Copyright © 2001 by Harcourt, Inc. All rights reserved.1 Chapter 15: Advanced Derivatives and Strategies The best advice may be this: treat exotic derivatives like powerful medicines, large doses of which can be harmful. Use them in moderation for a particular purpose (such as risk management) and only after having read the instructions on the bottle. Philippe Jorion Big Bets Gone Bad, 1995.

Copyright © 2001 by Harcourt, Inc. All rights reserved.2 Important Concepts in Chapter 15 n The concept of portfolio insurance and its execution using puts, calls, futures and t-bills n New and advanced derivatives and strategies such as equity forwards, warrants, equity-linked debt, equity swaps, variations of interest rate swaps, structured notes, and mortgage securities n Exotic options such as digital options, chooser options, Asian options, lookback options and barrier options

Copyright © 2001 by Harcourt, Inc. All rights reserved.3 Advanced Equity Derivatives and Strategies n Portfolio Insurance u We can insure a portfolio by holding one put for each share of stock. For a portfolio worth V, we should hold F N = V/(S 0 + P) puts and shares u This will establish a minimum of F V min = XV/(S 0 + P) where X is the exercise price u Example: On Sept. 26, market index is and Dec 485 put is $ Expiration is Dec. 19. Risk-free rate is 2.99 % continuously compounded. Volatility is.155.

Copyright © 2001 by Harcourt, Inc. All rights reserved.4 Advanced Equity Derivatives and Strategies (continued) n Portfolio Insurance (continued) F We hold 100,000 units of the index portfolio for V = $44,575,000. We have V min = (485)(44,575,000)/( ) = 44,637,585V min = (485)(44,575,000)/( ) = 44,637,585 N = 44,575,000/( ) = 92,036N = 44,575,000/( ) = 92,036 F This guarantees a minimum return of (365/84) - 1 =.0061 per year, which must be below the risk- free rate.

Copyright © 2001 by Harcourt, Inc. All rights reserved.5 Advanced Equity Derivatives and Strategies (continued) n Portfolio Insurance (continued) F Outcomes Index is 510 at expirationIndex is 510 at expiration –92,036 shares worth 510 = $46,938,360 –92,036 puts worth $0 = $0 –Total value = $46,938,360 (> V min ) Index is 450 at expirationIndex is 450 at expiration –Sell stock by exercising puts so you have 92,036(485) = $44,637,460 ( V min ) –Sell stock by exercising puts so you have 92,036(485) = $44,637,460 (  V min )

Copyright © 2001 by Harcourt, Inc. All rights reserved.6 Advanced Equity Derivatives and Strategies (continued) n Portfolio Insurance (continued) F See Figure 15.1, p u If calls and t-bills used, F N B = V min /B T (number of bills) F N C = V/(S 0 + P) (number of calls) F So N B = 44,637,585/100 = 446,376 F N C = 92,036

Copyright © 2001 by Harcourt, Inc. All rights reserved.7 Advanced Equity Derivatives and Strategies (continued) n Portfolio Insurance (continued) F Outcomes Index is 510 at expirationIndex is 510 at expiration –Bills worth $44,637,600 –92,036 calls worth $25 = $2,300,900 –Total value = $46,938,500 (> V min ) Index is 450 at expirationIndex is 450 at expiration –Bills worth $44,637,600 –92,036 calls worth $0 –Total value = $44,637,600 ( V min ) –Total value = $44,637,600 (  V min ) See Figure 15.2, p. 636.See Figure 15.2, p. 636.

Copyright © 2001 by Harcourt, Inc. All rights reserved.8 Advanced Equity Derivatives and Strategies (continued) n Portfolio Insurance (continued) u Dynamic hedging: A dynamically adjusted combination of stock and futures or stock and t-bills that can replicate the stock-put or call-tbill. F This can be easier because the futures and t-bill markets are more liquid than the options markets F The number of futures required is F See Appendix 15A for derivation.

Copyright © 2001 by Harcourt, Inc. All rights reserved.9 Advanced Equity Derivatives and Strategies (continued) n Portfolio Insurance (continued) u Alternatively, use stock and t-bills. u See Table 15.1, p. 638 for example of dynamic hedge

Copyright © 2001 by Harcourt, Inc. All rights reserved.10 Advanced Equity Derivatives and Strategies (continued) n Equity Forwards u Forward contracts on stock or stock indices u Work precisely like all other forward contracts we have covered. u Break forward is similar to an ordinary call but has no up-front cost. At expiration, however, its value can be negative, unlike an ordinary call. F See Table 15.2, p Note that K = compound future value of call with exercise price F plus compound future value of stock, which is forward price of stock.

Copyright © 2001 by Harcourt, Inc. All rights reserved.11 Advanced Equity Derivatives and Strategies (continued) n Equity Forwards (continued) F Example using AOL: S 0 = , T =.0959, r c =.0446, volatility =.83. F = e.0446(.0959) = F = e.0446(.0959) = Ordinary call with X = is worth K = e.0446(.0959) = Ordinary call with X = is worth K = e.0446(.0959) = See Figure 15.3, p. 641.See Figure 15.3, p F Note similarity to forward contract and call option.

Copyright © 2001 by Harcourt, Inc. All rights reserved.12 Advanced Equity Derivatives and Strategies (continued) n Equity Swaps u At least one counterparty pays the return, usually capital gains and dividends, on a stock or index F Based on a given notional principal F Party paying the equity return might have to pay both sides of the payments because of decreases in the index.

Copyright © 2001 by Harcourt, Inc. All rights reserved.13 Advanced Equity Derivatives and Strategies (continued) n Equity Swaps u See Table 15.3, p. 642 for swap with one side paying LIBOR. The interest payment is F $10,000,000(.09)(days/360) u The equity payment is F $10,000,000 x rate of return on index u See Table 15.4, p. 644 for swap with one side paying S&P 500 and other paying a foreign stock index. Both payments are based on the equity return on $25,000,000 notional principal.

Copyright © 2001 by Harcourt, Inc. All rights reserved.14 Advanced Equity Derivatives and Strategies (continued) n Equity Swaps (continued) u They allow easy changes among asset classes u They can be used to achieve better diversification for those whose portfolios are poorly diversified. u They are designed to replicate a transaction in a stock or index, but there are some differences F They do not bring or give up voting rights F A position in the stock has a variable notional principal as the value of the stock changes. The swap can be adjusted to help compensate for this.

Copyright © 2001 by Harcourt, Inc. All rights reserved.15 Advanced Equity Derivatives and Strategies (continued) n Equity Warrants u Warrants issued by firm u Warrants trading on over-the-counter markets and American Stock Exchange based on various securities and indices. u Many of these are quantos, which pay off based on the performance of a foreign stock index but payment is made in a different currency than the one associated with the country of the foreign stock index.

Copyright © 2001 by Harcourt, Inc. All rights reserved.16 Advanced Equity Derivatives and Strategies (continued) n Equity-Linked Debt u A bond that usually pays a minimum return plus a percentage of any increase in a stock index u Example: One-year zero coupon bond paying 1% interest and 50 percent of any gain on the S&P 500. F Currently one-year zero coupon bond offers 5 % compounded annually. S&P 500 is at 1500 and has a volatility of.12 and a yield of 1.5%. If you invest $10 you receive $10(1.01) = $10.10 for sure. The present value of this is 10.10/1.05 = 9.62 (5% is opportunity cost).If you invest $10 you receive $10(1.01) = $10.10 for sure. The present value of this is 10.10/1.05 = 9.62 (5% is opportunity cost). This amounts to a loss of $0.38.This amounts to a loss of $0.38.

Copyright © 2001 by Harcourt, Inc. All rights reserved.17 Advanced Equity Derivatives and Strategies (continued) n Equity-Linked Debt (continued) F Option payoff is $10(.5)Max(0,(S T )/1500). This can be written as (5/1500)Max(0,S T ), which is 5/1500th of a European call with exercise price 1500.(5/1500)Max(0,S T ), which is 5/1500th of a European call with exercise price F Plugging values into Black-Scholes model gives call value of $ Multiplying by 5/1500 gives a value of $0.32. This is less than the amount given up by accepting the lower rate on the bond ($0.38) but might be worth it to some investors.

Copyright © 2001 by Harcourt, Inc. All rights reserved.18 Advanced Interest Rate Derivatives n Variations of Interest Rate Swaps u Basis swaps: each side pays a floating rate. F Common basis swap: T-bill vs. Eurodollar (TED spread) F See Table 15.5, p. 648 for example of pricing. u Index amortizing swap: notional principal changes with level of interest rates to reflect a rate of prepayment on the underlying security. u Diff swap: Payoff based on interest rate of a given country with payment made in a different currency. Designed to speculate or hedge on a foreign interest rate without assuming currency risk.

Copyright © 2001 by Harcourt, Inc. All rights reserved.19 Advanced Interest Rate Derivatives (continued) n Variations of Interest Rate Swaps (continued) u Constant Maturity Swaps: One party pays floating such as LIBOR and another pays floating based on return with maturity longer than settlement period. Common rate is Constant Maturity Treasury (CMT), which is rate on U. S. Treasury note with a fixed maturity. For example, 5-year CMT is the rate on a five-year Treasury note. Although the maturity of a 5-year note decreases through time, new 5-year notes or interpolated values of other notes provide the CMT rate.

Copyright © 2001 by Harcourt, Inc. All rights reserved.20 Advanced Interest Rate Derivatives (continued) n Structured Notes u Definition: an intermediate term debt security issued by corporation with good credit rating in which the coupon is altered by the use of a derivative. Examples: F Floating coupon indexed to the CMT rate (e.g., 1.5 times the CMT rate). F Range floater, which pays interest only if a reference rate (e.g., LIBOR) stays within a given range over a period of time. If rate stays within range, coupon will be higher than otherwise. F Reverse (inverse) floater, where coupon moves opposite to interest rates, such as 12 - LIBOR

Copyright © 2001 by Harcourt, Inc. All rights reserved.21 Advanced Interest Rate Derivatives (continued) n Structured Notes (continued) Example: An issuer could hedge it by a swap paying LIBOR and receiving fixed rateExample: An issuer could hedge it by a swap paying LIBOR and receiving fixed rate –LIBOR < 12: -(12 - LIBOR) (note) + Fixed rate - LIBOR (swap) = Fixed rate - 12 –LIBOR  12: 0 (note) + Fixed rate - LIBOR (swap) = Fixed rate - LIBOR. Issuer could buy a cap to pay it LIBOR while it pays the strike rate if it wanted to make it risk-free. Many inverse floaters are extremely volatile due to leverage in the rate adjustment formula.Many inverse floaters are extremely volatile due to leverage in the rate adjustment formula.

Copyright © 2001 by Harcourt, Inc. All rights reserved.22 Advanced Interest Rate Derivatives (continued) n Mortgage-Backed Securities u Securities constructed by offering claims on a portfolio of mortgages, a process is called securitization. u Mortgage-backed securities subject to prepayment risk. u Mortgage pass-throughs and strips F Mortgage pass-through: a security in which the holder receives the principal and interest payments made on a portfolio of mortgages. F Mortgage strip: a claim on either the principal or interest on a mortgage pass-through. Called principal only (PO) or interest only (IO).

Copyright © 2001 by Harcourt, Inc. All rights reserved.23 Advanced Interest Rate Derivatives (continued) n Mortgage-Backed Securities (continued) u Example: Assume a mortgage-backed security representing a single $100,000 mortgage at 9.75 % for 30 years. Assume annual payments for simplicity. F See Table 15.6, p. 654 for amortization schedule. Annual payment would be $100,000/[(1-(1.0975) -30 )/.0975] = $10,387.$100,000/[(1-(1.0975) -30 )/.0975] = $10,387.

Copyright © 2001 by Harcourt, Inc. All rights reserved.24 Advanced Interest Rate Derivatives (continued) n Mortgage-Backed Securities (continued) F Assume a 7 percent discount rate and that the mortgage is paid off in year 12. Value of IO strip = 9,750(1.07) ,688(1.07) -2 + … + 8,614(1.07) -12 = 74,254.Value of IO strip = 9,750(1.07) ,688(1.07) -2 + … + 8,614(1.07) -12 = 74,254. Value of PO strip = 637(1.07) (1.07) -2 + … + (1, ,574)(1.07) -12 = 46,690.Value of PO strip = 637(1.07) (1.07) -2 + … + (1, ,574)(1.07) -12 = 46,690. Value of pass-through = $74,254 + $46,690 = $120,944Value of pass-through = $74,254 + $46,690 = $120,944

Copyright © 2001 by Harcourt, Inc. All rights reserved.25 Advanced Interest Rate Derivatives (continued) n Mortgage-Backed Securities (continued) F Let discount rate drop to 6 % and assume homeowner pays off two years from now. Value of IO = $9,750(1.06) -1 + $9,688(1.06) -2 = $17,820, loss of 76%Value of IO = $9,750(1.06) -1 + $9,688(1.06) -2 = $17,820, loss of 76% Value of PO = $637(1.06) -1 + ($699 + $98,663)(1.06) -2 = $89,033, gain of 91%Value of PO = $637(1.06) -1 + ($699 + $98,663)(1.06) -2 = $89,033, gain of 91% Value of pass-through = $17,820 + $89,034 = $106,854, loss of 12%Value of pass-through = $17,820 + $89,034 = $106,854, loss of 12%

Copyright © 2001 by Harcourt, Inc. All rights reserved.26 Advanced Interest Rate Derivatives (continued) n Mortgage-Backed Securities (continued) F If the discount rate rises to 8% and there is no change in the payoff date of year 12, Value of IO = $9,750(1.08 )-1 + $9,688(1.08 ) $8,614(1.08 )-12 = $ 70,532, a 5% lossValue of IO = $9,750(1.08 )-1 + $9,688(1.08 ) $8,614(1.08 )-12 = $ 70,532, a 5% loss Value of PO = $637(1.08 )-1 + $699(1.08 ) ($1,773 + $86,574)(1.08) -12 = $ 42,128, a 10% lossValue of PO = $637(1.08 )-1 + $699(1.08 ) ($1,773 + $86,574)(1.08) -12 = $ 42,128, a 10% loss Value of pass-through = $70,532 + $42,128 = $112,660, a loss of almost 7%.Value of pass-through = $70,532 + $42,128 = $112,660, a loss of almost 7%.

Copyright © 2001 by Harcourt, Inc. All rights reserved.27 Advanced Interest Rate Derivatives (continued) n Mortgage-Backed Securities (continued) F If rate goes to 8 % and prepayment moves back to year 14, Value of IO = $9,750(1.08 )-1 + $9,688(1.08 ) $8,250(1.08 )-14 = $ 76,445, a gain of 3%Value of IO = $9,750(1.08 )-1 + $9,688(1.08 ) $8,250(1.08 )-14 = $ 76,445, a gain of 3% Value of PO = $637(1.08 )-1 + $699(1.08 ) ($2,136 + $82,492)(1.08) -14 = $37,276, a loss of 20%.Value of PO = $637(1.08 )-1 + $699(1.08 ) ($2,136 + $82,492)(1.08) -14 = $37,276, a loss of 20%. Value of pass-through = $76,445 + $37,276 = $113,721, a loss of about 6%.Value of pass-through = $76,445 + $37,276 = $113,721, a loss of about 6%.

Copyright © 2001 by Harcourt, Inc. All rights reserved.28 Advanced Interest Rate Derivatives (continued) n Mortgage-Backed Securities (continued) u Mortgage-backed security values are very volatile. u Collateralized Mortgage Obligations (CMOs) F Mortgage-backed security in which payments are split into pieces called tranches with different claims reflecting different risks. F Some tranches are paid first, some receive only interest and some receive any residual after other tranches have been repaid.

Copyright © 2001 by Harcourt, Inc. All rights reserved.29 Advanced Interest Rate Derivatives (continued) n Mortgage-Backed Securities (continued) u Collateralized Mortgage Obligations (CMOs) (continued) F The different tranches receive interest, principal and prepayments according to different priorities. F Some CMO tranches are extremely volatile and others have low volatility. F A CMO is generally a fairly complex security.

Copyright © 2001 by Harcourt, Inc. All rights reserved.30 Exotic Options n Digital and Chooser Options u Digital options, sometimes called binary options, are of two types. F Asset-or-nothing options pay the holder the asset if the option expires in the money and nothing otherwise. F Cash-or-nothing options pay the holder a fixed amount of cash if the option expires in the money and nothing otherwise.

Copyright © 2001 by Harcourt, Inc. All rights reserved.31 Exotic Options (continued) n Digital and Chooser Options (continued) F See Table 15.7, p. 659 for example of long cash-or- nothing and short asset-or-nothing that pays off X dollars if in-the-money at expiration. This combination is equivalent to an ordinary European call. Values of options are

Copyright © 2001 by Harcourt, Inc. All rights reserved.32 Exotic Options (continued) n Digital and Chooser Options (continued) u Example: Asset-or-nothing option written on S&P 500 Total Return Index, at Exercise price of Risk-free rate is 4.88%, standard deviation is.11 and time to expiration is.5 years. We obtain F d 1 =.3526, N(.35) =.6368 F O aon = 1440(.6368) = 917 u For cash-or-nothing option, F d 2 =.2748, N(.27) =.6064 F O con = 1440e (.5) (.6064) =

Copyright © 2001 by Harcourt, Inc. All rights reserved.33 Exotic Options (continued) n Digital and Chooser Options (continued) u Chooser Options: Also known as as-you-like-it options, they enable the investor to decide at a specific time after purchasing the option but before expiration that the option will be a call or a put. F Assume that decision must be made at time t < T F The chooser option is identical to an ordinary call expiring at T with exercise price X plusan ordinary call expiring at T with exercise price X plus an ordinary put expiring at t with exercise price X(1+r) -(T-t)an ordinary put expiring at t with exercise price X(1+r) -(T-t) F Compare and contrast chooser with straddle.

Copyright © 2001 by Harcourt, Inc. All rights reserved.34 Exotic Options (continued) n Digital and Chooser Options (continued) u Example: AOL chooser in which choice must be made in 20 days. Call/put expires in 35 days. S 0 = , X = 125,  =.83, r c = T = 35/365 =.0959, t = 20/365 =.0548 so T - t = = Exercise price on put used to price the chooser is 125(1.0456) = u Using Black-Scholes model, put is worth 7.80 and call is worth for a total of Straddle is worth (call) (put) =

Copyright © 2001 by Harcourt, Inc. All rights reserved.35 Exotic Options (continued) n Path-Dependent Options u Path-dependent options are options in which the payoff is determined by the sequence of prices followed by the asset and not just by the price of the asset at expiration. u We shall price these options using a binomial framework. See Table 15.8, p. 662 which shows a three-period problem. Note 8 paths and the average, maximum and minimum prices of each path are computed. u Note how the probabilities are calculated. u In practice the binomial model is difficult to use for path-dependent options. Monte Carlo simulation (see Appendix 15B) is often used.

Copyright © 2001 by Harcourt, Inc. All rights reserved.36 Exotic Options (continued) n Path-Dependent Options (continued) u Asian option: an option in which the final payoff is determined by the average price of the asset during the option’s life. Some are average price options because the average price substitutes for the asset price at expiration. Others are average strike options because the average price substitutes for the exercise price at expiration. Can be calls or puts. Useful for hedging or speculating when the average is acceptable. Also useful for cases where market can be manipulated. u See Table 15.9, p. 663 for example of pricing Asian options.

Copyright © 2001 by Harcourt, Inc. All rights reserved.37 Exotic Options (continued) n Path-Dependent Options (continued) u Lookback option: Also called a no-regrets option, it permits purchase of the asset at its lowest price during the option’s life or sale of the asset at its highest price during the option’s life.

Copyright © 2001 by Harcourt, Inc. All rights reserved.38 Exotic Options (continued) n Path-Dependent Options (continued) u Lookback options (continued): F Four different types. lookback call: exercise price set at minimum price during option’s lifelookback call: exercise price set at minimum price during option’s life lookback put: exercise price set at maximum price during option’s life.lookback put: exercise price set at maximum price during option’s life. fixed-strike lookback call: payoff based on maximum price during option’s life (instead of final price) compared to fixed strikefixed-strike lookback call: payoff based on maximum price during option’s life (instead of final price) compared to fixed strike fixed-strike lookback put: payoff based on minimum price during option’s life (instead of final price) compared to fixed strikefixed-strike lookback put: payoff based on minimum price during option’s life (instead of final price) compared to fixed strike

Copyright © 2001 by Harcourt, Inc. All rights reserved.39 Exotic Options (continued) n Path-Dependent Options (continued) u Lookback options (continued): F See Table 15.10, p. 665 for example of pricing lookback options.

Copyright © 2001 by Harcourt, Inc. All rights reserved.40 Exotic Options (continued) n Path-Dependent Options (continued) u Barrier Options: Options that either terminate early if the asset price hits a certain level, called the barrier, or activate only if the asset price hits the barrier. The former are called knock-out options (or simply out- options) and the latter are called knock-in options (or simply in-options). If the barrier is above the current price, it is called an up-option. If the barrier is below the current price, it is called a down-option. F See Table 15.11, p. 667 for example of pricing. F Barrier options are normally cheaper than ordinary options because they provide payoffs for fewer outcomes than ordinary options.

Copyright © 2001 by Harcourt, Inc. All rights reserved.41 Exotic Options (continued) n Path-Dependent Options (continued) u Other Exotic Options: F compound and installment options F multi-asset options, exchange options, min-max options (rainbow options), alternative options, outperformance options F shout, cliquet and lock-in options F contingent premium, pay-later and deferred strike options F forward-start and tandem options

Copyright © 2001 by Harcourt, Inc. All rights reserved.42 Summary

Copyright © 2001 by Harcourt, Inc. All rights reserved.43 Appendix 15A: Derivation of the Dynamic Hedge Ratio for Portfolio Insurance n Stock-Futures Dynamic Hedge u Portfolio of N shares and N puts is worth F V = N(S + P) F So N = V/(S+P). u Change in portfolio value for a small change in stock price is

Copyright © 2001 by Harcourt, Inc. All rights reserved.44 Appendix 15A: Derivation of the Dynamic Hedge Ratio for Portfolio Insurance (continued) n Stock-Futures Dynamic Hedge (continued) u A portfolio of N S shares and N f futures is worth today V = N S S + N f V f F where V f is value of futures, which starts at zero. It follows that N S = V/S u Set change in portfolio value for small change in S to

Copyright © 2001 by Harcourt, Inc. All rights reserved.45 Appendix 15A: Derivation of the Dynamic Hedge Ratio for Portfolio Insurance (continued) n Stock-Futures Dynamic Hedge (continued) u Assuming no dividends, the futures price is u So

Copyright © 2001 by Harcourt, Inc. All rights reserved.46 Appendix 15A: Derivation of the Dynamic Hedge Ratio for Portfolio Insurance (continued) n Stock-Futures Dynamic Hedge (continued)  After substituting, setting the two partial derivatives of V with respect to S equal to other, recognizing that 1 + N(d 1 ) is we obtain the number of futures contracts as  After substituting, setting the two partial derivatives of V with respect to S equal to other, recognizing that 1 +  P/  S is  C/  S and N(d 1 ) is  C/  S, we obtain the number of futures contracts as

Copyright © 2001 by Harcourt, Inc. All rights reserved.47 Appendix 15A: Derivation of the Dynamic Hedge Ratio for Portfolio Insurance (continued) n Stock-Tbill Dynamic Hedge u A portfolio of stock and tbills is worth u Its sensitivity to a change in S is

Copyright © 2001 by Harcourt, Inc. All rights reserved.48 Appendix 15A: Derivation of the Dynamic Hedge Ratio for Portfolio Insurance (continued) n Stock-Tbill Dynamic Hedge (continued) u The t-bill price is not sensitive to the stock price. Setting the sensitivity of the stock-tbill portfolio to that of the stock-futures gives u This is the number of shares of stock to hold with t-bills to replicate the stock and put.

Copyright © 2001 by Harcourt, Inc. All rights reserved.49 Appendix 15B: Monte Carlo Simulation n A method of using random numbers designed to simulate the random observations of prices of an asset. A simulated series of asset prices at expiration is converted to an equivalent series of option prices at expiration. n Then the current option price is the discounted average of the option prices obtained at expiration from the simulation. Random prices can be simulated by drawing a standard normal random variable, Random prices can be simulated by drawing a standard normal random variable, , and inserting into the formula

Copyright © 2001 by Harcourt, Inc. All rights reserved.50 Appendix 15B: Monte Carlo Simulation (continued)  where  t is the length of the time interval over which the stock price change occurs. u Note: simulating a standard normal random variable can be done approximately as the sum of twelve unit uniform random numbers (in Excel, “=Rand( )”) minus 6.0. Each simulated stock price is treated as the stock price at expiration; thus, Each simulated stock price is treated as the stock price at expiration; thus,  t is the maturity in years of the option. u For each simulated stock price, compute the option price at expiration using the intrinsic value.

Copyright © 2001 by Harcourt, Inc. All rights reserved.51 Appendix 15B: Monte Carlo Simulation (continued) u Take the average of all of the option prices at expiration. u Discount the average over the life of the option at the risk-free rate. This is the estimate of the current option price. n This will probably require at least 50,000 random numbers for a standard option and more for exotic and complex options and derivatives.