Its hard to say for certain, but a few things come to mind. Second, to account for daily interest received on a replicating portfolio including lent funds in the amount of F(t0 , t), we treat the underlier as if it has a continuous dividend, using the risk-free interest rate for the dividend rate. While were here, notice how delta and vega are TAB LE 11 - 3 The Greeks as Ratios Greek Delta () Ratio Change in option price Positive change in underlier price Gamma () Change in delta Positive change in underlier price Vega (V ) Change in option price Positive 1% change in underlier volatility Theta () Change in option price One-day change in time-to-expiration Rho () Change in option price Positive 1% change in interest rates Example $0.50/$1.00 0.50 (expressed as 50) 3/$1.00 (expressed as 3) 0.35/1% 0.35 (expressed as 35) $0.05/day 0.05 (expressed as .05) 0.02/1% (expressed as .02) All About Derivatives 186 much greater numbers than the rest. Royal plans to buy 50,000 bushels of wheat in six months, and the company wants to lock in a price now. A one-day forecast is more likely to come true than a one-week forecast, and a one-hour forecast is probably even more reliable. What is the ultimate value of such marketization of credit risk? IQ1 100 0.015 1.5 IQ2 101.5 0.015 1.5225 IQ3 103.0225 0.015 1.5453 IQ4 104.5678 0.015 1.5685 FV PV IQ1 IQ2 IQ3 IQ4 106.14 We earn even more interest$6.14by compounding quarterly. A delta of 0.07 is seven one-hundredths of a share, a delta of 1.0 is exactly one share, and a delta of 0 is no shares. Bring your club to Amazon Book Clubs, start a new book club and invite your friends to join, or find a club thats right for you for free. Doing so is known euphemistically as taking a view of future prices, because taking a view sounds more legitimate than gambling. But speculating really is little more than gambling on an uncertain outcome. Instead, the nal payment would fall on February 29, and the nal coupon period would be longer than the three-month tenor by six days. The protection seller in a CLN may be some arm of an existing nancial institution, or it might be a specially created entity known as a special-purpose vehicle (SPV ). Positive for long calls; negative for long puts. (For our gold forward example, think of the numbers on both axes as multiples of $100.) While credit derivatives certainly cater to the inherent needs of lenders, they also afford participation in the credit market by parties with no preexisting exposure to a credit default. If you are author or own the copyright of this book, please report to us by using this DMCA report form. Refresh and try again. Whoohoo! How can you commit to buy something tomorrow you cant buy today? The point is, the better the model, the more reliable the facts. Now, everything weve said so far about pumpkin weights pertains only to this particular sampling. A rate index is just a price index like the ones we introduced earlier, providing an average price from a survey of related things. And how to price the darn things. Now, interest is simply the cost of borrowing money or, put another way, the payment for loaning or investing it. If your loan is nonamortizing, the notional remains constant over the life of the loan, and every payment is calculated from it. Derivatives are a natural nancial risk management tool for two key reasons that weve already seen. The short party still can lose money, of course, as soon as the spot price increases sufciently over strike. In our case, these are pumpkin weights. And ZED is trading for either $66 or $54. Here are both formulas, 8.15 and 8.16, together: FLong (FPNow FPPrevious Close ) Number of Contracts (Formula 8.15) FShort (FPPrevious Close FPNow ) Number of Contracts (Formula 8.16) This all leads us to something rather peculiar about futures: they are never literally canceled. Once you execute a contract, you are stuck with it till delivery. TAB LE 11 - 1 Lakewood-Cornelia Swap from Perspective of Floating-Payer Lakewood Fixed Rate = 4.1892% Payment Fixed Fixed Fixed Fixed 1 2 3 4 Notional Accrual Start Accrual End Accrual Days Payment Rate Year Days Payment Future Value $1,000,000 1,000,000 1,000,000 1,000,000 5/23/07 11/24/07 5/23/08 11/25/08 11/23/07 5/22/08 11/24/08 5/29/09 184 180 185 185 4.19% 4.19 4.19 4.19 365 365 365 365 $19,660.27 19,232.88 19,767.12 19,767.12 Spot Rate Days till Payment 3.53% 3.54 3.59 3.59 184 365 551 737 Payment Present Value $20,748.72 19,952.54 20,141.18 19,797.61 Net Fixed-Leg Present Value = $80,640.04 Floating Floating Floating Floating Floating Floating Floating Floating 1 2 3 4 5 6 7 8 ($1,000,000) (1,000,000) (1,000,000) (1,000,000) (1,000,000) (1,000,000) (1,000,000) (1,000,000) 5/23/07 8/23/07 11/24/07 2/23/08 5/23/08 8/23/08 11/25/08 2/24/09 8/22/07 11/23/07 2/22/08 5/22/08 8/22/08 11/24/08 2/23/09 5/29/09 91 92 90 89 91 93 90 94 3.53% 3.55 3.64 3.67 3.71 3.73 3.80 3.81 365 365 365 365 365 365 365 365 ($8,800.82) (8,947.95) (8,975.34) (8,948.77) (9,249.59) (9,503.84) (9,369.86) (9,812.05) 3.53% 3.54 3.59 3.63 3.67 3.70 3.75 3.78 91 184 275 365 457 551 642 737 ($8,724.04) (8,791.06) (8,738.97) (8,635.31) (8,843.24) (9,001.08) (8,790.08) (9,116.26) Net Floating-Leg Present Value = ($70,640.04) Swap Value = $10,000.00 Hedging a Derivatives Position FIGURE 183 11 - 6 Short Option Payoffs Now, an options delta changes with price, which changes all the time, illustrating why options require dynamic hedging. The payoff of a typical option, for instance, is proportional to the extent to which it expires in the money. After privatization and globalization financial market has entered into a new phase of global integration and liberalization. Well make some triple-As, whose holders will get rst dibs on subprime mortgage payments, and some double-As, and so on. In our one-step tree, the return is either 10 percent or 10 percent. Figure 11-7 shows payoff diagrams for both long and short stock positions to illustrate. In our option-pricing examples so far, weve relied heavily on no-arbitrage arguments to prove that our calculations are correct. A four-period amortization schedule, starting at $100 million and using a 7 percent mortgage rate, with annual payments, might look like the example in Table B-1. Consider the weight of a pumpkin--thats a perfectly autumnal RV. You want to borrow a million bucks to expand your business? And what of the writer, the short party to a call option? Upon execution of the forward and the simultaneous investing of $100, the long party can count on receiving $6 of interest on the same date he will buy the oil. Now, LTCM had hedged its long bond positions with short positions on the Russian currency the ruble, because in theory, those currency positions would increase in value when the long bond positions tanked. Like the dealer, they immediately establish a delta-neutral hedge position, which they adjust over time to remain delta-neutral. The binomial tree method constructs such a portfolio supposing that a price path follows a tree (from its trunk to one of its leaves) with a xed number of branching levels, where at each branch, the stock price can change to one of two possible new prices. Those traditional derivative contracts are price guarantees and are primarily traded on regulated exchanges, unlike credit derivatives, which are performance guarantees and are generally traded in the comparatively unsupervised and uncoordinated over-the-counter (OTC) markets. Pocket nothing. Over the life of this forward, there will be four payments for storage made at the start of each quarter in the amount of $7,500 (i.e., 5,000 barrels at $1.50 each). When that price is not specied up front, but instead is subject to change for reasons well get to later on, the loan is known as a oating-rate loan. Sure it is. At the end of the day, say your option holder exercises and you close out your stock position. It should. You make a dime no matter what. A share of stock represents a sliver of ownership in the company that issues it, and the stock market is, of course, a massive one. Because you have to buy something for $5 more than its worth, you lose $5. Say you buy an American call option on some companys stock. The McGraw-Hill Companies is not associated with any product or vendor mentioned in this book. He or she immediately establishes a delta-neutral hedge position in the underlier and then adjusts that hedge over time to remain delta-neutral. And their value derives in part from the value of the underlierbut only as that value is affected by a credit event such as bankruptcy. Cox, John C., and Mark Rubenstein. The number represented by e is an irrational one that starts out 2.7182 . Americans Versus Europeans Now lets turn back to the whole European-versus-American thing. Academic Press. This is not yet the case for performance guarantees or credit derivatives. In this new distribution, you can see how pumpkin weights are, on average, farther from the mean, which is again 16.11. Should the won-dollar exchange rate decrease over the next year, the dollar price of the purchase will increase. Does it change their moneyness? The Forward Contract 15 Forwards are also common in the energy commodity markets, where drillers, reneries, industrial consumers, and other participants often commit to large sales and purchases months or years in advance for commodities like oil and natural gas. (When dealing with foreign currency, just think of each unit of the other currency as something like a share of stock, which you buy with some amount of local currency.) Rather, an exchange has clearing members entitled to actually execute trades. Half-Truths about TOTAL RETURN SWAPS A total return swap (TRS), also known as a total rate of return swap, is a credit derivative and more. This page intentionally left blank CHAPTER 5 The Option Contract An option grants its holder the right, but not the obligation, to buy or sell something at a specied price, on or before a specied date. Could we not today execute a whole stack of FRAsa 3 9 and 6 12 and 9 15 and so on? and heres a diagram to prove it! Would that it were so. Figure 9-2 shows these dates on the calendar. The chief lesson from before 2008, which must not be ignored as time marches on, is that its the commonsense stuff that gets us in troublestuff Are Derivatives Any Good? You cannot tell future prices from futures prices. Market makers are the merchants of derivatives. In other words, its payoff is positive from your perspective. Full content visible, double tap to read brief content. b. Academia.edu no longer supports Internet Explorer. First, it is never wise to exercise early an American call option on a non-dividend-paying stock. With a spot price of $28 and contract price of $25, the farmer must sell those bushels to the tortilla maker for $3 per bushel less than the farmer could get in the spot market. ), which is a function that returns the greater of its argumentsin this case, 0 and S K. Stated in words, the long call option payoff is the greater of zero and the difference between strike price and spot price. Voil. And how is this a cost of carry? The company faced the prospect of corn prices rising excessively in the meantime and used a forward contract to mitigate that risk. But there are others. And as they are compensated primarily by bond issuers and dealers, one can reasonably question just how objective they can really be. Using Derivatives to Manage Risk 83 And thats a sampling of what hedging with derivatives is all about. For foreign currency, S is the spot price in local currency (i.e., the currency you want the option value in) of one unit of foreign currency. These NISM workbooks in PDF cover all the topics of the NISM test paper. A portfolio manager might use VAR to compare two nearly identical investment opportunities, to see which one has greater risk. Its essentially the value of the contract on its delivery date. So we basically iterate through the tree, working right to left, till we are at the starting node. 1993. Both Eurodollar futures and U.S. Treasuries are used a zillion times a day by swaps dealers around the world to hedge their exposures.1 So lets use futures to hedge the swap. Excellent question. Because with plain-vanilla swaps, the oating rate for a coupon is xed at the start of the period, which in this case is today. With a CDS-as-put-option, the holder may get no payoff when the spot price is far below the strike. Then when you see 25 basis points, you think of a quarter, and hearing ve bips brings to mind a nickel, which you know without thinking is onetwentieth of a dollar. Appendix C, More Binomial Option Pricing, illustrates the basic proof of risk neutrality. And so on. Bottom line, weve got this equation that says how changes to these factors are related to one another. Heres the math:2 FV PV (1 rt) 100 (1 [0.06 1]) 100 1.06 106 where: FV future value PV present value r annual rate of interest t time in years The key to how much money actually grows is not just the interest rate but also the compounding frequency. GENERAL BINOMIAL FORMULAS First, lets summarize in a key the symbols weve seen so far. Not unlike shmongers and sh, they buy derivative securities at one price and sell at a higher price, pocketing the difference as their prot. Quick example: If the three-month spot is 2.50 percent and the three-month rate three months forward is 2.60 percent, the implied six-month spot rate must be the arithmetic average, or 2.55 percent. Publisher McGraw-Hill Collection inlibrary; printdisabled; internetarchivebooks; china Digitizing sponsor Internet Archive Contributor Internet Archive Language English Access-restricted-item true Addeddate 2012-05-02 16:58:02 Bookplateleaf 0002 Boxid So, as you see in Figure 10-3, if ZED moved to $66, your payoff was $4, and if it moved to $54, your payoff was zero. If you exercise, you get to buy 10,000 shares of stock (remember, each option is to buy 100 shares) for $300,000 instead of the going price of $500,000, saving a whopping $200,000! If the buyers $10 million bond portfolio devalues to $2 million, the buyer receives a check for $8 million. The author is very up front about what the book covers and does not cover, and what is covered is covered very well. In many ways, then, hedging a future need with futures is like hedging it with a forward: you gure out how much of the underlier you want to buy on some future date and go long the appropriate number of contracts. Here are a few and what theyre all about in a nutshell: Q Q Q Q Q Portfolio CDS: This credit default swap has more than one reference entity. See Arbitrage/ arbitrageurs Model price, 18791 Model risk, 191, 194 Modied following (calendar convention), 241 Money, as an underlier, 11, 15, 30 Moneyness of an option, 3941, 43 Monte Carlo simulation, 193 Moodys ratings, 67, 199 Mortgage(s), 7 Mortgage crisis, 19597, 199211. The delta of a high-gamma option changes more rapidly than that of a low-gamma option. The notional FIGURE 4 -1 Floating-Rate and Fixed-Rate Borrowers All About Derivatives 32 FIGURE 4-2 Floating-Rate and Fixed-Rate Borrowers with a Swap amount is simply a number used when it comes time to calculate payments. But they do their best, analyzing things like the historical track record of issuers, the riskiness of their business, the quality of the issuers leadership, and who knows what else. . Mathematics comes into play not just in the pricing of derivatives but also in the area of nancial risk management. And different lag periods get their own forward curves. Boyle, Phelim, and Fiedhlim Boyle. They search for mispriced securities and attempt to prot from themtaking on no risk whatsoever if they do it right. The price of such a portfolio gives you the price of the option, because two things with the same payoff must have the same value to prevent arbitrage. While hanging out in the mild and enviable climes of the Carolina Piedmont, he also had the great 1 2 Mother Of All Trading Systems, managed by the one-of-a-kind Kurt PhillipsZabel. But at the time, it seemed perfectly reasonable, assuming, again, that default rates among the underlying pool of mortgages were uncorrelated. Reviewed in the United Kingdom on June 10, 2013, Reviewed in the United Kingdom on May 21, 2016, Reviewed in the United Kingdom on March 5, 2015, Reviewed in the United Kingdom on January 18, 2015, Reviewed in the United Kingdom on June 4, 2015, Learn more how customers reviews work on Amazon, McGraw Hill; 2nd edition (December 7, 2010). Compared with other hedges, All About Derivatives 76 TAB LE 7-3 Underlier Positions and Hedge Guidelines Long underlier Present or future ownership Hedge to sell Short underlier Present or future need Hedge to buy settling a forward hedge is uncomplicated: you simply take delivery of the goods and pay or receive the agreed-upon price. A delta of 0.50 (often expressed as simply 50 without the decimal point) tells us that for a $1 increase in the underlier price, the price of the option increases by 50 cents. And how. Wiley. In other words, a position in the option was equal to a position in the replicating portfolio: cZED62 ZED B Notice how the theoretical value of cZED62, $2.53, is consistent with the range we identied in the previous section of 0 to $60, the range between intrinsic value and stock price. And since a forward is zero-sum game, with one partys gain offset by the others pain, the only fair price of a forward upon execution must be zero. The term basis comes from the idea of basis risk. Other indexes include commercial paper rates, indicating rates at which businesses will pay to borrow funds, and numerous indexes analogous to LIBOR for other nancial centers besides London. We can combine the diagrams and see a nice illustration of this idea, as in Figure 5-11. Beyond the exchange and OTC markets, there are also derivatives markets where the traders dont even know they are trading derivatives. And if your hedge requires a short sale of stock, regulations in the United States may dictate that short selling is permitted only after an uptick (increase) in the stock price. Now back to the matter at hand. We just do things in reverse: write the option and buy the stock. So the value of the cZED62 at time 0 must have been $2.53, the cost of building the portfolio. Consider our previous example, wherein you executed a sixmonth forward contract with your brother to buy 100 shares of MGrove stock at a delivery price of $15.66 per share. Dividends6 mo. Imagine that Boris would prefer a xed rate and Chloe would prefer a oater. McGraw-Hill and its licensors do not warrant or guarantee that the functions contained in the work will meet your requirements or that its operation will be uninterrupted or error free. This is information worth getting to the top of your head if you plan to work with options. 3. She is a speculator with a view that the stock of ZED Corporation is undervalued and its price will rise. . The expected value of the stock is its future value using the risk-free rate of 6 percent. It might also have used a futures contract or even an option. This ubiquitous computational ingredient tells us how much of an annual interest rate to apply when calculating interest for some period of time. But because futures are settled daily, you need to consider the daily cash ow requirements and the present value of those settlements when calculating a futures price. How many hundreds are in a million? Figure 12-1 shows the cash ows associated with a typical bond. Thats all academic, of course. By virtue of the swap, the dealer has a short positionhence the idea that you are the seller of the swapon forward LIBOR at a delivery price of 3.5 percent (or $8,750, as we calculated previously) and a delivery date 12 months out. Thats compounding, which can be applied to either a xed- or oating-rate loan. The Lakewood-Cornelia swap differs from the Finch-Radley swap in three ways: it has a xed tenor of six months and oating tenor of three months, it calls for long stubs instead of short, and it will commence today, May 23, 2007, and run for two years. And the greater the time Pricing Options 131 before expiration, the more valuable this optionality becomes ergo, time value. So instead of swapping cash ows once a year, the parties do so four times per year, or quarterly. Pricing Options TAB LE 155 10 -2 Effect of Factor Changes on Option Prices Change in Factor Effect on Call Value Effect on Put Value Increase Increase Increase Increase Increase Increase Decrease Increase Increase Increase Decrease Increase Increase Increase Decrease S K t r Table 10-2 summarizes the effect of the ve pricing factors on the price of an option. The zero point represents a time at which you know the value of something. From day to day, the rates change unpredictably, but once xed, there is no quibble over the daily proclamations of those helpful British bankers. No matter what happens to the exchange rate in the meantime, we both end up with what we started with in terms of our local currency. Dividends 9 mo. Changing the composition of a portfolio means trading, and trading costs money (broker fees, etc. Boxwood wants to protect the portfolio against a large decline in value, and in particular does not want it to fall below $110 million. The Eurodollar, especially front-term Eurodollars for borrowing three, six, and nine months down the road, are highly Appendix A 226 FIGURE A-3 Beginnings of a LIBOR Spot Curve liquid.2 The prices at which they trade, then, can be seen as indicators of where people expect the BBA to publish LIBOR rates in the future. Bond dealers made triple-A (higher-priced) securities out of, essentially, triple-B (lower-priced) securities. Shares of Microsoft may be selling or trading on the New York Stock Exchange for $30.82 each, whereas shares of IBM may go for $130.68. For the mathematically curious, adventurous, or rusty, however, we will explain quite a lot of the math behind option pricing. Youve sold, or written, this call option, and any positive payoff to that option belongs to the other party. A CONDITIONAL SALES AGREEMENT An option is a price guarantee that may or may not result in a future sale. And a forward price is the delivery price of a theoretical new contract whose value is zero. Say the day after you sign your purchase agreement, some celebrity talks up the car on his or her TV show, and it becomes wildly popular, driving up the price consumers are willing to pay. Unlike investors in the real world, investors here require no compensation for risk. What is fair market forward price? The CDS differs in two key ways. Suppose a dealer mistakenly believes a model price to be $1.50 when its really $1.30, and the dealer buys 500 contracts at $1.40, expecting to make $5,000 in edge, when really the dealer will lose $5,000. Our payment security system encrypts your information during transmission. So you cover it with a short position in the stock, whose value will increase with the falling stock price, compensating you for your loss on the put. Say there is a 1 percent chance of the reference entity defaulting and youre writing a check for $10 million. And if you spend any time looking at the disasters, youll notice that not all of them had to do with the type of risk weve covered so far in this chapter. Having read this book might help you to understand a book on trading strategies though. You take out the collision insurancewith yourself as the beneciaryas a bet. The cash ows, of course, from the borrower to the lender.1 So cash ows are just payments, and payments involve a price. Now, short forward positions always make money when the spot price is below delivery price. Four of these (all except strike price) change all the time over the life of an option, so we say the option price is sensitive to changes in each of those factors. The lowest a strike price can go is zero, and the longest time to expiration is innity, and whats the value of a call option with a strike price of zero and no expiration? Makes sense, right? It may be optimal to exercise early an American call or put on a dividend-paying stock, depending mostly on the size of the dividend. Just look back to our earlier explanation of calculating present values. So Boxwood buys put options on a market index, struck such that the rm will be compensated for a market decline taking the portfolio below the $110 million oor. In this example, perhaps the terms of the CDS are such that the protection seller would have to pay only 60 percent of the $10 million, because thats the extent of the loss. We wont delve into these stealth markets in this book, but rest assured that the fundamentals of derivatives apply to those markets just as they do to the traditional exchange and OTC markets. A written option without a cover is known as a naked option. Writing naked options is rarely a good idea. Every Greek expresses a ratio of a change in one thing to a change in another thing, as depicted in Table 11-3. The credit event that triggers settlement can be anything the two parties agree to but is typically one or more of ve credit events dened by ISDA: bankruptcy, failure to pay, obligation default, obligation acceleration, or restructuring. 100 (0.06)(0.5) 100 0.03 3 I2nd 6 mo. And industry regulators can use VAR to set capital requirements (how much money a nancial institution is required to keep on hand to weather adverse market conditions). Well come across short selling again in Chapter 11, Hedging a Derivatives Position. HEDGING WITH FORWARD CONTRACTS In Chapter 1, Derivatives in a Nutshell, the tortilla maker bought forward contracts to lock in the price of corn for purchase in six months. As forwards-in-disguise, then, swaps are inherently less liquid than futures and carry more credit risk. Andwell, thats about it. Wiley Finance. And so on. Volatility ( ): Like time to expiration, increasing volatility increases the value of both calls and puts. Coupon date: On which day of the month are cash ows exchanged? As mentioned earlier, a futures pricethe guaranteed price at which the long party must buy and short party must sellis intuitively the same as an otherwise identical forward price when a contract is executed. The math should be familiar now, so well jump right to the solution. And heres the key to that: for a new forward or futures contract, the delivery price is the oneand only onesuch that the value of the contract is zero. What would they think of triple-A CDOs made of the very same triple-Bs? So the price of the contract would really be 100 $2.63, or $26.30, and the payoff 100 $0.10, or $10.00. And this trading activity itself adjusts prices such that the arbitrage opportunity goes away. What happens if we plot the combined payoff of the long call position and short put position on the same diagram? If your loan is amortizing, the notional changes, or amortizes. Because everyone settles up dailyrealizing any prot or loss before they go home, as it wereno partys obligation is permitted to grow unbridled for more than one day. Was it the ascension of home prices to unheard-of levelsand the forgetting that price bubbles are called price bubbles because they really do tend to burst now and again? Pretty cool, huh? The Black-Scholes method, which grows directly out of the binomial tree method, uses a tree with an innite number of branches. The option delta is 79. If the price of wheat decreases, they make less prot when it comes time to sell. LIBOR LIBOR None None None 6 months 3 months 22nd Long Actual/365 New York Modied following For the market, well use the same spot and forward curves from the previous swap. And because the composition of Boxwoods portfolio is nearly identical to the S&P 500 index portfolio, its value has likewise fallen to $105 million. Think back to how we priced an option by building a replicating portfolio on the one-step tree. Say you put a $100 wager on red at a roulette wheel for the chance to win double your bet if the wheel stops on a red number. Its payoff is negative from your perspective. Lots of it. Thats liquidity risk. Nobody can tell you the Pricing Options 149 current volatility of any stock, anywhere, at any time. Well, the rst thing you might wonder, as many do, is whether the problem is with Black-Scholes. For example, a bond paying its investors 5 percent when the prevailing interest rate is 4 percent clearly has positive value. These new mortgagees, the so-called subprime borrowers, were by denition likelier to default than were borrowers who could meet more stringent credit requirements. You cant be driving at both 60 and 65 mph at the same time, any more than you can weigh 195 and 205 at the same time. Learn how we and our ad partner Google, collect and use data. Short selling is traditionally used by the speculator who believes a stock price will fall.3 It just so happens that its also an indispensable tool for the options hedger. In the late 1990s, the hedge fund Long-Term Capital Management (LTCM) was having a grand time, showering its investors with annual returns of close to 40 percent. This powerful notion of position replication is everywhere in the world of derivative pricing and is All About Derivatives 52 FIGURE 5 -15 Option Payoff Patterns indeed like the glue that holds it all together. Reviewed in the United States on June 27, 2021. So we need to discount it to today. Say you begin with $10 million and decrease the notional by $1 million each period. These parties to a trade are known formally as counterparties. The positional ramications of a put option are just the reverse that of the call: the long party may have an interest in selling the underlier, in which case the short party is obligated to buy. The Delta Hedge Delta hedging is the basic routine for the option market maker, or dealer, and it goes like this: The dealer expresses a willingness to buy an option for some bid price less than model price, or to sell it for some ask or offer price greater than model price, where model price (MP) is simply the theoretical value produced by a pricing method such as Black-Scholes. Now, stock prices also change continuously and unpredictably, but options are all about probability. This binomial tree thing is bogus. No, its just that were still working with an abstraction and need to get more concrete. MORE ON RISK MANAGEMENT So what other fun stuff goes on in the heady world of nancial risk management? Tangling matters even more, an interest rate, whether for pricing or discounting, is only for some given length of time. First, though, just take a look at the Black-Scholes formula together with the binomial tree formula to see a resemblance: Black-Scholes: c SN(d1) Ke rtN(d2) Binomial tree: c S B The two formulas say basically the same thing! INDEX Abstraction, 1, 12, 35, 132, 144 Actual/360 calendar convention, 24344 Actual/365 calendar convention, 243 Additive amortization, 238 Adjustment, 247 reset, 248 AIG, 206, 208, 21011 American-style options, 39, 40, 4143 early exercise of, 17072 pricing, 133, 172 Amortization, 33, 23840, 246 Analytics, 166, 191 Annual rates of interest, 21920, 222, 242 Arbitrage/arbitrageurs, 7, 9, 10, 61, 216, 233. Heres All About Interest 233 the main thing: forward rates are derived from spot rates. Do we use the one-year LIBOR or oneyear U.S. Treasury rate? Forward rates converge to spot rates, so the forward rate for money borrowed now, on the spot, is just a spot rate. Please try again. An Introduction to the Mathematics of Financial Derivatives. They are neither risk-averse nor risk-inclined. How so? 2. 1998. It might look like the curve in Figure A-6. The party with an obligation to sell is the short party, as that party holds the short position. Because it is hedging an ownership. If it doesnt, then its volatility is said to be low. But wait, doesnt that describe the CDS buyer? The same idea is often applied in international economics in support of free trade (wherein countries are allowed to exploit their comparative advantage, making everyone theoretically better off) versus protectionism (where they are not). The number of option strategies one can cook up is limited only by ones imagination and tolerance for risk. Futures Contract Value The value of a futures contract is rather different from the value of an otherwise identical forward contract because the payoff from a futures contract is realized at the end of every trading day. Theres no need to go into details, but sufce it to say that exchange rates are inuenced in part by prevailing interest rates in the respective countries. Same math as before, but notice we multiply the rate by onequarter (0.25) instead of one-half (0.5) because our period is now one-quarter of a year (three months): P0 39.42e (0.04)(0.25) 39.03 Now we have values for two of three components of our portfolio and can solve for the third. Now zip forward a few happy-go-lucky years to 2008. And what does the short party think of this price? Then, every three months, no matter where LIBOR xes, we know our net payment would be based on a 3.25 percent annual rate, effectively converting our oating-rate obligation into a xed-rate obligation. But if we construct a model with a billion possible price paths, odds start getting pretty good that the actual price path will be one of, or very close to one of, those billion. Should the spot market move against the long party, he or she might simply refuse to buy. A forward is a contract wherein a buyer agrees to purchase the underlier from the seller at a specied price on a specied future date. An option like this, with a strike price equal to (or very close to) the underlier spot price, is said to be at the money (ATM ). Extending our curve with Eurodollar two-year, three-year, and four-year rates gives the growing curve in Figure A-4. Having so many different rates for a single index can be rather unwieldy, but thankfully we have an indispensable device known as a yield curve for dealing with term structure. It is sold with the understanding that neither the author nor the publisher is engaged in rendering legal, accounting, securities trading, or other professional services. The losing party in a forward deal can also default. And every month, MGRM would put on new futures hedges for future obligations, as part of its stack-and-roll hedge. This is another cost (an actual out-of-pocket cost this time, not an opportunity cost) to add to the interest we calculated. With respect to swaps and other interest rate derivatives, think of a rate index as an undisputed thermometer that monitors some aspect of an ever-changing environment and reports it as a number. There are two primary parties to a CDS: the 1 This begs a question, of course: How can a protection buyer be assured the protection seller wont default? Put-Call Parity So far, weve dealt only with call options. First is the Black-Scholes assumption about how stock prices change over time. Market makers publish indicative prices on some electronic forum or another, helping the prospective buyer or seller of protection to nd a counterparty. The coupon date is the 22nd of the month, the xed and oating tenors are both three months, and the swap maturity date is February 29, 2008. Recall rst that when we borrow money, we typically pay interest at regular intervals over the life of the loanonce a month for your mortgage, perhaps quarterly for a commercial loan, and so on. Clearly, with respect to interest, the fair delivery price for this deal is spot price adjusted for interest, or $106. The S&P 500 index has fallen steeply to 1,050. What is a cash ow, and what does it have to do with buying and selling? This attribute is expressed as a simple integer indicating a number of days preceding the start of the accrual period, and for LIBOR is typically two days. Next we turn to the valuation, or pricing, of derivatives. All About Derivatives 178 gain of $250 (i.e., 1,000,000 [0.035/4 0.034/4]). If you choose the option, you dont have that $50 to invest as the call holder does. And like so many things with options, the same logic in pseudo-reverse says a put can be worth no more than a short position in a stock. The two commonly used swaps are: Interest rate swaps: These entail swapping only the interest related cash flows between the parties in the same currency. Consider what youll do with your replicating portfolio: 1. Borrowing money costs money, and interest rates change all the time. If I am an oil driller and you are a renery, we might execute a forward contract for the sale of x barrels of crude oil at a price of y to be delivered z days from today. Heres a quick example: The Royal Mill buys wheat to make our. The extra $6 compensates her for the opportunity cost of selling forward versus selling spot. For ushering this second edition into existence, I must thank the forever resourceful, inventive, and persistent Morgan Ertel, my editor at McGraw-Hill. Parties to a forward, by contrast, are free to dene the terms of their contract however they mutually agree to. Choose any S value, and use the payoff table or payoff function to determine the corresponding P value. Derivative Securities: The Complete Investors Guide. Table 9-1 shows the spot and forward rates from the curves, going out two years, which well use in our calculations. You might wonder about the asymmetric shape of the lognormal distribution and how that ts with reality. Sell cZED62, and collect the $2.53 premium. Using real-world examples and simple language, it lucidly illustrates what derivatives are and why they are so powerful. Yes! The key, of course, is to run as many simulations as possible. Instead of buying options, this person could have simply bought up a bunch of stock and held it for six months, making the same basic upside when (and if) his prediction came true. Accrual start: Accrual end: Year difference: Month difference: Day difference: Total difference: Day fraction: Annual rate: Accrual rate: Accrual: 9/15/2006 12/14/2006 (2006 2006) 360 0 (12 9) 30 90 (14 15) 1 1 89 days 89/360 6 percent 0.06 89/360 $1,000,000 0.06 89/360 $14,833.33 SWAP ATTRIBUTE SUMMARY This section offers a few words on most of the features of a plainvanilla swap, starting with features that apply to a trade in its entirety, then zooming in on features of individual legs. Shouldnt I exercise and sell immediately if I cant sell the option, to at least lock in the intrinsic value? If you really believe the stock price will fall, then you should sell the stock short. Take a look at Table 7-1. Hedging a Derivatives Position TAB LE 191 11 - 6 Option Positions and Underlier Hedge Positions Option Position Hedge Position Short call Short put Long call Long put Long underlier Short underlier Short underlier Long underlier FIGURE 11 - 7 Long Underlier and Short Underlier Payoffs This is why the better your model, the better you will be as a dealer or arbitrageur. Maybe the option will go in the money. Notice that for any value of S, there is only one possible value for P. This makes drawing a payoff diagram quite easy. Your privacy is important to us. Guess which one? Sound familiar? The protection buyer pays a premium or CDS spread to the protection seller. THE IDEA BEHIND A FORWARD PRICE By piecing together what weve seen so far, you can see that the value of a forward at any time is basically the sum of two things: Pricing Forwards and Futures 93 the difference between the ever-changing spot price and neverchanging delivery price, plus the cost of carry (which can be a positive or negative value, as we will see in a bit). Options on simple underliers grant the holder the right to buy or sell some thing. If the thing is a commodity, it might be bushels of corn, barrels of crude oil, megawatts of electricity, or some other physical good. Forward CDS: This is a forward contract to buy or sell a CDS on a particular reference entity at a specied date in the future. Consider a currency swap with one leg denominated in Australian dollars and the other in U.S. dollars. The basic idea is to construct an imaginary portfolio of non-option instruments (whose prices are comparatively easy to obtain) such that the portfolio payoff mimics or replicates that of the option. And this illustrates one of the prime attractions of credit derivatives: they enable the decomposition or unbundling of overall risk into different types of risk, each of which can be dealt with on its own. Now were already familiar with the (really important) concept of delta. In this world, we deal in hundredths of percentage points, and one-hundredth of 1 percent is what we call a basis point. (If you have a coupon to buy milk at $2.00 per gallon, and milk is selling for $1.80 at the market, will FIGURE 5-7 Long Forward Payoff Function and Diagram The Option Contract FIGURE 47 5-8 Payoff Diagram for a Long Call Option TAB LE 5 -1 Payoff Table for Long Call Option S K PCall,Long max(0, S K ) $30 $50 $60 $40 $40 $40 0 $10 $20 you use the coupon?) Day basis: To apply an annual interest rate to an accrual period whose length is less than or greater than one year, we need a day fraction for converting the rate. There are two basic components. FIGURE 10 - 4 Two-Step Binomial Tree Pricing Options 145 Notice that our two-step tree is really just three one-step trees. An entirely new chapter, Chapter 12, Derivatives and the 2008 Financial Meltdown, is devoted to explaining the role of the credit default swap (CDS) during that painful economic period. See Replicating portfolio/synthetic option Tenors, 3233, 246 Term space, 15153 Theoretical value. The fundamental concept here is that money invested (or lent) grows with time. See also BlackScholes formula; Black-Scholes partial differential equation Black-Scholes partial differential equation (PDE), 14748, 185 Bond(s), 5960, 62, 63, 64, 137 collateralized debt obligations and, 20610 convertible, 7 nancial meltdown (2008) and, 197204 hedging a derivatives position and, 19293 mortgage-backed, 199204, 207, 208 price, 199, 208 ratings, 199 returns, 198 Russian crisis, 216 shorting, 134 super, 200202 swaps and, 177 yield, 198, 199, 202, 208 Bond basis calendar convention, 244 Borrower diversication, 60 263 264 British Bankers Association (BBA), 222, 226 Brownian motion, 164 Bull combos, 56 Butteries, 56 Buy-write strategy, 56 Calendar(s), 115, 24044, 247 business day conventions, 241 day count conventions, 24244 reset, 248 Calendar spreads, 19293 Call options, 3739, 40, 54 covered position, 5556 hedging a position in, 18889 moneyness and, 43 payoff, 4648 pricing, 130, 131, 13343, 153, 154, 15556, 16263, 16869, 17172, 173, 174, 25459 Caps, interest rate, 35, 80 Cash settlement, 12, 63 Central limit theorem, 164 Chicago Board of Trade (CBOT), 11, 25 Chicago Board Options Exchange, 11, 81 Chicago Mercantile Exchange (CME), 25, 45, 79 Citibank, 22223 Collar, interest rate, 35 Collateral, 15, 26 Collateralization, 77 Collateralized debt obligations (CDOs), 20610 Collateralized debt obligations (CDOs)squared, 208 Commodities futures, 213 Commodity underliers, 11, 15, 25 Compounding, 237, 245 Compounding frequency, 8890 Contango, 1089 Continuous dividends, 168, 16970, 172 Continuous-time models, 173 Contract price, 4, 14, 85. A 62-strike call option will be worth $4 if it goes up and 0 if it goes down. When any of these mortgages defaulted, they reasoned, it would be for factors unique to the individual mortgage holder, or at least to the region, and not due to something affecting all mortgages. Finch is using this swap to effectively convert a 3.9 percent xed-rate loan obligation into a oating rate, perhaps believing rates will on average oat below 3.9 percent over the life of the loan. AIG, one of the biggest of CDS casinos, all by itself required $180 billion of public assistance, primarily to pay off the gamblers. The payoff diagrams make the bad news perfectly clear: the exposure is hugetheoretically unlimited for written calls. Its analog for futures is the futures price. See also Hedging a derivatives position; Long position; Short position Premiums credit default swaps and, 5960, 61, 62, 6364 credit derivatives and, 68 credit-linked notes and, 66 options and, 37, 5253, 81, 82, 130 portfolio credit default swaps and, 67 Present value (PV), 9, 233 forward pricing and, 8791, 96, 97, 98, 99, 103, 104 of income, 98, 99 option pricing and, 14041, 250 of storage, 96, 97 swap pricing and, 112, 115 Previous (calendar convention), 241 Price, 3 bond, 199, 208 credit derivatives, 6869 delivery (see Delivery price) forwards (see Forward pricing) futures, 9, 27, 8591, 1057, 1089 indicative, 61 law of one, 9 market, 14950 mathematics of, 910 model, 18791 no-arbitrage, 1078, 134, 146 options (see Option pricing) overview, 78 spot (see Spot price) strike (see Strike price) swaps (see Swap pricing) taking a view of future, 4 underlier, 184, 185, 186, 192 value distinguished from, 78 Price bubbles, 195 Price discovery, 68 Price guarantees, 1, 125, 197 269 Price paths, 3941, 43 Price risk. Some underlier spot prices change almost continuously. The option, however, is the only one of the four with any inherent value upon inception. With each of these contracts, you are dealing with a future transaction you are certain will occur. Price Paths and Moneyness A price path is the course of an assets actual price as it changes over time. Strike Price (K): The strike price is just the stock price in reverse. And those arent just hypotheticals. So the interest is based on the remaining notional at the beginning of the period. In nancial parlance, a price path like this is known as a random walk. Reasonable? Under the terms of the loan, every three months for the next two years, Gondor will make an interest-only payment based on a oating rate of interest according to the London Inter-Bank Offered Rate (LIBOR) index (see Appendix A, All About Interest, for more about LIBOR). You can also trade options on ETFs. The price of such a portfolio gives you the price of the option, due to the so-called law of one price, which says two things with the same payoff must cost the same to prevent arbitrage. Option Value Versus Stock Value The rst thing to recognize when assessing the value of an option is that an option can be worth no more than a corresponding position in the underlier. Capital Ideas: The Improbable Origins of Modern Wall Street. TAB LE 9-3 Finch-Radley Swap from Perspective of Floating Value as of May 23, 2007, with 4 Periods Remaining Payment Fixed Fixed Fixed Fixed 5 6 7 8 Notional Accrual Start Accrual End $1,000,000 1,000,000 1,000,000 1,000,000 5/23/07 8/23/07 11/24/07 2/23/08 8/22/07 11/23/07 2/22/08 2/29/08 Accrual Days 91 92 90 6 Payment Rate Year Days Payment Future Value 3.90% 3.90 3.90 3.90 365 365 365 365 $9,723.29 9,830.14 9,616.44 641.10 Spot Rate 3.53% 3.54 3.59 3.59 Days till Payment 91 184 275 282 Payment Present Value $9,638.46 9,657.79 9,363.18 623.79 Net Fixed-Leg Present Value = $29,283.23 Floating Floating Floating Floating 5 6 7 8 ($1,000,000) (1,000,000) (1,000,000) (1,000,000) 5/23/07 8/23/07 11/24/07 2/23/08 8/22/07 11/23/07 2/22/08 2/29/08 91 92 90 6 3.53% 3.55 3.64 3.64 365 365 365 365 ($8,800.82) (8,947.95) (8,975.34) (598.36) 3.53% 3.54 3.59 3.59 91 184 275 282 ($8,724.04) (8,791.06) (8,738.97) 582.21) Net Floating-Leg Present Value = ($26,836.29) Swap Value = $2,446.94 Pricing Swaps 119 A New Swap Now lets work out another swap valuation problem but for a different purpose. 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