options and Futures The Binomial Model $u. S $S Cu 1/Δ – hedge ratio $C $d. S Cd u. S - (1/Δ)*Cu S – (1/Δ)*C d. S - (1/Δ)*Cd
Prof. Avner Kalay - Options and Futures Delta - Chose 1/Δ to hedge, thus; u. S - (1/Δ)*Cu = d. S - (1/Δ)*Cd 1/Δ = {u. S – d. S}/{Cu – Cd}
Prof. Avner Kalay - Options and Futures Delta $120 $20 =0 - $90 $0
Prof. Avner Kalay - Options and Futures The Binomial Model S – {1/Δ}*C u. S – {1/Δ}*Cu Investment Certain outcome {S – [1/Δ}*C}*R = u. S – {1/Δ}*Cu R = 1 + rf and u > R > d C = {S(R-u) + (1/Δ)Cu}/(1/Δ)R
Prof. Avner Kalay - Options and Futures The Binomial Model - Substitute for 1/Δ to get C = {P*Cu + (1 -P)*Cd}/R P = [R-d]/[u-d]
Prof. Avner Kalay - Options and Futures The Binomial Model - In our example: u=1. 2, d=0. 9, R=1. 1, u. S=120, ds=90, E = 100, S=100 - P =[R-d]/[u-d] = [1. 1 -0. 9]/[1. 2 -0. 9]=2/3 - C= {(2/3)*20 + (1/3)*0}/1. 1 = 12. 12
Prof. Avner Kalay - Options and Futures What is P? u>R>d 0<P<1 R=1. 1 ________________ d=0. 9 u=1. 2
Prof. Avner Kalay - Options and Futures What is P? - P cannot be a probability since we do not know the probability of a price increase – denoted q. - Since the valuation of C is true for any q we can assume (for our example) q = 0. 5 - Do you feel comfortable with q = 0. 5?
Prof. Avner Kalay - Options and Futures What is P? - - But if q=0. 5 we can compute the expected return of the stock. E(Rs) = 0. 5*20% + 0. 5*(10%) = 5% Hence, E(Rs) < rf
Prof. Avner Kalay - Options and Futures What is P? - Assume q=7/8=0. 875. - In our example P=[1. 1 -0. 9]/[1. 2 -0. 9] = 2/3 - E(Rs) = 0. 875*20% + 0. 125*(10%) = 16. 25% - Risk premium = 16. 25 – 10 = 6. 25%
Prof. Avner Kalay - Options and Futures What is P? - - Now reduce the risk aversion in the economy by reducing the risk premium to 1. 25%. Increase the risk free rate to 15%. P = [1. 15 -0. 9]/[1. 2 -0. 9] = 5/6 = 0. 833 - P gets closer to q - C=5/6*20/1. 15 = 14. 493
Prof. Avner Kalay - Options and Futures What is P? - Pushing it one step further, lets reduce the risk aversion in the economy to zero – R=1. 1625 P = [1. 1625 -0. 9]/[1. 2 -0. 9] = 7/8 - P is now equal to q - C = {7/8}*20/1. 1625 = 15. 054
Prof. Avner Kalay - Options and Futures P – the risk neutral probability P < q Risk Aversion P = q Risk neutral P > q Risk seeking
Prof. Avner Kalay - Options and Futures P – the risk neutral probability 6 5 7. 8 $20 0. 66 0 0. 3 5 12 33 0. $0 $0 0. 875*20=17. 5 0. 666*20=13. 333 17. 5/1. 1=15. 909 13. 333/1. 1=12. 12
Prof. Avner Kalay - Options and Futures Certainty equivalent - The difference 17. 5 – 13. 333 = 4. 167 is a correction for risk in the numerator - The option model is valuation by certainty equivalents. - Once we use P as if it is q we can take expectations and discount with the risk free rate
Prof. Avner Kalay - Options and Futures Two periods {0. 666*44+0. 333*8}/1. 1 144 44 120 29. 09 108 100 19. 08 90 81 8 4. 844 0 {0. 666*29. 09+0. 333*4. 844}/1. 1 {0. 666*8/1. 1
Prof. Avner Kalay - Options and Futures Two Periods - Cu = {P*Cuu + (1 -P)*Cud}/R - Cd = {P*Cud + (1 -P)*Cdd}/R - C = {P*Cu + (1 -P)*Cd}/R - C = {P 2 Cuu + 2 P(1 -P)Cud + (1 -P)2 Cdd}/R 2
Prof. Avner Kalay - Options and Futures Four periods 1 u 4 P 4 1 du 3 P 3 (1 -P) 4 d 2 u 2 P 2(1 -P)2 6 d 3 u (1 -P)3 P 4 d 4 (1 -P)4 1
Prof. Avner Kalay - Options and Futures The Binomial Distribution - The probability of a path with j ups and n-j downs is Pj(1 – P)n-j - The number of paths leading to a node is n!/{j!(n-j)!} - The probability to get to a node is {n!/j!(n-j)!}Pj(1 -P)n-j
Prof. Avner Kalay - Options and Futures The Binomial Distribution - The probability to get to any one of the nodes is Σj=0 [{n!/j!(n-j)!}Pj(1 -P)n-j] = 1 - The probability of at least a ups is Φ{a, n, P} = Σj=a{[n!/(j!(n-j)!]Pj(1 -P)n-j} < 1
Prof. Avner Kalay - Options and Futures The Binomial Option Pricing Model C = [Σj=0 {n!/j!(n-j)!}Pj(1 -P)n-j Max{0, ujdn-j. S – E}]/Rn Let a (number of ups) be the smallest integer such that the option will mature in the money
Prof. Avner Kalay - Options and Futures The Binomial Option Pricing Model C = [Σj=a {n!/j!(n-j)!} Pj(1 -P)n-j {ujdn-j. S – E}]/Rn = S[Σj=a {n!/j!(n-j)!} Pj(1 -P)n-j{ujdn-j/Rn} ER-n[Σj=a {n!/j!(n-j)!} Pj(1 -P)n-j]
Prof. Avner Kalay - Options and Futures The Binomial Option Pricing Model S[Σj=a {n!/j!(n-j)!} [u/R]j Pj (1 -P)n-j {d/R}n-j } Let P’ = [u/R]P than 1 – P’ = [u/R]{(R-d)/(u-d)} = [d/R](1 -P) S[Σj=a {n!/j!(n-j)!} P’j (1 -P’)n-j ]
Prof. Avner Kalay - Options and Futures The Binomial Option Pricing Model - C = S*Φ{a, n, P’} - E*R-n*Φ*{a, n, P} - Σj=0{[n!/(j!(n-j)!]Pj(1 -P)n-j}= 1 - Φ{a, n, P} = Σj=a{[n!/(j!(n-j)!]Pj(1 -P)n-j} < 1
Prof. Avner Kalay - Options and Futures The Binomial Option Pricing Model - C = S*Φ{a, n, P’} - E*R-n*Φ*{a, n, P} - P = [R-d]/[u-d] - P’ = [u/R]P
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