(BQ) Part 2 book Derivatives markets has contents: Financial Engineering and security design, corporate applications, the lognormal distribution, monte carlo valuation, brownian motion and itô’s lemma, the black scholes merton equation, interest rate and bond derivatives,... forwards and futures,...
PART Financial Engineering and Applications I n the preceding chapters we have focused on forwards, swaps, and options (including exotic options) as stand-alone financial claims In the next three chapters we will see that these claims can be used as financial building blocks to create new claims, and also see that derivatives pricing theory can help us understand corporate financial policy and the valuation of investment projects Specifically, in Chapter 15 we see how it is possible to construct and price bonds that make payments that, instead of being denominated in cash, are denominated in stocks, commodities, and different currencies Such bonds can be structured to contain embedded options We also see how such claims can be used for risk management and how their issuance can be motivated by tax and regulatory considerations Chapter 16 examines some corporate contexts in which derivatives are important, including corporate financial policy, compensation options, and mergers Chapter 17 examines real options, in which the insights from derivatives pricing are used to value investment projects 15 F Financial Engineering and Security Design orwards, calls, puts, and common exotic options can be added to bonds or otherwise combined to create new securities For example, many traded securities are effectively bonds with embedded options Individual derivatives thus become building blocks—ingredients used to construct new kinds of financial products In this chapter we will see how to assemble the ingredients to create new products The process of constructing new instruments from these building blocks is called financial engineering 15.1 THE MODIGLIANI-MILLER THEOREM The starting point for any discussion of modern financial engineering is the analysis of Franco Modigliani and Merton Miller (Modigliani and Miller, 1958) Before their work, financial analysts would puzzle over how to compare the values of firms with similar operating characteristics but different financial characteristics Modigliani and Miller realized that different financing decisions (for example, the choice of the firm’s debt-to-equity ratio) may carve up the firm’s cash flows in different ways, but if the total cash flows paid to all claimants is unchanged, the total value of all claims would remain the same They showed that if firms differing only in financial policy differed in market value, profitable arbitrage would exist Using their famous analogy, the price of whole milk should equal the total prices of the skim milk and butterfat that can be derived from that milk.1 The Modigliani-Miller analysis requires numerous assumptions: For example, there are no taxes, no transaction costs, no bankruptcy costs, and no private information Nevertheless, the basic Modigliani-Miller result provided clarity for a confusing issue, and it created a starting point for thinking about the effects of taxes, transaction costs, and the like, revolutionizing finance All of the no-arbitrage pricing arguments we have been using embody the ModiglianiMiller spirit For example, we saw in Chapter that we could synthetically create a forward contract using options, a call option using a forward contract, bonds, and a put, and so forth In Chapter 10 we saw that an option could also be synthetically created from a position in the stock and borrowing or lending If prices of actual claims differ from their synthetic equivalents, arbitrage is possible Standard corporate finance texts offer a more detailed discussion of the Modigliani-Miller results The original paper (Modigliani and Miller, 1958) is a classic 437 438 Chapter 15 Financial Engineering and Security Design Financial engineering is an application of the Modigliani-Miller idea We can combine claims such as stocks, bonds, forwards, and options and assemble them to create new claims The price for this new security is the sum of the pieces combined to create it When we create a new instrument in this fashion, as in the Modigliani-Miller analysis, value is neither created nor destroyed Thus, financial engineering has no value in a pure Modigliani-Miller world However, in real life, the new instrument may have different tax, regulatory, or accounting characteristics, or may provide a way for the issuer or buyer to obtain a particular payoff at lower transaction costs than the alternatives Financial engineering thus provides a way to create instruments that meet specific needs of investors and issuers To illustrate this, Box 15.1 discusses the application of financial engineering to satisfy religious restrictions As a starting point, you can ask the following questions when you confront new financial instruments: What is the payoff of the instrument? Is it possible to synthetically create the same payoffs using some combination of assets, bonds, and options? Who might issue or buy such an instrument? What problem does the instrument solve? We begin by discussing structured notes without and with options We then turn to examples of engineered products 15.2 STRUCTURED NOTES WITHOUT OPTIONS An ordinary note or bond has interest and maturity payments that are fixed at the time of issue.2 A structured note has interest or maturity payments that are not fixed in dollars but are contingent in some way Structured notes can make payments based on stock prices, interest rates, commodities, or currencies, and the payoffs may or may not contain options We first discuss bonds that make a single payment and then bonds that make multiple payments (such as coupon bonds), all without options In the next section we will introduce structures with options Single Payment Bonds A single payment bond is a financial instrument for which you pay today and that makes a single payment at time T The payment could be $1, a share of stock, an ounce of gold, or a bushel of corn A single payment bond is equivalent to a prepaid forward contract on the asset or commodity Because the price of a single payment bond is the value today of a future payment, it also is equivalent to a discount factor—a value that translates future payments into a value today This interpretation will play an important role in our discussion The most basic financial instrument is a zero-coupon bond that pays $1 at maturity As in Chapter 7, let rs (t , T ) represent the annual continuously compounded interest rate We will use the terms “bond” and “note” interchangeably, though in common usage a note has a medium time to maturity (2–10 years) and a bond has a longer maturity In earlier chapters we referred to this instrument as a zero-coupon bond In this chapter, “zero-coupon bond” will mean a single payment bond that pays in cash 15.2 Structured Notes without Options BOX 15.1: Islamic Finance* Shariah, the religious law of Islam, places re- Murabaha The bank purchases the home and resells it to the client at a markup, which can include financing cost strictions on financial transactions Four verses in the Qur’an, the holy book of Islam, prohibit the payment of interest By scholarly interpretation, the Qur’an also requires that business transactions must both pertain to real assets and have an ethical purpose These restrictions and requirements have given rise to a practice known as Islamic Finance The primary elements of Islamic Finance are No interest or usury No gambling No speculation Strive for fair and just business practices 439 Avoid prohibited goods and services (alcohol, weapons, hedonism) Obviously, standard financial practices in major financial markets may run afoul of one or more of these elements There is no religious objection to making a profit on a real asset, but interest as profit on money is prohibited Practitioners in Islamic Finance face the challenge of constructing transactions that serve a genuine business purpose and adhere to the tenets above The process of constructing such transactions is a form of financial engineering As an example, consider a residential mortgage There are at least three ways an owner can borrow money to finance a purchase: Musharaka The buyer and bank enter into a joint venture where the bank owns a percentage of the house, and as the client makes payments the bank’s ownership percentage declines Ijara The buyer can lease to own In all of these transactions, the bank owns the property for some period of time and can thus attribute gains to profit from ownership rather than as a return to the lending of money Islamic Finance also maintains a general objection to speculative uses of derivatives, stemming both from concerns about speculation and also from the idea that derivatives are removed from the primary underlying transaction, not directly furthering its real economic purpose Derivatives are acceptable, however, as a way to manage risk Calls are effected by the buyer making a down payment with the right to walk away, and puts with a third party guarantee against loss Derivatives on gold, silver, and currency are prohibited See Jobst and Sol´e (2012) for a comprehensive discussion of derivatives in Islamic Finance *I am grateful to Karen Hunt-Ahmed for her assistance with this box prevailing at time s ≤ t, for a loan from time t to time T Similarly, the price of a zerocoupon bond purchased at time t, maturing at time T , and quoted at time s is Ps (t , T ) Thus, we have Ps (t , T ) = e−rs (t , T )(T −t) When there is no risk of misunderstanding, we will assume that the interest rate is quoted at time t0 = 0, and the bond is also purchased then We will denote the rate r0(0, T ) = r(T ), or just r, and the corresponding bond price PT So we will write PT = e−r(T )T 440 Chapter 15 Financial Engineering and Security Design PT is the time price of a T -period zero-coupon bond We can describe PT as a bond price, as a discount factor and as the prepaid forward price for $1 delivered at time T : Zero-coupon bond price = Discount factor for $1 = Prepaid forward price for $1 A single payment bond that pays a unit of an asset or commodity is equivalent to a prepaid forward contract for that asset or commodity Thus, the time price of the bond is F0,P T It is helpful to keep in mind the link between the prepaid forward price, the forward price, and the spot price of the asset or commodity Because the only difference between a forward contract and a prepaid forward is the timing of the payment for the underlying asset, the prepaid forward price is the present value of the forward price, discounted at the risk-free rate: F0,P T = e−rT F0, T (15.1) The difference between the current spot price, S0, and the prepaid forward price can be expressed as a yield: F0,P T = S0e−δT (15.2) If S0 is the price of a financial asset, then δ represents a payment such as dividends or interest We saw in Chapter that if S0 is the price of a commodity, δ is the commodity lease rate Zero-coupon equity-linked bond From equation (15.2), the value of a single-payment bond that pays a share of stock at time T is F0,P T = S0e−δT Example 15.1 Suppose that XYZ stock has a price of $100 and pays no dividends, and that the annual continuously compounded interest rate is 6% In the absence of dividends, the prepaid forward price equals the stock price Thus, we would pay $100 to receive the stock in years We define an equity-linked bond as selling for par value if the bond price equals the maturity payment of the bond The bond in Example 15.1 is at par because the bond pays one share of stock at maturity and the price of the bond equals the price of one share of stock today If the stock pays dividends and the bond makes no coupon payments, the bond will sell at less than par because you are not entitled to receive dividends Example 15.2 Suppose the price of XYZ stock is $100, the quarterly dividend is $1.20, and the annual continuously compounded interest rate is 6% (the quarterly interest rate is therefore 1.5%) Using equation (5.3), the price of an equity-linked bond that pays one share in years is 20 $100 − $1.20e−0.015×i = $79.42 i=1 Zero-coupon commodity-linked bond If a bond pays a unit of a commodity for which there are traded futures contracts, it is possible to value the bond by using the futures price 15.2 Structured Notes without Options Example 15.3 Suppose the spot price of gold is S0 = $400/oz, the 3-year forward price is F0, = $455/oz, and the 3-year continuously compounded interest rate is 6.25% Then using equation (15.1), a zero-coupon note paying ounce of gold in years would sell for F0,P T = $455e−0.0625×3 = $377.208 The lease rate in this case is δl = r − 1 ln(F0, T /S0) = 0.0625 − ln(455/400) = 0.019556 T An alternative way to compute the present value uses equation (15.2): F0,P = S0e−δl = 400e−.019556×3 = 377.208 This amount is less than the spot price of $400 because the lease rate is positive Zero-Coupon Currency-Linked Bond From equation (5.17), a bond that pays one unit of foreign currency at time T has a time zero value of F0,P T = x0e−rf T where x0 is the time exchange rate denominated in units of the home currency per unit of the foreign currency, and rf is the foreign interest rate With a currency-linked bond, the foreign interest rate plays the same role as the dividend yield for stocks and the lease rate for commodities Multiple Payment Bonds You can easily construct and value multiple payment bonds as a portfolio of single payment bonds A common design question with multiple payment bonds (and structured products in general) is how to construct them so that they sell at par First we examine bonds that pay in cash Consider a bond that pays the coupon, c, n times over the life of the bond, makes the maturity payment M, and matures at time T We will denote the price of this bond as B(0, T , c, n, M) The time between coupon payments is T /n, and the ith coupon payment occurs at time ti = i × T /n We can value this bond by discounting its payments at the interest rate appropriate for each payment This bond has the price n B(0, T , c, n, M) = ce−r(ti )ti + Me−r(T )T i=1 n = i=1 (15.3) cPti + MPT This valuation equation shows us how to price the bond and also how to replicate the bond using zero-coupon bonds Suppose we buy c zero-coupon bonds maturing in year, c maturing in years, and so on, and c + M zero-coupon bonds maturing in T years This set of zero-coupon bonds will pay c in year, c in years, and c + M in T years We can say that the coupon bond is engineered from a set of zero-coupon bonds with the same maturities as the cash flows from the bond 441 442 Chapter 15 Financial Engineering and Security Design In practice, bonds are usually issued at par, meaning that the bond sells today for its maturity value, M The bond will sell at par if we set the coupon so that the price of the bond is M Using equation (15.3), B(0, T , c, n, M) = M if the coupon is set so that c=M (1 − PT ) n i=1 Pti (15.4) This formula appeared in Chapter and it was also the formula for the swap rate, equation (8.7) in Chapter In the special case of a constant interest rate, equation (15.4) becomes − e−rT n −rti i=1 e c=M (15.5) If a bond has payments denominated in stock, commodities, or a foreign currency instead of cash, simply replace the discount factor for cash, Pti , with the prepaid forward price, F0,P t , i which is the discount factor for a noncash payment If a bond makes some payments in cash and some in stock (for example), simply discount each payment using the appropriate discount factor For example, suppose a bond pays one share of stock at maturity, and that coupon payments are fractions of a share rather than a fixed number of dollars To price such a bond, we represent the number of fractional shares received at each coupon payment as c∗ The value at time of a share received at time t is F0,P t Thus, the formula for V0 the value of the note at time t0, is n V0 = c∗ i=1 F0,P t + F0,P T i The number of fractional shares that must be paid each year for the note to be initially priced at par, i.e., for V0 = S0, is c∗ = S0 − F0,P T n P i=1 F0, ti (15.6) When we pay coupons as shares rather than cash, the coupons have variable value Thus, it is appropriate to use the prepaid forward for the stock as a discount factor rather than the prepaid forward for cash The interpretation of equation (15.6) is the same as that of equation (15.4) The numerator is the difference between the current price of one unit of the underlying asset today and in the future The difference is amortized using the annuity factor for the underlying asset In the special case of a constant expected continuous dividend yield, δ, this equation becomes c∗ = − e−δT n −δti i=1 e (15.7) This expression resembles equation (15.4) Comparing the equations (15.5) and (15.7), we can see that the par coupon is determined from the lease rate on the underlying asset In the case of a bond denominated in cash, the lease rate is the interest rate, while in the case of a bond completely denominated in shares, the lease rate is the dividend yield 15.2 Structured Notes without Options Equity-linked bonds Example 15.2 illustrated a single payment equity-linked bond that sold for less than the stock price because the stock paid dividends It is possible for the bond to sell at par (the current stock price) if it makes coupon payments, compensating the holder for dividends not received To see this, if the bond pays cash coupons and also pays a share at maturity, the present value of the payments is n B(0, T , c, n, ST ) = c i=1 n =c i=1 Pti + F0,P t i n Pti + S0 − i=1 Pti Dti We can see that the price of the bond, B, will equal the stock price, S0, as long as the present value of the bond’s coupons (the first term on the right-hand side) equals the present value of the stock dividends (the third term on the right-hand side) Example 15.4 Consider XYZ stock as in Example 15.2 If the note promised to pay $1.20 quarterly—a coupon equal to the stock dividend—the note would sell for $100 A note that pays in shares of stock can be designed in different ways Coupon payments can be paid in cash or in shares of XYZ The instrument might be labelled either a stock or a bond, depending on regulatory or tax considerations Dividends may change unexpectedly over the life of the note, so the note issuer must decide whether the buyer or seller bears the dividend risk The coupon on the note could change to match the dividend paid by the stock, or the coupon could be fixed at the outset as in Example 15.4 Commodity-linked bonds Suppose a note pays one unit of a commodity at maturity In order for such a note to sell at par (which we take to be the current price of the commodity), the present value of coupon payments on the note must equal the present value of the lease payments on the commodity.4 The commodity lease rate plays the same role in a commoditylinked note as does the dividend yield when pricing an equity-linked note; both the lease rate and dividend yield create a difference between the prepaid forward price and the current spot price Example 15.5 Suppose the spot price of gold is $400/oz, the 3-year forward price is $455/oz, the 1-year continuously compounded interest rate is 5.5%, the 2-year rate is 6%, and the 3-year rate is 6.25% The annual coupon denominated in cash is c= $400 − $455e−0.0625×3 = $8.561 e−0.055 + e−0.06×2 + e−0.0625×3 The annual coupon on a 3-year gold-linked note is therefore about 2% of the spot price As we saw in Chapter 6, a lease rate can be negative if there are storage costs In this case, the holder of a commodity-linked note benefits by not having to pay storage costs associated with the physical commodity and will therefore pay a price above maturity value (in the case of a zero-coupon note) or else the note must carry a negative dividend, meaning that the holder must make coupon payments to the issuer 443 444 Chapter 15 Financial Engineering and Security Design A 2% yield in this example might seem inexpensive compared to the 5.5% risk-free rate, but this is only because the lease rate on gold is less than the lease rate on cash (the interest rate) Perpetuities A perpetuity is an infinitely lived coupon bond To illustrate, we can use equations (15.7) and (15.5) to consider two types: one that makes annual payments in dollars and another that makes payments in units of a commodity Suppose we want the dollar perpetuity to have a price of M and the commodity perpetuity to have a price of S0 Using standard perpetuity calculations, if we let T → ∞ in equation (15.5) (this also means that n → ∞), the coupon rate on the dollar bond is c=M e−r 1−e−r = M(er − 1) = rˆ M where rˆ is the effective annual interest rate, er − Similarly, for a perpetuity paying a unit of a commodity, equation (15.7) becomes c ∗ = S0 e−δ 1−e−δ ˆ = S0(eδ − 1) = δS where δˆ is the effective annual lease rate, eδ − Thus, in order for a commodity perpetuity to be worth one unit of the commodity, it must pay the lease rate in units of the commodity For example, if the effective annual lease rate is 2%, the bond pays 0.02 units of the commodity per year What if a bond pays one unit of the commodity per year, forever? We know that if it ˆ t in perpetuity it is worth S0 Thus, if it pays St it is worth pays δS S0 δˆ (15.8) This is the commodity equivalent of a perpetuity, with the lease rate taking the place of the interest rate Currency-linked bonds A bond completely denominated in a foreign currency will have a coupon given by equation (15.4), only using foreign zero-coupon bonds (and hence foreign interest rates): cF = M − PTF n F i=1 Pti The superscript F indicates that the price is denominated in the foreign currency If the bond has principal denominated in the home currency and coupons denominated in the foreign currency, we can discount the foreign currency coupon payments using the foreign interest rate, and then translate their value into dollars using the current exchange rate, x0 (denominated as $/unit of foreign currency) The value of the ith coupon is x0PiF c, and the value of the bond is B(0, T , cF , n, M) = x0cF n i=1 PtF + MPT i 934 Index Marking-to-market (continued) market-maker profits, 383 proceeds and margin balance, 162 Markopolos, Harry, 78 Married put, 59n Marshall & Ilsley security, 458–460 Martingale pricing, 663 binary options, 663–667 overview, 649–650 Martingale property, 655–657 Martingales Brownian motion, 604 LIBOR model, 782 money-market accounts, 660–662 Masters, M., 189 “Matched book” transactions, 237 Mathematics of delta-hedging, 389–395 Maturity default at, 817–820 effect on option prices, 280–281 profit diagrams before, 366–368 yield to, 196–197 MBS (mortgage-backed securities), 835, 841 McDonald, R L., 399n, 491n, 525n, 636n, 835 McGuire, William W., 490 McMurray, S., 455 Mean return estimates, 563–564 Mean reversion in arithmetic Brownian process, 608 Measures change of, 654–663 market size and activity, 4–5 Mergers Northrop Grumman-TRW, 499–502 price protection in, 469 Meriwether, John, 224 Merton, Robert C., 817 Brownian motion, 604n Itˆo process, 614n jump diffusion model, 736–737 jump pricing model, 623, 642–644 at LTCM, 224 option pricing, 275n, 350, 627n perpetual options, 372 Poisson distribution, 593, 595 volatility, 665n Merton default model, 817–820 Metallgesellschaft A G (MG), 187 Metrick, A., 223 Mexico, oil hedges in, 94 Microsoft compensation options, 487–488, 490–491 Millard, Charles E F., 672 Miller, H D., 604n, 639n Miller, Merton, 253, 350, 436–437, 473 Minimum price guarantees, 91–93 Mispriced options, arbitrage for, 297–298 Mitchell, M., 502n Mixture of normals models, 566 Modeling discrete dividends, 336–340 Modified duration, 213 Modigliani, Franco, 437 Modigliani-Miller theorem, 436–437, 473 Modules in VBA, 881 Money-market accounts, 659–662 Moneyness and exercise, 286 options, 44–45 Monotonicity, 811n Monte Carlo valuation, 492, 573 accuracy, 581–582 American options, 588–591 antithetic variate method, 587 arithmetic Asian options, 582–584 basket options, 711 computing random numbers, 577–578 control variate method, 584–587 correlated stock price simulation, 597–599 efficient, 584–588 European calls, 580–581 importance sampling, 587 Latin hypercube sampling, 587 low discrepancy sequences, 587 naive, 585, 587 nonlinear portfolios, 799–801 option prices as discounted expected value, 573–577 payoff formula, 580 Poisson distribution, 591–597 return distributions, 790n stratified sampling, 587 value at risk of two or more stocks, 795 written straddles after days, 802 Mood, A M., 550n, 861n Moody’s bond ratings, 821 KMV model, 818n Moon, M., 523, 525 Moore, David, 519n Index Moral hazard, 49n Morgan Stanley, CDS prices, 828, 830 Morgenson, G., 492n Mortgage-backed securities (MBS), 835, 841 Mortgage tranches, 451–452 Moving averages, exponentially weighted, 721–723 Mullins, David, 224 Multi-date swaps, 233 Multi-period options, 329–330 Multiple consumption in portfolio selection, 679 Multiple debt issues, 477–478 Multiple payment bonds, 441–445 Multiplication rules for geometric Brownian motion, 610–612 Multivariate Black-Scholes analysis, 646 Multivariate Itˆo’s Lemma, 616–617 Multivariate options basket options, 710–711 exchange options as, 708 options on best of two assets, 709–710 Murabaha transactions, 439 Musharaka transactions, 439 Myers, S., 103n, 327n Myers, S C., 477n Naik, V., 642n Naive Monte Carlo valuation, 585 Naked writing, 66 National Securities Clearance Corporation (NSCC), Nationally Recognized Statistical Rating Organizations (NRSROs), 821 Natural disasters, 789 Natural gas markets, 180–182 swaps, 241–243 Natural resources and commodity extraction, 525–531 Neftci, S N., 606n, 679n Nelson, D B., 724n Net payoff, 32 Net present value (NPV) correct use of, 511 investments, 509–512 static, 510–511 Netscape, 456–457 Neuberger, A., 733 New York Mercantile Exchange (NYMEX) contracts traded annually at, gold futures contracts, 176 light oil contracts, 182–183 natural gas futures, 181–182 Night wind storage, 180n Nikkei 225 index, 710 and futures contracts, 145–146, 256 investing in, 697–704 put warrants, 705 No-arbitrage bounds, 136–137 No-arbitrage pricing, 71 Noise term for geometric Brownian motion, 610 Nonconvertible bonds, callable, 482–483 Nondiversifiable risk, 11 Nonfinancial firms, derivatives used by, 104 Nonfinancial risk management, 103 Noninvestment assets in Black-Scholes equation, 635–637 Nonlinear portfolios, value at risk for, 796–801 Nonmonetary return, convenience yield as, 174 Nonrecombining trees, 307 Nonstandard options See Exotic options Nontraded assets, 890 Normal distribution, 545–548 cumulative function, 546–547 cumulative inverse, 578 density, 545–546 standard, 378 variable conversions, 548–549 variable sums, 549–550 Normal probability plots jumps and, 595–596 overview, 566–569 Normal random variables converting to standard normal, 548–549 sums, 549–550 Northrop Grumman-TRW merger, 499–502 Notes credit-linked, 834 with embedded options, 462–463 gold-linked, 460–462 structured, 438, 451 Treasury note futures, 217–220 Notional amount swaps, 235 time-varying, 252 Notional principal, 203–204 Notional value, Novation, 935 936 Index NPV (net present value) correct use of, 511 investments, 509–512 static, 510–511 NQLX (OneChicago), 140 NRSROs (Nationally Recognized Statistical Rating Organizations), 821 NSCC (National Securities Clearance Corporation), Nth to default baskets, 842–845 Numeraires, 639–642 change of, 654–663 money-market accounts as, 659–662 risky assets as, 662, 681 zero coupon bonds as, 662–663 NYMEX See New York Mercantile Exchange (NYMEX) O’Brien, J., 689 OCC (Options Clearing Corporation), 57, 400 Off-market forwards, 70 Off-the-run bonds, 196, 222 Offer price, 15 Offer structures, 499 Oil barrel, definition, 167 hedging, 187 hedging by Mexico, 94 hedging jet fuel with, 106, 187–188 markets, 182–184 spreads, 184–185 Oil extraction, 525–531 with shut-down and restart options, 531–532 valuing infinite oil reserves, 530–531 OIS (overnight indexed swaps), 250–251 On-the-run bonds, 196, 222 On-the-run/off-the-run arbitrage, 222 One-period binomial example, 328–329 One-period binomial trees, 293–300, 305–306 OneChicago exchange, 140 Onion futures, 165 Open interest, 5, 141 Open outcry, 138 Operational risk, 791 Optimal investment decision, solving for, 517–518 Option backdating, 490 Option-based model of debt, 477 Option elasticity, 362 Option Greeks, 356 in binomial model, 407–408 bull spreads, 363 definition, 356–357 delta units, 356–358 elasticity, 362–366 formulas, 379–380 gamma, 358 psi, 360–361 rho, 360–361 theta, 359–360 vega, 359 Option overwriting, 66 Option premiums, 367 Option prices for commodity extraction, 527 delta- and delta-gamma approximations of, 393 as discounted expected value, 327–330, 573–577 jumps, 737 jumps and, 642–644 probabilities, 326–330, 576 Option pricing model binomial model See Binomial model Black-Scholes equation See Black-Scholes equation Black-Scholes formula See Black-Scholes formula evidence of volatility skew in, 742–745 Option risk in absence of delta hedging, 382–383 Option writers, 38–40 Options, 263–264 all-or-nothing, 633–634 American-style See American-style options Asian See Asian options backdating, 490 barrier See Barrier options basket, 710–711 binomial See Binomial option pricing Black-Scholes formula See Black-Scholes formula bonds See Bonds buying, 37, 57–59 call See Call options capped, 696 commodities, 315–316 compensation See Compensation options currency, 269, 273–275, 312–314, 354–355 debt and equity as, 469–477 with discrete dividends, 354 distribution of returns in portfolios, 796–797 European See European options exchange, 270–275, 424–425, 432–433 exercise of, 57–58, 286 Index exercise styles, 36 exotic See Exotic options futures, 314–315, 355 gamma to approximate change in, 390–391 gap, 421–423, 497, 633–634, 686–687 as insurance, 47–50 interest rates, 753–754 ladder, 715 long and short positions, 45 martingale pricing, 663–667 maximum and minimum prices, 276–277 moneyness, 44–45 multi-period example, 329–330 one-period binomial example, 328–329 outperformance, 424, 667 overpriced arbitrage of, 297–298 path-dependent, 410 payoff and profit diagrams, 38 perpetual, 372–374, 665 power, 636 projects as, 511–512 put See Put options rainbow, 709–710 real See Real options risk premium, 363–365 Sharpe ratio, 365 shout, 714 spreads, 71–74 on stock indexes, 312 on stocks, 266–269 style, maturity, and strike of, 275–286 summary of, 53 swaptions, 256–257 synthetic, 268–269 terminology, 35–36 underpriced, 298 volatility, 363 warrants, 478–479 written, 38–40, 42–44 zero-coupon bonds, 667 Options Clearing Corporation (OCC), 57, 400 Order statistics, 567 Ordinary options, 686–687 Ornstein-Uhlenbeck process, 608, 761 Out-of-the-money options, 44 Outperform stock options (OSOs), 495 Outperformance feature, valuing, 496–497 Outperformance options, 424, 667 937 Outputs for VBA arrays, 878–879 Over-the-counter contracts and credit risk, 34 forward contracts, 139 Over-the-counter market, 3–4, Over-the-counter options, 399n Overhedging, 117 Overnight indexed swaps (OIS), 250–251 Paddock, J L., 525n Palladium, 105 Par bonds, 199 Par coupons, 199, 246 Par value of bonds, 458 Parameter estimates in lognormal distribution, 562–564 Paris Interbank Offer Rate (PIBOR), 156 Parity, 265 bounds for American options, 291–292 compound options, 419 generalized, and exchange options, 270–275 options style, maturity, and strike, 275–286 put-call, 68–71, 265–270 Partial expectation, 560–561 Pass-through structures, 835 Path-dependent options, 410 barrier options as, 415 Monte Carlo valuation, 582–584 Paulson, John, 841 Payer swaptions, 257 Paylater strategies, 111–112, 422n Payment bonds, multiple, 441–445 Payments for credit default swaps, 827 Payoff diagrams, 32 covered calls, 68 covered puts, 68 long forward contracts, 33 purchased and written call options, 36–38 zero-coupon bonds, 33–34 Payoff tables for arbitrage opportunity, 271 Payoffs, 29 Asian options, 631n call options, 37 on CD, 50–51 CDO-squareds, 840–841, 843–844 combined index position and put, 61–64 comparison of long position vs forward contract, 31 distribution of, 573 at expiration, 39, 65, 67 938 Index Payoffs (continued) on forward contracts, 29–30 for future values of index, 29 graphing on forward contracts, 30–31 net, 32 purchased and written call options, 36–40 put options, 41–44 supershare, 689 Payout-protected options, 57n Peak-load electricity generation, 519–523 Peak-load manufacturing at Intel, 520 Pension Benefit Guarantee Corporation (PBGC), 672, 835 PEPS (Premium Equity Participating Shares), 445n, 457–458 Per-share earnings, 486 Percentage risk of option, 362–363 PERCS (Preferred Equity Redeemable for Common Stock), 445n Perpetual options, 372–374, 695 Perpetuities, 444 Perrow, C., 792 Petersen, M A., 106 Pharmaceutical investments, 523–525 Physical measure of default at maturity, 818 Physical probability, 344, 346–347 PIBOR (Paris Interbank Offer Rate), 156 Pindyck, R S., 525 Plots of normal probability, 566–569 Poisson distribution overview, 591–593 simulating jumps with, 593–597 Poisson process, 819 Ponzi schemes, 78 Porter, William, 212 Portfolio insurance for long run, 282, 560 Portfolio selection first-order condition for, 652–654 multiple consumption and investment periods, 679 one-period problem, 676–678 risk premium of assets, 678 Portfolios elasticity of, 365, 477–478 Greek measures for, 361 risk assessment for, 789 risk premium of, 365 self-financing, 389 value at risk for nonlinear, 796–801 Positive-definite correlation, 598–599 Positive homogeneity, 811n Posting collateral, 3n Power options, 636 Prediction intervals in lognormal distribution, 557–559 Prediction markets, 12, 17 Prediction of extraordinary events, 792 Premium Equity Participating Shares (PEPS), 445n, 457–458 Premiums, 35 call options, 41, 62 conversion, 481 default, 209 forward, 132 for forward contracts and forward price, 132–133 liquidity, 210 options, 267, 282n put options, 41 Prepaid forward contracts (prepays), 126, 631–632 on gold, 177 pricing by analogy, 126–131 pricing by arbitrage, 127–128 pricing by discounted present value, 127 pricing with dividends, 129–130 on stock, 126–131 Prepaid forward price, 177 Prepaid forwards, 704 binomial trees with, 339–340 currency prepaid, 150–151 Prepaid swaps, 257 description, 234 Enron, 239 Prepaid variable forwards, 445–446, 452–453 Present value See also Net present value (NPV) barrier values, 374 calculations, 631 cap payments, 772 pricing by discounted, 127 projects, 509 traded, 512 Price ask, 15 bid, 15 bond conventions, 228–231 credit default swaps, 828–831 exotic options, 430–433 futures and forwards, 143 guaranteeing with put options, 91–93 Index and interest rate swaps, 244–246 lognormal distribution, 564–569 offer, 15 strike, 35, 95–96 volatility, 733–736 Price discovery in electricity markets, 180 Price limit, 139 Price participation of notes, 446–447 Price value of basis point (PVBP), 211–212 Pricing kernel, 677 Primary commodities, 167 Pro forma arbitrage calculations, 136n Probabilities bankruptcy, 822–824 binomial option pricing, 326–330 CDO-squareds, 840–841, 843–844 defaults at maturity, 818 distributions See Distributions in high and low states of economy, 347 Hull-White model, 774–776 pricing options, 343 real, 327–330 risk-neutral, 300, 326–327, 344–347, 574–575 in value at risk assessment, 789, 790n Procter & Gamble, swap with Bankers Trust, 252–253 Producers, risk management by, 89–96 Production, seasonality in, 178–179 Profit, 32 call, 37 from delta-hedging, 385–387 at expiration, 39, 65, 67 hedged, 91 from insurance on house, 48 for long positions, 47 market-maker, 389, 394–395 overnight market-maker, 388 for purchased and written call options, 36–40 put options, 41–44 on short forward position, 90–91 and spark spreads, 521 unhedged, 90–91 from written straddle, 81 Profit diagrams, 32, 54 bull spreads, 73, 85 butterfly, 85 collars, 76, 85 covered calls, 68 covered puts, 68 939 insured houses, 64 for no arbitrage, 71n ratio spreads, 85 straddle, 80, 85 strangle, 85 unhedged buyer, long forward, and buyer hedged with long forward, 98 zero-coupon bonds in, 33–34 Profit diagrams before maturity, 366–368 Projects with infinite investment horizons, 519 as options, 511–512 Proprietary trading, 382 Protection buyers and sellers in credit default swaps, 826–827 Psi formula, 380 option Greeks, 356, 360–361 Pulvino, T., 502n Pumped storage hydroelectricity, 180n Purchase of stock alternative methods, 125–126 delta-hedging, 386 Purchased call options, 366–367 gamma for, 358 payoff and profit for, 36–38 profit diagram for, 47 Purchased options, Greek for, 356, 358 Purchased put options, 46–47 gamma for, 358 payoff and profit for, 41–42 Put-call parity, 68–72, 265–266 bond options, 269 currency options, 269 dividend forward contracts, 270 stock options, 266–269 versions, 287 Put options, 41–45 adjusting insurance with, 95–96 binomial option pricing, 309–310 Black-Scholes formula for, 350, 352 buying, 70 calls as, 272–273 cash-or-nothing, 685 and collars, 74 covered, 66–68 down-and-in cash, 692 early exercise, 278–279, 324–325 940 Index Put options (continued) gamma for, 358 and insurance, 48, 61, 91–93 interest rate call equivalents, 754 long-maturity, 667–671 payoff and profit for, 41–44 premium for, 41 risk of, 49 strike price properties, 286 summary of, 53 up-and-in cash, 693 up-and-out cash, 694 Put premium for gold options, 107 Put strikes, 95–96 Put warrants Nikkei, 705 overview, 486–487 PutPerpetual function, 374n, 533 Puttable bonds, 485 PVBP (price value of basis point), 211–212 Quadratic variation Brownian motion, 606–607 realized, 729–731 Quantiles, 567 Quantity uncertainty, 114–117, 640 Quantos, 115n, 145–146, 697–698 binomial model, 701–704 dollar perspective, 699–701 foreign calls, 707–708 yen perspective, 697–699 Quasi-arbitrage, 137–138 Qur’an, 439 Rainbow options, 709–710 RainbowCall function, 709–710 Random number computations, 577–578 Random walks, 330–332, 604 Rate computations for swaps, 240–243 Rate of return, 16 Ratings transitions, 822–824 Ratio, hedge, 112–117 Ratio spreads description, 74 profit diagram for, 85 Rational option pricing theory, 275n Re-hedging frequency, 397–398 Real assets, 509 Real options, 509 commodity extraction, 525–531 commodity extraction with shut-down and restart options, 531–538 investment under uncertainty, 513–519 peak-load electricity generation, 519–523 research and development, 523–525 Realized quadratic variation, 729–731 Realized returns and Sharpe ratio, 365n Realized volatility, 730n Rebate options, 415, 694–695 Rebates, short, 19 Rebonato, R., 769n Recalculation speed in VBA, 881–882 Receiver swaptions, 257 Recombining trees, 307 Recovery rate bonds, 824–825 defaults, 816 Reduced form bankruptcy models, 824–826 Reference assets, 827 Reference obligations, 827 Reference price, 233 Regression in hedges beta, 149 linear, 113n Regulatory arbitrage, 12–14 Regulatory capital and value at risk, 791 Regulatory considerations, 453 deferred capital gains, 454–458 Marshall & Ilsley security, 458–460 Rehypothecation, 223 Reiner, E., 688n, 704n Reinsurance insurance companies, 403 market, 11 Reinvestment of dividends, 130 Reload options, 493–495 Rendleman, R J., Jr., 293, 765n Rendleman-Bartter model, 760–761 Renewable commodities, 166–167 Rennie, A., 679n Repo rate in bond markets, 19 implied, 133 Repricing compensation options, 492–493 Repurchase agreements (repos), 220–223 in financial crisis, 223 Index in Long-Term Capital Management crisis, 224 Repurchase programs, 486 Research and development as capital expenditure, 523–525 Residential mortgage-backed securities (RMBS), 836n Restart options for oil production, 531–537 Return distributions, bootstrapping, 806 Returns, continuously compounded, 301, 553 Reverse cash-and-carry, 136 Reverse cash-and-carry arbitrage, 169–171, 173–174 Reverse conversion, 269 Reverse convertible bonds, 445–446, 449–451 Reverse repos, 220 Revlon stock, constructive sale of, 455 Rho formula, 380 option Greeks, 356, 360–361 Risk, basis, 114, 185–187 bond price, 802 coupon bonds, 213 credit See Credit risk currency, 703 diversifiable, 11 dividend, 400 extreme price moves, 398–399 foreign stock index, 697–698 and insurance, 95 jump, 737–738 market, 791 market-maker, 382–384 nondiversifiable, 11 operational, 791 pooling, 402 put options, 49 in short-selling, 19 and swaps, 251 value at risk See Value at risk (VaR) volatility See Volatility Risk arbitrageurs, 502 Risk-averse investors, 103 and declining marginal utility, 344–345 utility functions of, 651 Risk aversion, 650–652 Risk-free bonds, valuing, 347 Risk-free rate of return, 574 Risk management, 12, 89 buyer perspective, 96–99 941 cash-and-carry as, 135 nonfinancial, 103 producer perspective, 89–96 reasons for, 99–107 for stock-pickers, 150 Risk measures, subadditive, 811–812 Risk-neutral distribution and VaR, 810–811 Risk-neutral investors, 326–327 Risk-neutral measure, 659–663 Risk-neutral pricing, 299–300, 323, 326–330 binomial model for, 344–348 Black-Scholes equation, 637–639 first-order condition for portfolio selection, 652–654 as forward price, 514 long-maturity put options, 667–671 measure and numeraire change, 654–663 overview, 649–650 risk aversion and marginal utility, 650–652 Risk-neutral probability, 300, 326–327 cash calls, 691 cash puts, 692 quantos, 702–704 Risk-neutral valuation Brownian motion, 618–622 stocks, 300, 347–348 Risk premium assets, 678 options, 363–365 portfolios, 365 Sharpe ratio, 617 and VaR, 810–811 Risk reversals collars, 74n implied volatility, 372 Risk-sharing, 10–11 Risky assets as numeraires, 662, 681 RMBS (Residential mortgage-backed securities), 836n Roosevelt, D., 843n Rosansky, V I., 190n Ross, S A., 336, 516, 637, 762–763 Rouwenhorst, K G., 190n Royal Bank of Scotland, CDS purchases by, 832 Rubinstein, M barrier options, 688n Cox-Ross-Rubinstein binomial trees, 335–336, 516 option characteristics, 275n options on best of two assets, 710n quanto pricing, 702n, 704 942 Index Rubinstein, M (continued) rainbow options, 709n supershares, 689 Ryan, M D., 705 S&P 500 futures contract, 139–140 S&P 500 index and arbitrage, 143–145 volatility estimates for, 722–724 S&P Depository Receipts (SPDRs), 145n S&P GSCI index, 189–190 Sales, constructive, 454–455, 457 Saly, P J., 493, 494n Scarcity in short-selling, 19 Schedules, call, 482 Scheinkman, J., 760 Scholes, Myron, 224, 349–350, 469 See also BlackScholes formula Schultz, Howard, 453n Schwartz, E S American option valuation, 588–589 commodity extraction with shutdown, 533n convertible bond valuation, 481n drug R&D, 523, 525 Scott, L O., 741n Seasonality corn forward markets, 178–179 in dividend payments, 130n natural gas, 180–182 Secondary commodities, 167 Securities and Exchange Commission (SEC), 821 reporting requirements, 13 structured notes, 451 Securitization, 835 Security design, 14, 437 Self-financing portfolios, 387, 389 Seller, short as, 29 Senior tranches, 477 Seniorities of debt-holders, 477 Settlements forward rate agreements, 203–204 swaps, 234–236 SFAS (Statement of Financial Accounting Standards) 123R, 104, 133, 489–490, 494 Shao, J., 806n Shapiro, A C., 222 Share-equivalent of option, delta as, 357 Share-equivalent position, 362 Shares convertible bond exchanged for, 478 repurchases, 475, 486 Shariah law, 439 Sharpe, W F., 293 Sharpe ratio, 365n, 617–618 bond pricing model, 758 Cox-Ingersoll-Ross model, 762 money-market accounts, 661 of options, 365 Shimko, David, 85n Short (seller), 29 Short-against-the-box sales, 455 Short call profit, 54 Short calls, 46 Short forward contracts and collars, 75 Short forward position, 46–47 Short forward profit, 54 Short positions, 16, 46–47, 53 bonds and repos, 221 insuring with cap, 64–65 Short put profit, 54 Short puts, 46 Short rebate, 19 Short-sales, 16–18 cash flows, 18 commodity forwards, 170–171 risk and scarcity in, 19 wine, 17–18 Short-term interest rates, 754–755, 758–759 Shout options, 714 Shoven, J B., 492 Shreve, S E., 604n, 606n, 614n, 679n Shumway, T., 744 Shut-down of oil production, 531–537 Siegel, D R., 113n, 525n, 636n Siegel, J J., 282 Simulation correlated stock prices, 597–599 jumps, 593–597 lognormal stock prices, 578–579 Single-barrel extraction of oil under certainty, 525–528 under uncertainty, 528–530 Single-name credit default swaps, 826–828 Single-payment bonds, 438–441 Single-payment swaps, 233 Single stock futures, 140 Index Singleton, K J., 825n Skewness, 601 Smith, C W., 102n, 105n Smith, D., 230n Snyder, Betsy J., 106 Soci´et´e G´en´erale, 792 Solanki, R., 671n Solnik, B., 247n Sorkin, Andrew Ross, 460n Southwest Airlines, jet fuel hedging by, 106 Spain, CDS prices in, 828, 831 Spark spreads, 521 SPDRs (S&P Depository Receipts), 145n Special collateral repurchase agreements, 221 Speculation, 12 financing with repos, 221 on foreign index, 705–706 and short-selling, 16 on volatility using options, 79–84 Speculative grade bonds, 821 Split-strike conversions, 78 Spot curves, 893 Spot price, 27 Spread options, 522 Spreads, 71 bear, 73 bid-ask, 14–15 box, 73–74 bull, 72 butterfly, 82–85 calendar, 367–368 commodity, 184–185 crack, 185 credit, 816 credit default swaps, 827 crush, 185n ratio, 74 spark, 521 swap, 248–249 vertical, 72 Spring-loading, 490 Stable distributions, 549 Stack and roll, 186 Stack hedge, 186 Staged investment, 524 Standard and Poor’s 500 index and arbitrage, 143–145 volatility estimates for, 722–724 943 Standard deviation binomial option pricing, 302–303 estimate of, 563–564 Standard normal density, 545–546 Standard normal distribution, 378 Standard normal probability density function, 378 Standard normal variables, converting normal random variables to, 548–549 State price, 344 Statement of Financial Accounting Standards (SFAS) 123R, 489–490, 494 Statement of Financial Accounting Standards (SFAS) 133, 104 States, event, 651 Static CDOs, 837 Static NPV, 510–511, 516 Static option replication, 399 Stiglitz, J E., 128n Stochastic differential equations, 603 Stochastic discount factor, 654 Stochastic processes, 604, 740–741 Stochastic volatility, 720, 736, 744 Stock alternative purchase methods, 125–126 calls on non-dividend-paying, 277–278 cash flows for, 266 with discrete dividends, 336–340 forward contracts on, 131–138 options on, 266–269 options to exchange, 272 prepaid forward contracts on, 126–131 risk management for picking, 150 risk-neutral valuation of, 347–348 short-selling, 18 synthetic, 268 value at risk for, 793–796 Stock index futures, 27 Stock indexes, 26–27 forward contracts vs options, 38 options on, 312 synthetic, 189–191 Stock markets, 5–6 Stock options See Compensation options; Options Stock price trees, 407 Stock prices Black-Scholes formula assumptions, 603–604 bond valuation based on, 485 conditional expected price, 559–561 944 Index Stock prices (continued) current price as present value of future price, 127 jumps in, 623–624 lognormal model of, 552–555 portfolio value as function of, 803 as random walk, 330–332 simulating correlated, 597–599 simulating jumps, 593–597 simulating sequence of, 578–579 and standard deviation correspondence, 559 Stock purchase contracts, 458–459 Storage commodities, 166, 172–174 corn, 178–179 electricity, 180 natural gas, 180–182 Straddle rules, 59 Straddles, 79 at-the-money written, 799–800 profit diagram for, 85 on single stock, 800 and strangles, 79–81 written, 80 Strangles profit diagram for, 85 for straddles, 79–81 Strategic options, 519 Stratified sampling, 587 Street name, 15 Stressed value-at-risk, 791 Strike, foreign equity calls in domestic currency, 706 in foreign currency, 705–706 Strike price, 35, 520 average as, 412 and convexity, 286 effect on option price, 281–285 investment costs as, 509 risk management, 95–96 volatility, 742 Strike price convexity, 283 Strip hedge, 186 STRIPS, 196 Strips, Eurodollar, 155 Structured finance, 835–836 Structured notes description, 438 warnings about, 451 Structured notes with options, 445–446 convertible bonds, 446–448 reverse convertible bonds, 449–451 tranched payoffs, 451–452 variable prepaid forwards, 452–453 Structured notes without options, 438 multiple payment bonds, 441–445 single payment bonds, 438–441 Stulz, R., 790n Subadditivity, 811–812 Subrahmanyam, M G., 247n subroutines in VBA, 865–866 Sumitomo Corporation, 792 Sums of normal random variables, 549–550 Sundaram, R K., 827n Supershares, 689 Supply of corn, 179 Swap curves, 247–248 Swap-rate calculations, 260 Swap spreads, 248–249 Swap tenor, 244 Swap term, 244 Swap writers, 826 Swaps, 123, 233, 263–264, 802 amortizing and accreting, 252 asset, 244 commodity See Commodity swaps counterparties, 237–238, 244–246 credit default See Credit default swaps (CDS) currency, 252–256 default, 259 deferred, 249–250 dividend, 270 as forward rate agreements, 247n implicit loan balance of, 248–249 interest rate See Interest rate swaps market value of, 238–240 physical vs financial settlement, 234–236 prepaid, 234, 257 price, 236–237 rate computations, 240–243 swaptions, 256–257 total rate of return, 834 total return, 257–259 variance, 731–732 volatility, 732, 803–805 Swaptions, 256–257 Synthetic CDOs, 837, 839–840 Index Synthetic commodities, 189–191 Synthetic forwards, 68–70, 110–111, 157 and box spreads, 73–74 creating, 133–134 in market-making and arbitrage, 135–136 Synthetic FRAs, 204–206 Synthetic Nikkei investments, 697 Synthetic options, 268–269 Synthetic stock, 268 Synthetic T-bills, 268–269 T-bills See Treasury bills T-notes, 217–220 T+3 settlements, 25n Tail VaR, 807, 810–811 Tailing, 130 Taleb, N N., 792 Tanker-based arbitrage, 184 Tax-deductible equities, 458–460 Tax strategies, 453 deferred capital gains, 454–458 Marshall & Ilsley security, 458–460 Taxes and box spreads, 75 for derivatives, 58–59 on employee options, 491n hedging and, 102, 104 Taylor, H M., 639n Taylor, J B., 251 Taylor series approximation, 392, 406–407 Taylor series expansion of bond price, 216n Tenor, 244 Term repos, 220 Term sheets for swaps, 235 Terrorism, futures on, 28 Texas Hedge, 672 Theta delta-hedging, 392–393 formula, 379 hedging, 388 market-maker profits, 394 option Greeks, 356, 359–360, 408 Thiagarajan, S R., 106 Three-period interest rate trees, 766, 768 TIBOR (Tokyo Interbank Offer Rate), 156 Time decay, 359–360 Time-varying volatility, 723–727 Times-Mirror Co., 456–458 945 Total rate of return swaps, 834 Total return payers, 257 Total return swaps, 257–259 Total variation in Brownian process, 607 Tourre, Fabrice, 841 Traded present value, 512 Trading and arbitrage, 137 financial assets, 2–4 proprietary, 382 Trading volume, Tranched payoffs, 445–446, 451–452 Tranched structures, 834–835 CDO-squareds, 840–844 CDOs, 836–840 seniorities, 477–478 Transaction costs, 12, 14–15 bonds, 219 future overlays, 147 hedging, 104 no-arbitrage bounds with, 136–137 Transferable stock options (TSOs), 493 Translation invariance, 811n Treasury bills historical monthly rate changes, LIBOR vs 3-month T-bills, 209–211 quotations for, 230–231 rates, 6–7 and stocks, 146 synthetic, 268–269 yield on, 144 Treasury-bond futures, 217–220 Treasury-note futures, 217–220 Trees binomial See Binomial trees in Hull-White model, 773–779 interest rate, 765–769 Tri-party repos, 221 Trinomial trees, 773–779 Troy ounce, 167 True probabilities pricing options with, 343 valuation with, 575–577 Trust securities and tax-deductible equities, 458–460 TRW-Northrop Grumman merger, 499–502 Tsiveriotis, K., 485 TSOs (transferable stock options), 493 Tu, D., 806n 946 Index Tufano, P., 106 Turnbull, S M., 825n Twin securities, 512 Two-parameter distribution, 545–546 Two-period European calls, 306–309 Uncertainty in binomial option pricing, 303 discounted cash flow, 513–514 investment under, 513–519 quantity, 114–117 single-barrel extraction under, 528–530 Underlying assets, 25 Unfunded CDS, 833 Uniformly distributed random variables, 578 Units of denomination, 639–642 Up-and-in cash calls, 694 Up-and-in cash puts, 693 Up-and-in options, 415 Up-and-out cash calls, 694 Up-and-out cash puts, 693 Up-and-out options, 415, 417 Upfront payments for credit default swaps, 827 Upper DECS, 460 Upward-sloping yield curves, 107n URDeferred function, 694 US Airways, credit guarantees for, 835 Utility, marginal, 650–652 Utility-based valuation, 344–345 Utility weights in high and low states of economy, 345–346 Value at risk (VaR), 789 alternative risk measures, 807–810 bonds, 801–805 Monte Carlo simulation for, 799–801 nonlinear portfolios, 796–801 one stock, 793–795 regulatory capital, 791 risk-neutral distribution, 810–811 subadditive risk measures, 811–812 two or more stocks, 795–796 uses, 790–791 volatility in, 805–806 van Binsbergen, J H., 270n Vanilla options, 895 VaR See Value at risk (VaR) Variable prepaid forwards (VPFs), 445–446, 452–453, 456–457 Variable quantity swaps, 241–243 Variables in VBA, 868–870 Variance binomial option pricing, 302 constant elasticity of, 736–740 estimates of, 564 Variance swaps, 731–732 Vasicek, O., 756–757 Vasicek model, 761–762 Vassalou, M., 820 VBA See Visual Basic for Applications (VBA) Vega formula, 380 option Greeks, 356, 359 Vertical spreads, 72 Visual Basic for Applications (VBA), 863 add-ins, 882 arrays, 874–875, 878–880 Black-Scholes formula computation with, 871–872 calculations without, 863–864 conditions, 873–874 creating button to invoke subroutine, 866–867 debugging, 881–882 Excel functions from within, 871–873 function creation, 864–865 functions calling functions, 867 functions vs subroutines, 867–868 illegal function names, 867 iterations, 875–877 learning, 864 macros, 880–881 modules, 881 object browser, 872–873 recalculation speed, 881–882 storing and retrieving variables, 868–870 subroutines, 865–866 VIX volatility index, 719, 731–732, 735–736 Volatility, 717 asymmetric butterfly spreads, 82–84 averaging, 412–413 binomial option pricing, 302–305 Black model, 780–781 Black-Derman-Toy model, 770–772 Black-Scholes model, 736–745 bond pricing, 766–770 bonds and swaps, 803–805 Index butterfly spreads, 80–82 constant elasticity of variance, 737–740 deterministic changes over time, 744–745 early exercise of options, 323–325 electricity and natural gas, 521–523 and equity-holders, 475 estimating, 805–806 exponentially weighted moving averages for, 721–723 GARCH model, 727–731 hedging, 731–736 Heston model, 740–742 historical, 720–721 IBM and S&P 500 index, 722–724 implied, 369–372, 718–720 martingale pricing, 665 measurement and behavior, 720 option pricing model evidence, 742–745 options, 363, 514n pricing, 733–736 speculating on, 79–84 stochastic, 720 and straddles, 79–81 time-varying, 723–727 various positions, 85 zero, 303 Volatility clustering, 724 Volatility frowns, 718 Volatility skew, 370–371, 742–745 Volatility smiles, 370, 718 Volatility smirks, 718 Volatility surfaces, 718 Volatility swaps, 732 Vorst, A C F., 585n Vorst, T., 830n VPFs (variable prepaid forwards), 445–446, 452–453, 456–457 Warrants description, 478–479 put, 486–487 Washington Mutual, 476 Watkinson, L., 843n Weather derivatives, 188–189 White, A., 741n, 773 White, Gregory L., 105 Wiggins, Jenny B., 486n, 741n Williams, J C., 251 Wind storage, 180n Writing for insurance, 66 Written call options, 46–47, 94 with different underlying stocks, 801 payoff and profit for, 38–40 Written options Greek for, 356 margins for, 58 Written puts and covered calls, 66 with different underlying stocks, 801 payoff and profit for, 42–44 profit diagrams, 43, 47 with same underlying stock, 799–800 Written straddles, 80 at-the-money, 799–800 and butterfly spreads, 80–82 Wu, L., 744n Wyden, Ron, 28 Xing, Y., 820 XYZ debt issue, 827 Yen-based investors, 697–699 Yen forward contracts, 152 Yermack, D., 490 Yield curves, 197 Black-Derman-Toy model, 769 upward-sloping, 107n Vasicek and CIR models, 763 Yield to maturity, 196–197, 228–231 Yields bond, 229–230 continuously compounded, 144n, 202 convenience, 174–175 dividend, 130 effective annual rate, 144n expected interest rates, 768 Yurday, E C., 832, 843n Zero-cost collars, 76–77, 108–109, 455–456 financing of, 77–78 forward contract as, 109–110 Zero-coupon bond prices calculating, 766–769 Hull-White model, 776–778 inferring, 200–201 Zero-coupon bonds, 195–197 Black-Scholes equation, 631 947 948 Index Zero-coupon bonds (continued) commodity-linked, 440–441, 443 currency-linked, 441, 444–445 and defaults, 815–817 equilibrium equations, 759 equity-linked, 440, 443 Macaulay duration and, 216n movement of, 804 as numeraires, 662–663 options, 667 payoff and profit, 33–34, 63 STRIPS as, 196 value at risk for, 802–804 Zero-coupon debt, 477 Zero-coupon yield curves, 197 Zero premium of forward contracts, 70 Zero volatility, 303 Zuckerman, G., 832 ... Bondholders ($) Net Cash Flow ($) 350 400 450 500 −30 20 70 120 425 .25 425 .25 425 .25 425 .25 −355 .25 −405 .25 −455 .25 −505 .25 40 40 40 40 461 4 62 Chapter 15 Financial Engineering and Security Design... −30 20 70 120 TABLE 15.4 Price of Gold ($) 350 400 450 500 FV(Gross Bond Proceeds) ($) 425 .25 425 .25 425 .25 425 .25 Payment to Bondholders ($) −416.04 −416.04 −446.04 −496.04 Net Cash Flow ($) 20 .79... Proceeds) ($) 425 .25 425 .25 425 .25 425 .25 Payment to Bondholders ($) −364.46 −414.46 −434.46 −434.46 Net Cash Flow ($) 30.79 30.79 60.79 110.79 $ 425 .25 − $434.46 + (S1 − $380) = S1 − $380 − $9 .21 If gold