If you borrow money at LIBOR, the exact amounts of your future payment obligations are unpredictable at the time you take out the loan. Currency Swap So far, we have only considered swaps where both legs are denominated in the same currency. From day to day, the rates change unpredictably, but once xed, there is no quibble over the daily proclamations of those helpful British bankers. This one manages to expose the necessary math without getting too in the weeds with the proofs. Derivatives: The Tools That Changed Finance. (The yields were too low, remember, meaning the prices were too high.) After privatization and globalization financial market has entered into a new phase of global integration and liberalization. That translates to a lower price, meaning a lower commission (more or less) for the bond dealer. That would be using the third-party collision policy for hedging, which as you recall is one of only two reasons most people use derivatives. How is VAR calculated? MLX exists only in my head. Genres Finance Business. After all, plenty of the comparatively plain-vanilla subprime mortgage bonds went bust all by themselves with nary a CDO or CDS in sight. Fungible just means one is as good as another, as with barrels of oil and dollar bills, and liquid in this sense means there are large numbers of active buyers and sellers at any given time. Bonds are also sensitive to changes in interest rates, so they can be used as well for rho hedging. A Now is just like a forward except that the delivery price is spot and the delivery date is right now. 1998. Reset offset: For an accrual period commencing on some given day, which business days rate xing should we use? (In reality, disagreements in model prices are generally not as wide as these simple examples.) Lets choose 0.15. If you think about it, the U.S. stock market over many decades has been a bull market. Jorion, Philippe. Now, you generally cant buy or sell on a spot market the average stock represented by an equity index. CF CF9mo PV 7,500 7,500e 0.03(0.25) 7,500e 0.03(0.5) 7,500e 0.03(0.75) 7,500 7,443.96 7,388.34 7,333.13 29,665.43 Now we have everything in present-value terms, so we just plug the following amounts into the formula to get the future value of the whole enchilada: S 32 5,000 160,000 U 29,665.43 r 0.03 t1 Putting it all together we have: F (S U)e rt (160,000 29,665.43) e 0.03(1) 195,441.60 The fair-market delivery price for this contract is $195,441.60, or just about $39 per barrel. Use the date as-is. Option Market Making: Trading and Risk Analysis for the Financial and Commodity Option Markets. Some of these are more tedious than others, but all are well understood, and thats all that matters. Morris, Charles R. 2008. 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 The point is, to get out of a position, you may have to buy at a price higher than you would like or sell for a price lower than youd like, and you may even have trouble nding a bid or offer with the quantity (size) you need. Question for extra credit: What if these options The Option Contract FIGURE 41 5-3 European Call Option Strike Points were puts instead of calls? For understanding the math behind swap pricing, it helps to also understand the distinction between a spot rate and a forward rate and what a yield curve is all about. Parties to a forward, by contrast, are free to dene the terms of their contract however they mutually agree to. And for reasons soon to be revealed, it is crucial for transaction counterparties to agree on which days are business days and which are not. But with a forward, the long party gets to hang onto that cash and can invest it, earning interest. How might we arbitrage a mispriced forward? Youll never get more than $5 because the stock price cant go below zero. They also tend to be iso- FIGURE 6-3 Credit-Linked Note Credit Derivatives 67 lated from the credit woes of any other entity. And this makes sense, right? In the Gondor-Marlow swap example from the rst page of this chapter, one of the parties (Marlow Securities) is a swaps dealer or nancial institution that makes a market in swapsthat is, provides them to parties who need them. This is the cost of maintaining or carrying a forward position over time. Keep the remaining $0.10 as your prot. The Gondor example, where one party was a swaps dealer, is far more common than the Boris-Chloe example. A market price of $10,000 implies a yield of 5 percent (i.e., $500/$10,000), and a market price of $9,000 implies a yield of 6 percent (lower price, higher yield). Hedging a Derivatives Position Greek Theta () Ratio 187 Facts Option price Time to expiration Technically expresses rate of loss with passage of time, so is always positive. Like any bond investor, he or she makes an investment of principal in return for future interest payments, all with the implicit understanding the investor may never see that principal should the issuer default. It was crazy. We can execute a pay-LIBOR, receive-prime swap with the same notional and payment schedule as our loan. Vega, theta, and rho also tell us how the option price changes per unit change in underlier volatility (1 percent change), time to expiration (one day), and the risk-free interest rate (1 percent), respectively. Now, this might seem backward at rst, with the protection seller making payments, but stay with me. The two legs of a swap need not have the same The Swap Contract 33 tenor. payoffs (pay close attention to these pay-offs, since all the strategies in the book are derived out of these basic payoffs). For example, a present value of $100 (think of this as the investment or principal All About Derivatives 88 FIGURE 8-1 Future Valuation and Present Valuation amount) in an account that pays 6 percent annually grows to $106 (a future value) in one year. Appendix A 228 FIGURE A-4 Partial LIBOR Spot Curve Going out Four Years once issued, they are heavily traded on secondary markets. Here are the payoff functions for a forward: PFwd,Long S K (Formula 2.1) PFwd,Short K S (Formula 2.2) Formulas 2.1 and 2.2 say the payoff for the long party (PFwd,Long) is spot (S) minus delivery price (K), and the payoff for the short party (PFwd,Short) is delivery price (K) minus spot (S). Still, owing to widespread acceptance of Black-Scholes, we can say clearly the markets believe that these assumptions are reasonable. With such an FRA in place, we are obligated to pay 3.25 percent, and the short party is obligated to accept 3.25 percent for a six-month loan commencing in three months. In our formulas, it is just some proportional quantity of stock. This characteristic is known as risk neutrality, and well explain it later in Appendix C, More Binomial Option Pricing. For now, sufce it to say we can disregard risk preferences when pricing derivatives and can therefore eliminate the risk premium component of an interest rate and use whats left over. Sorry, preview is currently unavailable. This trader can buy both a call and a put, both with the same strike price and expiration and different premiums, resulting in a net payoff that looks roughly like Figure 5-18. While waiting to exercise the option, that savings earns interest. Because there are two alternatives to exercise in which you are better off. First lets handle storage. Leeson spent a few years in jail, after which he gave speeches on the dangers of rogue traders for a reported $100,000 a crack. Its holder could theoretically exercise at one of the times it is in the money and therefore make some money. Black-Scholes depends on calculus, but you certainly dont need to know calculus to use it. A nancial instrument is just a standard type of agreement, or contract if you will, that bestows certain nancial rights and/or responsibilities to its parties. MORE ON RISK MANAGEMENT So what other fun stuff goes on in the heady world of nancial risk management? Thats their power. PDF | This is a module writing in Financial derivatives. That opportunity for upside and shield from downside is worth something, and that something is known as time value. 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). Interest Rate Options The family of basic interest rate derivative also includes the cap, oor, collar, and swaption. (The spot price changes over the life of a forward, but the delivery price does not; the difference between them must change as well. So the value of the replicating portfolio at time zero is $2.53. I2nd 6 mo. Its no wonder that rms in the derivatives business, especially market makers and arbitrageurs, spend vast sums of money on ever-faster computer systems for pricing and trading derivatives. Here you dont mind giving up remaining dividends and optionality, because youll make more by selling the stock and investing the proceeds. Someone comes along and buys 20 calls from you. Recall that a futures is just an exchange-traded forward whose value is affected by the daily marking to market, and with some math, we can back out that effect. If a rm knows it will need to borrow money in the future, it can lock in the interest rate with a forward rate agreement (FRA, pronounced so it rhymes with ahh). In the example in Table B-1, then, the rstperiod interest is 7 percent of the full $100 million notional, or the notional as of the beginning of the rst period. And how is this a cost of carry? You can see where this is going, of course. A third way we can organize the universe of options is by underlier. Table 9-1 shows the spot and forward rates from the curves, going out two years, which well use in our calculations. Here it is for our previous example: FV PV e rt 100 e 0.06(1) 106.18 All About Derivatives 90 So the most interest we can earn is $6.18, for a maximum future value of $106.18. Volatility, you should know, is the most interesting price factor to option traders. Bottom line, oating-rate debt is easier to come by, so theres a lot of it out there. In practice, there are a couple of ways one can obtain a probability of default. The over-the-counter (OTC ) market is where two parties nd each other and then work directly with each otherand nobody elseto formulate, execute, and enforce a derivative transaction. And how do we do that? The payoff diagram is a visual representation of the not-sovisual payoff function. A hedge fund combines short Eurodollar futures and long U.S. Treasury bonds to arbitrage pricing discrepancies between those two instruments. Remember why? See also Fixed-oating interest rate swaps Floor, interest rate, 35 Following (calendar convention), 241 Foreign-exchange (FX) forwards, 13, 14, 61, 7476, 77 Foreign-exchange (FX) market, 11 Foreign-exchange (FX) rates, 35 Forward(s), 3, 4, 6, 1321 as an advance sales agreement, 1314 common, 1415 dened, 2 foreign-exchange, 13, 14, 61, 7476, 77 futures compared with, 14, 2324, 27, 7677 hedging with, 14, 7476 key benet of, 14 obligations and, 1516 payoff, 1621, 46, 9192 Forward contract value, 87, 1035 Forward credit default swaps (CDSs), 67 Forward curves, 113, 119, 22425, 23235 Forward price of currency (formula), 101 Forward pricing, 9, 27, 85105, 1078 formulas, 95102 idea behind, 9294 option pricing compared with, 145 Forward rate(s), 9, 125, 22425, 23235 dened, 111 swap pricing and, 111, 113, 114, 116, 117 Forward rate agreements (FRAs), 15, 79 swaps as a portfolio of, 12028 three by nine, 125 Forward with income (formula), 98101 Index Forward with storage (formula), 9698 Freddie Mac, 210 Free boundary problem, 172 Frictionless markets, 165, 166 Front-term/-month contracts, 226n2, 227 Fungibility, 11, 45 Future(s), 4, 7, 1516, 2328 cancelation and, 25, 106 commodities, 213 common, 2526 daily settlement and, 24, 2627 dened, 2 Eurodollar, 79, 177, 178, 193, 214, 22527 forwards compared with, 14, 2324, 27, 7677 hedging with, 72, 7678 option pricing on, 17374 pricing, 9, 27, 8591, 1057, 1089 Future(s) contract value, 87, 1057 Future values, 9 forward pricing and, 8791, 96, 97, 98, 100 swap pricing and, 112 Future(s)-forward adjustment, 226 Gamma (), 184, 185, 186, 192 Geometric Brownian motion, 164 Greeks, 18493. Mathematically speaking, they assumed that the default rates of the individual subprime mortgages were statistically uncorrelated. Lets dissect and expand it a bit to see what this deriving is all about. Figure 5-3 superimposes the option strikes atop the price path. But not for long. Apples are currently selling wholesale for $6 per bushel. 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. Youll learn more about that when we get into valuation in Chapter 9, Pricing Swaps.) Payments are made only on the payment dates prescribed by the swapin this case, every 12 months for 10 years. Now, short forward positions always make money when the spot price is below delivery price. Say the index is currently at 1,200, the current value of Boxwoods stock portfolio is $120 million, and the rm doesnt want it to fall below $110 million. Imagine holding 100 American call options with a strike price of $30 on a stock currently trading for $50, and the options dont expire for another three months. Why, yes, it does. Any other rate would permit arbitrage. 194 All About Derivatives Its no wonder, of course, why risk management is so important in the world of derivatives. If some of the 1,000 homeowners fail to make their mortgage payment (not unlikely), the triple-B holders wont get their full interest, or perhaps will get none at all. The good news is, each disaster provides both a wake-up call to the importance of risk management and valuable real-worldnot hypotheticalexamples of what can happen. If not, he or she loses the amount paid for the option100 percent of the investment. The currency market, also known as the foreign-exchange (FX ) market, is in fact the worlds largest spot market of any kind. Well see, for example, that a forward contract is like a highly customizable futures contract. Recall that long futures positions lose money when prices decline, and the MGRM positions were huge. Even though the market for credit contracts is impressive and growing every day, compared with the market for forwards, futures, swaps, and options, it is still rather small. Wed love your help. In our rst climb up the binomial tree, we created a replicating portfolioor synthetic option, as it is sometimes knownby borrowing money and buying stock. And what of the writer, the short party to a call option? Its pretty cool stuff. For position value, we need to know the number of contracts involved and whether one is long or short. Because forward prices and futures prices are calculated intuitively the same way, and to simplify the text, whenever you read forward price, you can safely assume it applies to a futures price, too. They sold lots and lots and lots of these things, grinning all the way to the bank, thinking they were collecting bets that would never pay off. Lets see if that bears out. You are a dealer and offer it for $2.00, for a theoretical edge of $1.50 per contract (recall each contract is for 100 shares of the underlier). And due to the law of no-arbitrage, two positions with the same payoff must cost the same. 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. A little bonus? Say you are a seller of a CDS where the reference entity is the corporate issuer of a bond with a face value of $10 million. A U.S. savings bond wont make you wealthy, but you can count on Uncle Sam to return your investment and at least a little bit more. And a short position in a bond, recall, is just the borrowing of some money. With a real put option, the holder is entitled to a payoff simply if the strike price is above spot price on or before some expiration date. They are something of a fudge factor to make the equality between forward prices and their other input factors hold true. Daily settlement: This is the biggie. (The risk-free interest rate r has not changed.) All About Derivatives 162 entire earth for exactly one day. Swap Conventions 239 Straight-Line Amortization Straight-line amortization is a special case of additive amortization where the notional steps down by the same amount each period, and the nal notional amount is the stepping amount. CHAPTER 12 Derivatives and the 2008 Financial Meltdown There was, of course, no single cause of the economic earthquake of 2008. Say were interested in the 50-strike call. This leads to liquidity risk, which is simply the chance that you may not nd a trading opportunity at a desirable price when you are ready to get out of a position. Normally distributed random variables can take values less than zero, and stock prices are always positive, so they cant be normally distributed. A million bucks from Citibank might go for $65,000 of interest per year. As the futures did what futures do, their payoff neutralized any loss or gain from buying oil spot to make good on their forwards. But what interest rate should we use? As a result, most option hedgers spend most of their time worrying about these. 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! Now back on the bus. The Black-Scholes method, which grows directly out of the binomial tree method, uses a tree with an innite number of branches. Any errors that might have slipped into this one will be noted at www.michaelpdurbin.com. 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. But when you cant nd someone, it sure is nice to have those liquid futures markets. Or the wheat specied in the contract may not be exactly the type of wheat Royal needs. HEDGING WITH FUTURES CONTRACTS In Chapter 3, The Futures Contract, Royal Mill used wheat futures to lock in the price of wheat for a purchase in six months. And by the law of no arbitrage, two positions with the same payoff must have the same value. And where does credit risk protection t into this picture? For x 26, the area is 96 percent. (To the short party, interest truly is a cost of carry; to the long party, its actually a benet, but we still call it cost of carry.) And by applying a futures-forward adjustment involving math we wont delve into, we can deduce a forward price from a futures price.3 And from such forward rates, we can deduce spot rates with which to extend our spot curve past the 12-month term available from the BBA.4 Following are some spot rates we might infer from the Eurodollars market: 2 3 4 Youll see this lingo used all the time in the securities markets. But, of course, they can and do. Then at ve months, the price drops back to around $62, and so on. By the same reasoning and math, they will lose $250 if rates increase to 3.6 percent. After one year, the dollar-won exchange rate decreases to 1,000 KRW/USD. One thing you might hear about is the idea of value at risk (VAR). Using Black-Scholes in this way, modied this way for future options, is known as Blacks model. The left slope is steeper than Pricing Options FIGURE 165 10 -13 Normal and Lognormal Distributions the right, which says there are fewer price decreases than increases. Not so for Royal Mill. Forward contract value and futures contract value differ due to daily settlement, and youll see further on in this chapter exactly why. If youve ever purchased a car and agreed to buy it before the date you actually took delivery, youve already been long a forward contract. Heres another quick example involving a future purchase of foreign currency: The U.S.-based Gizmo Company agrees to purchase 100,000 circuit boards from a South Korean manufacturer in one year, at a per-unit price of 24 won (the currency of South Korea, abbreviated KRW). Krinkets are currently trading for $0.35. As we saw in Chapter 1, for example, you can buy stock options to replicate a stock portfolios payoff for much less money than buying some desired amount of the stock itself. 12 mo. And if you dont pay me that interest, it will cost you more interest, and so on, until you nally fork over the dough. 2 The convention I describe here is known technically in some quarters as actual/365 (xed) and is not to be confused with a different actual/365, which is also known as actual/actual. Now you see why Im giving only a few examples. And as with forwards and futures, parties to a swap are obligated to buy or sell at the predetermined rates, no matter what happens in the spot markets. And how can you tell? Well learn more about LIBOR in Appendix A, All About Interest. The risk-free rate is almost always lower than a rate of return that carries risk. Figure 5-5 depicts the changing moneyness of CZED62 against our imaginary price path. You would expect the implied volatility from two market options that differ only in expiration to be the same. This isnt literally necessary. But wait, doesnt that describe the CDS buyer? One way is to reduce uncertainty about a single future interest rate payment based on a oating rate of interest. Using our previous examples, if a trade is based on a New York calendar, then September 18 is a business day. All rights reserved. The return of stock with a volatility of 0.05 changes 5 percent. 18 mo. Another tool is the spread trade, an option strategy in which you take simultaneous positions in different options (say, both November and December options with otherwise identical features), such that the value of the combined position is sensitive to changes in volatility. Invest the remaining $17.57 at 6 percent. You cant make toast without bread, and you cant make a forward curve without a spot curve. Wiley. 1997. CHAPTER 4 The Swap Contract A swap contract is an agreement to exchange future cash ows. We can express this mathematically by using Su for the stock price after an up move, Sd for the stock price after a down move, Cu for the option value after an up move in the stock, and Cd for the option value after a down move in the stock: Su 66 Cu 4 Sd 54 Cd 0 (Cu Cd)/(Su Sd) (4 0)/(66 54) 4/12 1/3 0.3333 Pricing Options 137 We can generalize this delta formula, thus making it more useful later on, by denoting 1 plus stock return after an up move by the letter u and 1 plus the stock return after a down move by the letter d, and replacing the (Su Sd) factor with the product of S and the difference between u and d (see Formula 10.1). Figure 9-2 shows these dates on the calendar. This fact is illustrated in the payoff function by max(. For periods longer than one year, we simply increase the t factor. If you hold that futures option and the futures price at expiration is $34, your option expires worthless. For ushering this second edition into existence, I must thank the forever resourceful, inventive, and persistent Morgan Ertel, my editor at McGraw-Hill. The short party does. A swaption grants the right, but not the obligation, to enter into a swap in the future. Such obligations are sometimes known as contingent liabilities. So while it does not explain all the fancy stuff, it does explain the basic contracts forwards, futures, swaps, and optionsfrom which nearly every derivative is derived. The bank uses long Eurodollar futures to hedge the swap. The number of option strategies one can cook up is limited only by ones imagination and tolerance for risk. Common selections here are actual/365, actual/360, 30/360, and others. Delta Underlier price Gamma Underlier price Vega Underlier volatility Theta Time to expiration Rho Interest rates Hedging a Derivatives Position 185 kappa, which you might see in some academic texts. COMMON FORWARDS Among the most common forwards are those on currency. They might also eat one now and again (not always by choice), but mostly they act as sellers to want-to-be buyers and as buyers to want-to-be sellers, and they like to do so by taking on as little risk as possible. LIQUIDITY RISK While an exchange provides liquidity by always having buyers for prospective sellers and vice versa, some contracts are more liquid than others. 1997. If the value of the thing you buy should fall below what you owe, you can get into a heap of trouble right quick. Is there any way to avoid that? The formulas for present valuation and future valuation are rather intuitive, and well explain them fully before we actually use them. Sure. a. We wont go further into arbitrage, but just know its out there in a big way. 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