(BQ) Part 2 book “Options, futures, and other derivatives” has contents: The greek letters, volatility smiles, basic numerical procedures, credit derivatives, estimating volatilities and correlations, real options, equilibrium models of the short rate,… and other contents.
www.downloadslide.net 19 The Greek Letters C H A P T E R A financial institution that sells an option to a client in the over-the-counter markets is faced with the problem of managing its risk If the option happens to be the same as one that is traded actively on an exchange or in the OTC market, the financial institution can neutralize its exposure by buying the same option as it has sold But when the option has been tailored to the needs of a client and does not correspond to the standardized products traded by exchanges, hedging the exposure is far more difficult In this chapter we discuss some of the alternative approaches to this problem We cover what are commonly referred to as the ‘‘Greek letters’’, or simply the ‘‘Greeks’’ Each Greek letter measures a different dimension to the risk in an option position and the aim of a trader is to manage the Greeks so that all risks are acceptable The analysis presented in this chapter is applicable to market makers in options on an exchange as well as to traders working in the over-the-counter market for financial institutions Toward the end of the chapter, we will consider the creation of options synthetically This turns out to be very closely related to the hedging of options Creating an option position synthetically is essentially the same task as hedging the opposite option position For example, creating a long call option synthetically is the same as hedging a short position in the call option 19.1 ILLUSTRATION In the next few sections we use as an example the position of a financial institution that has sold for $300,000 a European call option on 100,000 shares of a non-dividendpaying stock We assume that the stock price is $49, the strike price is $50, the risk-free interest rate is 5% per annum, the stock price volatility is 20% per annum, the time to maturity is 20 weeks (0.3846 years), and the expected return from the stock is 13% per annum.1 With our usual notation, this means that S0 ¼ 49; K ¼ 50; r ¼ 0:05; ¼ 0:20; T ¼ 0:3846; ¼ 0:13 The Black–Scholes–Merton price of the option is about $240,000 (This is because the As shown in Chapters 13 and 15, the expected return is irrelevant to the pricing of an option It is given here because it can have some bearing on the effectiveness of a hedging procedure 397 www.downloadslide.net 398 CHAPTER 19 value of an option to buy one share is $2.40.) The financial institution has therefore sold a product for $60,000 more than its theoretical value But it is faced with the problem of hedging the risks.2 19.2 NAKED AND COVERED POSITIONS One strategy open to the financial institution is to nothing This is sometimes referred to as a naked position It is a strategy that works well if the stock price is below $50 at the end of the 20 weeks The option then costs the financial institution nothing and it makes a profit of $300,000 A naked position works less well if the call is exercised because the financial institution then has to buy 100,000 shares at the market price prevailing in 20 weeks to cover the call The cost to the financial institution is 100,000 times the amount by which the stock price exceeds the strike price For example, if after 20 weeks the stock price is $60, the option costs the financial institution $1,000,000 This is considerably greater than the $300,000 charged for the option As an alternative to a naked position, the financial institution can adopt a covered position This involves buying 100,000 shares as soon as the option has been sold If the option is exercised, this strategy works well, but in other circumstances it could lead to a significant loss For example, if the stock price drops to $40, the financial institution loses $900,000 on its stock position This is also considerably greater than the $300,000 charged for the option.3 Neither a naked position nor a covered position provides a good hedge If the assumptions underlying the Black–Scholes–Merton formula hold, the cost to the financial institution should always be $240,000 on average for both approaches.4 But on any one occasion the cost is liable to range from zero to over $1,000,000 A good hedge would ensure that the cost is always close to $240,000 A Stop-Loss Strategy One interesting hedging procedure that is sometimes proposed involves a stop-loss strategy To illustrate the basic idea, consider an institution that has written a call option with strike price K to buy one unit of a stock The hedging procedure involves buying one unit of the stock as soon as its price rises above K and selling it as soon as its price falls below K The objective is to hold a naked position whenever the stock price is less than K and a covered position whenever the stock price is greater than K The procedure is designed to ensure that at time T the institution owns the stock if the option closes in the money and does not own it if the option closes out of the money In the situation illustrated in Figure 19.1, it involves buying the stock at time t1 , selling it at time t2 , buying it at time t3 , selling it at time t4 , buying it at time t5 , and delivering it at time T A call option on a non-dividend-paying stock is a convenient example with which to develop our ideas The points that will be made apply to other types of options and to other derivatives Put–call parity shows that the exposure from writing a covered call is the same as the exposure from writing a naked put More precisely, the present value of the expected cost is $240,000 for both approaches assuming that appropriate risk-adjusted discount rates are used www.downloadslide.net 399 The Greek Letters Figure 19.1 A stop-loss strategy Stock price, S(t) K Buy t1 Sell Buy t2 t3 Sell Buy Deliver t4 t5 T Time, t As usual, we denote the initial stock price by S0 The cost of setting up the hedge initially is S0 if S0 > K and zero otherwise It seems as though the total cost, Q, of writing and hedging the option is the option’s initial intrinsic value: Q ẳ maxS0 K; 0ị 19:1ị This is because all purchases and sales subsequent to time are made at price K If this were in fact correct, the hedging procedure would work perfectly in the absence of transaction costs Furthermore, the cost of hedging the option would always be less than its Black–Scholes–Merton price Thus, a trader could earn riskless profits by writing options and hedging them There are two key reasons why equation (19.1) is incorrect The first is that the cash flows to the hedger occur at different times and must be discounted The second is that purchases and sales cannot be made at exactly the same price K This second point is critical If we assume a risk-neutral world with zero interest rates, we can justify ignoring the time value of money But we cannot legitimately assume that both purchases and sales are made at the same price If markets are efficient, the hedger cannot know whether, when the stock price equals K, it will continue above or below K As a practical matter, purchases must be made at a price K þ and sales must be made at a price K À , for some small positive number Thus, every purchase and subsequent sale involves a cost (apart from transaction costs) of 2 A natural response on the part of the hedger is to monitor price movements more closely, so that is reduced Assuming that stock prices change continuously, can be made arbitrarily small by monitoring the stock prices closely But as is made smaller, trades tend to occur more frequently Thus, the lower cost per trade is offset by the increased frequency of trading As ! 0, the expected number of trades tends to infinity.5 As mentioned in Section 14.2, the expected number of times a Wiener process equals any particular value in a given time interval is infinite www.downloadslide.net 400 CHAPTER 19 Table 19.1 Performance of stop-loss strategy The performance measure is the ratio of the standard deviation of the cost of writing the option and hedging it to the theoretical price of the option Át (weeks) 0.5 0.25 Hedge performance 0.98 0.93 0.83 0.79 0.77 0.76 A stop-loss strategy, although superficially attractive, does not work particularly well as a hedging procedure Consider its use for an out-of-the-money option If the stock price never reaches the strike price K, the hedging procedure costs nothing If the path of the stock price crosses the strike price level many times, the procedure is quite expensive Monte Carlo simulation can be used to assess the overall performance of stop-loss hedging This involves randomly sampling paths for the stock price and observing the results of using the procedure Table 19.1 shows the results for the option considered in Section 19.1 It assumes that the stock price is observed at the end of time intervals of length Át.6 The hedge performance measure in Table 19.1 is the ratio of the standard deviation of the cost of hedging the option to the Black–Scholes–Merton price (The cost of hedging was calculated as the cumulative cost excluding the impact of interest payments and discounting.) Each result is based on one million sample paths for the stock price An effective hedging scheme should have a hedge performance measure close to zero In this case, it seems to stay above 0.7 regardless of how small Át is This emphasizes that the stop-loss strategy is not a good hedging procedure 19.3 GREEK LETTER CALCULATION Most traders use more sophisticated hedging procedures than those mentioned so far These hedging procedures involve calculating measures such as delta, gamma, and vega The measures are collectively referred to as Greek letters They quantify different aspects of the risk in an option position This chapter considers the properties of some of most important Greek letters In order to calculate a Greek letter, it is necessary to assume an option pricing model Traders usually assume the Black–Scholes–Merton model (or its extensions in Chapters 17 and 18) for European options and the binomial tree model (introduced in Chapter 13) for American options (As has been pointed out, the latter makes the same assumptions as Black–Scholes–Merton model.) When calculating Greek letters, traders normally set the volatility equal to the current implied volatility This approach, which is sometimes referred to as using the ‘‘practitioner Black–Scholes model,’’ is appealing When volatility is set equal to the implied volatility, the model gives the option price at a particular time as an exact function of the price of the underlying asset, the implied volatility, interest rates, and (possibly) dividends The only way the option price can change in a short time period is if one of these variables changes A trader naturally feels confident if the risks of changes in all these variables have been adequately hedged The precise hedging rule used was as follows If the stock price moves from below K to above K in a time interval of length Át, it is bought at the end of the interval If it moves from above K to below K in the time interval, it is sold at the end of the interval; otherwise, no action is taken www.downloadslide.net 401 The Greek Letters In this chapter, we first consider the calculation of Greek letters for a European option on a non-dividend-paying stock We then present results for other European options Chapter 21 will show how Greek letters can be calculated for American-style options 19.4 DELTA HEDGING The delta (Á) of an option was introduced in Chapter 13 It is defined as the rate of change of the option price with respect to the price of the underlying asset It is the slope of the curve that relates the option price to the underlying asset price Suppose that the delta of a call option on a stock is 0.6 This means that when the stock price changes by a small amount, the option price changes by about 60% of that amount Figure 19.2 shows the relationship between a call price and the underlying stock price When the stock price corresponds to point A, the option price corresponds to point B, and Á is the slope of the line indicated In general, Á¼ @c @S where c is the price of the call option and S is the stock price Suppose that, in Figure 19.2, the stock price is $100 and the option price is $10 Imagine an investor who has sold call options to buy 2,000 shares of a stock The investor’s position could be hedged by buying 0:6  2,000 ¼ 1,200 shares The gain (loss) on the stock position would then tend to offset the loss (gain) on the option position For example, if the stock price goes up by $1 (producing a gain of $1,200 on the shares purchased), the option price will tend to go up by 0:6  $1 ¼ $0:60 (producing a loss of $1,200 on the options written); if the stock price goes down by $1 (producing a loss of $1,200 on the shares purchased), the option price will tend to go down by $0.60 (producing a gain of $1,200 on the options written) In this example, the delta of the trader’s short position in 2,000 options is 0:6  ðÀ2,000Þ ¼ À1,200 This means that the trader loses 1,200ÁS on the option position when the stock price Figure 19.2 Calculation of delta Option price Slope = Δ = 0.6 B Stock price A www.downloadslide.net 402 CHAPTER 19 increases by ÁS The delta of one share of the stock is 1.0, so that the long position in 1,200 shares has a delta of ỵ1,200 The delta of the traders overall position in our example is, therefore, zero The delta of the stock position offsets the delta of the option position A position with a delta of zero is referred to as delta neutral It is important to realize that, since the delta of an option does not remain constant, the trader’s position remains delta hedged (or delta neutral) for only a relatively short period of time The hedge has to be adjusted periodically This is known as rebalancing In our example, by the end of day the stock price might have increased to $110 As indicated by Figure 19.2, an increase in the stock price leads to an increase in delta Suppose that delta rises from 0.60 to 0.65 An extra 0:05  2,000 ¼ 100 shares would then have to be purchased to maintain the hedge A procedure such as this, where the hedge is adjusted on a regular basis, is referred to as dynamic hedging It can be contrasted with static hedging, where a hedge is set up initially and never adjusted Static hedging is sometimes also referred to as ‘‘hedge-and-forget.’’ Delta is closely related to the Black–Scholes–Merton analysis As explained in Chapter 15, the Black–Scholes–Merton differential equation can be derived by setting up a riskless portfolio consisting of a position in an option on a stock and a position in the stock Expressed in terms of Á, the portfolio is 1: option ỵ: shares of the stock Using our new terminology, we can say that options can be valued by setting up a deltaneutral position and arguing that the return on the position should (instantaneously) be the risk-free interest rate Delta of European Stock Options For a European call option on a non-dividend-paying stock, it can be shown (see Problem 15.17) that the Black–Scholes–Merton model gives ÁðcallÞ ¼ Nðd1 Þ Figure 19.3 Variation of delta with stock price for (a) a call option and (b) a put option on a non-dividend-paying stock (K ¼ 50, r ¼ 0, ¼ 25%, T ¼ 2) 1.0 0.0 0.8 -0.2 0.6 -0.4 0.4 -0.6 20 40 60 Stock price ($) 0.2 -0.8 Stock price ($) 0.0 -1.0 20 40 60 (a) 80 100 (b) 80 100 www.downloadslide.net 403 The Greek Letters Figure 19.4 Typical patterns for variation of delta with time to maturity for a call option (S0 ¼ 50, r ¼ 0, ¼ 25%) 1.0 Delta 0.8 0.6 0.4 In the money (K = 40) At the money (K = 50) Out of the money (K = 60) 0.2 Time to maturity (years) 0.0 10 where d1 is defined as in equation (15.20) and NðxÞ is the cumulative distribution function for a standard normal distribution The formula gives the delta of a long position in one call option The delta of a short position in one call option is ÀNðd1 Þ Using delta hedging for a short position in a European call option involves maintaining a long position of Nðd1 Þ for each option sold Similarly, using delta hedging for a long position in a European call option involves maintaining a short position of Nðd1 Þ shares for each option purchased For a European put option on a non-dividend-paying stock, delta is given by putị ẳ Nd1 ị Delta is negative, which means that a long position in a put option should be hedged with a long position in the underlying stock, and a short position in a put option should be hedged with a short position in the underlying stock Figure 19.3 shows the variation of the delta of a call option and a put option with the stock price Figure 19.4 shows the variation of delta with the time to maturity for in-the-money, at-the-money, and out-of-the-money call options Example 19.1 Consider again the call option on a non-dividend-paying stock in Section 19.1 where the stock price is $49, the strike price is $50, the risk-free rate is 5%, the time to maturity is 20 weeks (¼ 0:3846 years), and the volatility is 20% In this case, d1 ẳ ln49=50ị ỵ 0:05 ỵ 0:22 =2ị 0:3846 p ẳ 0:0542 0:2 0:3846 Delta is Nd1 ị, or 0.522 When the stock price changes by ÁS, the option price changes by 0:522ÁS www.downloadslide.net 404 CHAPTER 19 Dynamic Aspects of Delta Hedging Tables 19.2 and 19.3 provide two examples of the operation of delta hedging for the example in Section 19.1, where 100,000 call options are sold The hedge is assumed to be adjusted or rebalanced weekly and the assumptions underlying the Black–Scholes– Merton model are assumed to hold with the volatility staying constant at 20% The initial value of delta for a single option is calculated in Example 19.1 as 0.522 This means that the delta of the option position is initially À100,000  0:522, or À52,200 As soon as the option is written, $2,557,800 must be borrowed to buy 52,200 shares at a price of $49 to create a delta-neutral position The rate of interest is 5% An interest cost of approximately $2,500 is therefore incurred in the first week In Table 19.2, the stock price falls by the end of the first week to $48.12 The delta of the option declines to 0.458, so that the new delta of the option position is À45,800 This means that 6,400 of the shares initially purchased are sold to maintain the delta-neutral hedge The strategy realizes $308,000 in cash, and the cumulative borrowings at the end of Week are reduced to $2,252,300 During the second week, the stock price reduces to $47.37, delta declines again, and so on Toward the end of the life of the option, it becomes apparent that the option will be exercised and the delta of the option approaches 1.0 By Week 20, therefore, the hedger has a fully covered position The Table 19.2 Simulation of delta hedging Option closes in the money and cost of hedging is $263,300 Week Stock price Delta Shares purchased Cost of shares purchased ($000) Cumulative cost including interest ($000) Interest cost ($000) 10 11 12 13 14 15 16 17 18 19 20 49.00 48.12 47.37 50.25 51.75 53.12 53.00 51.87 51.38 53.00 49.88 48.50 49.88 50.37 52.13 51.88 52.87 54.87 54.62 55.87 57.25 0.522 0.458 0.400 0.596 0.693 0.774 0.771 0.706 0.674 0.787 0.550 0.413 0.542 0.591 0.768 0.759 0.865 0.978 0.990 1.000 1.000 52,200 (6,400) (5,800) 19,600 9,700 8,100 (300) (6,500) (3,200) 11,300 (23,700) (13,700) 12,900 4,900 17,700 (900) 10,600 11,300 1,200 1,000 2,557.8 (308.0) (274.7) 984.9 502.0 430.3 (15.9) (337.2) (164.4) 598.9 (1,182.2) (664.4) 643.5 246.8 922.7 (46.7) 560.4 620.0 65.5 55.9 0.0 2,557.8 2,252.3 1,979.8 2,966.6 3,471.5 3,905.1 3,893.0 3,559.5 3,398.5 4,000.7 2,822.3 2,160.6 2,806.2 3,055.7 3,981.3 3,938.4 4,502.6 5,126.9 5,197.3 5,258.2 5,263.3 2.5 2.2 1.9 2.9 3.3 3.8 3.7 3.4 3.3 3.8 2.7 2.1 2.7 2.9 3.8 3.8 4.3 4.9 5.0 5.1 www.downloadslide.net 405 The Greek Letters Table 19.3 Simulation of delta hedging Option closes out of the money and cost of hedging is $256,600 Week Stock price Delta Shares purchased Cost of shares purchased ($000) Cumulative cost including interest ($000) Interest cost ($000) 10 11 12 13 14 15 16 17 18 19 20 49.00 49.75 52.00 50.00 48.38 48.25 48.75 49.63 48.25 48.25 51.12 51.50 49.88 49.88 48.75 47.50 48.00 46.25 48.13 46.63 48.12 0.522 0.568 0.705 0.579 0.459 0.443 0.475 0.540 0.420 0.410 0.658 0.692 0.542 0.538 0.400 0.236 0.261 0.062 0.183 0.007 0.000 52,200 4,600 13,700 (12,600) (12,000) (1,600) 3,200 6,500 (12,000) (1,000) 24,800 3,400 (15,000) (400) (13,800) (16,400) 2,500 (19,900) 12,100 (17,600) (700) 2,557.8 228.9 712.4 (630.0) (580.6) (77.2) 156.0 322.6 (579.0) (48.2) 1,267.8 175.1 (748.2) (20.0) (672.7) (779.0) 120.0 (920.4) 582.4 (820.7) (33.7) 2,557.8 2,789.2 3,504.3 2,877.7 2,299.9 2,224.9 2,383.0 2,707.9 2,131.5 2,085.4 3,355.2 3,533.5 2,788.7 2,771.4 2,101.4 1,324.4 1,445.7 526.7 1,109.6 290.0 256.6 2.5 2.7 3.4 2.8 2.2 2.1 2.3 2.6 2.1 2.0 3.2 3.4 2.7 2.7 2.0 1.3 1.4 0.5 1.1 0.3 hedger receives $5 million for the stock held, so that the total cost of writing the option and hedging it is $263,300 Table 19.3 illustrates an alternative sequence of events such that the option closes out of the money As it becomes clear that the option will not be exercised, delta approaches zero By Week 20 the hedger has a naked position and has incurred costs totaling $256,600 In Tables 19.2 and 19.3, the costs of hedging the option, when discounted to the beginning of the period, are close to but not exactly the same as the Black–Scholes– Merton price of $240,000 If the hedging worked perfectly, the cost of hedging would, after discounting, be exactly equal to the Black–Scholes–Merton price for every simulated stock price path The reason for the variation in the hedging cost is that the hedge is rebalanced only once a week As rebalancing takes place more frequently, the variation in the hedging cost is reduced Of course, the examples in Tables 19.2 and 19.3 are idealized in that they assume that the volatility is constant and there are no transaction costs Table 19.4 shows statistics on the performance of delta hedging obtained from one million random stock price paths in our example The performance measure is calculated, similarly to Table 19.1, as the ratio of the standard deviation of the cost of hedging the option to the Black–Scholes–Merton price of the option It is clear that delta hedging is a www.downloadslide.net 406 CHAPTER 19 Table 19.4 Performance of delta hedging The performance measure is the ratio of the standard deviation of the cost of writing the option and hedging it to the theoretical price of the option Time between hedge rebalancing (weeks): 0.5 0.25 Performance measure: 0.42 0.38 0.28 0.21 0.16 0.13 great improvement over a stop-loss strategy Unlike a stop-loss strategy, the performance of delta-hedging gets steadily better as the hedge is monitored more frequently Delta hedging aims to keep the value of the financial institution’s position as close to unchanged as possible Initially, the value of the written option is $240,000 In the situation depicted in Table 19.2, the value of the option can be calculated as $414,500 in Week (This value is obtained from the Black–Scholes–Merton model by setting the stock price equal to $53 and the time to maturity equal to 11 weeks.) Thus, the financial institution has lost $174,500 on its short option position Its cash position, as measured by the cumulative cost, is $1,442,900 worse in Week than in Week The value of the shares held has increased from $2,557,800 to $4,171,100 The net effect of all this is that the value of the financial institution’s position has changed by only $4,100 between Week and Week Where the Cost Comes From The delta-hedging procedure in Tables 19.2 and 19.3 creates the equivalent of a long position in the option This neutralizes the short position the financial institution created by writing the option As the tables illustrate, delta hedging a short position generally involves selling stock just after the price has gone down and buying stock just after the price has gone up It might be termed a buy-high, sell-low trading strategy! The average cost of $240,000 comes from the present value of the difference between the price at which stock is purchased and the price at which it is sold Delta of a Portfolio The delta of a portfolio of options or other derivatives dependent on a single asset whose price is S is @Å @S where Å is the value of the portfolio The delta of the portfolio can be calculated from the deltas of the individual options in the portfolio If a portfolio consists of a quantity wi of option i (1 i n), the delta of the portfolio is given by n X Á¼ wi Á i i¼1 where Ái is the delta of the ith option The formula can be used to calculate the position in the underlying asset necessary to make the delta of the portfolio zero When this position has been taken, the portfolio is delta neutral www.downloadslide.net 749 HJM, LMM, and Multiple Zero Curves The swap volatility to be substituted into the standard market model for valuing a swap option proves to be (see Problem 33.13) vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u ð T p NÀ1 M u X X X k;m k;m;q tịGk;m 0ị k 0ị2 t 33:19ị dt ỵ k;m Gk;m 0ị T0 tẳ0 qẳ1 kẳn mẳ1 Here j;m;q tị is the qth component of the volatility of Gj;m ðtÞ It is the qth component of the volatility of a cap forward rate when the time to maturity is from t to the beginning of the mth subperiod in the Tj ; Tjỵ1 ị swap accrual period The expressions (33.18) and (33.19) for the swap volatility involve the approximations that Gj tị ẳ Gj 0ị and Gj;m tị ẳ Gj;m ð0Þ Hull and White compared the prices of European swap options calculated using (33.18) and (33.19) with the prices calculated from a Monte Carlo simulation and found the two to be very close Once the LIBOR market model has been calibrated, (33.18) and (33.19) therefore provide a quick way of valuing European swap options Analysts can determine whether European swap options are overpriced or underpriced relative to caps As we shall see shortly, they can also use the results to calibrate the model to the market prices of swap options The analysis can be extended to cover OIS discounting Calibrating the Model The variable Ãj is the volatility at time t of the forward rate Fj for the period between tk and tkỵ1 when there are j whole accrual periods between t and tk To calibrate the LIBOR market model, it is necessary to determine the Ãj and how they are split into j;q The Ã’s are usually determined from current market data, whereas the split into ’s is determined from historical data Consider first the determination of the ’s from the Ã’s A principal components analysis (see Section 22.9) on forward rate data can be used The model is ÁFj ¼ M X j;q xq q¼1 where M is the total number of factors (which equals the number of different forward rates), ÁFj is the change in the j th forward rate Fj , j;q is the factor loading for the jth forward rate and the qth factor, xq is the factor score for the qth factor Define sq as the standard deviation of the qth factor score If the number of factors used in the LIBOR market model, p, is equal to the total number of factors, M, it is correct to set j;q ¼ j;q sq for j;q M When, as is usual, p < M, the j;q must be scaled so that rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Xp 2 Ãj ¼ q¼1 j;q This involves setting Ãj sq j;q j;q ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Pp 2 qẳ1 sq j;q 33:20ị www.downloadslide.net 750 CHAPTER 33 Consider next the estimation of the Ã’s Equation (33.11) provides one way that they can be theoretically determined so that they are consistent with caplet prices In practice, this is not usually used because it often leads to wild swings in the Ã’s and sometimes there is no set of Ã’s exactly consistent with cap quotes A commonly used calibration procedure is similar to that described in Section 32.6 Suppose that Ui is the market price of the ith calibrating instrument (typically a cap or European swaption) and Vi is the model price The Ã’s are chosen to minimize X ðUi Vi ị2 ỵ P i where P is a penalty function chosen to ensure that the Ã’s are ‘‘well behaved.’’ Similarly to Section 32.6, P might have the form X X Pẳ w1;i iỵ1 i ị2 ỵ w2;i iỵ1 ỵ i1 2i ị2 i i When the calibrating instrument is a European swaption, formulas (33.18) and (33.19) make the minimization feasible using the Levenberg–Marquardt procedure Equation (33.20) is used to determine the ’s from the Ã’s Volatility Skews Brokers provide quotes on caps that are not at the money as well as on caps that are at the money In some markets a volatility skew is observed, that is, the quoted (Black) volatility for a cap or a floor is a declining function of the strike price This can be handled using the CEV model (See Section 27.1 for the application of the CEV model to equities.) The model is p X i;q ðtÞFi ðtÞ dzq ð33:21Þ dFi tị ẳ ỵ qẳ1 where is a constant ð0 < < 1Þ It turns out that this model can be handled very similarly to the lognormal model Caps and floors can be valued analytically using the cumulative noncentral 2 distribution There are similar analytic approximations to those given above for the prices of European swap options.10 Bermudan Swap Options A popular interest rate derivative is a Bermudan swap option This is a swap option that can be exercised on some or all of the payment dates of the underlying swap Bermudan swap options are difficult to value using the LIBOR market model because the LIBOR market model relies on Monte Carlo simulation and it is difficult to evaluate early exercise decisions when Monte Carlo simulation is used Fortunately, the procedures described in Section 27.8 can be used Longstaff and Schwartz apply the least-squares approach when there are a large number of factors The value of not exercising on a particular payment date is assumed to be a polynomial function of the 10 For details, see L B G Andersen and J Andreasen, ‘‘Volatility Skews and Extensions of the LIBOR Market Model,’’ Applied Mathematical Finance, 7, (2000): 1–32; J C Hull and A White, ‘‘Forward Rate Volatilities, Swap Rate Volatilities, and the Implementation of the LIBOR Market Model,’’ Journal of Fixed Income, 10, (September 2000): 46–62 www.downloadslide.net HJM, LMM, and Multiple Zero Curves 751 values of the factors.11 Andersen shows that the optimal early exercise boundary approach can be used He experiments with a number of ways of parameterizing the early exercise boundary and finds that good results are obtained when the early exercise decision is assumed to depend only on the intrinsic value of the option.12 Most traders value Bermudan options using one of the one-factor no-arbitrage models discussed in Chapter 32 However, the accuracy of one-factor models for pricing Bermudan swap options has been a controversial issue.13 33.3 HANDLING MULTIPLE ZERO CURVES The label ‘‘LIBOR Market Model’’ dates back to the time when LIBOR was used as a proxy for the risk-free rate By modeling LIBOR, analysts were able to value all derivatives that provided payoffs dependent on LIBOR There was assumed to be a single LIBOR zero curve and rates of all maturities could be determined, as indicated in Section 31.1, from the assumed behavior of the LIBOR short rate Since the 2007 to 2008 crisis, derivatives practitioners have switched to using OIS as a proxy for the risk-free rate and have built credit risk into their LIBOR estimates Prior to the crisis, the 12-month LIBOR–OIS spread was sometimes greater and sometimes less than the 1-month LIBOR–OIS spread Since the crisis, as indicated in Figure 33.2, this has not been the case The 12-month spread has been greater than the 6-month spread, which has been greater than the 3-month spread, which in turn has been greater than the 1-month spread This ranking is simply a reflection of the fact that the longer an unsecured loan is made to a AA-rated bank, the greater is the spread necessary to compensate for credit risk Analysts have to keep track of 1-month, 3-month, 6-month, and 12-month LIBOR forward rates, as well as OIS rates, in all the currencies in which they transact interest rate derivatives The HJM or LMM models can be used to model OIS rates in the way described in this chapter (The LIBOR market model should then be called the OIS market model.) Suppose that a derivative of interest has a payoff dependent on 3-month LIBOR Once OIS rates have been modeled, one simple approach would be to assume that the future 3-month LIBOR–OIS spreads equal to forward spreads observed in the market today An alternative is to base a model for forward spreads on one or more factors similarly to the models in equations (33.10) and (33.15) for forward interest rates, so that p X dFk tị ẳ ỵ k;q tị dzq Fk tị qẳ1 where, for the purposes of this equation, we define Fk ðtÞ as the forward spread between 11 See F A Longstaff and E S Schwartz, ‘‘Valuing American Options by Simulation: A Simple Least Squares Approach,’’ Review of Financial Studies, 14, (2001): 113–47 12 L B G Andersen, ‘‘A Simple Approach to the Pricing of Bermudan Swaptions in the Multifactor LIBOR Market Model,’’ Journal of Computational Finance, 3, (Winter 2000): 5–32 13 For opposing viewpoints, see ‘‘Factor Dependence of Bermudan Swaptions: Fact or Fiction,’’ by L B G Andersen and J Andreasen, and ‘‘Throwing Away a Billion Dollars: The Cost of Suboptimal Exercise Strategies in the Swaption Market,’’ by F A Longstaff, P Santa-Clara, and E S Schwartz Both articles are in Journal of Financial Economics, 62, (October 2001) www.downloadslide.net 752 CHAPTER 33 Figure 33.2 Post-crisis LIBOR–OIS spreads for different tenors (basis points) 200 150 100 12-month 6-month 50 3-month 1-month Jan 09 Jan 11 Jan 13 Jan 15 times tk and tkỵ1 as seen at time t and k;q as the qth component of the volatility of this forward spread All the results we have given in Section 33.2 for calculating the process followed by interest rates under the rolling risk-neutral measure then apply to spreads Correlations between the factors driving OIS rates and the factors driving spreads can be built into the simulation model 33.4 AGENCY MORTGAGE-BACKED SECURITIES One application of the models presented in this chapter is to the agency mortgagebacked security (agency MBS) market in the United States An agency MBS is similar to the ABS considered in Chapter except that payments are guaranteed by a government-related agency such as the Government National Mortgage Association (GNMA) or the Federal National Mortgage Association (FNMA) so that investors are protected against defaults This makes an agency MBS sound like a regular fixed-income security issued by the government In fact, there is a critical difference between an agency MBS and a regular fixed-income investment This difference is that the mortgages in an agency MBS pool have prepayment privileges These prepayment privileges can be quite valuable to the householder In the United States, mortgages typically last for 30 years and can be prepaid at any time This means that the householder has a 30-year American-style option to put the mortgage back to the lender at its face value Prepayments on mortgages occur for a variety of reasons Sometimes interest rates fall and the owner of the house decides to refinance at a lower rate On other occasions, a mortgage is prepaid simply because the house is being sold A critical element in valuing an agency MBS is the determination of what is known as the prepayment function This is a function describing expected prepayments on the underlying pool of mortgages at a time t in terms of the yield curve at time t and other relevant variables www.downloadslide.net HJM, LMM, and Multiple Zero Curves 753 A prepayment function is very unreliable as a predictor of actual prepayment experience for an individual mortgage When many similar mortgage loans are combined in the same pool, there is a ‘‘law of large numbers’’ effect at work and prepayments can be predicted more accurately from an analysis of historical data As mentioned, prepayments are not always motivated by pure interest rate considerations Nevertheless, there is a tendency for prepayments to be more likely when interest rates are low than when they are high This means that investors require a higher rate of interest on an agency MBS than on other fixed-income securities to compensate for the prepayment options they have written Collateralized Mortgage Obligations The simplest type of agency MBS is referred to as a pass-through All investors receive the same return and bear the same prepayment risk Not all mortgage-backed securities work in this way In a collateralized mortgage obligation (CMO) the investors are divided into a number of classes and rules are developed for determining how principal repayments are channeled to different classes A CMO creates classes of securities that bear different amounts of prepayment risk in the same way that the ABS considered in Chapter creates classes of securities bearing different amounts of credit risk As an example of a CMO, consider an agency MBS where investors are divided into three classes: class A, class B, and class C All the principal repayments (both those that are scheduled and those that are prepayments) are channeled to class A investors until investors in this class have been completely paid off Principal repayments are then channeled to class B investors until these investors have been completely paid off Finally, principal repayments are channeled to class C investors In this situation, class A investors bear the most prepayment risk The class A securities can be expected to last for a shorter time than the class B securities, and these, in turn, can be expected to last less long than the class C securities The objective of this type of structure is to create classes of securities that are more attractive to institutional investors than those created by a simpler pass-through MBS The prepayment risks assumed by the different classes depend on the par value in each class For example, class C bears very little prepayment risk if the par values in classes A, B, and C are 400, 300, and 100, respectively Class C bears rather more prepayment risk in the situation where the par values in the classes are 100, 200, and 500 The creators of mortgage-backed securities have created many more exotic structures than the one we have just described Business Snapshot 33.1 gives an example Valuing Agency Mortgage-Backed Securities Agency MBSs are usually valued by modeling the behavior of Treasury rates using Monte Carlo simulation The HJM/LMM approach can be used Consider what happens on one simulation trial Each month, expected prepayments are calculated from the current yield curve and the history of yield curve movements These prepayments determine the expected cash flows to the holder of the agency MBS and the cash flows are discounted at the Treasury rate plus a spread to time zero to obtain a sample value for the agency MBS An estimate of the value of the agency MBS is the average of the sample values over many simulation trials www.downloadslide.net 754 CHAPTER 33 Business Snapshot 33.1 IOs and POs In what is known as a stripped MBS, principal payments are separated from interest payments All principal payments are channeled to one class of security, known as a principal only (PO) All interest payments are channeled to another class of security known as an interest only (IO) Both IOs and POs are risky investments As prepayment rates increase, a PO becomes more valuable and an IO becomes less valuable As prepayment rates decrease, the reverse happens In a PO, a fixed amount of principal is returned to the investor, but the timing is uncertain A high rate of prepayments on the underlying pool leads to the principal being received early (which is, of course, good news for the holder of the PO) A low rate of prepayments on the underlying pool delays the return of the principal and reduces the yield provided by the PO In the case of an IO, the total of the cash flows received by the investor is uncertain The higher the rate of prepayments, the lower the total cash flows received by the investor, and vice versa Option-Adjusted Spread In addition to calculating theoretical prices for mortgage-backed securities and other bonds with embedded options, traders also like to compute what is known as the option-adjusted spread (OAS) This is a measure of the spread over the yields on government Treasury bonds provided by the instrument when all options have been taken into account To calculate an OAS for an instrument, it is priced as described above using Treasury rates plus a spread for discounting The price of the instrument given by the model is compared to the price in the market A series of iterations is then used to determine the value of the spread that causes the model price to be equal to the market price This spread is the OAS SUMMARY The HJM and LMM models provide approaches to valuing interest rate derivatives that give the user complete freedom in choosing the volatility term structure The LMM model has two key advantages over the HJM model First, it is developed in terms of the forward rates that determine the pricing of caps, rather than in terms of instantaneous forward rates Second, it is relatively easy to calibrate to the price of caps or European swap options The HJM and LMM models both have the disadvantage that they cannot be represented as recombining trees In practice, this means that they must usually be implemented using Monte Carlo simulation and require much more computation time than the models in Chapter 32 Since the credit crisis that started in 2007, the OIS rate has been used as the risk-free discount rate for collateralized derivatives This means that it is sometimes necessary to model the joint evolution of the OIS zero curve and the LIBOR–OIS spreads The agency mortgage-backed security market in the United States has given birth to many exotic interest rate derivatives: CMOs, IOs, POs, and so on These instruments provide cash flows to the holder that depend on the prepayments on a pool of www.downloadslide.net HJM, LMM, and Multiple Zero Curves 755 mortgages These prepayments depend on, among other things, the level of interest rates Because they are heavily path dependent, agency mortgage-backed securities usually have to be valued using Monte Carlo simulation These are, therefore, ideal candidates for applications of the HJM and LMM models FURTHER READING Andersen, L B G., ‘‘A Simple Approach to the Pricing of Bermudan Swaption in the MultiFactor LIBOR Market Model,’’ The Journal of Computational Finance, 3, (2000): 5–32 Andersen, L B G., and J Andreasen, ‘‘Volatility Skews and Extensions of the LIBOR Market Model,’’ Applied Mathematical Finance, 7, (March 2000): 1–32 Andersen, L B G., and V Piterbarg, Interest Rate Modeling, Vols I–III New York: Atlantic Financial Press, 2010 Brace A., D Gatarek, and M Musiela ‘‘The Market Model of Interest Rate Dynamics,’’ Mathematical Finance, 7, (1997): 127–55 Duffie, D and R Kan, ‘‘A Yield-Factor Model of Interest Rates,’’ Mathematical Finance 6, (1996), 379–406 Heath, D., R Jarrow, and A Morton, ‘‘Bond Pricing and the Term Structure of Interest Rates: A Discrete Time Approximation,’’ Journal of Financial and Quantitative Analysis, 25, (December 1990): 419–40 Heath, D., R Jarrow, and A Morton, ‘‘Bond Pricing and the Term Structure of Interest Rates: A New Methodology,’’ Econometrica, 60, (1992): 77–105 Hull, J C., and A White, ‘‘Forward Rate Volatilities, Swap Rate Volatilities, and the Implementation of the LIBOR Market Model,’’ Journal of Fixed Income, 10, (September 2000): 46–62 Jamshidian, F., ‘‘LIBOR and Swap Market Models and Measures,’’ Finance and Stochastics, (1997): 293–330 Jarrow, R A., and S M Turnbull, ‘‘Delta, Gamma, and Bucket Hedging of Interest Rate Derivatives,’’ Applied Mathematical Finance, (1994): 21–48 Mercurio, F., and Z Xie, ‘‘The Basis Goes Stochastic,’’ Risk, 25, 12 (December 2012): 78–83 Miltersen, K., K Sandmann, and D Sondermann, ‘‘Closed Form Solutions for Term Structure Derivatives with Lognormal Interest Rates,’’ Journal of Finance, 52, (March 1997): 409–30 Rebonato, R., Modern Pricing of Interest Rate Derivatives: The LIBOR Market Model and Beyond Princeton Umiversity Press, 2002 Practice Questions (Answers in Solutions Manual) 33.1 Explain the difference between a Markov and a non-Markov model of the short rate 33.2 Prove the relationship between the drift and volatility of the forward rate for the multifactor version of HJM in equation (33.6) 33.3 ‘‘When the forward rate volatility sðt; T Þ in HJM is constant, the Ho–Lee model results.’’ Verify that this is true by showing that HJM gives a process for bond prices that is consistent with the Ho–Lee model in Chapter 32 33.4 ‘‘When the forward rate volatility, sðt; T Þ, in HJM is eÀaðT ÀtÞ , the Hull–White model results.’’ Verify that this is true by showing that HJM gives a process for bond prices that is consistent with the Hull–White model in Chapter 32 www.downloadslide.net 756 CHAPTER 33 33.5 What is the advantage of LMM over HJM? 33.6 Provide an intuitive explanation of why a ratchet cap increases in value as the number of factors increase 33.7 Show that equation (33.10) reduces to (33.4) as the i tend to zero 33.8 Explain why a sticky cap is more expensive than a similar ratchet cap 33.9 Explain why IOs and POs have opposite sensitivities to the rate of prepayments 33.10 ‘‘An option adjusted spread is analogous to the yield on a bond.’’ Explain this statement 33.11 Prove equation (33.15) 33.12 Prove the formula for the variance V ðT Þ of the swap rate in equation (33.17) 33.13 Show that the swap volatility expression (33.19) in Section 33.2 is correct Further Questions 33.14 In an annual-pay cap, the Black volatilities for at-the-money caplets which start in 1, 2, 3, and years and end year later are 18%, 20%, 22%, and 20%, respectively Estimate the volatility of a 1-year forward rate in the LIBOR Market Model when the time to the start of the period covered by the forward rate is (a) to year, (b) to years, (c) to years, and (d) to years Assume that the zero curve is flat at 5% per annum (annually compounded) Use DerivaGem with LIBOR discounting to estimate flat volatilities for 2-, 3-, 4-, 5-, and 6-year at-the-money caps 33.15 In the flexi cap considered in Section 33.2 the holder is obligated to exercise the first N in-the-money caplets After that no further caplets can be exercised (In the example, N ¼ 5.) Two other ways that flexi caps are sometimes defined are: (a) The holder can choose whether any caplet is exercised, but there is a limit of N on the total number of caplets that can be exercised (b) Once the holder chooses to exercise a caplet all subsequent in-the-money caplets must be exercised up to a maximum of N Discuss the problems in valuing these types of flexi caps Of the three types of flexi caps, which would you expect to be most expensive? Which would you expect to be least expensive? www.downloadslide.net 34 C H A P T E R Swaps Revisited Swaps have been central to the success of over-the-counter derivatives markets They have proved to be very flexible instruments for managing risk Based on the range of different contracts that now trade and the total volume of business transacted each year, swaps are arguably one of the most successful innovations in financial markets ever Chapter discussed how plain vanilla LIBOR-for-fixed interest rate swaps can be valued The standard approach can be summarized as: ‘‘Assume forward rates will be realized.’’ The steps are as follows: Calculate the swap’s net cash flows on the assumption that LIBOR rates in the future equal the forward rates calculated from instruments trading in the market today Set the value of the swap equal to the present value of the net cash flows This chapter describes a number of nonstandard swaps Some can be valued using the ‘‘assume forward rates will be realized’’ approach; some require the application of the convexity, timing, and quanto adjustments we encountered in Chapter 30; some contain embedded options that must be valued using the procedures described in Chapters 29, 32, and 33 34.1 VARIATIONS ON THE VANILLA DEAL Many interest rate swaps involve relatively minor variations to the plain vanilla structure discussed in Chapter In some swaps the notional principal changes with time in a predetermined way Swaps where the notional principal is an increasing function of time are known as step-up swaps Swaps where the notional principal is a decreasing function of time are known as amortizing swaps Step-up swaps could be useful for a construction company that intends to borrow increasing amounts of money at floating rates to finance a particular project and wants to swap to fixed-rate funding An amortizing swap could be used by a company that has fixed-rate borrowings with a certain prepayment schedule and wants to swap to borrowings at a floating rate 757 www.downloadslide.net 758 CHAPTER 34 Business Snapshot 34.1 Hypothetical Confirmation for Nonstandard Swap Trade date: Effective date: Business day convention (all dates): Holiday calendar: Termination date: Fixed amounts Fixed-rate payer: Fixed-rate notional principal: Fixed rate: Fixed-rate day count convention: Fixed-rate payment dates Floating amounts Floating-rate payer Floating-rate notional principal Floating rate Floating-rate day count convention Floating-rate payment dates 4-January, 2016 11-January, 2016 Following business day U.S 11-January, 2021 Microsoft USD 100 million 6% per annum Actual/365 Each 11-July and 11-January commencing 11-July, 2016, up to and including 11-January, 2021 Goldman Sachs USD 120 million USD 1-month LIBOR Actual/360 11-July, 2016, and the 11th of each month thereafter up to and including 11-January, 2021 The principal can be different on the two sides of a swap Also the frequency of payments can be different Business Snapshot 34.1 illustrates this by showing a hypothetical swap between Microsoft and Goldman Sachs where the notional principal is $120 million on the floating side and $100 million on fixed side Payments are made every month on the floating side and every months on the fixed side These type of variations to the basic plain vanilla structure not affect the valuation methodology The ‘‘assume forward rates are realized’’ approach can still be used The floating reference rate for a swap is not always LIBOR For instance, in some swaps it is the commercial paper (CP) rate or the OIS rate A basis swap involves exchanging cash flows calculated using one floating reference rate for cash flows calculated using another floating reference rate An example would be a swap where the 3-month OIS rate plus 10 basis points is exchanged for 3-month LIBOR with both being applied to a principal of $100 million A basis swap could be used for risk management by a financial institution whose assets and liabilities are dependent on different floating reference rates Swaps where the floating reference rate is not LIBOR can usually be valued using the ‘‘assume forward rates are realized’’ approach The forward rate is calculated so that swaps involving the reference rate have zero value (This is similar to the way forward LIBOR is calculated when OIS discounting is used.) www.downloadslide.net 759 Swaps Revisited Business Snapshot 34.2 Hypothetical Confirmation for Compounding Swap Trade date: Effective date: Holiday calendar: Business day convention (all dates): Termination date: Fixed amounts Fixed-rate payer: Fixed-rate notional principal: Fixed rate: Fixed-rate day count convention: Fixed-rate payment date: Fixed-rate compounding: Fixed-rate compounding dates Floating amounts Floating-rate payer: Floating-rate notional principal: Floating rate: Floating-rate day count convention: Floating-rate payment date: Floating-rate compounding: Floating-rate compounding dates: 4-January, 2016 11-January, 2016 U.S Following business day 11-January, 2021 Microsoft USD 100 million 6% per annum Actual/365 11-January, 2021 Applicable at 6.3% Each 11-July and 11-January commencing 11-July, 2016, up to and including 11-July, 2020 Goldman Sachs USD 100 million USD 6-month LIBOR plus 20 basis points Actual/360 11-January, 2021 Applicable at LIBOR plus 10 basis points Each 11-July and 11-January commencing 11-July, 2016, up to and including 11-July, 2020 34.2 COMPOUNDING SWAPS Another variation on the plain vanilla swap is a compounding swap A hypothetical confirmation for a compounding swap is in Business Snapshot 34.2 In this example there is only one payment date for both the floating-rate payments and the fixed-rate payments This is at the end of the life of the swap The floating rate of interest is LIBOR plus 20 basis points Instead of being paid, the interest is compounded forward until the end of the life of the swap at a rate of LIBOR plus 10 basis points The fixed rate of interest is 6% Instead of being paid this interest is compounded forward at a fixed rate of interest of 6.3% until the end of the swap The ‘‘assume forward rates are realized’’ approach can be used at least approximately for valuing a compounding swap such as that in Business Snapshot 34.2 It is straightforward to deal with the fixed side of the swap because the payment that will be made at maturity is known with certainty The ‘‘assume forward rates are realized’’ approach for the floating side is justifiable because there exist a series of forward rate agreements www.downloadslide.net 760 CHAPTER 34 (FRAs) where the floating-rate cash flows are exchanged for the values they would have if each floating rate equaled the corresponding forward rate.1 Example 34.1 A compounding swap with annual resets has a life of years A fixed rate is paid and a floating rate is received The fixed interest rate is 4% and the floating interest rate is 12-month LIBOR The fixed side compounds at 3.9% and the floating side compounds at 12-month LIBOR minus 20 basis points All LIBOR forward rates are 5% OIS rates are all 4% and are used for discounting The notional principal is $100 million On the fixed side, interest of $4 million is earned at the end of the first year This compounds to  1:039 ¼ $4:156 million at the end of the second year A second interest amount of $4 million is added at the end of the second year bringing the total compounded forward amount to $8.156 million This compounds to 8:156  1:039 ¼ $8:474 million by the end of the third year when there is the third interest amount of $4 million The cash flow at the end of the third year on the fixed side of the swap is therefore $12.474 million On the floating side we assume all future interest rates equal the corresponding forward LIBOR rates This means that all future interest rates are assumed to be 5% with annual compounding The interest calculated at the end of the first year is $5 million Compounding this forward at 4.8% (forward LIBOR minus 20 basis points) gives  1:048 ¼ $5:24 million at the end of the second year Adding in the interest, the compounded forward amount is $10.24 million Compounding forward to the end of the third year, we get 10:24  1:048 ¼ $10:731 million Adding in the final interest gives $15.731 million The swap can be valued by assuming that it leads to an inflow of $15.731 million and an outflow of $12.474 million at the end of year The value of the swap is therefore 15:731 À 12:474 ¼ 2:895 1:043 or $2.895 million (This analysis ignores day count issues and makes the approximation indicated in footnote 1.) 34.3 CURRENCY SWAPS Currency swaps were introduced in Chapter They enable an interest rate exposure in one currency to be swapped for an interest rate exposure in another currency Usually two principals are specified, one in each currency The principals are exchanged at both the beginning and the end of the life of the swap as described in Section 7.7 Suppose that the currencies involved in a currency swap are U.S dollars (USD) and British pounds (GBP) In a fixed-for-fixed currency swap, a fixed rate of interest is specified in each currency The payments on one side are determined by applying the See Technical Note 18 at www-2.rotman.utoronto.ca/$hull/TechnicalNotes for the details The ‘‘assume forward rates are realized’’ approach works exactly if the spread used for compounding, sc , is zero or if it is applied so that Q at time t compounds to Q1 ỵ R ị1 ỵ sc ị at time t ỵ , where R is LIBOR If, as is more usual, it compounds to Qẵ1 ỵ R þ sc Þ , then there is a small approximation www.downloadslide.net 761 Swaps Revisited fixed rate of interest in USD to the USD principal; the payments on the other side are determined by applying the fixed rate of interest in GBP to the GBP principal In a floating-for-floating currency swap, the payments on one side are determined by applying USD LIBOR (possibly with a spread added) to the USD principal; similarly the payments on the other side are determined by applying GBP LIBOR (possibly with a spread added) to the GBP principal In a cross-currency interest rate swap, a floating rate in one currency is exchanged for a fixed rate in another currency Floating-for-floating and cross-currency interest rate swaps can be valued using the ‘‘assume forward rates are realized’’ rule Future LIBOR rates in each currency are assumed to equal today’s forward rates This enables the cash flows in the currencies to be determined The USD cash flows are discounted at the USD zero rate The GBP cash flows are discounted at the GBP zero rate The current exchange rate is then used to translate the two present values to a common currency It is important to ensure that valuation procedures are such that transactions that could be negotiated today (at the midpoint of bid and offer) have zero value The discount rates that are used are often adjusted to ensure that this is the case.2 34.4 MORE COMPLEX SWAPS We now move on to consider some examples of swaps where the simple rule ‘‘assume forward rates will be realized’’ does not work In each case, it must be assumed that an adjusted forward rate, rather than the actual forward rate, is realized This section builds on the discussion in Chapter 30 LIBOR-in-Arrears Swap A plain vanilla interest rate swap is designed so that the floating rate of interest observed on one payment date is paid on the next payment date An alternative instrument that is sometimes traded is a LIBOR-in-arrears swap In this, the floating rate paid on a payment date equals the rate observed on the payment date itself Suppose that the reset dates in the swap are ti for i ¼ 0; 1; ; n, with i ẳ tiỵ1 ti Dene Ri as the LIBOR rate for the period between ti and tiỵ1 , Fi as the forward value of Ri , and i as the volatility of this forward rate (The value of i is typically implied from caplet prices.) In a LIBOR-in-arrears swap, the payment on the floating side at time ti is based on Ri rather than RiÀ1 As explained in Section 30.1, it is necessary to make a convexity adjustment to the forward rate when the payment is valued The valuation should be based on the assumption that the oating rate paid is Fi ỵ and not Fi Fi2 i2 i ti ỵ Fi i ð34:1Þ Example 34.2 In a LIBOR-in-arrears swap, the principal is $100 million A fixed rate of 5% is received annually and LIBOR is paid Payments are exchanged at the ends of If a bank’s system does not value deals consistently with the market, its traders will be able to arbitrage the system ... 2, 877.7 2, 299.9 2, 224 .9 2, 383.0 2, 707.9 2, 131.5 2, 085.4 3,355 .2 3,533.5 2, 788.7 2, 771.4 2, 101.4 1, 324 .4 1,445.7 526 .7 1,109.6 29 0.0 25 6.6 2. 5 2. 7 3.4 2. 8 2. 2 2. 1 2. 3 2. 6 2. 1 2. 0 3 .2 3.4 2. 7 2. 7... 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