Appendix 1: Option Valuation

We have mentioned at several points in this text the importance of option pricing and valuation in the work of derivative product and credit profession• als. The seminal work in the area of option valuation is the 1973 article by Black and Scholes entitled 'The Pricing of Options and Corporate Liabili• ties', where the authors develop an analytic solution to the valuation of Euro• pean options on non-dividend paying stocks. The simplifying assumptions used to develop the equation include unlimited borrow and lending at a constant risk-free rate, full access to short-selling, an underlying asset which follows a continuous time stochastic process, and frictionless markets with no taxes or transactions costs.' It is not our intent in this appendix to develop a detailed explanation of the mathematics of the Black-Scholes model; there are many works which do an excellent job of explaining the quantitative aspects of the model. Instead, we provide a very brief overview of the main points of the model and illustrate, with a few simple examples, the computation of option prices and comparative statics based on the model. To commence, we define in advance several of the terms used in the equa• tions below (note that we make a few changes to the notation from earlier in the text; most notably, CMP becomes S, and SP becomes X):

c is the value of a European call p is the value of a European put S is the stock price S(T) is the terminal stock price X is the strike price t is the time to maturity r is the risk-free rate J.L is the expected return of the stock a is the standard deviation of the return

We begin our outline by defining the value of a European call at expiry as

C(ST, X, 't) == max(O, ST - X) (Al.I)

The value of the call today is simply the discounted value of the expected payoff at expiry:

342 Option Valuation

C(ST. X. t) = e-rt E(max(O. ST - X» (A 1.2)

As indicated above, stock prices under the Black-Scholes model follow a stochastic process; in this case prices follow a geometric Brownian motion, where the returns of the stock prices are lognormally distributed (see Chapter 2 for additional detail). Under the lognormal distribution the expected value of the stock at maturity of the option is:

E(ST) = Se(p-!f)t (A 1.3)

where the price of the stock, S, increases by the instantaneously compounded expected return J1 less half the variance of the stock price. In the equation above, 0 is the standard deviation of the expected return. Under risk neutral• ity, one of the key elements of the Black-Scholes framework. the value of the expected return can be ignored. This occurs because an investor can cre• ate a risk-free portfolio by combining a short position in a call option (-c) with a delta equivalent amount of stock (AS): the perfect correlation between the call and the stock over a very small time period means gains on the call are offset by losses on the stock and vice versa. Therefore, the value of the option is not influenced by the expected return. and investors' risk prefer• ences need not be considered; such simplifies the pricing framework consid• erably. Knowing that stock prices are distributed lognormally. we return to the equation presented in Chapter 2 (Le. equation to in note 6):

In(ST) ~ • [In(S) + (J1 - ~ 't),

Under risk neutrality. J1 is replaced with the risk-free rate, r, yielding:

In(ST) ~ • [In(S) + (r - 0; t). o-ft] (A 1.5)

Returning to equation A 1.2, it is necessary to solve for the value of c (where g is the distribution of SeT) under risk-neutrality):

C(ST, X. t) = e-rt i

After a rather extensive mathematical exercise. equation (A1.6) yields the familiar Black-Scholes call valuation equation:

343 Appendix J

(Al.7)

where N(dl) and N(d2), defined below, represent the cumulative probability distribution function for a standardized normal variable. The N(d) values can be obtained from standard tables of probability functions. The values of d 1 and d2 are shown as:

(Al.B)

and:

d _ In( ~) + (r - ~)~ (Al.9) 2- a..f.t where d2 may also be expressed as:

(Al.lO)

Through this formula, one can value a European call option on a non-dividend paying stock, under the constraints detailed above. Knowing the above, and employing put-call parity, which effectively says that due to boundary conditions and arbitrage:

c + Xe- rc == p + S (AUt) it is relatively easy to solve for the Black-Scholes equation for puts:

(Al.12)

Let us consider a numerical example in order to demonstrate the model. Assume we want to purchase a European call on a stock with a current price of 20 and a strike price of 22 (i.e. it is 9 per cent out-of-the-money); expiry is in three months (90 days). Recall that ~ is adjusted to a fraction of a year if the volatility measure is annualized; in this case ~ equals 0.25 years. The risk free rate is 4 per cent and the volatility of the stock is 20 per cent. The fair value of the call, based on the equations above, is:

dl = On(20122) + (0.04 + (0.201\212) * 0.25»/(0.20 * "0.25)

344 Option Valuation

= «-0.0953) + (0.015»/.10 = -0.803

d2 = -0.803 - 0.10 = -0.903

c = 20 '" N( -0.803) - 22 '" (0.99) '" N( -0.903)

Using statistical tables we detennine the probabilities associated with N(dl) and N(d2):

c = 20 '" (1 - 0.7967) - 22 '" 0.99 '" (l - 0.8238) = 4.066 - 3.837 = .229

Thus, one caIl on the stock is worth nearly $0.23. If we apply the same procedure to an in-the-money call (i.e. the price S is now $24, instead of $20, with the strike remaining at $22), the value of the call rises:

dl = (In(24/22) + (0.04 + (0.2()A2/2) '" 0.25»/(0.20 '" ",,0.25) = «0.087) + (0.015»/0.10 = 1.020

d2 = 1.020 - 0.10 = 0.920

c = 24 '" N(1.020) - 22 '" (0.99) '" N(0.920) = 24 '" 0.8461 - 22 '" (0.99) '" 0.8212 = 20.306 - 17.885 = 2.421

As one might expect, the value of the in-the-money caIl is much higher than the out-of-the-money caIl (at $2.42 vs. $0.23) as it already has intr!nsic value. In Chapter 3 we have described the comparative statics delta, gamma, vega and theta; these statics measure an option's sensitivity to changes in the stock price, volatility and time. Utilizing the Black-Scholes model, we highlight the equations which measure these statics (note that we shaIl not consider rho, sensitivity of the option's value to a change in the risk-free rate, as the effects of rho on a stock option are generaIly negligible when compared with the other statics). It can be shown that delta, which is the change in the option's price for a unit change in the stock price, is equal to the first derivative of the option price with respect to the stock price:

345 Appendix J

(A 1.13)

The quantity we obtain from N(dl) above is precisely equal to delta. We shall not explore the mathematics behind this, but we may interpret the above as saying that a short call is delta hedged by an amount of long stock equal to N(dl); conversely, a long call is delta hedged by an amount of short stock equal to N(dl). The equation for puts is:

(A1.l4)

Returning to the second example above. we note that the result obtained from N(dl) is equal to 0.846. This means that a portfolio is delta hedged when a short call is offset by 0.846 shares of long stock; alternatively, a long call is hedged by shorting 0.846 shares of stock. This also means that if the stock rises by $1, the value of the call increases by $0.846. Gamma, which is the change in the delta of the option for a unit change in the stock price, is the first derivative of delta with respect to the stock price, or the second derivative of the option price with respect to the stock price. This may be shown as:

(A1.15)

where

(AI.l6) The first derivative of N(dl) is the standard normal density function. The gamma of a put is:

(AU7)

which is precisely equal to the gamma of a call. Returning to the example above. we calculate the gamma of the call as

r = 1/(24 * (0.20 * "0.25» * N'(dl) = 0.4167 * N'(dl)

346 Option Valuation

N'(dl) = 11(2.507) * el\-(0.520) = 0.2371 = 0.4167 * 0.2371 = 0.0988

Thus, for a $1 increase in the stock price, the delta of the option rises by $0.099. Theta, which is the change in the option's price for a change in time, is simply the derivative of the option price with respect to time until maturity. This is shown as:

(A 1.18)

Where N'(dl) and N(d2) are defined above. Note the derivative of the call option price with respect to the price, if positive, yields a value greater than O. The positive value of the derivative says that an option with a longer time to maturity, 't, has more value than an option with a shorter 'to However, we are interested, via theta, in determining how the price of the option deteriorates as time to maturity decreases (through some small interval t), hence use of a negative in the equation above. The theta for puts is simply:

()P Sa - - = e = - - N'(d) + rXe-rt N(-d) (A 1.19)

In the case of an at-the-money European put, theta may not be negative; much depends on the interaction between 't and the present value of the strike price. In our example, the theta of the call is equal to

e = -«24 * 0.20)/(2 * VO.25» * 0.2371 - (0.04 * 22 * 0.99 * 0.8212) = -(4.8) * 0.2371 - 0.7154 = -1.138 - 0.7154 = -1.853

This says that if the time to maturity for the call increases by one year, the value of the call increases by $1.853. We may also say that the actual daily decay in the value of the call is simply equal to $1.853/365, or $0.005. For one day's passage of time, the value of a long call option declines by $0.005 (holding volatility, the underlying stock price and the risk free rate constant);

347 Appendix 1

this is precisely equal to the time decay discussed throughout the text. Vega, or lambba, measures the change in the option's price for a change in the volatility of the underlying stock. The vega calculation for calls and puts is shown as:

(A 1.20)

If vega is high, the option's value is very sensitive to small changes in vola• tility; if vega is low, the option's value is less sensitive to such changes. In our example, we can show the value of vega as

V = 24 ... -.10.25 ... 0.2371 = 2.845

Thus, for an increase in the volatility of the underlying stock from 20 to 21, the price of the call increases by $0.028; for an increase from 20 to 30 per cent, the price of the call increases by $0.2845, and for an increase from 20 to 120 per cent, the price rises by $2.845. This appendix is intended only as a brief primer on Black-Scholes. Read• ers are strongly urged to consult other references - Haley and Schall, 1979; Hull, 1989; Cox and Rubinstein, 1985; Merton, 1990; Gibson, 1991; Dubofsky, 1992; Eades, 1992; Stoll and Whalley, 1993; and Wilmott, Dewynne and Howison, 1995; among many others - for a comprehensive review of the Black-Scholes framework.

Note

1. Many subsequent articles have sought to introduce new equations Qr models which allow certain assumptions of the original model to be relaxed (for instance, the 'no dividend' restriction was dealt with by Merton in 1973. the 'no taxes/trans• action costs' by Ingersoll in 1976, the 'continuous stock prices' by Cox and Ross in 1976, the 'European exercise only' by Roll in 1977. Geske in 1979 and Whaley in 1981. and so on).

348 Appendix 2: A Model for Calculating Swap Risk

In Chapter 6 we have outlined in general fonn a number of ways by which swap risk may be calculated. As we have seen, there is no single correct way to value the fractional exposure portion of a given swap; much is dependent on the individual views of individual institutions (as well as any regulatory constraints particular institutions may face). We present below one method of calculating swap risk. I

The Model

We have seen in Chapter 6 that the main components of any simulation• based potential market exposure model are (a) a method by which to simulate or project future interest rate movements and (b) an equation by which to compute the replacement cost based on simulated future rates. The combina• tion of the two allows the development of a risk factor for a swap with par• ticular maturity and rate parameters; it is, effectively, a percentage measure of fractional exposure for a swap with those parameters. When RF is applied to the notional of a swap, we obtain an REE which, on trade date, is com• prised solely of fractional exposure. Over time, as RFs decline, fractional exposure declines (though potential exposure may increase if the swap's mark• to-market value, or actual exposure, increases). Our goal in this appendix is to develop a method by which to value poten• tial market exposure; the process begins by establishing a method of simulat• ing future interest rates. The binomial option pricing model developed by Cox, Ross and Rubinstein, 1979, suggests 'up' and 'down' movements of a stochastic variable (such as an interest rate) which can be described by

(A2.l) and

349 Appendix 2

(A2.2)

with probabilities defined by

q = 1/2 + 1/2«Jlla)-fi/n) (A2.3)

where a is the standard deviation Jl is the expected return n is a time increment 't is the time to maturity

(Note that it can be shown that the mean and variance of the binomial pro• cess converge to the mean and variance of the lognormal diffusion process described in Chapter 2 as n moves to infinity (see Gibson, 1991». Rendleman and Bartter, 1980, - in 'The Pricing of Options on Debt Secu• rities' - have adapted the above equations for modelling interest rate move• ments by incorporating several simple assumptions: namely, the growth rate of the stochastic variable (the interest rate) is constant, the volatility of the variable is constant and the market price of ri* is constant; these assump• tions, for our purposes, are reasonable.2 The Rendleman and Bartter equa• tions are given by:

(A2A)

and

(A2.5)

where s is the volatility .1t is a small time increment

With this model we focus on the upward movement of rates for fixed rate payers (which is when risk increases), and the downward movement of rates for fixed rate receivers (which is when risk increases). Adapting these equa• tions to terms covered in Chapter 2, we can state that .1t represents the time change in small intervals. In a Markov process a long interval becomes T, which is precisely equal to the T we utilize in Chapter 2 (and represents a cumulative, not interval. time frame). In Chapter 2 we define the standard deviation of rates via historical volatility HV. after an adjustment to the ap• propriate confidence level. Utilizing this terminology, we adjust the two equations as follows:

350 Calculating Risk Swap

Upward movements:

u = e(HV.JT) (A2.6)

Downward movements:

d = e(-HV{f) (A2.7)

With these equations providing information on the possible direction of future rates, we set new rates equal to original rates times the upward (or downward) movement estimated by the equations above. For a fixed rate payer this equals:

(A2.S)

And, for a fixed rate receiver, equals:

(A2.9)

If we assume original rates are 10 per cent and historical volatility of interest rates is 16 per cent (Iognormally distributed and adjusted to a 90 per cent confidence level), we obtain the results highlighted in Table A2.1. This becomes the first component of the potential market exposure frame• work for swaps. There are, of course, many other means by which to model interest rates; this is but one approach an institution may consider. Note that the above can be incorporated into a broad Monte Carlo simulation, where a large number of realizations yields an average value for the future path of interest rates at select points in time. We now tum our attention to the replacement cost calculation. As indicated in Chapter 6, a standard replacement cost equation sums the difference between the new coupon on the swap (r(n» and the original coupon on the swap (r(0» over the number of periods remaining until maturity, and discounts back by the new coupon rate (i.e. the coupon rate in existence at the time of default). The equa• tion provides an estimate of how much it will cost to replace a swap in the future (at the time of default) utilizing simulated future rates. The use of replacement cost equation tends to be fairly standard within the financial industry, and may be defined as: (rn - ro) p t rd = 1: (A2.10) x=1 (I + fY

351 Appendix 2

Table Al.l. Calculated Up/Down Rate Movements (percentages)

Tim. r(O, r(n' (up' r(n' (down' 0.5 10.00 11.19 8.93 1.0 10.00 11.74 8.52 1.5 10.00 12.16 8.22 2.0 10.00 12.54 7.97 2.5 10.00 12.88 7.76 3.0 10.00 13.19 7.57 3.5 10.00 13.49 7.41 4.0 10.00 13.77 7.26 4.5 10.00 14.04 7.12 5.0 10.00 14.30 7.00 5.5 10.00 14.55 6.87 6.0 10.00 14.80 6.76 6.5 10.00 15.00 6.65 7.0 10.00 15.27 6.55 7.5 10.00 15.50 6.45 8.0 10.00 15.72 6.36 8.5 10.00 15.94 6.27 9.0 10.00 16.16 6.19 9.5 10.00 16.37 6.10 10.0 10.00 16.58 6.02

where r(n) is the new rate in period x r(O) is the original rate p is the number of periods until maturity t is the frequency of payments under the swap

We may also wish to show the results of the equation above (which we term the calculated replacement cost) in today's terms. This is accomplished by discounting the result above using r(O):

(A2.ll)

where q is the period in which the default occurs We term this equation the discounted replacement cost. The maximum re• sult obtained through equation A2.II is the risk factor RF for a swap with a specific final maturity and original starting rate; this may also be thought of as the maximum replacement cost likely to be encountered during the life of the swap. Since we now have a framework for calculating upward and downward

352 Calculating Risk Swap

Table A2.2. Calculated Rate Movements (percentages)

Time r(OI r(n) (up)

0.5 10.00 11.19 1.0 10.00 11.74 1.5 10.00 12.16 2.0 10.00 12.54 2.5 10.00 12.88 3.0 10.00 ·13.19 3.5 10.00 13.49 4.0 10.00 13.77 4.5 10.00 14.04 5.0 10.00 14.30

Table A2.3. Discounted Replacement Costs

(%) No. of periods remaining in swap life n 1 2 3 4 5 6 7 8 9

0.5 0.37 0.39 0.41 0.43 0.46 0.48 0.51 0.54 0.57 1 0.55 0.58 0.62 0.65 0.69 0.73 0.77 0.82 1.5 0.72 0.76 0.81 0.86 0.91 0.96 1.00 2 0.88 0.94 0.99 1.06 1.12 1.20 2.5 1.05 1.12 1.19 1.27 1.35 3 1.24 1.32 1.41 1.50 3.5 1.44 1.53 1.64 4 1.65 1.76 4.5 1.89 5 0.00 movements in rates and a replacement cost equation, we can complete an example which demonstrates the full development of a swap risk factor. High• lighted in Table A2.2 are the calculated rate movements for a five-year fixed• floating semi-annual interest rate swap, based on 16 per cent historical volatility and a to per cent original rate for the fixed rate payer. Utilizing the calculated rates from Table A2.2 in equation A2.10 (recalling that we compare r(n) with r(O) at each semi-annual period), we obtain a dis• counted calculation for each semi-annual period of the five year swap. The results are shown in Table A2.3. Adding the individual calculated replacement cost terms above for each time period yields r(d), the total calculated replacement cost. Discounting back

353 Appendix 2

Table A2.4. Calculated and Discounted Replacement Costs

n red) (%) r (%)

0.5 4.1 4.0 1 5.4 4.9 1.5 6.0 5.2 2 6.2 5.1 2.5 6.0 4.7 3 5.5 4.1 3.5 4.6 3.3 4 3.4 2.3 4.5 1.9 1.2 5 0.0 0.0

using r(O) yields the discounted replacement cost, r. These results are given in Table A2.4. From the table we note the maximum value of r is 5.2 per cent, occurring in year 1.5. This indicates that the maximum replacement cost (and, hence, the maximum risk) likely to be encountered under the above scenario is 5.2 per cent of the notional amount of the swap. We can therefore say the risk factor RF for a five-year, 10 per cent fixed payer swap is 5.2 per cent. The reader may note that the maximum risk under this transaction occurs in year 1.5, rather than in year four or five; this occurs as a result of the 'diffusion and amortization' effect. If we plot the maturity of the swap trans• action on the x-axis and the replacement cost along the y-axis, we observe that maximum risk occurs approximately one-third to one-half through the transaction. This interaction is highlighted in Figure A2.l, where we illus• trate swap replacement curves for five- and ten-year transactions (based on 16 per cent volatility and a 10 per cent starting rate). The reason for the 'diffusion and amortization' effect is relatively easy to explain: simulated fu• ture rates used in the calculation of replacement cost will not have had a chance to move sufficiently in the early periods to pose the greatest economic loss (this is the 'diffusion' effect); and, insufficient payments remain to be made towards the end of the swap to pose the greatest economic loss (this is the 'amortization' effect). As a result, the greatest economic loss occurs at some point in between, when rates have had a chance to move and when there are still a number of swap payments remaining to be made. Depending on the historical volatility used, the maximum replacement cost occurs one• third to one-half through the life of the swap. We can see in the example above that the maximum point on the five-year swap under 16 per cent vola• tility is reached in year 1.5, precisely one-third through the life of the transaction. Once the equations above have been used to produce the series of replace-

354 Calculating Risk Swap

Replacement Maximum replacement cost C08lWhlchequals lhe ~ risk 'aclor 'or a swap 16 wtth fixed rale x and - .....-~~ maximum maturity y 14

12

10

8

6

2

O+-'-'-~-r-r-r~~~~~~~~~~+-+-+-~~ o 0.5 1 1 .5 2 2.5 3 3 .5 4 4 .5 5 5 .5 6 6 .5 7 7 .5 8.5 8 9 9 .5 10 Years

Figure A2.1. Swap Replacement Cost Curves

ment costs (discounted to today's value) it is relatively easy to create a table of risk factors for use in determining swap risk. The procedure generally en• tails creating data tables with a range of fixed rates, say 5 to 12 per cent, and a range of maturities, from one year to ten or 15 years (we can also replicate the procedure for fixed receiver swaps where simulated rates are based on equation (A.2.9); this yields a slightly different set of RF's). Such tables should encompass a normal range of likely swap rates and maturities; for each rate and each maturity, the maximum replacement cost calculated by the method above becomes the risk factor for that maturity. This risk factor (or replace• ment cost) is then multiplied by the notional to obtain REE. Over the course of the swap, it is entirely appropriate for the risk factors to be lowered to reflect the fractional exposure attributable to the remaining maturity of the transaction. Highlighted in Table A2.5 are sample risk factors based on the model dis• cussed above. The factors are based on Treasury rates with a 16 per cent historical volatility (after being adjusted to a 90 per cent confidence level, lognormal distribution). Factors are shown for fixed rates from 5 to 12 per cent, and for maturities from one year to ten years. As detailed earlier, the risk factor for a given swap with fixed rate x and final maturity y is simply the maximum replacement cost for a swap with those characteristics when

355 Appendix 2

Table A2's. Sample Risk Factor for Interest Rate Swaps (%) (Fixed Payer)

Y ••r: 1 2 3 4 6 8 7 8 8 10

5% 0.3 0.8 1.0 1.5 2.3 3.9 4.9 5.8 7.2 7.6 6% 0.3 1.0 1.8 2.7 3.6 4.6 5.6 6.6 8.1 8.5 7% 0.4 1.1 2.1 3.0 4.1 5.1 6.2 7.3 8.8 9.3 8% 0.4 1.3 2.3 3.4 4.5 5.6 6.7 7.8 9.4 9.9 9% 0.5 1.4 2.5 3.7 4.9 6.1 7.2 8.3 9.9 10.4 10% 0.5 1.5 2.7 3.9 5.2 6.4 7.6 8.7 10.3 10.7 11% 0.6 1.7 2.9 4.2 5.5 6.8 8.0 9.1 10.6 11.1 12% 0.6 1.8 3.1 4.5 5.8 7.1 8.3 9.4 10.9 11.3

the adjusted historical volatility is 16 per cent. For example, in applying the model presented in the appendix, a ten-year, 10 per cent fixed rate swap will have a maximum replacement cost of 10.7 per cent, occurring in year three. Therefore, the relevant factor, encompassing the maximum point of risk, is 10.7 per cent. An analyst or credit committee seeking to approve a $IOOm swap with such parameters would assign an REE of $1O.7m to the transaction.

Notes

I. This appendix is drawn largely from a previous work by this author (Banks. 1993). 2. In addition to the original paper by Rendleman and Bartter. readers may wish to reference Hull, 1989. for a detailed review of the model; in addition. readers may wish to review other interest rate pricing models which model the stochastic movement of interest rates through different processes (with varying degrees of sophistication and assumptions), such as those by Black, Derman and Toy, Vasicek, Ho and Lee, Cox, Ingersoll and Ross, Jamshidian, and Hull and White. among others.

356 Appendix 3: Twenty Questions for the Derivatives Desk

Listed below is a series of 20 questions that can be explored with a deriva• tives specialist seeking credit approval for a specific derivative transaction. Review of these questions helps identify unique credit (or other) risk charac• teristics which may exist.

Swap-related Derivatives

I. What is the size of the transaction, in notional terms? Does principal decrease or increase at any time and, if so, on what basis? Is the notional ever exchanged (e.g. as in a currency swap)? 2. What is the maturity of the transaction? Are there periodic options to terminate? 3. What determines the payment of interim coupons/flows? Are there pay• ment mismatches in either direction? 4. Are any of the swap payments or receipts structured in an off-market fashion? If so, how and why? 5. Is there a leverage component to the payments and/or receipts? How is the leverage defined? What is the impact of the leverage under different market conditions? 6. Does the swap contain any embedded options? How do such options af• fect or influence payments and/or receipts? 7. Is the payoff of the swap currency-protected?

Option-related Derivatives

8. What is the size of the transaction, in notional or contract terms? 9. What is the maturity of the transaction?

357 Appendix 3

10. How is the option exercised (e.g. European, American, Bermudan)? II. If the option is being purchased, how is premium being paid (e.g. upfront, amortized, contingent)? 12. What is the relative starting price and strike price of the structure? Is the strike price set in a non-standard fashion (e.g. deferred, forward, floating)? 13. What determines the payoff of the option? Does the option feature a pay• off profile which adds or extinguishes exposure rapidly (e.g. reverse knock• in/knock-out) or otherwise locks-in exposure (e.g. cliquet, ladder, shout)? ]4. Is there a leverage component to the payoff and, if so, how is it defined? What is the impact of the leverage under different market scenarios? 15. Is the payoff of the option currency-protected?

General

16. What is the client's motivation for entering into this transaction? Does the transaction require preparation and review of scenario analyses? 17. Are term sheets available and designed to convey the particulars of the transaction in a clear fashion? 18. Has the client signed, or will it sign, standard documentation (including a master agreement)? 19. Does this transaction carry any special legal, operational, risk manage• ment or credit considerations or does it require any special attention? 20. Does this transaction warrant the allocation of supplemental credit re• serves and/or credit capital for any reason?

358 Glossary

Accreting swap - a swap featuring a notional principal balance which increases on a preset schedule or on the basis of a trigger event. See also reverse index principal swap.

Accrual note - see range fioater note.

Adual exposure - the amount of true credit exposure inherent in a derivative trans• action; it is generally included as one of two components (along with fractional ex• posure) in the determination of total credit exposure. If actual exposure is positive at the time of default a credit loss is sustained; if actual exposure is negative at the time of default no credit loss is sustained. Actual exposure is also known as replacement cost, actual market risk or mark-to-market value.

Adual market risk - see actual exposure.

All-or-nothing option - see binary option.

American binary option - see binary-barrier option.

American option - an option which is exercisable at any time up until, and includ• ing, maturity.

Amortizing swap - a swap featuring a notional principal balance which amortizes on either a preset schedule or on the basis of a trigger event. See also mortgage swap and index principal swap.

Asian option - an option which generates a payoff based on the average price of the underlying market reference over a predetermined averaging period; the average path can be applied to either the price of the underlying market reference (average price option) or the strike price (average strike option) to determine the payoff. Asian op• tions are classified as path dependent derivatives. See also average price option, aver• age strike option.

Asset-at-expiry option - a binary option which generates a payoff at expiry equal to a fixed asset M if the price of the underlying market reference exceeds the strikel barrier at any time during the transaction. Asset-at-expiry options are members of the binary-barrier option family. See also binary option, binary-barrier option.

359 Glossary

Asset-at-hit option - a binary option which generates an immediate payoff equal to a fixed asset M if the price of the underlying market reference exceeds the strikel barrier. Asset-at-hit options are members of the binary-barrier option family. See also binary option, binary-barrier option.

Asset-or-nothing option - a binary option which generates a payoff equal to a fixed asset M if the price of the underlying market reference exceeds the strike price at expiry.. Asset-or-nothing options are members of the binary option family. See also binary option.

Allset swap - a swap which exchanges fixed coupons from an underlying asset into ftoating coupons, giving the investor a synthetic ftoating rate asset in a single pack• age. An asset swap can also be structured to convert ftoating rate coupons into fi xed rate coupons.

Assignment - the process of transferring a derivative exposure from the original counterparty to a third party as a means of reducing exposure to the original counterparty.

At-the-money - a term applied to an option whose strike price (SP) is equal to the current market price (CMP). An at-the-money option has no intrinsic value but has maximum time value.

Average expected risk exposure - potential credit exposure based on the average maturity of the transaction and the expected market movement of the underlying market reference; average expected exposure is the smallest of four classes of credit ex• posure. See also average worst case risk exposure, terminal worst case risk exposure, terminal expected risk exposure.

Average price option - an option which generates a payoff equal to the difference between an average price on an underlying market reference and a predefined strike price. Average price options are members of the Asian option family. See also aver• age strike option, Asian option.

Average strike option - an option which generates a payoff equal to the difference between an average strike price and the terminal value of the underlying market ref• erence. Average strike options are members of the Asian option family. See also average price option, Asian option.

Average worst case risk exposure - potential credit exposure based on the average maturity of the transaction and the worst case market movement of the underlying market reference; average worst case exposure is the third largest of four classes of credit exposure. See also average expected risk exposure, terminal worst case risk exposure and terminal expected risk exposure.

Backspread - an option spread designed to take advantage of volatility. Backspreads are created through the sale of a smaller quantity of closer-to-the-money puts or calls 360 Glossary and the purchase of a larger quantity of farther-from-the-money puts or calls. A short backspread is known as a ratio vertical spread (see below).

Barrier option - an option which extinguishes or creates an underlying European option as the price of the underlying market reference moves through a prespecified barrier. There are four versions of the barrier option, including the down and in op• tion (an option created when the underlying market references falls through a barrier on the downside), the down and out option (an option which is extinguished when the underlying market reference falls through a barrier on the downside), the up and in option

Basis risk - risk generated from two floating indexes which track, but do not dupli• cate, one another. Basis risk is often assumed as a means of minimizing directional risk and can arise from hedging an asset with a futures contract on the same (or similar) asset, or hedging an asset with a second, similar asset.

Basis swap - an interest rate swap which involves the exchange of two floating rate indexes, such as Libor and commercial paper rates or Libor and Federal Funds.

Basket option - an option which generates a payoff based on the difference between the price of a basket of assets and a predefined strike price; the basket of assets may be from identical or different asset classes/markets. Basket options are classified as path independent derivatives and are members of the multi-index option/rainbow op• tion family. See also multi-index option.

Bear spread - an option spread which attempts to take advantage of bearish (falling) markets; spreads can be structured as bearish vertical call spreads (purchase of a call and sale of a second call, where the short call is struck c1oser-to-the-money) or bear• ish vertical put spreads (purchase of a put and sale of a second put, where the long put is struck c1oser-to-the-money). Also known as vertical spreads, price spreads, money spreads.

Bermudan option - an option which can be exercised only on specific dates prior to maturity. A Bermudan option is thus more flexible than a European option, but less flexible than an American option. Also known as a Mid-Atlantic option.

Best/worst of n-assets option - options which generate a payoff based on the best or worst performing of a group of assets against a predefined strike price; assets may be from identical or different asset classes/markets. There are four different forms of options, including a call on the best of n-assets, a call on the worst of n-assets, a put on the best of n-assets, and a put on the worst of n-assets. These options are classified 361 Glossary as path independent derivatives and are members of the multi-index option/rainbow option family. See also multi-index option, call on the best of n-assets, call on the worst of n-assets, put on the best of n-assets, put on the worst of n-assets.

Best/worst of n-assets and casb option - options which generate a payoff based on the best or worst performing of a group of assets and cash; assets may be from identical or different asset classes/markets. Best-worst options generate a minimum payoff. equal to a predefined cash amount C and do not carry strike prices; as such, they never end out-of-the-money. These options are classified as path independent derivatives and are members of the multi-index option/rainbow option family. See al~o multi-index option.

Binary option - an option which generates a payoff equal to a prespecified amount M if the price of the underlying market reference exceeds the strike price; this de• rivative is also known as a digital option or an all-or-nothing option. A binary option can be structured to pay oft" in cash (a cash-or-nothing option) or the underlying asset (asset-or-nothing option) and may have a European exercise (e.g. it must end in-the• money) or American exercise (e.g. it must move in-the-money). An American exer• cise binary option can payoff as soon as the underlying crosses the strike price (an at-hit option) or at the expiration of the option (an at-expiry option). European binary options are classified as path independent derivatives, while American binary options are generally classified as path dependent derivatives. See also asset-or-nothing op• tion. cash-or-nothing option, binary-barrier option.

Binary-barrier option - an option which generates a payoff equal to a prespecified cash or asset amount M if the price of the underlying market reference exceeds the strike price. This derivative, also known as an American binary option, a one-touch option or an at-hit/at-expiry option, is a hybrid of the barrier and binary options and generates a payoff if the underlying market reference triggers the barrier (which is also the strike) at any time during the life of the transaction. See also cash-at-hit option, cash-at-expiry option, asset-at-hit option, asset-at-expiry option.

Black-Seholes option pricing model - the fundamental closed-form option pricing model of the derivative markets, developed in 1973 by Black and Scholes to value European options on non-dividend paying stocks. The Black-Scholes framework re• lies on a series of assumptions (e.g. continuous movement of the underlying refer• ence, unlimited borrowing at a risk free rate, no transactions costs/taxes) to generate a price for the option.

Bull spread - an option spread which attempts to take advantage of bullish (rising) markets; spreads can be structured as bullish vertical call spreads (purchase of a call and sale of a second call, where the long call is struck c1oser-to-the-money) or bull• ish vertical put spreads (purchase of a put and sale of a second put, where the short put is struck closer-to-the-money). Also known as vertical spreads, price spreads, money spreads.

362 Glossary

Butterfly spread - an option spread designed to take advantage of volatility. Butter• flies are always created with the same ratio of options (one low strike, two middle strikes, one high strike) which expire at the same time; spreads can be based on puts or calls. Short butterflies consist of short low and high strike options and long middle strike options and long butterflies consist of long low and high strike options and short middle strike options.

Calendar spread - see time spread.

Call - a derivative granting the buyer the right, but not the obligation, to purchase an underlying reference security at a predefined strike price.

Call on a call - an option which grants the buyer the right to purchase an underlying call. Calls on calls are members of the compound option family. See also compound option.

Call on a put - an option which grants the buyer the right to purchase an underlying put. Calls on puts are members of the compound option family. See also compound option.

Call on the best of n-assets - an option which generates a payoff based on the difference between a predefined strike price and the best performing asset in a port• folio; the option effectively permits the holder to purchase the best performing of a group of assets. Calls on the best of n-assets are members of the multi-index/rainbow option family. See also best/worst of n-assets option.

Call on the maximum - an option which generates a payoff based on the difference between a predefined strike price and the highest price achieved by the underlying reference asset over the life of the transaction. Calls on the maximum are members of the lookback option family. See also options on the maximum/minimum.

Call on the worst of n-assets - an option which generates a payoff based on the difference between a predefined strike price and the worst performing asset in a port• folio; the option effectively permits the holder to purchase the worst performing of a group of assets. Calls on the worst of n-assets are members of the multi-index/rain• bow option family. See also best/worst of n-assets option.

Call spread - an option spread which is created to express a bullish or bearish direc• tional market view; the spread requires the sale and purchase of two separate calls with different strikes; see bull spreads, bear spreads.

Callable bond - a bond with embedded calls which permits the issuer to call the bond at a particular price level on select call dates.

Callable swap - a swap structure where the fixed rate payer has the option to enter into an underlying swap at a future date. See also putable swap, cancellable swap.

363 Glossary

Cancellable swap - a swap structure where the fixed rate receiver has the option to terminate the underlying swap at a future date. See also putable swap, callable swap.

Cap - an option which generates a payoff when an underlying interest rate reference exceeds a predefined strike level.

Caption - an option on a cap; the caption grants the purchaser the right to buy a cap.

Cash-at-expiry option - a binary option which generates a payoff at expiry equal to a fixed cash amount M if the price of the underlying market reference exceeds the strike/barrier at any time during the transaction. Cash-at-expiry options are members of the binary-barrier option family. See also binary-barrier option.

Cash-at-hit option - a binary option which generates an immediate payoff equal to a fixed cash amount M once the price of the underlying market reference exceeds the strike/barrier. Cash-at-hit options are members of the binary-barrier option family. See also binary-barrier option.

Cash on delivery option - see contingent premium option.

Cash-or-nothing option - a binary option which generates a payoff equal to a fixed cash amount M if the price of the underlying market reference exceeds the strike price at expiry. Cash-or-nothing options are members of the binary option family. See also binary option.

Chooser option - an option which permits the purchaser to choose between an under• lying call and put (with equal strikes and maturities) between trade date and choice date. A chooser option, also known as a regular chooser option or preference option, is classified as a path independent derivative. See also complex chooser option.

Cliquet option - an option which locks in gains at prespecified evaluation intervals if the option is in-the-money at such intervals. If the option is out-of-the-money on an evaluation date, the strike resets at-the-money based on the new market level. Cliquet options, also known as ratchet options, are classified as path dependent derivatives.

Collar - a spread consisting of a purchased cap/call and a sold fioor/put, or vice versa. The sale of one option reduces the premium payable on the second option. See also zero-cost collar.

Commodity swap - a swap which involves the exchange of two different commodity references. A conventional commodity swap exchanges a fixed commodity price for a fioating commodity price, in price and unit terms most common to the underlying reference. Commodity swaps can be written on virtually any physical commodity (e.g. energy products, base metals, precious metals, grains, and so on) and can be struc• tured to settle in cash or physical.

364 Glossary

Complex chooser option - an option which permits the buyer to choose between an underlying call (with a certain strike and maturity) and an underlying put (with a different strike and maturity) between trade date and choice date. See also chooser option.

Compound option - an option which can be exercised into an underlying option. There are four different types of compound options, including a call on a call (the right to buy an underlying call), a call on a put (the right to buy an underlying put), a put on a call (the right to sell an underlying call), and a put on a put (the right to sell an underlying put). Compound options are classified as path dependent deriva• tives. See also call on a call, call on a put, put on a call, put on a put.

Condor spread- an option spread designed to take advantage of volatility. Condors are always created with the same ratio of options (one low strike, one middle low strike, one middle high strike, one high strike) which expire at the same time; spreads can be based on puts or calls. Short condors consist of short low and high strike options and long middle low and high strike options, and long condors consist of long low and high strike options and short middle low and high strike options.

Contingent premium option - an option where the purchaser is only obliged to pay the seller premium if the option ends in-the-money; if the option ends out-of-the• money the seller receives no premium payment. If the option ends in-the-money but the intrinsic value is less than the premium due to the seller, the purchaser is still obligated to exercise the option and pay the seller premium. Contingent premium options, also known as when-in-the-money options, pay later options or cash on de• livery options, are classified as path independent derivatives.

Convertible bond - a hybrid debt/equity security which consists of a coupon-paying bond and an embedded equity option which allows the holder to convert into a speci• fied number of shares once a conversion level is attained.

Convexity - a mathematical measure which quantifies the price sensitivity of a bond to large changes in a bond's yield; convexity measures the non-linear effects of price/ yield changes.

Correlation - a statistical measure indicating the extent to which two or more under• lying assets move in identical, or opposite, directions. Correlation is a key element in the pricing and risk management of various complex derivatives, including multi• index derivatives.

Credit derivative - a derivative which generates payoff or protection based on an underlying risky debt reference. Credit derivatives include credit forwards, credit options, default swaps, and total return swaps (see each).

Credit forward - a single period forward contract which generates a payoff based on the difference between an agreed credit spread/price and a terminal credit spread/ price of a risky debt reference. 365 Glossary

Credit option - an option which generates a payoff based on the difference between a risky debt credit spread/price and a predefined strike price. In standard form credit options generate a continuum of payoffs based on credit appreciation or depreciation; in binary form credit options generate a payoff based solely on default by the refer• ence credit. Also known as default options.

Credit reserve - reserves allocated in support of expected credit losses.

Credit risk - the risk of economic loss if a counterparty to a transaction fails to perform.

Currency swap - a swap involving the exchange of two different currencies. A typi• cal currency swap involves the exchange of a fixed payment in one currency for a floating payment in a second currency. although the exchange of two fixed or two floating payments can also be structured. Currency swaps involve the initial and final exchange of principal. which results in a high degree of credit exposure.

Current exposure method - a regulatory method of computing swap credit risk which focuses on the sum of future credit exposure and actual exposure.

Day count note - see range floater note.

Deemed risk - see fractional exposure.

Default option - see credit option.

Default swap - a swap based on the exchange of deferred premium (often in floating rate form) for a lump-sum payment if an underlying reference credit defaults. The payoff associated with this structure is similar to a binary credit option.

Deferred payment American option - an American exercise option which permits the seller to utilize option proceeds from the time of exercise until the original ma• turity of the option. In exchange for relinquishing use of proceeds until maturity. the buyer pays the seller a lower premium.

Deferred strike option - an option with a strike price set at a future time period as a specific function of the spot value of the underlying market reference at that time.

Delta - a change in the value of an option for a small change in the value of the underlying market reference.

Delta hedging - the dynamic process of reducing or neutralizing the exposure of an option to market direction. Delta hedging is accomplished by establishing a delta• equivalent long or short position in the underlying reference against a long or short position in the option. Long calls are hedged with short underlying. long puts with long underlying. short calls with long underlying. and short puts with short underlying.

366 Glossary

Ditrerence option - see spread option.

Ditrerentlal swap - a swap based on floating rate references in two distinct cur• rencies, payable in a single currency.

Digital option - see binary option.

Directional risk - exposure to the direction of a particular market or reference. Also known as delta risk.

Discount swap - an off-market swap where one party pays a second party an upfront payment in exchange for paying below-market coupons during the life of the transaction.

Discrete barrier - see partial barrier.

Down and in option - an option which is created if the price of the underlying market reference declines through a predefined barrier. Down and in options are members of the barrier option family. See also barrier option.

Down and out option - an option which is extinguished if the price of the underly• ing market reference declines through a predefined barrier. Down and out options are members of the barrier option family. See also barrier option.

Duration - a mathematical measure which quantifies the price sensitivity of a bond to small changes in yield.

Embedded option - an option which is incorporated into a note or swap to provide the purchaser or investor with a unique payoff profile.

Equity derivative - a broad category of derivatives based on single stocks, equity baskets and equity indexes. Equity derivatives are available as forwards, options and swaps with embedded options.

European barrier option - see point barrier option.

European option - an option which is only exercisable at maturity.

Exotic option - an option which provides a specific payoff or protection profile not attainable through a conventional American or European option. The option may con• tain unique features related to the determination of a payoff path, payment of pre• mium, establishment of a strike price, and so on.

Expected credit loss - an average, or mathematically expected, credit loss, most often determined through a combination of expected credit exposure, probability of default, and recoveries. See also unexpected credit loss, worst case credit loss.

367 Glossary

Exploding option - a bullish or bearish vertical spread which generates a payoff once the two strikes defining the spread are breached. See also bull spread, bear spread.

Extendible option - an option which enables the purchaser either to exercise an option on a particular reset date or reset the strike price to the current market level and extend the option for a subsequent reset period. This derivative is a variation of the reset, or partial look back, option. See also partial look back option.

Fixed strike ladder option - an option which locks in gains prior to expiry as the price of the underlying market reference exceeds prespecified market levels (or 'rungs'); gains are not relinquished if the market subsequently retraces. This version of the option generates gains by comparing the terminal price and ladder rungs against a_ predefined strike price and allocating a gain to the larger of the two. Fixed strike ladder options are members of the ladder option family. See also ladder options.

Fixed strike lookback option - see options on the maximum/minimum.

Fixed strike shout option - an option which locks in gains at a particular market level when the purchaser 'shouts' for gains to be locked in; gains are not relinquished if the market subsequently retraces. This version of the option generates gains by comparing the terminal price and shout level against a predefined strike price and allocating a gain to the larger of the two. Fixed strike shout options are members of the shout option family. See also shout options.

Floating strike ladder option - an option which locks in gains prior to expiry as the price of the underlying market reference exceeds prespecified market levels (or 'rungs'); gains are not relinquished if the market subsequently retraces. This version of the option carries no preset strike price and generates gains by comparing the terminal price and ladder rungs at maturity. Floating strike ladder options are members of the ladder option family. See also ladder options.

Floating strike lookback option - an option which generates a maximum gain by 'looking back' over the price path of the asset and determining the point which cre• ates the greatest economic gain. This version of the option carries no preset strike price and generates a gain by comparing the terminal price against the lowest buying price (for calls) or highest selling price (for puts) achieved by the asset. Floating strike lookback options are members of the lookback option family. See also lookback options.

Floating strike shout option - an option which locks in gains at a particular market level once the purchaser 'shouts' for gains to be locked in; gains are not relinquished if the market subsequently retraces. This version of the option carries no predefined strike price and generates gains by comparing the terminal price and shout level at maturity. Floating strike shout options are members of the shout option family. See also shout options.

368 Glossary

Floor - an option which generates a payoff when an underlying interest rate refer• ence falls below a lower strike level.

Floortlon - an option on a floor; the floortion grants the purchaser the right to buy a floor.

Forward - an over-the-counter derivative which permits the purchaser to buy an asset at a predetermined forward price on a forward date and the seller to sell an asset at a predetermined forward price on a forward date. Forwards are available on any underlying security/asset from the fixed income, equity, currency, commodity or credit markets and can be settled in cash or physical.

Forward rate agreement - a single period interest rate forward contract.

Forward start option - an option which is contracted on trade date t and com• mences on forward date tl, with the forward start date, strike price, and final ma• turity parameters established on trade date. Once the forward date is reached and the transaction begins the contract assumes the characteristics of a conventional Euro• pean option. Forward start options are classified as path independent derivatives.

Forward swap - a swap which is contracted on trade date t and commences on forward date II, with the rate and final maturity parameters established on trade date. Once the forward date is reached and the transaction begins the contract assumes the characteristics of a conventional swap.

Fractional exposure - the amount of future credit exposure inherent in a derivative transaction, typically included as one of two components (along with actual, or mark• to-market, exposure) in the determination of total credit exposure. Fractional exposure is generally estimated through statistical or simulation methods and is utilized as a measure of the potential credit exposure which might be encountered in a transaction. Fractional exposure is positive at the inception of a trade and declines as maturity approaches. Fractional exposure is also known as presettlement risk, time-to-decay risk, or deemed risk.

Future - an exchange-listed contract which permits the purchaser to buy an asset at a predetermined forward price on a forward date and the seller to sell an asset at a predetermined forward price on a forward date. Futures are available on assets/secur• ities from the fixed income, equity, currency and commodity markets. Contracts are marked-to-market on a daily basis by the exchange clearing-house and variation mar• gins are posted to cover daily market movement.

Futures option - an exchange-listed option on a futures contracts.

Gamma - a change in the value of delta for a change in the price of the underlying market reference. Gamma, which measures the convexity or non-linearity of option

369 Glossary

prices, is often used to measure sensitivity to large market movements.

Guaranteed exchange rate option - see quanto.

High-low option - an option which generates a payoff based on the difference be• tween the high and low prices of the underlying reference asset during the life of the transaction. High-low options are classified as path dependent derivatives.

Historical volatility - a statistical measure of the price movement of an asset based on historical data, generally denominated in terms of variance or standard deviation. Historical volatility is often used to estimate future market moves (e.g. as in a risk equivalency or value-at-risk framework). See also implied volatility.

Implied volatility - a measure of the price movement of an asset based on data implied by traded option prices. Volatility is one of the central inputs of option pric• ing models but is not directly observable in the market; as such, traders generally utilize observed option prices to derive the volatility implied by the option prices. See also historical volatility.

In-the-money - a term applied to an option whose strike price (SP) is above the current market price (CMP) for puts and below the CMP for calls. An in-the-money option has maximum intrinsic value.

Index amortizing rate swap - see index principal swap.

Index principal swap - a swap with a notional principal which amortizes as a float• ing rate reference declines through pre-specified barrier levels. As the notional de• clines, fixed and floating rate payments associated with the swap become smaller. See also reverse index principal swap.

Instalment option - an option where the purchaser is permitted to pay the seller premium in instalments, rather than upfront; under the terms of the option the pur• chaser can cancel the contract at any time by suspending instalment payments. If the purchaser completes all required payments, the seller grants a conventional European option with contract details as specified on trade date. Instalment options are classi• fied as path dependent derivatives.

Interest rate swap - a swap based on the exchange of two interest rate references. A typical interest rate swap involves the exchange of a fixed interest rate for a /loating interest rate, although two floating rates can also be exchanged. Interest rate swaps do not involve the initial and final exchange of principal and. as such. feature less credit exposure than currency swaps. See also basis swap.

Intermediation - the process of executing a derivative transaction through another intermediary rather than directly with the end-user; the intermediary. rather than the arranger, is thus exposed to the credit risk of the end-user.

370 Glossary

Intrinsic value - one of two components (along with time value) which determines the value of an option. Intrinsic value measures the current value, or moneyness, of an option; options which are in-the-money can be exercised for an immediate gain and thus have intrinsic value equal to the difference between the current market price and the strike price. Options which are either out-of-the-money or at-the-money have no intrinsic value. See also time value.

Inverse floater note - a structured note which provides the investor with a coupon based on an inverse rate generally defined as (x per cent fixed rate minus floating rate); thus, rising rates generate a lower coupon and falling rates generate a higher coupon.

Inverse floater swaps - a swap which exchanges a fixed rate for an inverse rate generally defined as (x per cent fixed rate minus floating rate); this structure has the effect of magnifying upward or downward movements in rates since both legs of the swap move in the same direction.

Jump process - a mathematical process used to describe the movement of asset prices which are impacted by sudden, discontinuous moves. Certain option pricing models utilize a jump process, rather than a continuous stochastic process, to generate option values.

Kappa - see vega.

Kick-in option - see reverse knock-in option.

Kick-out option - see reverse knock-out option.

Knock-In option - an option which is created when the price of the underlying mar• ket reference moves above or below a barrier. Knock-in options, which include down and in options and up and in options, are members of the barrier option family. See also barrier option.

Knock-out option - an option which is extinguished when the price of the underly• ing market reference moves above or below a barrier. Knock-out options, which in• clude down and out options and up and out options, are members of the barrier option family. See also barrier option .

.Ladder option - an option which locks in gains prior to expiry as the price of the underlying market reference exceeds prespecified market levels (or 'rungs'); gains are not relinquished if the market subsequently retraces. Ladder options, which are classified as path dependent derivatives, are available in fixed and floating strike form (see each).

Leveraged note - a structured note which provides the investor with an enhanced coupon based on a predefined leverage formula which multiplies the effects of market movements. 371 Glossary

Leveraged option - see power option.

Leveraged swap - a swap which generates a payoff by multiplying one side of the transaction by a leverage factor; the use of leverage compounds the upward or down• ward movement of the underlying market reference. Leverage can be applied to any swap and can be defined in any fashion. Also known as power swap, ratio swap.

Lognormal distrIbution - a statistical distribution widely used for characterizing the distribution of asset prices. A lognormal distribution, which is skewed to the right, faces a lower bound of zero and an upper bound of infinity.

Lookback option - an option which generates a maximum gain by 'looking back' over the price path of the asset and determining the point which creates the greatest economic gain. Lookback options, which are classified as path dependent derivatives, are available in fixed and floating strike form (see each).

Mark-to·market - the process of revaluing a derivative transaction based on current market prices or rates. Marking-to-market is widely used in the quantification of ac• tual credit exposure as it provides an accurate indication of the current value, and hence replacement cost, of a derivative.

Market risk - the risk of economic loss due to adverse movement in the market price/rate of an underlying reference security or market.

Mean reversion - the tendency for interest rates, and the volatility of interest rates, to return to some mean or average level over the long term. Mean reversion is incor• porated into certain bond option pricing models.

Mid-Atlantic optIon - see Bermudan option.

Modlfted ladder option - see fixed strike ladder option.

Modified shout option - see fixed strike shout option.

Money spread - see bear spread, bull spread.

Monte Carlo simUlation - a computer-intensive statistical process which generates asset paths based on user-defined inputs and drawings from a random number gener• ator. Monte Carlo simulation is widely used in the pricing of derivatives and the development of portfolio measurement techniques.

Mortple swap - an amortizing swap which synthetically replicates the investment performance of a physical mortgage-backed security. A mortgage swap amortizes at a rate equal to the amortization of the reference mortgage backed security.

Multl·factor option - see multiple index option.

372 Glossary

Multi-index note - a structured note which pays an enhanced coupon based on the performance of multiple indexes drawn from the same, or different, asset c1assesl markets. See also structured note.

Multi-index option - an option which generates a payoff based on the difference between two or more reference assets and a predefined strike price; assets may be drawn from the same, or different, asset classes/markets. These derivatives, also known as rainbow options, encompass a series of individual derivatives including basket options, spread options, multiple strike options, best/worst of n-assets options and best/worst of n-assets and cash options (see each). Multi-index options are classified as path independent derivatives.

Multiple strike option - an option which generates a payoff based on the best or worst performing of a series of assets, each with a specific strike price. Option refer• ences may be drawn from the same, or different, asset classes/markets. Multiple strike options are classified as path independent derivatives and are members of the multi• index option/rainbow option family. See also multi-index option.

Negative gamma - risk exposure to large market moves generated through the sale of puts or calls. See also gamma.

Netting - the process of offsetting exposures and reducing payments to a single How. In order for netting to be employed a master agreement governing all transactions must be signed by both parties; in addition, netting must have legal basis in the jurisdiction of operation.

Normal distribution - a statistical distribution characterized by the familiar 'bell• shaped' curve. A normal faces no lower or upper bounds (e.g. both are infinite).

Notional - a common method of denominating the size of a derivative transaction, particularly a swap. In most instances notional is used only as a reference to compute amounts payable and/or receivable, though for currency swaps the full notional is typically exchanged.

One-factor interest rate models - option pricing models which value bond options by generating an entire yield curve through a single interest rate reference, generally a short rate. Though such models are thought to be less accurate than two-factor models, they are far simpler to implement. See also two-factor interest rate models.

One-touch option - see binary-barrier option.

Option - a derivative granting the bolder the right, but not the obligation, to buy or sell a given reference security at a predefined strike price. In exchange for the right, the buyer pays the seller a premium. Options are available on any underlying secur• ity/asset from the fixed income, equity, currency, commodity or credit markets, and

373 Glossary can be bought or sold in over-the-counter or listed form.

Options on the maximum/minimum - options which generate a maximum gain by 'looking back' over the price path of the asset and determining the points which create the greatest economic value. This version of the lookback option, also known as a fixed strike look back, carries a preset strike price and generates a payoff based on the difference between the strike price and the maximum price (for calls on the maximum) or minimum price (for puts on the minimum) achieved by the asset. Op• tions on the maximum/minimum are members of the lookback option family. See also lookback options.

Original exposure method - a regulatory method of computing swap credit risk which focuses solely on future credit exposure (rather than the sum of future credit exposure and actual exposure).

Out-of-the-money - a term applied to an option whose strike price (SP) is below the current market price (CMP) for puts and above the CMP for calls. An out-of-the• money option has no intrinsic value.

Outperformanee option - an option which generates a payoff based on the out• performance of a spread against a predefined strike price. See also spread option.

Outside barrier option - a multivariate option with a barrier which is triggered by a reference other than the one defining the underlying option. See also barrier option.

Partial barrIer option - a barrier option where the barrier is only in effect during a portion of the option's life. Partial barrier options are members of the barrier option family. See also point barrier option, barrier option.

Partial lookback option - an option with a strike price which can be reset on a particular evaluation date if the option is out-of-the-money. Partiallookback options, also known as a reset options, are classified as path dependent derivatives.

Path dependent option - a broad category of options whose payoff at expiry is dependent on the price path of the underlying reference asset at previous points in time.

Path independent option - a broad category of options whose payoff at expiry is dependent solely on the price of the underlying reference asset at expiry.

Pay later option - see contingent premium option.

Point barrier option - a barrier option where the barrier is only in effect during a single point in time, generally maturity. Point barrier options, also known as Euro• pean barrier options, are members of the barrier option family. See also partial bar• rier option, barrier option.

374 Glossary

Potential exposure - the sum of actual exposure and fractional exposure; potential exposure is a key measure of the current and future credit exposure impacting a de• rivative transaction. See also actual exposure, fractional exposure.

Potential market risk - see fractional exposure.

Power option - an option which generates a payoff on an exponential basis. A power option raises the price of the underlying market reference to a prespecified exponent (or power) and compares the result against a predefined strike price to generate an economic gain. Power options, also known as leveraged options or turbo options, are classified as path independent derivatives.

Power swap - see leveraged swap.

Preference option - see chooser option.

Premium - the payment made by the purchaser of an option to seller of an option in exchange for potential gains under the option contract.

Premium swap - an off-market swap where one party pays a second party an upfront payment in exchange for receiving above-market coupons during the life of the transaction.

Presettlement risk - see fractional exposure.

Price spread - see bear spread. bull spread.

Put - a derivative granting the holder the right, but not the obligation, to sell an underlying reference security at a prespecified strike price.

Put on a call - an option which grants the buyer the right to sell an underlying call. Puts on calls are members of the compound option family. See also compound option.

Put on a put - an option which grants the buyer the right to sell an underlying put. Puts on puts are members of the compound option family. See also compound option.

Put on the best or n-assets - an option which generates a payoff based on the differ• ence between a predefined strike price and the best performing asset in a portfolio; the option effectively permits the holder to sell the best performing of a group of assets. Puts on the best of n-assets are members of the multi-index/rainbow option family. See also best/worst of n-asset options.

Put on the minImum - an option which generates a payoff based on the difference between a predefined strike price and the lowest price achieved by the underlying

375 Glossary reference asset over the life of the transaction. Puts on the minimum are members of the lookback option family. See also options on the maximum/minimum.

Put on the worst of n-assets - an option which generates a payoff based on the difference between a predefined strike price and the worst performing asset in a port• folio; the option effectively permits the holder to sell the worst performing of a group of assets. Puts on the worst of n-assets are members of the multi-index/rainbow op• tion family. See also best/worst of n-assets options.

Put spread - an option spread which is created to express a bullish or bearish direc• tional market view; the spread requires the sale and purchase of two separate puts with different strikes; see bull spreads, bear spreads.

Putable bond - a bond with embedded puts giving the investor the right to sell the bond back to the issuer at a predetermined price.

Putable swap - a swap structure where the fixed rate payer has the option to termin• ate the underlying swap at a future date. See also callable swap, cancellable swap.

Quanto option - an option which converts gains from an underlying foreign cur• rency derivative into a home currency, at a predetermined foreign exchange rate; use of a quanto option protects an investor from exchange rate risk. Quanto options, also known as quantity adjusted options or guaranteed exchange rate options, are classi• fied as path independent derivatives.

Rainbow option - see multi-index option.

Range Hoater note - a structured note which provides the investor with an enhanced coupon if the floating rate reference trades within a predefined range, or band; for every day the reference falls outside the band the investor loses one day's interest. The security is effectively a standard floating rate note with a strip of embedded binary options. Also known as accrual notes, day count notes.

Ratchet option - see cliquet option.

Ratio horizontal spread - see time spread.

Ratio swap - see leveraged swap.

Ratio vertical spread - an option spread designed to take advantage of volatility. Ratio vertical spreads are created through the purchase of a smaller quantity of closer• to-the-money puts or calls and the sale of a larger quantity of farther-from-the-money puts or calls. A long ratio vertical spread is known as a backspread (see above).

Recouponing - the process of marking-to-market and settling a portfolio of deriva-

376 Glossary

tives as a means of reducing actual exposure. During the process a net cash settle• ment is paid by the party holding the out-of-the-money contracts to the party holding the in-the-money contracts and the derivatives are then rewritten, or recouponed, at current market levels.

Regular chooser option - see chooser option.

Replacement cost - see actual exposure.

Reset option - see partial barrier option.

Reverse barrier option - see reverse knock-in option, reverse knock-out option.

Reverse Index principal swap - a swap with a notional principal which increases as a floating rate reference declines through prespecified barrier levels. As the notional increases, fixed and floating rate payments associated with the swap become larger. See also index principal swap, accreting swap.

Reverse knock-in option - a barrier option which knocks-in in-the-money. Reverse knock-in options are members of the barrier option family. See also barrier option.

Reverse knock-out option - a barrier option which knocks-out in-the-money. Re• verse knock-out options are members of the barrier option family. See also barrier option.

Rho - a change in the value of an option for a change in the risk-free rate.

Risk-adjusted capital - capital allocated in support of unexpected credit losses.

Risk equivalent exposure - the potential credit risk generated by a derivative trans• action. Risk equivalent exposure generally includes actual exposure (also known as mark-ta-market exposure, replacement cost or actual market risk) and fractional ex• posure (also known as potential market risk, deemed risk, time-to-decay risk or presettlement risk). Risk equivalent exposure is often estimated through a statisticall volatility framework or a simulation process. See also actual exposure, fractional exposure, potential exposure.

Shout option - an option which locks in gains at a particular market level when the purchaser 'shouts' for gains to be locked in; gains are not relinquished if the market subsequently retraces. Shout options, which are available in fixed and floating strike form (see each). are classified as path dependent derivatives.

Spread option - an option which generates a payoff based on the difference between two reference assets and a predefined strike price; the assets may be drawn from similar or different asset classes/markets. Spread options. also known as difference

377 Glossary options, out performance options, or underperformance options, are classified as path independent derivatives and are members of the multi-index option/rainbow option family. See also multi-index options, yield curve options, outperformance options, underperformance options.

Stochastic process - a mathematical process used to describe continuous and dy• namic movement of asset prices. Many option pricing models utilize a stochastic process to generate option prices.

Straddle - an option spread designed to take advantage of volatility. Straddles are created through the purchase or sale of options with identical strikes and expiry dates. Long straddles consist of long puts and calls struck at the same level and short strad• dles consist of short puts and calls struck at the same level.

Strangle - an option spread designed to take advantage of volatility. Strangles are created through the purchase or sale of options with different strikes but identical expiry dates. Long strangles consist of long puts and calls struck at different levels and short strangles consist of short puts and calls struck at different levels.

Structured note - a note or bond with an embedded derivative which generates a specific payoff. Broad categories of structured notes include range floater notes, in• verse floater notes, mUltiple index notes, and leveraged notes (see each).

Swap - an over-the-counter derivative calling for the periodic exchange of payments between two parties. Swaps are available on any underlying security/asset from the fixed income, equity, currency, commodity or credit markets.

Swaption - an option on a swap. A receiver swaption grants the buyer the right to receive fixed in swap while a payer swaption grants the buyer the right to pay fixed in a swap.

Synthetic option - a long or short option created synthetically through the combina• tion of a long or short option on an underlying security and a long or short position in the same underlying security.

Synthetic underlying - a long or short position in an underlying security created synthetically through the combination of two long or short options on the underlying security.

Terminal expected risk exposure - potential credit exposure based on the final maturity of the transaction and the expected movement of the underlying market reference; terminal expected exposure is the second largest of four classes of credit exposure. See also average expected risk exposure, average worst case risk exposure, terminal worst case risk exposure.

378 Glossary

Terminal worst case risk exposure - potential credit exposure based on the final maturity of the transaction and the worst case movement of the underlying market reference; terminal worst case exposure is the largest of four classes of credit ex• posure. See also average expected risk exposure, average worst case risk exposure, terminal expected risk exposure.

Termination option - an option to terminate a derivative transaction based on the passage of time or the occurrence of a credit event (e.g. ratings downgrade).

Theta - a change in the value of an option for a change in the passage of time. See also time decay, time value.

Time decay - daily gain or loss experienced in the time value component of option premium due to the passage of time; often used as a reference to theta. See also theta, time value.

Time spread - an option spread designed to take advantage of volatility as related to time. Time spreads are created through the purchase or sale of options with identical strikes but different expiry dates. Long time spreads consist of short near month puts or calls and long far month puts or calls and short time spreads consist of long near month puts or calls and short far month puts or calls. Also known as a calendar spread or ratio horizontal spread.

Tlme-to-decay risk - see fractional exposure.

Time value - one of two components (along with intrinsic value) which determines the value of an option. In purchasing an option, a buyer is purchasing time for the underlying reference to move in-the-money (or further in-the-money); lime value measures the remaining economic value of the option which is attributable to time. Longer term options have greater time value than shorter time options, while in-the-money and out-of-the-money options have less time value than at-the-money options. See also intrinsic value.

Total return swap - a swap which synthetically replicates the economic flows of a risky bond position. Under a total return swap one party pays periodic fixed or float• ing coupons from a reference bond plus any resulting appreciation in the price of the bond at maturity of the transaction, while the second party pays a periodic nominal floating coupon plus any resulting depreciation in the price of the bond at maturity of the transaction.

Turbo option - see power option.

Two-'actor interest rate models - option pricing models which value bond options by generating an entire yield curve through two interest rate references, generally a short rate and a long rate. Though such models are more complex to implement than

379 Glossary one-factor models, they are generally thought to be more accurate. See also one• factor interest rate models.

Underperformance option - an option which generates a payoff based on the underperformance of a spread against a predefined strike price. See also spread option.

Unexpected credit loss - the difference between expected and worst case credit losses; alternately, the difference between the mean of the credit loss distribution function and a point represented by multiple standard deviations from the mean. See also ex• pected credit loss, worst case credit loss.

Up-and-In option - an option which is created once the price of the underlying market reference rises above a predefined barrier. Up and in options are members of the barrier option family. See also barrier option.

Up-and-out option - an option which is extinguished once the price of the under• lying market reference rises above a predefined barrier. Up and out options are mem• bers of the barrier option family. See also barrier option.

Value-at-rlsk - a measure of the worst case loss expected over a prespecified hold• ing period, adjusted to a particular confidence level. Value-at-risk is implemented by using correlation and volatility matrices and can be quantified using historical or simu• lation methods.

Variable strike option - see deferred strike option.

Vega - a change in the value of an option for a change in volatility.

Vertical spread - see bear spread, bull spread.

Volatility skew - the skewness obtained in comparing the implied volatility of out• of-the-money calls and puts. Out-of-the-money calls generally have lower implied volatilities than out-of-the-money puts as there are typically more call sellers than put sellers in the market.

Volatility smile - the shape of a curve which plots implied volatility (on the y-axis) against strike price (on the x-axis). A conventional smile attributes greater implied volatility to in- or out-of-the-money options.

Warrant - an 'option-like' derivative which is often attached to a bond issue and then detached and traded separately. The most common attached warrants provide the investor with a call on the issuer's stock, but individual warrants are also available on fixed income, equity, currency and commodity references.

When-In-the-money option - see contingent premium option.

380 Glossary

Worst case credit loss - a potential credit loss represented by a point multiple standard deviations from the mean (or expected) value of the credit loss distribution function. See also expected credit loss, unexpected credit loss.

Yield curve option - a spread option which generates a payoff based on the differ• ence between two points on a yield curve against a predefined strike price. See also spread option.

Zero-coupon swap - a swap where one party makes a single payment at maturity, in bullet form. In most cases the second party makes a series of interim payments up to and including maturity.

Zero cost collar - a spread consisting of a purchased cap/call and a sold floor/put, or vice versa. The sale of one option offsets the premium payable on the second option resulting in 'zero cost'. See also collar.

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387 Index

ability v. willingness to pay 339 besl/worst of n-assets 146, 182-3 abritraging, transactions 337 credit risk 184 accrued interest receivable, swap credit options 146-7, 183-5 risk 217-18 best/worst of n-assets and cash actual exposure (AE), swap credit credit risk 182-3 risk 215-19 options 145-7, 182-5 actual market risk (AMR) 44-5, 66, 77 bilateral netting 29-30 Air Products and Chemicals 25, 27 binary options 143--4, 180-1 Allied Lyons 24 Black-Scholes model, option valuation all-or-nothing options see binary options 342-8 American options, deferred payment bonds 156-7, 195 convertible 36 amortizing swaps 232--4, 249-51 with embedded credit derivatives 5, credit risk 249-50 36,327-8 Arak, M. 224 with warrants 5, 36 Asian/average price options 131, 172 Bretton Woods agreement 35 credit risk 172-3 British and Commonwealth Merchant Asian/average strike options 132, Bankers 26 172-3 butterflies, spreads 97-9, 114-17 credit risk 173 credit risk 97-8, 116 asset-or-nothing options 143--4 assignment calendar spreads see time spreads credit exposures 321-2 call options 73 derivative exposure 31 credit risk 76 average expected credit exposure 60-2 vertical spreads 92-3, 107-10 average worst case credit capital, covering credit losses exposure 60-2 309-11 cash-or-nothing options 143--4 backspreads 101-2, 120-2 Chemical Bank 24 credit risk 120-1 China International Petroleum and Bank for International Settlements 9, Chemical Company 25 30, 209, 211-12, 213 China National Metals and Minerals Bank of England 223--4 Import and Export Company 25 Bankers Trust 270-1 chooser options 153-5, 192-3 bankruptcy-remoteness, DPCs 31-2 credit risk 192-3 Barings 5, 25, 27, 37 CITIC Shanghai 25 barrier options 127-30 cliquet options 138--40, 177-9 credit risk 166-71 credit risk 178 multiple 163--4 Codelco 24 basket options 151-2, 188-90 collateral credit risk 189 derivative exposure 30 Berjaya International 25 risk reduction 318-20

388 Index

comparative statics. option pricing credit rating. cohort techniques 298 models 82-6 credit rating. migrations 298 complex chooser options 154-5 credit risk 4-5. 40-1. 73-5 complex swaps 228--43 alternative methodologies 81-6 credit risk 244-5 ancillary issues 335-8 compound options 152-3. 190-2 complex options 4. 81. 126-203 credit risk 190-2 complex swaps 4. 228--44 condors. spreads 99-100. 117-18 exposure 293--4 credit risk 118 management 162 confidence levels. statistical 53-5. 69. mitigation 317-28 301 on-line calculation 334-5 contingent premium options 155-6. options 73-86 194-5 option strategies 81, 88-125 credit risk 194 quantifying 75-81. 106-10 continuation vehicles. DPCs 32-3 credit simulation/scenario correlation 284-5 analysis 328-30 convertible bonds 36 credit swaps 237--41, 326-7 counterparties credit risks 254-61 default 294-9 current exposure method (CEM), swap portfolios 291 credit risk 209-14 willingness and ability to current market price 75. 76, 77-8 perform 339--41 credit charge structures 332 Daiwa Bank 27,38 credit derivatives 324-8 databases. historical, credit as products 6. 9 systems 334 as risk mitigating instruments 324 default credit exposure 66 options 326 average expected 60. 62. 63 probability 294-9. 314 average worst case 60. 61. 63 recoveries in 299-30 I calculating 276-8 swaps 238-9. 257-9. 326-7 information. real-time access 333--4 deferred payment. premiums 155 management 33 deferred payment American technological base 332-5 options 156-7. 195 terminal expected 60-3 credit risk 195 terminal worst case 60-3 deferred strike options 158-9 types 60-3 delivery risk credit forwards 254. 324-5 delta. option prices 83. 345-6 credit risk 324 derivative product companies credit swaps 237-8. 254-7. 326-7 (DPCs) 31-3 credit risk 255. 326 derivatives credit limits 292 long-term transactions 338-9 dynamic allocation 330-2 losses 23-6 credit losses market volatility 8-20 computing 302-9 notional amounts 45-6 distributions 293. 302. 303 risk and return considerations expected and unexpected 292-301 20-3 risk adjusted profitability 309-13 risk management 26-34 credit migration 297-8 transaction motivation 335-8 credit options 161-3, 201-3. 326 Derivatives Policy Group (DPG) 27. credit risk 326 28.29.315

389 Index

Development Finance Corporation fractronal exposure (FE), swap credit (DFC) 26 risks 219-21 Dharmala Sakti Sejahtara 25 futures, trading 5 difference options see spread options differential swaps 232, 247-9 gamma, option prices 83, 346-7 credit risk 248 geometric Brownian motion 48, 64-5 diffusion process, price behaviour Giberti, D. 226 49-50 Gibson Greetings 25, 27, 270 digital" options see binary options glossary 359-81 discrete barrier options 129 Goldman Sachs 28 down and in options 128 Goodman, L. 224 f:redit risk 169-70 Group of 30 27, 313 down and out options 128 guarantee vehicles, DPes 32-3 credit risk 170-1 guaranteed exchange rate options, see downgrade quanto options credit exposure 322-3 guarantees, derivative exposures 317-18 derivative exposure 31 Drexel Burnham Lambert 26 Hammersmith and Fulham 24 Handjinicolaou, G. 225 equity-index swaps 242-4 hedging, transactions 336-7 credit risks 261-70 high-low options 135-6, 174-5 exchange-traded products 4-5 credit risk 135 expected credit losses 302-5, 310-11, historical databases, credit systems 334 335 historical rate movements, fractional exploding options 159-60, 199 exposure 219-20 credit risk 127, 159 historical volatility, price changes 51-3,350,354-5 Federal Reserve 223-4 Hull, J. 225 Federated Paper Board 25 hybrid approach, portfolios 290-1 Ferron, M. 225 financial institutions, development of implied volatility 52 derivatives 8-9 incremental summation approach, financial intermediaries 21-2 portfolios 278-89 risks 24 index amortizing rate swaps see index financial markets, volatility 8-20, principal swaps 58-9 index-principal swaps 235-6, 252-4 First Boston 24, 28 credit risk 253 fixed strike ladder options 138, 176-7 instalment options 141-3, 179-80 credit risk 176 credit risk 142 fixed strike lookback options 135 intermediation fixed strike shout options 140, 179 credit exposures 322 floating strike ladder options 138 derivative exposure 31 credit risk 138 International Swaps and Derivatives floating strike lookback options 132-4, Association 29-30, 38 173, 178,204 intrinsic value of premium 156 credit risk 173 inverse floater swaps 228-9, 245-6 floating strike shout options 140 credit risk 129 floors forward start options 158-9, 197-9 Kanematsu Corporation 25 credit risk 197 Kashima Oil 25

390 Index

Kidder Peabody 27, 37-8 Nippon Sanso 25 Knock-in. knock-out options see barrier Nippon Steel Chemical 24 options Olympia and York 26 ladder options 137-8, 176-7 one-way collateral 319 credit risk 138 One-touch options. see binary options fixed strike 138 option method. fractional exposure 220 floating strike 138 options Lehman Brothers 28 complex 4. 357-8 leveraged options see power options credit risk see credit risk leveraged swaps 230-1, 246-7, 270-1 strategies 88, III credit risk 246 valuation 342-8 leveraged transactions 337-8 options on the maximum/minimum local authorities 24 126. 134-5. 173-4 lognormal distribution, pricing credit risk 126 movements 48-51 original exposure method. swap credit look back options risk 209-14 fixed strike 135 outsider barrier options 129. 130 floating strike 132-4, 173 over-the-counter (OTC) derivatives partial 136-7. 175-6, 204 4-5.73 Overseas Chinese Bank 25 mark -to-market derivative exposures 320-1 partial barrier options 129 swaps 215-17, 280 partiallookback options 136-7. 175-6.204 market risk 42-5 credit risk Mentini, M. 226 path dependent options 126. 127-42 Merrill Lynch 24. 28 path independent options 126-7. Merrill Lynch Derivative Products 32 143-64 Metallgesellschaft 24 periodic collateral 318-19 money spreads see vertical spreads point barrier options 129 Morgan, JP 24 pooled portfolio collateral 319-20 Morgan Stanley 28 portfolio credit vectors 290 mortgage swaps 234-5, 251-2 portfolios credit risk 228 counterparties 291 mortgage-backed securities 234-5 credit exposure management 276-90 Muffet. M. 225 potential exposure (PE) swap credit risk multi-index options 145-52. 181-2 potential market risk (PMR) 43-5. 66 credit risk 145 power barrier options 164 multiple barrier options 163-4 power options 160-1. 199-201 multiple limit structure. credit power swaps see leveraged swaps limits 331-2 preference options see chooser options multiple strike options 147-8. 185-6 premium. credit risk 141-2 credit risk premiums. deferred payment 155 price changes. historical volatility 51-3 nested options see compound options price sensitivities method 20 I net present value (NPV). swaps 215-17 price spreads see vertical spreads netting pricing. collateral 320 bilateral 29-30 pricing movements. lognormal credit exposure 212-13. 289. 323 distribution 48-51 Nikkei 225 index 56 Procter and Gamble 25. 27. 37. 270-1

391 Index

puts 73 credit risk 140 credit risk 73, 79 fixed strike 140 floating strike 140 quanto options 157-8, 196-7 Showa Shell 25 credit risk 196-7 simulation method credit risk 328-30 rainbow options 145-52, 181-2 fractional exposure 220-1 credit risk 127 portfolios 289-90 ratchet options see cliquet options risk equivalency 46, 63"'(', 82 ratio horizontal spreads see time spreads simulation-scenario engine 334 ratio swaps see leveraged swaps Sinar Mas Group 25 ratio vertical spreads 102-3, 122-5 single limit structure, credit limits credit risk 125 331 recouponing speculation, transactions 337-8 credit exposures 320-1 spread options 148-51, 187-8 derivative exposure 30 credi t risk 165 regular chooser options 154-5 square root rule 51, 55 regulators, derivative trading 23 statics, comparative, option pricing replacement costs, calculation 351-5 models 82",(, reserves, covering credit losses 309-13 stochastic variables, pricing reverse index-principal swaps movements 48 (IPS) 236, 253-4 stock market, voltatility 9 credit risk 253 straddles 93-5, 110-13 reverse knock in/knock-out options see credit risk 110-11 barrier options strangles 95-7, 113-14 reverse swaps see inverse Hoater swaps credit risk 113 risk, see market risk straps, long and shon 103 risk adjusted profitability 309-13 strike price 75 risk equivalency 45-7 strips, long and short 103 framework 58-9 structured notes, credit exposure 33, risk equivalent exposure (REE) 47, 58, 327-8 75,76-81 Sumitomo Corporation 5, 25, 27, 38 calculation 334-5 summation approach, portfolios multiple derivative transactions 275",(, 278-9 refining 59-63 swap credit risk risk factors, derivative complete framework 221-2 transactions 47-57 individual institution approach risk management 8 214-21 derivatives 26-34 model 349-56 internal controls 27-8 regulatory approach 209-14 risk mitigation 30-1,317-28 research findings 222-6 risk-return mechanism 20-3 see also complex swaps Rones, A. 224 swaps, complex 4, 357 synthetic options 104-6, 125 Salomon Brothers 28 synthetic positions 105 Salomon Swapco 32 synthetic underlyings lOS, 106 Scabellone, P. 226 scenario analysis, credit risk 328-30 terminal expected credit exposure 60 sensitivities approach, credit risk 82-6 terminal worst case credit shout options 140-1, 179 exposure 59-63

392 Index termination options, credit unexpected credit losses 305,308-9,335 exposures 322-3 up and in options 128 termination vehicles, DPCs 32-3 credit risk 166-7 theta, option prices 84-5, 347-8 up and out options 128 third-party enhancements 317-18 credit risk 167-9 third-party support, derivative upfront collateral 318-19 exposure 30 time spreads 100-1, 118-20 vega 84.348 credit risk 119 vertical spreads 92-3, 107-10 time value premium 170, 175 credit risk 108 Tokai Corporation 24 volatility-based method. risk Tokyo Securities 25 equivalency 46 total return swaps 239-41, 259-61, 326-7 Weiner process 51 TPI Polene 25 Whittaker. J.O. 224-5 transaction-specific collateral 319-20 worst case credit losses 302-8 transactions, maturity 47-8, 55 turbo options see power options yield curve options 150-1 two-way colateral 319 yield enhancing. transactions 337

393