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Chapter 14: Advanced Derivatives and Strategies

Chapter 14: Advanced Derivatives and Strategies. We look at everything. We don’t get scared because of the complexity involved. But we examine it to death. Arvind Sodhani, treasury, Intel Business Week , October 21, 1994, p. 95. Important Concepts in Chapter 14.

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Chapter 14: Advanced Derivatives and Strategies

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  1. Chapter 14: Advanced Derivatives and Strategies We look at everything. We don’t get scared because of the complexity involved. But we examine it to death. Arvind Sodhani, treasury, Intel Business Week, October 21, 1994, p. 95 An Introduction to Derivatives and Risk Management, 6th ed.

  2. Important Concepts in Chapter 14 • The concept of portfolio insurance and its execution using puts, calls, futures and t-bills • New and advanced derivatives and strategies such as equity forwards, warrants, equity-linked debt, structured notes, and mortgage securities • Exotic options such as digital options, chooser options, Asian options, lookback options, and barrier options • Derivatives on electricity and weather An Introduction to Derivatives and Risk Management, 6th ed.

  3. Advanced Equity Derivatives and Strategies • Portfolio Insurance • We can insure a portfolio by holding one put for each share of stock. For a portfolio worth V, we should hold • N = V/(S0 + P) puts and shares • This will establish a minimum of • Vmin = XV/(S0 + P) where X is the exercise price • Example: On Sept. 26, market index is 445.75 and Dec 485 put is $38.57. Expiration is Dec. 19. Risk-free rate is 2.99 % continuously compounded. Volatility is .155. An Introduction to Derivatives and Risk Management, 6th ed.

  4. Advanced Equity Derivatives and Strategies (continued) • Portfolio Insurance (continued) • We hold 100,000 units of the index portfolio for V = $44,575,000. We have • Vmin = (485)(44,575,000)/(445.75 + 38.57) = 44,637,585 • N = 44,575,000/(445.75 + 38.57) = 92,036 • This guarantees a minimum return of 1.0014(365/84) - 1 = .0061 per year, which must be below the risk-free rate. An Introduction to Derivatives and Risk Management, 6th ed.

  5. Advanced Equity Derivatives and Strategies (continued) • Portfolio Insurance (continued) • Outcomes • Index is 510 at expiration • 92,036 shares worth 510 = $46,938,360 • 92,036 puts worth $0 = $0 • Total value = $46,938,360 (> Vmin) • Index is 450 at expiration • Sell stock by exercising puts so you have 92,036(485) = $44,637,460 (» Vmin) An Introduction to Derivatives and Risk Management, 6th ed.

  6. Advanced Equity Derivatives and Strategies (continued) • Portfolio Insurance (continued) • See Figure 14.1, p. 503. • If calls and t-bills used, • NB = Vmin/BT (number of bills) • NC = V/(S0 + P) (number of calls) • So NB = 44,637,585/100 = 446,376 • NC = 92,036 An Introduction to Derivatives and Risk Management, 6th ed.

  7. Advanced Equity Derivatives and Strategies (continued) • Portfolio Insurance (continued) • Outcomes • Index is 510 at expiration • Bills worth $44,637,600 • 92,036 calls worth $25 = $2,300,900 • Total value = $46,938,500 (> Vmin) • Index is 450 at expiration • Bills worth $44,637,600 • 92,036 calls worth $0 • Total value = $44,637,600 (» Vmin) • See Figure 14.2, p. 505. An Introduction to Derivatives and Risk Management, 6th ed.

  8. Advanced Equity Derivatives and Strategies (continued) • Portfolio Insurance (continued) • Dynamic hedging: A dynamically adjusted combination of stock and futures or stock and t-bills that can replicate the stock-put or call-tbill. • This can be easier because the futures and t-bill markets are more liquid than the options markets • The number of futures required is • See Appendix 14.A for derivation. An Introduction to Derivatives and Risk Management, 6th ed.

  9. Advanced Equity Derivatives and Strategies (continued) • Portfolio Insurance (continued) • Alternatively, use stock and t-bills (see Appendix 14.A again for derivation). • See Table 14.1, p. 507for example of dynamic hedge An Introduction to Derivatives and Risk Management, 6th ed.

  10. Advanced Equity Derivatives and Strategies (continued) • Equity Forwards • Forward contracts on stock or stock indices • Precisely like all other forward contracts we have covered. • 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. • See Table 14.2, p. 510. Note that K = compound future value of call with exercise price F plus compound future value of stock, which is forward price of stock. An Introduction to Derivatives and Risk Management, 6th ed.

  11. Advanced Equity Derivatives and Strategies (continued) • Equity Forwards (continued) • Example using AOL: S0 = 125.9375, T = .0959, rc = .0446, volatility = .83. • F = 125.9375e.0446(.0959) = 126.48 • Ordinary call with X = 126.48 is worth 12.88. K = 126.48 + 12.88e.0446(.0959) = 139.41 • See Figure 14.3, p. 511. • Note similarity to forward contract and call option. • To determine the value of a break forward at time t during its life, we simply value it as a call and a loan: An Introduction to Derivatives and Risk Management, 6th ed.

  12. Advanced Equity Derivatives and Strategies (continued) • Equity Forwards (continued) • For example, 15 days later, AOL is at 115.75, T – t = 20/365 = 0.0548, and the other inputs are unchanged. We obtain • A more general version of a break forward is a pay-later option. In this case, the buyer simply borrows the premium and has to pay it back at expiration. This option is just an ordinary call plus a loan of the call premium Ce(S0,T,X). At expiration, the buyer decides whether to exercise the call and in either case pays back An Introduction to Derivatives and Risk Management, 6th ed.

  13. Advanced Equity Derivatives and Strategies (continued) • Equity Warrants • Warrants issued by firm • Warrants trading on over-the-counter markets and American Stock Exchange based on various securities and indices. • 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. An Introduction to Derivatives and Risk Management, 6th ed.

  14. Advanced Equity Derivatives and Strategies (continued) • Equity-Linked Debt • A bond that usually pays a minimum return plus a percentage of any increase in a stock index • Example: One-year zero coupon bond paying 1% interest and 50 percent of any gain on the S&P 500. • Currently one-year zero coupon bond offers 5 % compounded annually. S&P 500 is at 1500 with 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). • This amounts to a loss of $0.38. An Introduction to Derivatives and Risk Management, 6th ed.

  15. Advanced Equity Derivatives and Strategies (continued) • Equity-Linked Debt (continued) • Option payoff is $10(.5)Max(0,(ST - 1500)/1500). This can be written as • (5/1500)Max(0,ST - 1500), which is 5/1500th of a European call with exercise price 1500. • Plugging values into Black-Scholes model gives call value of $96.81. 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 worthwhile to some investors. An Introduction to Derivatives and Risk Management, 6th ed.

  16. Advanced Interest Rate Derivatives • Structured Notes • Definition: an intermediate term debt security issued by a corporation with a good credit rating in which the coupon is altered by the use of a derivative. Examples: • Floating coupon indexed usually to LIBOR or the CMT rate (e.g., 1.5 times the rate). • 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. • Reverse (inverse) floater, where coupon moves opposite to interest rates, such as 12 - LIBOR An Introduction to Derivatives and Risk Management, 6th ed.

  17. Advanced Interest Rate Derivatives (continued) • Structured Notes (continued) • Example: 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. An Introduction to Derivatives and Risk Management, 6th ed.

  18. Advanced Interest Rate Derivatives (continued) • Mortgage-Backed Securities • Securities constructed by offering claims on a portfolio of mortgages, a process called securitization. • Mortgage-backed securities are subject to prepayment risk. • Mortgage pass-throughs and strips • Mortgage pass-through: a security in which the holder receives the principal and interest payments made on a portfolio of mortgages. • Mortgage strip: a claim on either the principal or interest on a mortgage pass-through. Called principal only (PO) or interest only (IO). An Introduction to Derivatives and Risk Management, 6th ed.

  19. Advanced Interest Rate Derivatives (continued) • Mortgage-Backed Securities (continued) • Example: Assume a mortgage-backed security representing a single $100,000 mortgage at 9.75 % for 30 years. Assume annual payments for simplicity. • See Table 14.3, p. 516 for amortization schedule. Annual payment would be • $100,000/[(1-(1.0975)-30)/.0975] = $10,387. An Introduction to Derivatives and Risk Management, 6th ed.

  20. Advanced Interest Rate Derivatives (continued) • Mortgage-Backed Securities (continued) • Assume a 7 percent discount rate and that the mortgage is paid off in year 12. • Value of IO strip = 9,750(1.07)-1 + 9,688(1.07)-2 + … + 8,614(1.07)-12 = 74,254. • Value of PO strip = 637(1.07)-1 + 699(1.07)-2 + … + (1,773 + 86,574)(1.07)-12 = 46,690. • Value of pass-through = $74,254 + $46,690 = $120,944 An Introduction to Derivatives and Risk Management, 6th ed.

  21. Advanced Interest Rate Derivatives (continued) • Mortgage-Backed Securities (continued) • 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 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% An Introduction to Derivatives and Risk Management, 6th ed.

  22. Advanced Interest Rate Derivatives (continued) • Mortgage-Backed Securities (continued) • 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)-2 + . . . + $8,614(1.08)-12 = $70,532, a 5% loss • Value of PO = $637(1.08)-1 + $699(1.08)-2 + . . . + ($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%. An Introduction to Derivatives and Risk Management, 6th ed.

  23. Advanced Interest Rate Derivatives (continued) • Mortgage-Backed Securities (continued) • If rate goes to 8 % and prepayment moves back to year 14, • Value of IO = $9,750(1.08)-1 + $9,688(1.08)-2 + . . . . + $8,250(1.08)-14 = $76,445, a gain of 3% • Value of PO = $637(1.08)-1 + $699(1.08)-2 + . . . + ($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%. An Introduction to Derivatives and Risk Management, 6th ed.

  24. Advanced Interest Rate Derivatives (continued) • Mortgage-Backed Securities (continued) • Mortgage-backed security values are typically very volatile. • Collateralized Mortgage Obligations (CMOs) • Mortgage-backed security in which payments are split into pieces called tranches with different claims reflecting different risks. • Some tranches are paid first, some receive only interest and some receive any residual after other tranches have been repaid. An Introduction to Derivatives and Risk Management, 6th ed.

  25. Advanced Interest Rate Derivatives (continued) • Mortgage-Backed Securities (continued) • Collateralized Mortgage Obligations (CMOs) (continued) • The different tranches receive interest, principal and prepayments according to different priorities. • Some CMO tranches are extremely volatile and others have low volatility. • A CMO is generally a fairly complex security. An Introduction to Derivatives and Risk Management, 6th ed.

  26. Exotic Options • Digital and Chooser Options • Digital options, sometimes called binary options, are of two types: • Asset-or-nothing options pay the holder the asset if the option expires in the money and nothing otherwise. • Cash-or-nothing options pay the holder a fixed amount of cash (usually $1) if the option expires in the money and nothing otherwise. An Introduction to Derivatives and Risk Management, 6th ed.

  27. Exotic Options (continued) • Digital and Chooser Options (continued) • See Table 14.4, p. 520 for example of a portfolio long cash-or-nothings and short X asset-or-nothings. This combination is equivalent to an ordinary European call. The values of the options are An Introduction to Derivatives and Risk Management, 6th ed.

  28. Exotic Options (continued) • Digital and Chooser Options (continued) • Example: Asset-or-nothing option written on S&P 500 Total Return Index, at 1440. Exercise price of 1440. Risk-free rate is 4.88%, standard deviation is .11 and time to expiration is 0.5 years. We obtain • d1 = .3526, N(.35) = .6368 • Oaon = 1440(.6368) = 917 • For 1,440 cash-or-nothing options, • d2 = .2748, N(.27) = .6064 • (1,440)Ocon = 1440e-.0488(.5)(.6064) = 852.17. An Introduction to Derivatives and Risk Management, 6th ed.

  29. Exotic Options (continued) • Digital and Chooser Options (continued) • A variation of the previously covered pay-later option is the contingent-pay option. Here the premium is paid at expiration but only if the option expires in-the-money. Table 14.5, p. 521 shows that this option is a combination of a standard option and Ccp cash-or-nothing calls. The value must be zero today so An Introduction to Derivatives and Risk Management, 6th ed.

  30. Exotic Options (continued) • Digital and Chooser Options (continued) • Solving for Ccp gives • For the example we have been using • Now move forward two months where St = 1440 and T-t = 4/6 = 0.333. The value is An Introduction to Derivatives and Risk Management, 6th ed.

  31. Exotic Options (continued) • Digital and Chooser Options (continued) • Chooser Options: Also called 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. • Assume the decision must be made at time t < T • The chooser option is identical to • an ordinary call expiring at T with exercise price X plus • an ordinary put expiring at t with exercise price X(1+r)-(T-t) • Compare and contrast chooser with straddle. An Introduction to Derivatives and Risk Management, 6th ed.

  32. Exotic Options (continued) • Digital and Chooser Options (continued) • Example: AOL chooser in which choice must be made in 20 days. Call/put expires in 35 days. S0 = 125.9375, X = 125,  = .83, rc = .0446. T = 35/365 = .0959, t = 20/365 = .0548 so T - t = .0959 - .0548 = .0411. Exercise price on put used to price the chooser is 125(1.0456)-.0411 = 124.77. • Using Black-Scholes model, put is worth 7.80 and call is worth 13.21 for a total of 21.01. Straddle is worth 13.21 (call) + 12.09 (put) = 25.30. An Introduction to Derivatives and Risk Management, 6th ed.

  33. Exotic Options (continued) • Path-Dependent Options • 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. • We shall price these options using a binomial framework. See Table 14.6, p. 523 which shows a three-period problem. Note eight paths, and the average, maximum, and minimum prices of each path are computed. • Note how the probabilities are calculated. • In practice the binomial model is difficult to use for path-dependent options. Monte Carlo simulation (see Appendix 15.B) is often used. An Introduction to Derivatives and Risk Management, 6th ed.

  34. Exotic Options (continued) • Path-Dependent Options (continued) • 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 as a measure of the underlying risk. Also useful for cases where market can be manipulated. • See Table 14.7, p. 525 for example of pricing Asian options. An Introduction to Derivatives and Risk Management, 6th ed.

  35. Exotic Options (continued) • Path-Dependent Options (continued) • 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. An Introduction to Derivatives and Risk Management, 6th ed.

  36. Exotic Options (continued) • Path-Dependent Options (continued) • Lookback options (continued): • Four different types. • lookback call: exercise price set at minimum 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 strike • fixed-strike lookback put: payoff based on minimum price during option’s life (instead of final price) compared to fixed strike An Introduction to Derivatives and Risk Management, 6th ed.

  37. Exotic Options (continued) • Path-Dependent Options (continued) • Lookback options (continued): • See Table 14.8, p. 526 for example of pricing lookback options. An Introduction to Derivatives and Risk Management, 6th ed.

  38. Exotic Options (continued) • Path-Dependent Options (continued) • 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. • See Table 14.9, p. 528 for example of pricing. • Barrier options are normally cheaper than ordinary options because they provide payoffs for fewer outcomes than ordinary options. An Introduction to Derivatives and Risk Management, 6th ed.

  39. Exotic Options (continued) • Path-Dependent Options (continued) • Other Exotic Options: • compound and installment options • multi-asset options, exchange options, min-max options (rainbow options), alternative options, outperformance options • shout, cliquet and lock-in options • contingent premium, pay-later and deferred strike options • forward-start and tandem options An Introduction to Derivatives and Risk Management, 6th ed.

  40. Some Important New Derivatives • Electricity Derivatives • Electricity is a non-storable asset • These derivatives are difficult to price • Weather Derivatives • Measures of weather activity • Heating degree days and cooling degree days • Quantity of rain or snow • Financial loss caused by weather • Pricing is difficult but not impossible; a lot of data are available on weather An Introduction to Derivatives and Risk Management, 6th ed.

  41. Summary An Introduction to Derivatives and Risk Management, 6th ed.

  42. Appendix 14A.: Derivation of the Dynamic Hedge Ratio for Portfolio Insurance • Stock-Futures Dynamic Hedge • Portfolio of N shares and N puts is worth • V = N(S + P) • So N = V/(S+P). • Change in portfolio value for a small change in stock price is An Introduction to Derivatives and Risk Management, 6th ed.

  43. Appendix 14.A: Derivation of the Dynamic Hedge Ratio for Portfolio Insurance (continued) • Stock-Futures Dynamic Hedge (continued) • A portfolio of NS shares and Nf futures is worth today V = NSS + NfVf • where Vf is value of futures, which starts at zero. It follows that NS = V/S • Set change in portfolio value for small change in S to An Introduction to Derivatives and Risk Management, 6th ed.

  44. Appendix 14.A: Derivation of the Dynamic Hedge Ratio for Portfolio Insurance (continued) • Stock-Futures Dynamic Hedge (continued) • Assuming no dividends, the futures price is • So An Introduction to Derivatives and Risk Management, 6th ed.

  45. Appendix 14.A: Derivation of the Dynamic Hedge Ratio for Portfolio Insurance (continued) • Stock-Futures Dynamic Hedge (continued) • 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(d1) is ¶C/ ¶S, we obtain the number of futures contracts as An Introduction to Derivatives and Risk Management, 6th ed.

  46. Appendix 14.A: Derivation of the Dynamic Hedge Ratio for Portfolio Insurance (continued) • Stock-Tbill Dynamic Hedge • A portfolio of stock and tbills is worth • Its sensitivity to a change in S is An Introduction to Derivatives and Risk Management, 6th ed.

  47. Appendix 14.A: Derivation of the Dynamic Hedge Ratio for Portfolio Insurance (continued) • Stock-Tbill Dynamic Hedge (continued) • 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 portfolio gives • This is the number of shares of stock to hold with t-bills to replicate the stock and put. An Introduction to Derivatives and Risk Management, 6th ed.

  48. Appendix 14.B: Monte Carlo Simulation • A method of using random numbers designed to simulate the random observations of prices of an asset. The simulated series of asset prices at expiration is then converted to an equivalent series of option prices at expiration. • 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, e, and inserting into the formula An Introduction to Derivatives and Risk Management, 6th ed.

  49. Appendix 14.B: Monte Carlo Simulation (continued) • where Dt is the length of the time interval over which the stock price change occurs. • 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, Dt is the maturity in years of the option. • For each simulated stock price, compute the option price at expiration using the intrinsic value. An Introduction to Derivatives and Risk Management, 6th ed.

  50. Appendix 14.B: Monte Carlo Simulation (continued) • Take the average of all of the option prices at expiration. • Discount the average over the life of the option at the risk-free rate. This is the estimate of the current option price. • This procedure will probably require at least 50,000 random numbers for a standard option and more for exotic and complex options and derivatives. An Introduction to Derivatives and Risk Management, 6th ed.

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