© K.Cuthbertson, D. Nitzsche1 Version 1/9/2001 FINANCIAL ENGINEERING: DERIVATIVES AND RISK MANAGEMENT (J. Wiley, 2001) K. Cuthbertson and D. Nitzsche LECTURE Options Pricing BOPM

© K.Cuthbertson, D. Nitzsche Topics Pricing: One Period BOPM Pricing: Two-Period BOPM Delta Hedging

© K.Cuthbertson, D. Nitzsche Pricing: One Period BOPM

© K.Cuthbertson, D. Nitzsche Binomial Model Pricing an option by creating a synthetic risk free portfolio of “stock + written call”. Risk free, synthetic portfolio must earn the (known) risk free rate, r =5% - otherwise riskless arbitrage profits could be made. One period call option (expires at t=1) with payoff max (S T - K, 0)

© K.Cuthbertson, D. Nitzsche Figure 3: Payoffs from the 1-Period BOPM A. Long One Share (U=1.1, D=0.9) 100 S u = SU = 110 S d = SD = 90 B. Long One Call K = $ 100 C C u = 10 = (S T - K) C d = 0 Difference =20 Difference =10 Try : Long 1/2 share and write(sell) one call Hedge Ratio = 1/2 (=10 / 20) = C u - C d / S u - S d

© K.Cuthbertson, D. Nitzsche Portfolio =Long 1/2 share and write one call Note: payoff to written(sold) call is “opposite” of long call Up V u = (0.5) = 45 Down V d = (0.5) = 45 Hence we have a Risk free portfolio

© K.Cuthbertson, D. Nitzsche Portfolio =Long 1/2 share and write one call Cost of risk free portfolio = (1/2) S - C Portfolio must earn the risk free rate: (Value at t=1) / (Cost at t=0) = (1+r) 45 / [ (1/2) (100) - C)] = (1+r) If r = 0.05 = risk free rate then “fair/corrrect/ no-arbitrage price, C = Note: Cost of portfolio at t=0 for correctly priced call is (1/2) S - C = =

© K.Cuthbertson, D. Nitzsche Overpriced Call C a = 7.5 Can I make a riskless profit ? Overpriced call, therefore sell (write) the call AND go long (ie. Buy ) 1/2 share Cost of strategy(t=0) = (1/2) S - C a = = 42.5 Borrow 42.5 from bank at r=5% to execute the strategy Owe the bank at t=1 : 42.5 (1.05) = $ Value of your portfolio at t=1 = 45 (regardless of “up” or “down”) Net $ profit (at t=1) = $ Note: ( 0.375/42.5) = %

© K.Cuthbertson, D. Nitzsche Overpriced Call C a = 7.5 Alternative presentation: Cost of strategy(t=0) = 42.5 Value of your portfolio at t=1 = 45 (regardless of “up” or “down”) Return = (45/42.5)- 1 = 5.882% (ie. greater than r=5%, cost of borrowing from bank, or return on a riskless deposit in the bank)

© K.Cuthbertson, D. Nitzsche Under-priced Call C a = 7.0 Can I make a riskless profit ? Underpriced call, therefore buy the call AND short sell 1/2 share Receipts at t=0 = (1/2) S - C a = = 43 Invest at r giving, at t=1, 43(1.05) = Value ( t=1) = - 45 regardless of “up” or “down” (ie. You have to buy back the share and take the payoff from your long call) Net profit = = $0.15 Return = 0.15/ = 0.349%

© K.Cuthbertson, D. Nitzsche Complete Algebra V u = h. (SU) - C u = V d = h. (SD) - C d Hence h =1/2 = ( C u - C d ) / (SU - SD) Riskless Portfolio earns risk free rate of return: [ Value at t=1 / Cost at t=0 ] = (1+r) V u or V d / [ h S - C)] = (1+r) C = (1 / R) [ q C u +(1-q) C d ] where q= [R - D] / ( U-D ) and R=(1+r) q ~ risk neutral probability of “up” move

© K.Cuthbertson, D. Nitzsche Pricing: Two-Period BOPM

© K.Cuthbertson, D. Nitzsche Figure5: Payoffs from the 2-Period BOPM A. Long One Share 100 SU = 110 SD = 90 SD 2 = 81 SD = 99 SU 2 = 121 B. Long One Call K = $ 100 C C u = 15 C d = 0 C dd = 0 C ud = 0 C uu = 21

© K.Cuthbertson, D. Nitzsche Figure 6 : Payoffs from the 2-Period BOPM Option Value and Hedge Ratios C = (h = 0.75) C u = 15 (h u =0.9545) C d = 0 (h d =0) C dd = 0 C ud = 0 C uu = 21

© K.Cuthbertson, D. Nitzsche Two Period :Algebra

© K.Cuthbertson, D. Nitzsche Delta Hedging with the (Two-period) BOPM

© K.Cuthbertson, D. Nitzsche Delta Hedging with the (Two-period) BOPM At t=0 ( h = 0.75, C = , S=100) Write 1000 calls and buy 750 shares Buy = $75,000 Write 1000 calls at = $10,714 Net investment= $64,285 C u node ( h u =0.9545, C u = 15, S =110 ) Value of portfolio V u = 750 (110) – 1000(15) = $67,500 Return over period-1 = /64,285 = 1.05 (5%) New hedge ratio h u = = N s / N c

© K.Cuthbertson, D. Nitzsche Delta Hedging New hedge ratio h u = = N s / N c : Either, A)Write 1000 calls and hold shares or,B) Write calls (= 750/0.9545) and hold the ‘original’ 750 shares ‘A’ involves increasing the number of shares held from 750 to (at a price of $110 per share) ‘B’ appears to be the cheaper alternative since it involves buying back calls ( = 1,000 – 785.7) at C u = 15 per contract. Assume we take ‘B’. We do not increase our “own funds” in the hedge so we borrow the funds required at the risk free rate r = 5%.

© K.Cuthbertson, D. Nitzsche Delta Hedging Case B)Buy back = $3,214 (= borrowed funds) Case B): C uu node ( C u = 21, S =121, r= 0.05 ) Value of shares = 750 (121) = 90,750 Less call payoff = (21)= 16,500 Less loan outstanding 3,214(1.05)= 3,375 Value of portfolio V uu (case B)= 70,875 Return over period-2(case B) = 70,875/67,500 = 1.05 (5%)

© K.Cuthbertson, D. Nitzsche Case-A: Outcome is again a return of 5%. Case A)Continue to hold 1000 written options and increase shares to Buy additional ( ) 110 = $22,495 (= borrowed funds) Case A): C uu node ( C u = 21, S =121, r= 0.05 ) Value of shares = (121) = 115, Less call payoff = 1000 (21)= 21,000 Less loan outstanding 22,495(1.05)= 23, Value of portfolio V uu (case A) = 70,875 Return over period-2(case A) = 70,875/67,500 = 1.05 (5%) R=(1+r)

© K.Cuthbertson, D. Nitzsche Figure4 : Binomial Tree : Dividend Payments S SU SD SD 2 (1-  ) SUD(1-  ) SU 2 (1-  ) SU(1-  ) SD(1-  ) Note: Dividend payments are proportional The tree still recombines

© K.Cuthbertson, D. Nitzsche End of Slides