The best advice may be this: treat exotic derivatives like The best advice may be this: treat exotic derivatives like powerful medicines, large doses of which can be harmful. powerful medicines, large doses of which can be harmful. Use them in moderation for a particular purpose (such as Use them in moderation for a particular purpose (such as risk management) and only after having read the risk management) and only after having read the instructions on the bottle.instructions on the bottle.
The concept of portfolio insurance and its execution using The concept of portfolio insurance and its execution using puts, calls, futures and t-billsputs, calls, futures and t-bills
New and advanced derivatives and strategies such as New and advanced derivatives and strategies such as equity forwards, warrants, equity-linked debt, equity equity forwards, warrants, equity-linked debt, equity swaps, variations of interest rate swaps, structured notes, swaps, variations of interest rate swaps, structured notes, and mortgage securitiesand mortgage securities
Exotic options such as digital options, chooser options, Exotic options such as digital options, chooser options, Asian options, lookback options and barrier optionsAsian options, lookback options and barrier options
Portfolio InsurancePortfolio Insurance We can insure a portfolio by holding one put for each We can insure a portfolio by holding one put for each
share of stock. For a portfolio worth V, we should holdshare of stock. For a portfolio worth V, we should hold N = V/(SN = V/(S00 + P) puts and shares + P) puts and shares
This will establish a minimum ofThis will establish a minimum of VVminmin = XV/(S = XV/(S00 + P) where X is the exercise price + P) where X is the exercise price
Example: On Sept. 26, market index is 445.75 and Dec Example: On Sept. 26, market index is 445.75 and Dec 485 put is $38.57. Expiration is Dec. 19. Risk-free rate 485 put is $38.57. Expiration is Dec. 19. Risk-free rate is 2.99 % continuously compounded. Volatility is .155.is 2.99 % continuously compounded. Volatility is .155.
Advanced Equity Derivatives and Strategies (continued)
Portfolio Insurance (continued)Portfolio Insurance (continued) We hold 100,000 units of the index portfolio for V = We hold 100,000 units of the index portfolio for V =
• N = 44,575,000/(445.75 + 38.57) = 92,036N = 44,575,000/(445.75 + 38.57) = 92,036 This guarantees a minimum return of 1.0014This guarantees a minimum return of 1.0014(365/84)(365/84) - -
1 = .0061 per year, which must be below the risk-1 = .0061 per year, which must be below the risk-free rate.free rate.
– Total value = $46,938,360 (> VTotal value = $46,938,360 (> Vminmin))
• Index is 450 at expirationIndex is 450 at expiration
– Sell stock by exercising puts so you have Sell stock by exercising puts so you have 92,036(485) = $44,637,460 (92,036(485) = $44,637,460 ( V Vminmin))
Dynamic hedging: A dynamically adjusted combination of Dynamic hedging: A dynamically adjusted combination of stock and futures or stock and t-bills that can replicate the stock and futures or stock and t-bills that can replicate the stock-put or call-tbill.stock-put or call-tbill. This can be easier because the futures and t-bill markets This can be easier because the futures and t-bill markets
are more liquid than the options marketsare more liquid than the options markets The number of futures required isThe number of futures required is
See Appendix 15A for derivation.See Appendix 15A for derivation.
Advanced Equity Derivatives and Strategies (continued) Equity ForwardsEquity Forwards
Forward contracts on stock or stock indicesForward contracts on stock or stock indices Work precisely like all other forward contracts we have Work precisely like all other forward contracts we have
covered.covered. Break forward is similar to an ordinary call but has no up-Break forward is similar to an ordinary call but has no up-
front cost. At expiration, however, its value can be front cost. At expiration, however, its value can be negative, unlike an ordinary call.negative, unlike an ordinary call. See See Table 15.2, p. 640Table 15.2, p. 640. Note that K = compound . Note that K = compound
future value of call with exercise price F plus future value of call with exercise price F plus compound future value of stock, which is forward compound future value of stock, which is forward price of stock.price of stock.
Advanced Equity Derivatives and Strategies (continued)
Equity Forwards (continued)Equity Forwards (continued) Example using AOL: SExample using AOL: S00 = 125.9375, T = .0959, r = 125.9375, T = .0959, rcc = =
.0446, volatility = .83. .0446, volatility = .83.
• F = 125.9375eF = 125.9375e.0446(.0959).0446(.0959) = 126.48 = 126.48
• Ordinary call with X = 126.48 is worth 12.88. K Ordinary call with X = 126.48 is worth 12.88. K = 126.48 + 12.88e= 126.48 + 12.88e.0446(.0959).0446(.0959) = 139.41 = 139.41
• See See Figure 15.3, p. 641Figure 15.3, p. 641.. Note similarity to forward contract and call option.Note similarity to forward contract and call option.
Advanced Equity Derivatives and Strategies (continued) Equity SwapsEquity Swaps
At least one counterparty pays the return, usually At least one counterparty pays the return, usually capital gains and dividends, on a stock or indexcapital gains and dividends, on a stock or index Based on a given notional principalBased on a given notional principal Party paying the equity return might have to pay Party paying the equity return might have to pay
both sides of the payments because of decreases in both sides of the payments because of decreases in the index.the index.
Advanced Equity Derivatives and Strategies (continued) Equity SwapsEquity Swaps
See See Table 15.3, p. 642Table 15.3, p. 642 for swap with one side paying for swap with one side paying LIBOR. The interest payment isLIBOR. The interest payment is $10,000,000(.09)(days/360)$10,000,000(.09)(days/360)
The equity payment isThe equity payment is $10,000,000 x rate of return on index$10,000,000 x rate of return on index
See See Table 15.4, p. 644Table 15.4, p. 644 for swap with one side paying for swap with one side paying S&P 500 and other paying a foreign stock index. Both S&P 500 and other paying a foreign stock index. Both payments are based on the equity return on payments are based on the equity return on $25,000,000 notional principal.$25,000,000 notional principal.
Advanced Equity Derivatives and Strategies (continued)
Equity Swaps (continued)Equity Swaps (continued) They allow easy changes among asset classesThey allow easy changes among asset classes They can be used to achieve better diversification for They can be used to achieve better diversification for
those whose portfolios are poorly diversified.those whose portfolios are poorly diversified. They are designed to replicate a transaction in a stock They are designed to replicate a transaction in a stock
or index, but there are some differencesor index, but there are some differences They do not bring or give up voting rightsThey do not bring or give up voting rights A position in the stock has a variable notional A position in the stock has a variable notional
principal as the value of the stock changes. The principal as the value of the stock changes. The swap can be adjusted to help compensate for this. swap can be adjusted to help compensate for this.
Advanced Equity Derivatives and Strategies (continued)
Equity WarrantsEquity Warrants Warrants issued by firmWarrants issued by firm Warrants trading on over-the-counter markets and Warrants trading on over-the-counter markets and
American Stock Exchange based on various securities American Stock Exchange based on various securities and indices.and indices.
Many of these are quantos, which pay off based on the Many of these are quantos, which pay off based on the performance of a foreign stock index but payment is performance of a foreign stock index but payment is made in a different currency than the one associated made in a different currency than the one associated with the country of the foreign stock index.with the country of the foreign stock index.
Advanced Equity Derivatives and Strategies (continued) Equity-Linked DebtEquity-Linked Debt
A bond that usually pays a minimum return plus a percentage of A bond that usually pays a minimum return plus a percentage of any increase in a stock indexany increase in a stock index
Example: One-year zero coupon bond paying 1% interest and 50 Example: One-year zero coupon bond paying 1% interest and 50 percent of any gain on the S&P 500.percent of any gain on the S&P 500. Currently one-year zero coupon bond offers 5 % compounded Currently one-year zero coupon bond offers 5 % compounded
annually. S&P 500 is at 1500 and has a volatility of .12 and a annually. S&P 500 is at 1500 and has a volatility of .12 and a yield of 1.5%.yield of 1.5%.
• If you invest $10 you receive $10(1.01) = $10.10 for sure. 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 The present value of this is 10.10/1.05 = 9.62 (5% is opportunity cost).opportunity cost).
• This amounts to a loss of $0.38.This amounts to a loss of $0.38.
Advanced Equity Derivatives and Strategies (continued)
Equity-Linked Debt (continued)Equity-Linked Debt (continued) Option payoff is $10(.5)Max(0,(SOption payoff is $10(.5)Max(0,(STT - 1500)/1500). - 1500)/1500).
This can be written asThis can be written as
• (5/1500)Max(0,S(5/1500)Max(0,STT - 1500), which is 5/1500th of - 1500), which is 5/1500th of
a European call with exercise price 1500.a European call with exercise price 1500. Plugging values into Black-Scholes model gives call Plugging values into Black-Scholes model gives call
value of $96.81. Multiplying by 5/1500 gives a value of $96.81. Multiplying by 5/1500 gives a value of $0.32. This is less than the amount given value of $0.32. This is less than the amount given up by accepting the lower rate on the bond ($0.38) up by accepting the lower rate on the bond ($0.38) but might be worth it to some investors.but might be worth it to some investors.
Advanced Interest Rate Derivatives Variations of Interest Rate SwapsVariations of Interest Rate Swaps
Basis swaps: each side pays a floating rate.Basis swaps: each side pays a floating rate. Common basis swap: T-bill vs. Eurodollar (TED spread)Common basis swap: T-bill vs. Eurodollar (TED spread) See See Table 15.5, p. 648Table 15.5, p. 648 for example of pricing. for example of pricing.
Index amortizing swap: notional principal changes with Index amortizing swap: notional principal changes with level of interest rates to reflect a rate of prepayment on the level of interest rates to reflect a rate of prepayment on the underlying security.underlying security.
Diff swap: Payoff based on interest rate of a given country Diff swap: Payoff based on interest rate of a given country with payment made in a different currency. Designed to with payment made in a different currency. Designed to speculate or hedge on a foreign interest rate without speculate or hedge on a foreign interest rate without assuming currency risk.assuming currency risk.
Variations of Interest Rate Swaps (continued)Variations of Interest Rate Swaps (continued) Constant Maturity Swaps: One party pays floating Constant Maturity Swaps: One party pays floating
such as LIBOR and another pays floating based on such as LIBOR and another pays floating based on return with maturity longer than settlement period. return with maturity longer than settlement period. Common rate is Constant Maturity Treasury (CMT), Common rate is Constant Maturity Treasury (CMT), which is rate on U. S. Treasury note with a fixed which is rate on U. S. Treasury note with a fixed maturity. For example, 5-year CMT is the rate on a maturity. For example, 5-year CMT is the rate on a five-year Treasury note. Although the maturity of a 5-five-year Treasury note. Although the maturity of a 5-year note decreases through time, new 5-year notes or year note decreases through time, new 5-year notes or interpolated values of other notes provide the CMT interpolated values of other notes provide the CMT rate.rate.
Definition: an intermediate term debt security issued by Definition: an intermediate term debt security issued by corporation with good credit rating in which the coupon is corporation with good credit rating in which the coupon is altered by the use of a derivative. Examples:altered by the use of a derivative. Examples: Floating coupon indexed to the CMT rate (e.g., 1.5 times Floating coupon indexed to the CMT rate (e.g., 1.5 times
the CMT rate).the CMT rate). Range floater, which pays interest only if a reference rate Range floater, which pays interest only if a reference rate
(e.g., LIBOR) stays within a given range over a period of (e.g., LIBOR) stays within a given range over a period of time. If rate stays within range, coupon will be higher than time. If rate stays within range, coupon will be higher than otherwise.otherwise.
Reverse (inverse) floater, where coupon moves opposite to Reverse (inverse) floater, where coupon moves opposite to interest rates, such as 12 - LIBORinterest rates, such as 12 - LIBOR
(swap) = Fixed rate - LIBOR. Issuer could buy (swap) = Fixed rate - LIBOR. Issuer could buy a cap to pay it LIBOR while it pays the strike a cap to pay it LIBOR while it pays the strike rate if it wanted to make it risk-free.rate if it wanted to make it risk-free.
• Many inverse floaters are extremely volatile due to Many inverse floaters are extremely volatile due to leverage in the rate adjustment formula.leverage in the rate adjustment formula.
Securities constructed by offering claims on a portfolio of Securities constructed by offering claims on a portfolio of mortgages, a process is called securitization.mortgages, a process is called securitization.
Mortgage-backed securities subject to prepayment risk.Mortgage-backed securities subject to prepayment risk. Mortgage pass-throughs and stripsMortgage pass-throughs and strips
Mortgage pass-through: a security in which the holder Mortgage pass-through: a security in which the holder receives the principal and interest payments made on a receives the principal and interest payments made on a portfolio of mortgages.portfolio of mortgages.
Mortgage strip: a claim on either the principal or Mortgage strip: a claim on either the principal or interest on a mortgage pass-through. Called principal interest on a mortgage pass-through. Called principal only (PO) or interest only (IO)only (PO) or interest only (IO)..
Example: Assume a mortgage-backed security Example: Assume a mortgage-backed security representing a single $100,000 mortgage at 9.75 % for representing a single $100,000 mortgage at 9.75 % for 30 years. Assume annual payments for simplicity. 30 years. Assume annual payments for simplicity. See See Table 15.6, p. 654Table 15.6, p. 654 for amortization schedule. for amortization schedule.
Assume a 7 percent discount rate and that the Assume a 7 percent discount rate and that the mortgage is paid off in year 12.mortgage is paid off in year 12.
• Value of IO strip = 9,750(1.07)Value of IO strip = 9,750(1.07)-1-1 + 9,688(1.07) + 9,688(1.07)-2-2 + … + 8,614(1.07)+ … + 8,614(1.07)-12-12 = 74,254. = 74,254.
• Value of PO strip = 637(1.07)Value of PO strip = 637(1.07)-1-1 + 699(1.07) + 699(1.07)-2-2 + + … + (1,773 + 86,574)(1.07)… + (1,773 + 86,574)(1.07)-12-12 = 46,690. = 46,690.
• Value of pass-through = $74,254 + $46,690 = Value of pass-through = $74,254 + $46,690 = $120,944$120,944
Let discount rate drop to 6 % and assume Let discount rate drop to 6 % and assume homeowner pays off two years from now. homeowner pays off two years from now.
• Value of IO = $9,750(1.06)Value of IO = $9,750(1.06)-1-1 + $9,688(1.06) + $9,688(1.06)-2-2 = = $17,820, loss of 76%$17,820, loss of 76%
• Value of PO = $637(1.06)Value of PO = $637(1.06)-1-1 + ($699 + $98,663) + ($699 + $98,663)(1.06)(1.06)-2-2 = $89,033, gain of 91% = $89,033, gain of 91%
• Value of pass-through = $17,820 + $89,034 = Value of pass-through = $17,820 + $89,034 = $106,854, loss of 12%$106,854, loss of 12%
If the discount rate rises to 8% and there is no If the discount rate rises to 8% and there is no change in the payoff date of year 12,change in the payoff date of year 12,
• Value of IO = $9,750(1.08Value of IO = $9,750(1.08)-1)-1 + $9,688(1.08 + $9,688(1.08)-2)-2 + . . . + $8,614(1.08+ . . . + $8,614(1.08)-12)-12 = = $$70,532, a 5% loss70,532, a 5% loss
• Value of PO = $637(1.08Value of PO = $637(1.08)-1)-1 + $699(1.08 + $699(1.08)-2)-2 + . . . + . . . + ($1,773 + $86,574)(1.08)+ ($1,773 + $86,574)(1.08)-12-12 = = $$42,128, a 10% 42,128, a 10% lossloss
• Value of pass-through = $70,532 + $42,128 = Value of pass-through = $70,532 + $42,128 = $112,660, a loss of almost 7%.$112,660, a loss of almost 7%.
If rate goes to 8 % and prepayment moves back to If rate goes to 8 % and prepayment moves back to year 14, year 14,
• Value of IO = $9,750(1.08Value of IO = $9,750(1.08)-1)-1 + $9,688(1.08 + $9,688(1.08)-2)-2 + . . . . + $8,250(1.08+ . . . . + $8,250(1.08)-14)-14 = = $$76,445, a gain of 3%76,445, a gain of 3%
• Value of PO = $637(1.08Value of PO = $637(1.08)-1)-1 + $699(1.08 + $699(1.08)-2)-2 + . . . + . . . + ($2,136 + $82,492)(1.08)+ ($2,136 + $82,492)(1.08)-14-14 = $37,276, a loss = $37,276, a loss of 20%.of 20%.
• Value of pass-through = $76,445 + $37,276 = Value of pass-through = $76,445 + $37,276 = $113,721, a loss of about 6%.$113,721, a loss of about 6%.
Mortgage-backed security values are very volatile.Mortgage-backed security values are very volatile. Collateralized Mortgage Obligations (CMOs)Collateralized Mortgage Obligations (CMOs)
Mortgage-backed security in which payments are Mortgage-backed security in which payments are split into pieces called tranches with different claims split into pieces called tranches with different claims reflecting different risks.reflecting different risks.
Some tranches are paid first, some receive only Some tranches are paid first, some receive only interest and some receive any residual after other interest and some receive any residual after other tranches have been repaid.tranches have been repaid.
(continued)(continued) The different tranches receive interest, principal and The different tranches receive interest, principal and
prepayments according to different priorities.prepayments according to different priorities. Some CMO tranches are extremely volatile and Some CMO tranches are extremely volatile and
others have low volatility.others have low volatility. A CMO is generally a fairly complex security.A CMO is generally a fairly complex security.
Digital and Chooser OptionsDigital and Chooser Options Digital options, sometimes called binary options, are of Digital options, sometimes called binary options, are of
two types.two types. Asset-or-nothing options pay the holder the asset if Asset-or-nothing options pay the holder the asset if
the option expires in the money and nothing the option expires in the money and nothing otherwise.otherwise.
Cash-or-nothing options pay the holder a fixed Cash-or-nothing options pay the holder a fixed amount of cash if the option expires in the money amount of cash if the option expires in the money and nothing otherwise.and nothing otherwise.
Digital and Chooser Options (continued)Digital and Chooser Options (continued) See See Table 15.7, p. 659Table 15.7, p. 659 for example of long cash-or- for example of long cash-or-
nothing and short asset-or-nothing that pays off X nothing and short asset-or-nothing that pays off X dollars if in-the-money at expiration. This dollars if in-the-money at expiration. This combination is equivalent to an ordinary European combination is equivalent to an ordinary European call. Values of options arecall. Values of options are
Digital and Chooser Options (continued)Digital and Chooser Options (continued) Example: Asset-or-nothing option written on S&P 500 Example: Asset-or-nothing option written on S&P 500
Total Return Index, at 1440. Exercise price of 1440. Total Return Index, at 1440. Exercise price of 1440. Risk-free rate is 4.88%, standard deviation is .11 and time Risk-free rate is 4.88%, standard deviation is .11 and time to expiration is .5 years. We obtainto expiration is .5 years. We obtain dd11 = .3526, N(.35) = .6368 = .3526, N(.35) = .6368
Exotic Options (continued) Digital and Chooser Options (continued)Digital and Chooser Options (continued)
Chooser Options: Also known as as-you-like-it options, they Chooser Options: Also known as as-you-like-it options, they enable the investor to decide at a specific time after enable the investor to decide at a specific time after purchasing the option but before expiration that the option will purchasing the option but before expiration that the option will be a call or a put.be a call or a put. Assume that decision must be made at time t < TAssume that decision must be made at time t < T The chooser option is identical to 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)an ordinary put expiring at t with exercise price X(1+r)--
(T-t)(T-t)
Compare and contrast chooser with straddle.Compare and contrast chooser with straddle.
Exotic Options (continued) Digital and Chooser Options (continued)Digital and Chooser Options (continued)
Example: AOL chooser in which choice must be made Example: AOL chooser in which choice must be made in 20 days. Call/put expires in 35 days. Sin 20 days. Call/put expires in 35 days. S00 = 125.9375, = 125.9375,
X = 125, X = 125, = .83, r = .83, rcc = .0446. T = 35/365 = .0959, t = = .0446. T = 35/365 = .0959, t =
20/365 = .0548 so T - t = .0959 - .0548 = .0411. 20/365 = .0548 so T - t = .0959 - .0548 = .0411. Exercise price on put used to price the chooser is Exercise price on put used to price the chooser is 125(1.0456)125(1.0456)-.0411-.0411 = 124.77. = 124.77.
Using Black-Scholes model, put is worth 7.80 and call Using Black-Scholes model, put is worth 7.80 and call is worth 13.21 for a total of 21.01. Straddle is worth is worth 13.21 for a total of 21.01. Straddle is worth 13.21 (call) + 12.09 (put) = 25.30.13.21 (call) + 12.09 (put) = 25.30.
Path-dependent options are options in which the payoff is Path-dependent options are options in which the payoff is determined by the sequence of prices followed by the asset and determined by the sequence of prices followed by the asset and not just by the price of the asset at expiration.not just by the price of the asset at expiration.
We shall price these options using a binomial framework. See We shall price these options using a binomial framework. See Table 15.8, p. 662Table 15.8, p. 662 which shows a three-period problem. Note which shows a three-period problem. Note 8 paths and the average, maximum and minimum prices of 8 paths and the average, maximum and minimum prices of each path are computed.each path are computed.
Note how the probabilities are calculated.Note how the probabilities are calculated. In practice the binomial model is difficult to use for path-In practice the binomial model is difficult to use for path-
dependent options. Monte Carlo simulation (see Appendix dependent options. Monte Carlo simulation (see Appendix 15B) is often used.15B) is often used.
Path-Dependent Options (continued)Path-Dependent Options (continued) Asian option: an option in which the final payoff is Asian option: an option in which the final payoff is
determined by the average price of the asset during the determined by the average price of the asset during the option’s life. Some are average price options because the option’s life. Some are average price options because the average price substitutes for the asset price at expiration. average price substitutes for the asset price at expiration. Others are average strike options because the average price Others are average strike options because the average price substitutes for the exercise price at expiration. Can be calls substitutes for the exercise price at expiration. Can be calls or puts. Useful for hedging or speculating when the average or puts. Useful for hedging or speculating when the average is acceptable. Also useful for cases where market can be is acceptable. Also useful for cases where market can be manipulated.manipulated.
See See Table 15.9, p. 663Table 15.9, p. 663 for example of pricing Asian options. for example of pricing Asian options.
Path-Dependent Options (continued)Path-Dependent Options (continued) Lookback option: Also called a no-regrets option, it Lookback option: Also called a no-regrets option, it
permits purchase of the asset at its lowest price during permits purchase of the asset at its lowest price during the option’s life or sale of the asset at its highest price the option’s life or sale of the asset at its highest price during the option’s life. during the option’s life.
Lookback options (continued):Lookback options (continued): Four different types.Four different types.
• lookback call: exercise price set at minimum price during lookback call: exercise price set at minimum price during option’s lifeoption’s life
• lookback put: exercise price set at maximum price during lookback put: exercise price set at maximum price during option’s life.option’s life.
• fixed-strike lookback call: payoff based on maximum price fixed-strike lookback call: payoff based on maximum price during option’s life (instead of final price) compared to fixed during option’s life (instead of final price) compared to fixed strikestrike
• fixed-strike lookback put: payoff based on minimum price fixed-strike lookback put: payoff based on minimum price during option’s life (instead of final price) compared to fixed during option’s life (instead of final price) compared to fixed strikestrike
Barrier Options: Options that either terminate early if the Barrier Options: Options that either terminate early if the asset price hits a certain level, called the barrier, or activate asset price hits a certain level, called the barrier, or activate only if the asset price hits the barrier. The former are called only if the asset price hits the barrier. The former are called knock-out options (or simply out-options)knock-out options (or simply out-options) and the latter are and the latter are called knock-in optionscalled knock-in options (or simply in-options). If the barrier (or simply in-options). If the barrier is above the current price, it is called an up-option. If the 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. barrier is below the current price, it is called a down-option. See See Table 15.11, p. 667Table 15.11, p. 667 for example of pricing. for example of pricing. Barrier options are normally cheaper than ordinary options Barrier options are normally cheaper than ordinary options
because they provide payoffs for fewer outcomes than because they provide payoffs for fewer outcomes than ordinary options.ordinary options.
Appendix 15A: Derivation of the Dynamic Hedge Ratio for Portfolio Insurance (continued) Stock-Futures Dynamic Hedge (continued)Stock-Futures Dynamic Hedge (continued)
A portfolio of NA portfolio of NSS shares and N shares and Nff futures is worth today futures is worth today
V = NV = NSSS + NS + NffVVff
where Vwhere Vff is value of futures, which starts at zero. It is value of futures, which starts at zero. It follows that Nfollows that NSS = V/S = V/S
Set change in portfolio value for small change in S toSet change in portfolio value for small change in S to
Appendix 15A: Derivation of the Dynamic Hedge Ratio for Portfolio Insurance (continued) Stock-Futures Dynamic Hedge (continued)Stock-Futures Dynamic Hedge (continued)
Assuming no dividends, the futures price isAssuming no dividends, the futures price is
Appendix 15A: Derivation of the Dynamic Hedge Ratio for Portfolio Insurance (continued) Stock-Futures Dynamic Hedge (continued)Stock-Futures Dynamic Hedge (continued)
After substituting, setting the two partial derivatives of After substituting, setting the two partial derivatives of V with respect to S equal to other, recognizing that 1 + V with respect to S equal to other, recognizing that 1 + P/ S is C/ S and N(d N(d11) is ) is C/ S, we obtain the we obtain the
number of futures contracts asnumber of futures contracts as
Appendix 15A: Derivation of the Dynamic Hedge Ratio for Portfolio Insurance (continued) Stock-Tbill Dynamic Hedge (continued) Stock-Tbill Dynamic Hedge (continued)
The t-bill price is not sensitive to the stock price. The t-bill price is not sensitive to the stock price. Setting the sensitivity of the stock-tbill portfolio to that Setting the sensitivity of the stock-tbill portfolio to that of the stock-futures givesof the stock-futures gives
This is the number of shares of stock to hold with t-This is the number of shares of stock to hold with t-bills to replicate the stock and put.bills to replicate the stock and put.
A method of using random numbers designed to simulate A method of using random numbers designed to simulate the random observations of prices of an asset. A simulated the random observations of prices of an asset. A simulated series of asset prices at expiration is converted to an series of asset prices at expiration is converted to an equivalent series of option prices at expiration.equivalent series of option prices at expiration.
Then the current option price is the discounted average of Then the current option price is the discounted average of the option prices obtained at expiration from the the option prices obtained at expiration from the simulation.simulation.
Random prices can be simulated by drawing a standard Random prices can be simulated by drawing a standard normal random variable, normal random variable, , and inserting into the formula
where where t is the length of the time interval over which t is the length of the time interval over which the stock price change occurs. the stock price change occurs.
Note: simulating a standard normal random variable Note: simulating a standard normal random variable can be done approximately as the sum of twelve unit can be done approximately as the sum of twelve unit uniform random numbers (in Excel, “=Rand( )”) minus uniform random numbers (in Excel, “=Rand( )”) minus 6.0.6.0.
Each simulated stock price is treated as the stock price at Each simulated stock price is treated as the stock price at expiration; thus, expiration; thus, t is the maturity in years of the option. For each simulated stock price, compute the option For each simulated stock price, compute the option
price at expiration using the intrinsic value.price at expiration using the intrinsic value.
Take the average of all of the option prices at Take the average of all of the option prices at expiration.expiration.
Discount the average over the life of the option at the Discount the average over the life of the option at the risk-free rate. This is the estimate of the current option risk-free rate. This is the estimate of the current option price.price.
This will probably require at least 50,000 random numbers This will probably require at least 50,000 random numbers for a standard option and more for exotic and complex for a standard option and more for exotic and complex options and derivatives.options and derivatives.
