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© K. Cuthbertson and D. Nitzsche Figures for Chapter 8 OPTIONS PRICING (Financial Engineering : Derivatives and Risk Management)
© K. Cuthbertson and D. Nitzsche Figure 8.1 : Put-call parity Profit (at expiry) STST $100 K = 100 110 Long share Synthetic long call plus K = $100 at T Long put 0 +1 At t = 0 : long share + long put = long call + cash of $Ke -rT 0 +1
© K. Cuthbertson and D. Nitzsche Figure 8.2 : Payoffs from the one-period BOPM A. Long one share 100 SU = 110 SD = 90 B. Long one call K = $100 C C u = 10 C d = 0
© K. Cuthbertson and D. Nitzsche Figure 8.3 : Payoffs from the two-period BOPM A. Long one share 100 SU = 110 SD = 90 SU 2 = 121 SUD = 99 SD 2 = 81 B. Long one call K = $100 C = 10.714 (h = 0.75) C u = 15 (h u = 0.9545) C d = 0 (h d = 0) C uu = 21 C ud = 0 C dd = 0
© K. Cuthbertson and D. Nitzsche Figure 8.4 : Stock + bond replicates long call Value of call at t = 0 equals the value of the replication portfolio at t = 0 C = NS + B = [qC u + (1 - q)C d ] / R (NS + B) NS u + BR = C u NS d + BR = C d Hedge ratio and bonds held at t = 0 N = (C u - C d ) / (S u - S d ) B = (C u - NS) / R At t = 1, values of replication portfolio equal to outcomes for the long call Value of replication portfolio at t = 0
© K. Cuthbertson and D. Nitzsche Figure 8.5 : Notation for nodes tt + 1 (t, k) (t+1, k) (t+1, k+1) Value of call at t is given by the recursion C(t, k) = [qC(t + 1, k) + (1 - q) C(t + 1, k + 1)] / R
© K. Cuthbertson and D. Nitzsche Figure 8.6 : Binomial tree : dividend payout S SU SD SU 2 (1 - ) SU 3 (1 - ) SU 2 D(1 - ) SUD(1 - ) SUD 2 (1 - ) SD 2 (1 - ) SD 3 (1 - ) Ex-dividend date just before 2nd node Single dividend payable with dividend yield known, then the tree recombines
© K. Cuthbertson and D. Nitzsche Figure 8.7 : Standard normal distribution Probability -1.96+1.96+0.7475 2.5% of the area Probability for N(0, 0.7475) = 0.7721
© K. Cuthbertson and D. Nitzsche Figure 8.8 : Black-Scholes Stock price Call premium 0 S* C A B C K Value of call prior to expiry Payoff from call at expiry BC = time value D CD = intrinsic value X CXCX
© K. Cuthbertson and D. Nitzsche Figure 8.9 : Call premium and BOPM model N
© K. Cuthbertson and D. Nitzsche Figure 8.10 : Brownian motion
Binomial Option Pricing Model (BOPM) References: Neftci, Chapter 11.6 Cuthbertson & Nitzsche, Chapter 8 1.
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