# Interest Rate Options

### The Black-Scholes Option Pricing Model

Fisher Black and Myron Scholes developed their option pricing model under the assumption that assets prices adjust to prevent arbitrage, that stock prices must change continuously, and that stock returns follow a log-normal distribution. Also, their model holds for European call options on stocks with no dividends. Further, they assume that the interest rate and the volatility of the stock remain include stochastic calculus. In this section, we present their model and illustrate the basic intuition that underlies it. We show that the form of the Black-Scholes model parallels the bounds on option pricing that we have already observed. In fact, the form of Black-Scholes model is very close to the binomial model.

### The Black-Scholes Call Option Pricing Model

The following expression gives the Black-Scholes option pricing model for a call option:

Equation (13.13)

c =St x N(d1) - Xexp-r(T-t)N(d2)

where N() is the cumulative normal distribution function, and

Equation (13.14)

d1 = ln (St/X) + (r + 0.5σ2)(T-t) / σ √T-t

d2 = d1 - σ √T-t

This model has the general form that we have long considered - the value of a call must equal or exceed the stock price minus the present value of the exercise price:

ct ≥ St - Xexp-r(T-t)

To adapt this formula to account for risk, as in the Black-Scholes model, we multiple the stock price and the exercise price by some factors to account for risk, giving the general form:

ct ≥ St x risk factor 1 - Xexp-r(T-t) x risk factor 2:

Equation (13.2)

ct = j=0 (n!/j!(n-j)!)[U&PI;j,DΠn-1]MAX{0,UjDn-jSt-X}/Rn

The binomial model shares this general form with the Black-Scholes model. With the binomial model, the risk adjustment factors where the large bracketed expressions in Equation 13.2. With the Black-Scholes model, the risk factors are N(d1) and N(d2), In the Black-Scholes model, these risk adjustment factors are the continuous time equivalent of the bracketed expression in the binomial model.

### Computing Black-Scholes Option Pricing

In this section, we show how to compute Black-Scholes option prices. Assume that a stock trades at \$100 ("X") and the risk-free interest rate is 6 percent ("r"). A call option on the stock has an exercise price of \$100 ("S") and expires in one year ("t"). The standard deviation ("σ2") of the stock's returns is 0.10 per year. We compute the values of d1 and d2 as follows:

d1 = ln(St/X) + (r + 0.5σ2(T-t) σ√(T-t)

d1 = ln(100/100) + ((.06 + 0.5 x 0.10) x 1 / 0.10 x √1 = 0.65

d2 = d1 - σ√(T-t)

d2 = 0.65 - 0.1 x 1 = 0.55

Next we find the cumulative normal values associated with d1 and d2. These values are the probability that a normally distributed variable with a zero mean and a standard deviation of 1.0 will have a value equal to or less than the d1 or d2 term we are considering. Figure 13.8 shows a graph of a normally distributed variable with a zero mean and a standard deviation of 1.0. It shows the values of d1 and d2 for our example. For illustrations, we focus on d1, which equals 0.65. In finding N(d1), we want to know which portion of the area under the curve lies to the left of 0.65. This is the value of N(d1). Clearly, the value we seek is larger than 0.5, because d1 is above the mean of zero. We can find the exact value by consulting a table of the cumulative normal distribution for this variable (or use excel NORMDIST function). For a value of 0.65, we find our probability on the interior: N(0.65) = 0.7422. Similarly, N(d2) = N(0.55) = 0.7088. We now have the following:

N(0.65) =NORMDIST(0.65,0,1,TRUE) = .7422

N(0.55) =NORMDIST(0.55,0,1,TRUE) = .7098

-r*(T-t) is rate of 6 percent with 1 year to maturity = exp(-.06)*1 = .9418

c =St x N(d1) - Xexp-r(T-t)N(d2)

Call value = ct = \$100 x 0.7422 - \$100 x 0.9418 x 0.7088 = \$7.46

We chose these values for our example because they parallel the values from our original binomial example. There we also assumed that the stock traded for \$100 and that the risk-free rate was 6.0 percent. We assumed an up factor of 10 percent and a down factor of -10 percent. With a single-period binomial model and these values, we found that a call must be priced at \$7.55. The two results are close. However, if we use more periods in the binomial model and use up and down factors that are consistent with a log-normal distribution of stock returns, the binomial model will converge to the Black-Scholes model price.

### The Black-Scholes Put Option Pricing Model

Black and Scholes developed their option pricing model for calls only. However, we can find the Black-Scholes model for European puts by applying put-call parity:

pt = ct - S + Xexp-r(T-t)

Substituting the Black-Scholes call formula in the put-call parity equation gives the following:

pt =StN(d1) -Xexp-r(T-t)N(d2) -St + Xexp-r(T-t)

Collecting like terms simplifies the equation to the following:

pt =St[N(d1) -1] + Xexp-r(T-t)[1 - N(d2)]

If we consider the cumulative distribution of all values from -∞ to +∞, the maximum value is 1.0. For any value of d1 that we consider, part of the whole must lie at or below the value and the remainder must lie above it. For example, if N(d1) is 0.7422, for d1 = 0.65, then 25.78 percent of the total area under the curve must lie at values greater than 0.65. Now we apply a principle of normal distributions. The normal distribution is symmetrical, so the same percentage of the area under the curve that lies above the curve that lies above d1 must lie below -d1. Therefore, for any symmetrical distribution and any arbitrary value, w:

N(w) + N(-w) = 1

Following this pattern and substituting for N(d1) and N(d2) gives the equation Black-Scholes value for a put option:

pt = Xexp-r(T-t)N(-dF2) - StN(-d2)

### Inputs for the Black-Scholes Model

We have seen that the Black-Scholes model for the price on an option depends of five variables: the stock price, the exercise price, the time until expiration, the risk-free rate, and the standard deviation of the stock. Of these, the stock price is observable in the financial press and online. The exercise price and the time until expiration can be known with certainty. We want to consider how to obtain the other two parameters: the risk-free interest rate and the standard deviation of the stock.

### Estimating the Risk-Free Rate of Interest

Estimates of the risk-free interest rate are widely available and are usually quite reliable. There are still a few points to consider, however. First we need to select the correct rate. Because the Black-Scholes model uses a risk-free rate, we can use the Treasury bill rate as a good estimate. Quoted interest rates for T-bills are expressed as discount rates. We need to convert these to regular interest rate and express them as continuously compounded rates. As a second consideration, we should select the maturity of the T-bill carefully. If the yield curve has a steep slope, yields for different maturities can differ significantly. With T-bills maturing each week, we choose the bill that matures closest to the option expiration.

We illustrate the computation with the following example. Consider a T-bill with 84 days until maturity. Its bid yield is 8.83 percent, and its asking yield is 8.77 percent. Letting BID and ASK be the bid and asked yields, the following formula gives the price of a T-bill as a percentage of its face value:

PTB = 1 - 0.01 x (BID + ASK)/2 (days until maturity/360)

= 1 - 0.01 x (8.83 + 8.77)/2 (84/360) = 0.97947

In this formula, we average the bid and ask yields to estimate the unobservable true yield that lies between the observerable bid and asked yields. For our example, the price of the T-bill is 97.947 percent of the face value. To find the corresponding compounded rate, we solve the following: equation for r:

expr(T-t) = 1/PTB

T-t = 84/360

expr(0.23) = 1/0.97947

0.23r = ln(1.02096) = .0207

r = 0.0902

In the equation, T - t = 0.23, because 84 days is 23 percent of the year. Thus, the appropriate interest rate in this example is 9.02 percent. It is fairly easy to secure good estimates of the risk-free interest rate. However, it is not critical to have an exact estimate, as option prices are not very sensitive to the interest rate.

### Estimating the Stock's Standard Deviation [viewable here in Excel]

Estimating the standard deviation of the stock's returns is more difficult and more important than estimating the risk-free rate. The Black-Scholes model takes as its input the current, instantaneous standard deviation of the stock. In other words, the immediate volatility of the stock is the riskiness of the stock that affects the option price. The Black-Scholes model also assumes that the volatility is constant over the life of the option. There are two basic ways to estimate the volatility. The first method uses historical data, while the second technique employs fresh data from the options market itself. The second method uses option prices to find the option market's estimate of the stock's standard deviation. An estimate of the stock's standard deviation that is drawn from the options market is call an implied volatility. We consider each method in turn.

### Historical Data

To estimate volatility using historical data, we compute the price relatives, the logarithmic price relatives, and the mean and standard deviation of the logarithmic price relatives. Letting PR, indicate the price relative for day t, so that PRt = Pt/Pt-1, we give the formulas for the mean and variance of the logarithmic price relatives as follows:

PR = 1/TTΣt=1ln PRt

VAR(PR) = PR = 1/T - 1TΣt=1ln PRt - PR)2

As an example, we apply these formulas to data in Table 13.1, which gives 11 days of price information for a stock. With 11 price observations, we compute ten daily returns. The first column tracks the day, while the second column records the stock's closing price for the day. The third column computes the price relative from the prices in the second column. The forth column gives the log of the price relative in the third column. The last column contains the result of subtracting the mean of the logarithmic price relatives from each observation and squaring the result. The mean, variance, and standard deviation that we have calculated are all based on our sample of daily data. We use the sample standard deviation as an input to the Black-Scholes model.

 Table 13.1 Historical volatility computations Day Pt PRt ln(PRt) [lnPRt - PRμ]2 0 100.00 1 101.50 1.0150 0.014888612 0.000154240 2 98.00 0.9655 -0.03509132 0.001410797 3 96.75 0.9872 -0.012837147 0.000234286 4 100.50 1.0388 0.038027396 0.001264381 5 101.00 1.0050 0.004962789 0.000006218 6 103.25 1.0223 0.022032715 0.000382729 7 105.00 1.0169 0.016807118 0.000205574 8 102.75 0.9786 -0.021661497 0.000582293 9 103.00 1.0024 0.002430135 0.000000002 10 102.50 0.9951 -0.00486619 0.000053809 Sums 0.02469 0.00429 Sample μ 0.00247 Sample σ2 0.000477 Sample σ 0.02184371 Annualized Standard Deviation 0.34537938 Annualized Standard Deviation = Sample σ x √250

Three inputs to the Black-Scholes model depend on the unit of time. These inputs are the interest rate, the time until expiration, and the standard deviation. We can use any single measure we wish, but we need to express all three variables in the same time units. For example, we can use days as our time unit and express the time until expiration as the number of days remaining. Then we must also use daily standard deviation into a comparable yearly estimate. We have estimated our daily standard deviation of ten days. However, these are ten trading days, not calendar days. Accordingly, we recognize that we are working in trading time, not calendar time. Deleting weekend days and holidays, each year has about 250-252 trading days. We use 250 trading days per year.

We have already seen that stock prices are distributed with a standard deviation that increases as the square root of time. Accordingly, we can adjust the time dimension of our volatility estimate by multiplying it by the square root of time. For example, we convert from our daily standard deviation estimate to an equivalent yearly value by multiplying the daily estimate time the square root of 250:

Equation (13.16) annualized σ = daily σ x √250

For our daily estimate of 0.021843, the estimated standard deviation in annual terms is 0.3454.

In our example, we have used ten days of data. In actual practice, we face a trade-off between using the most recent possible data and using more data. In statistics, we almost always get more reliable estimates by using more data. However, the Black-Scholes model takes the instantaneous standard deviation as an input. This gives great importance to using current data. If we use the last year of historical data, then we have a rich data set for estimating the old volatility. Using just ten days, as we did in our example, emphasizes current data, but it is really not very much data for getting a reliable estimate.

To emphasize the importance of using current data, consider the Crush of 1987. On Black Monday, October 19, 1987, the market lost about 22 percent of its value. If we used a full year of daily data to estimate a stock's historical volatility the next day, out estimate would be too low. In light of Black Monday, the instantaneous volatility had surely increased.

### Implied volatility (see "Imp-Volatility" worksheet)

To overcome the limitations inherent in using historical data to estimate standard deviations, some scholars have turned to techniques of implied volatility. In this section, we show how to use market data and the Black-Scholes model to estimate a stock's volatility. There are five inputs to the Black-Scholes model, which the model relates to a sixth variable, the call price. With a total of six variables, any five imply a unique value for the sixth. The technique of implied volatility uses known values of five variables to estimate the standard deviation. The estimated standard deviation is an implied volatility, because it is the value implied by the other five variables in the model.

To find the volatilities, we begin with established values for the stock price, the exercise price, the interest rate, the time until expiration, and the call price. We use these to find the implied standard deviation. However, the standard deviation enters the Black-Scholes model through the values for d1 and d2, which are used to determine the values of the cumulative normal distribution. As a result, we cannot solve for the standard deviation directly. Instead, we must search for the volatility that makes the Black-Scholes equation hold. To do this, we need a computer. Otherwise, we would have to try an estimate of the standard deviation, make all of the Black-Scholes computations by hand, and adjust the standard deviation for the next try. This would be cumbersome and time consuming. Therefore, implied volatilities are almost always found using a computer.

For most stocks with options, several options with different expiration trade at once. Some researchers have argued that all of these options should be used to find the volatility implied by each. The result-ing estimates are then given weights and averaged to find a single volatility estimate. The single estimate is known as a weighted implied standard deviation. In principle, this is a good idea because it uses more information. Other things being equal, estimates based on more information should dominate estimates based on less information. However, some options trade infrequently, which makes their prices less reliable for computing implied volatilities. In addition, options way out-of-the-money tend to give the least biased volatility estimates, and may options traders derive implied volatilities by focusing on at-the-money options.

Considering the following example of an implied standard deviation based on a call option. We assume that X = \$100 and the option is at-the-money, so that S = \$100. We also assume that the option has 90 days remaining until expiration and that the risk-free interest rate is 10 percent, so we have T - t = 90 days and r = 0.10. The call price is \$5.00. To find the implied standard deviation, we need to find the standard deviation that is consistent with these other values. To do this, we can compute the Black-Scholes model price for alternative standard deviations. We adjust the standard deviation to make the option price coverage to its actual price of \$5.00. The sequence of standard deviations and corresponding call prices below show this relationship. In our example, we first try σ = 0.10, which gives a call price of \$3.41. This price is too low. Thus, we know the correct standard deviation must be larger, because the call price varies directly with the standard deviation. Next, σ = 0.5 results in a call price of \$11.03 which is too high. Now we know that the standard deviation must be greater than 0.10, but less than 0.50. The task is to find the standard deviation that gives a call value equal to the specified \$5.00. This happens with σ = 0.185.

We turn to analytic techniques for valuing options. The first of these is the option-adjusted spread. The option-adjusted spread (OAS) is the yield differential between corporate bond (or mortgage instrument) with an embedded option and a Treasury bond. The Treasury bond is chosen so that the two instruments will have essentially similar maturity and coupon characteristics. So defined, the yield spread provides a measure of the option's value and the default risk differential between the Treasury and the bond (or mortgage instrument). Some-times the OAS is defined between two securities of the same basic default risk, coupon, and maturity. In this case, the OAS would embrace only the yield differential due to the option feature inherent in one of the two securities. Because it is difficult to find comparable bond or mortgage instruments, it is more common to compute the OAS relative to a Treasury instrument.

The OAS is often applied to corporate bonds with call provisions and to home mortgages with the option for the home-owner to prepay the mortgage. In each case, the spread is typically calculated as a spread between the yield on the instrument with the embedded option and a Treasury instrument with similar cash flows. The OAS is the number of basis points that must be added to the yield on the Treasury instrument so that the spread-adjusted Treasury and the corporate bond or mortgage will have the same price.

As an illustration of this technique, consider a 20-year corporate semiannual coupon bond with a coupon rate of 10 percent. The bond is callable in five years and is currently priced at 97.25 per-cent of par. The corresponding yield to maturity for this bond is 10.3278 percent. The call feature embedded in this bond obviously has a value to the issuer, but the value of that option is unclear. Under the OAS technology, a spread to Treasury is computed to provide a metric of that value. Consider also a T-bond with a 10 percent coupon and a 20-year remaining maturity. The T-bond is priced at 108.533 percent of par and yields 9.0681 percent. Thus we have the following:

 Instrument Price Yield to maturity Callable corporate bond 97.25 10.3278% Treasury bond 108.533 9.0681% 1.2597%

The promised cash flows on the two instruments are the same. The difference in the yield to maturity stems from two sources: the difference in default risk and the presence of the call feature on the corporate bond.

Table 19.5 presents detailed information on these two bonds. The first column of the table lists the 40 periods of the life of each bond, while the second column sets out the semiannual cash flows that are common to each bond. The third column shows the zero-coupon annual rate appropriate to each period, which reflects a term structure assumed for this example. The short-term rate is 8 percent, rising to a rate of 10.7 percent over 20 years of remaining maturity on the bonds. Column 4 shows the zero-coupon factor for each semiannual period, while column 5 presents the zero-coupon factor appropriate to the cash flow for each period's cash flow. The zero-coupon rates in column 3 give a unique discounting rate for each period's cash flow. This reflects the shape of the yield curve, instead of discounting flows from various times at a single yield to maturity. The zero-coupon factors of column 5 are the Z0,t terms for this particular term structure environment that we saw how to find earlier.

Column 6 shows the present value of each of the cash flows from the T-bond, which total 108.533. (Each element in column 6 is just the corresponding cash flow from column 2 divided by the appropriate factor from column 5. (Given the first six columns of information in Table 19.5, we are now prepared to find the OAS. Specifically, we need to find the number of basis points to add to each zero-coupon semiannual factor in column 4 that will equate the price of the T-bond and the price of the corporate bond. Essentially it is a Trial & Error approach, for example, 0.007447 generates a cash flow of \$95.67 and 0.005447 generates a cash flow of \$98.87.

This is found by an interactive search. When 64.46 basis points are added to the zero-coupon semiannual discount rate for each period, the present value of the Treasury cash flows will be 97.25, which is the same price as the corporate bond. Column 7 of Table 19.5 shows the zero-coupon factor that results from adding the 64.46 basis points to each semiannual discount rate, while column 8 shows the resulting present value of each cash flow. With 64.46 basis points being the semiannual OAS, the annualized OAS is just twice that, or 128.02 basis points. Figure 19.4 (graph not shown) shows the yield curves for the Treasury and the callable corporate bonds. They have the same shape, but the callable bond yield curve lies 128.92 basis points above the T-bond curve.

The basic procedure for finding the OAS can be elaborated more fully. For example, one might consider the effect of varying interest rate environments on the instruments, such as a steeping yield curve or a downward-sloping yield curve. Also, the potential exercise of the embedded option under different interest rate scenarios can affect the relative spread of the instrument with the embedded option versus Treasuries. For home mortgages, prepayment can occur at any time, so the incentives to prepay under different interest rate regimes can be analyzed. These problems are often tackled using various kinds of simulation analysis. For example, one might specify a probability distribution for possible changes in short-term interest rates and then conduct Monte Carlo analyses to generate numerous interest rate paths and the corresponding price paths for the bonds. In fact, there is quite an active industry in analyzing instruments with embedded options, particularly in the mortgage arena.

Table 19.5 Data for the callable corporate bond and the Treasury bond

 https://www.youtube.com/watch?v=M0R8-rYed0I Trial and Error Table 19.5 Data for the callable corporate bond and the Treasury bond 0.006446 0.012892 (1) (2) (3) (4) (5) (6) (7) (8) Period Cash flow Zero-coupon Treasury Rate Zero-coupon semiannual Treasury discount factor Zero-coupon annual Treasury discount factor PV of T-bond cash flows OAS -adjusted discount factor PV of OAS -adjusted Treasury cash flows 1 5 0.080000 1.04000 1.040000 \$4.81 1.04645 \$4.78 2 5 0.081631 1.04082 1.08245 \$4.62 1.09590 \$4.56 3 5 0.082096 1.04105 1.12688 \$4.44 1.14795 \$4.36 4 5 0.083242 1.04162 1.17378 \$4.26 1.20313 \$4.16 5 5 0.084439 1.04222 1.22334 \$4.09 1.26168 \$3.96 6 5 0.084882 1.04244 1.27526 \$3.92 1.32336 \$3.78 7 5 0.085351 1.04268 1.32968 \$3.76 1.38837 \$3.60 8 5 0.086052 1.04303 1.38689 \$3.61 1.45705 \$3.43 9 5 0.087164 1.04358 1.44734 \$3.45 1.52994 \$3.27 10 5 0.087926 1.04396 1.51096 \$3.31 1.60707 \$3.11 11 5 0.089075 1.04454 1.57826 \$3.17 1.68900 \$2.96 12 5 0.090063 1.04503 1.64933 \$3.03 1.77595 \$2.82 13 5 0.090536 1.04527 1.72399 \$2.90 1.86779 \$2.68 14 5 0.091434 1.04572 1.80281 \$2.77 1.96522 \$2.54 15 5 0.092050 1.04603 1.88578 \$2.65 2.06834 \$2.42 16 5 0.092425 1.04621 1.97293 \$2.53 2.17725 \$2.30 17 5 0.092640 1.04632 2.06432 \$2.42 2.29214 \$2.18 18 5 0.093304 1.04665 2.16062 \$2.31 2.41384 \$2.07 19 5 0.093427 1.04671 2.26155 \$2.21 2.54216 \$1.97 20 5 0.094332 1.04717 2.36822 \$2.11 2.67845 \$1.87 21 5 0.095423 1.04771 2.48121 \$2.02 2.82351 \$1.77 22 5 0.096137 1.04807 2.60048 \$1.92 2.97743 \$1.68 23 5 0.096262 1.04813 2.72564 \$1.83 3.13993 \$1.59 24 5 0.096791 1.04840 2.85755 \$1.75 3.31213 \$1.51 25 5 0.097595 1.04880 2.99699 \$1.67 3.49510 \$1.43 26 5 0.098334 1.04917 3.14434 \$1.59 3.68948 \$1.36 27 5 0.099000 1.04950 3.29999 \$1.52 3.89589 \$1.28 28 5 0.099261 1.04963 3.46377 \$1.44 4.11436 \$1.22 29 5 0.100178 1.05009 3.63727 \$1.37 4.34696 \$1.15 30 5 0.100982 1.05049 3.82091 \$1.31 4.59446 \$1.09 31 5 0.101899 1.05095 4.01559 \$1.25 4.85817 \$1.03 32 5 0.102798 1.05140 4.22199 \$1.18 5.13919 \$0.97 33 5 0.103538 1.05177 4.44055 \$1.13 5.43837 \$0.92 34 5 0.103917 1.05196 4.67128 \$1.07 5.75599 \$0.87 35 5 0.104388 1.05219 4.91509 \$1.02 6.09352 \$0.82 36 5 0.105145 1.05257 5.17349 \$0.97 6.45315 \$0.77 37 5 0.105323 1.05266 5.44593 \$0.92 6.83458 \$0.73 38 5 0.105691 1.05285 5.73373 \$0.87 7.23981 \$0.69 39 5 0.106441 1.05322 6.03888 \$0.83 7.67179 \$0.65 40 105 0.106995 1.05350 6.36194 \$16.50 8.13166 \$12.91

### The Black Model

In addition to options embedded in debt instruments, a number of different types of "free-standing" interest rate options exist, and these can be valued directly. This section explores the Black model, which serves as an industry benchmark for the valuation of many types of interest rate options As we will see, it is a direct extension of the Black-Scholes-Merton model. We introduce Merton's extension of the Black-Scholes model to pertain to options on underlying goods paying a dividend at a continuous rate:

In additions to calculating theoretical prices for mortgage-backed securities and other bonds with embedded options, traders also like to compute what is known as the option-adjusted spread (OAS). This is the measure of the spread over the yields on government Treasury bonds provided by the instrument when all options have been taken into account.

All input to any term structure model is the initial zero-coupon yield curve. Usually this is the LIBOR zero curve. However, to calculate an OAS for an instrument, it is first priced using the zero-coupon government Treasury curve. The price of the instrument given by the model is compared to the price in the market. A series of iterations is then used to determine the parallel shift to the imputed Treasury curve that causes the model price to be equal to the market price. This parallel shift is the OAS.

To illustrate the nature of the calculations, suppose that the market price is \$102.00 and that the price calculated using the Treasury curve is \$103.27. As a first trial we might chose to try a 60-basis-point parallel shift to the Treasury zero curve. Suppose that this gives a price of \$101.20 for the instrument. That is less than the market price of \$102.00 and means that a parallel shift somewhere between 0 and 60 basis points will lead to the model price being equal to the market price. We could use linear interpolation to calculate.

60 x 103.27 - 102.00/103.37 - 101.20 = 36.81

or 36.81 basis points as the next trial shift. Suppose that this give a price of \$101.95. This indicates that the OAS is slightly less than 36.81 basis points. Linear interpolation suggest that the next trial shift be:

36.81 x 103.27 - 102.00/103.37 - 101.95 = 35.41

or 35.41 basis points, and so on.

The Merton Model Equation (19.4)

ct = exp-δ(T-t)[St x N(d1M) - X exp-r(T-t)N(d2M)]

Equation (19.4)

d1M = ln (St/X) + (r -δ + 0.5σ2)(T-t) / σ√T-t

d2M = d1M - σ√T-t

where δ is the continuous dividend rate on the stock.

Previously, we noted that Fischer Black extends the Black-Scholes-Merton model to options on futures contracts. The rate at which the spot price grows, r, takes the place of \$delta;, and the futures price takes the place of the stock price in Merton's model. In other words, for futures on commodities that conform to the cost of carry model, δ = r. When we make this substitution in Equation 19.4, the formula becomes considerably simpler due to the equivalence of δ and r. For European futures options, the price of the futures call, cFt , and put, pFt , are given by the Black model:

The Black Model Equation (19.5)

cFt = exp-r(T-t)[Ft x N(d1F) - X N(d2F)]

Equation (19.5)

pFt = exp-r(T-t)[X x N(-d2F) - FtN(-d2F)]

d1F = ln (Ft/X) + 0.5σ2 / σ√T-t

d2F = d1F - σ√T-t

Equation 19.5 employs the standard deviation of the futures price. If we compare the Merton model and the Black model carefully, we see that there are only two differences. The first difference is that the futures price Ft takes the place of the stock price, St. The second difference concerns the interest rate. We consider each difference in turn.

Futures and forward prices are not necessarily identical. Differences in the two prices are due to varying interest rate patterns and their effect on the daily settlement cash flows of futures. These differences tend to be statistically significant, but economically small. In the Black model, the futures price appears as a proxy for the forward price of the good at the expiration date of the futures. This is tantamount to assuming one of two things. First, it may be regarded as assuming that interest rates are nonstochastic, so that differences between the futures price and forward price is not economically significant, so that the futures price is a good proxy for the forward price. In practical terms, the assumptions are not onerous; the Black model works extremely well for many interest rate options.

The Black model, as developed and as specified above, pertains specifically to options on futures. However, because the futures price is being interpreted as a forward price, the model price of a bond could take the role of F, in the Black model. When the terminal price of the underlying good at the expiration of the option is distributed log-normally, the Black model applies.

We will consider a variety of applications and extensions of the Black model. Because the good underlying the futures contract is assumed to obey the cost-of-carry relationship, the interest rate principally drops out of the equation. Comparing Equations 19.4 and 19.5, we see that r appears in both the dM1 and dM2 terms. In the Black model, r does not appear in either the dF1 term or the dF2 term. In fact, r appears in the Black model only in the discounting of the payoffs from the option expiration to the present.

The second difference between the Merton model and the Black model concerns the interest rate term, r. In the Black model, r is still assumed to be constant, but this assumption has little effect on the model's applicability to options on interest rate futures. The payoffs on the option depend largely on the N(dF1) and N(dF2) terms, but the interest rate has dropped out of these terms. Assuming that r is constant for the purposes of discounting the payoffs from expiration to the present, it is a small matter that is unlikely to affect the model value of the option to any significant degree. We may also use the zero-coupon factor for computing the present value of the option payoff. In this case, 1/Zt,T would replace the term exp-r(T-t) in Equation 19.5, giving the following:

Equation (19.6)

cFt = 1/Zt,T[Ft x N(d1F) - X N(d2F)]

pFt = 1/Zt,T [XN(-d2F) - FtN(-d2F)]

Because of its wide applicability, accuracy in matching market prices, and relative mathematical simplicity, the Black model is widely used in industry to price options on interest rate futures as well as other interest rate options. In fact, it provides an industry benchmark against which more complicated models are measured. However, we should bear in mind that the Black model, as an outgrowth of the Black-Scholes-Merton model, is a model for European options.

### Applications of the Black model

In this section, we apply the Black model to a variety of interest rate options. As discussed earlier in this section, a large market exists for options on interest rate futures, so we will consider an application of the Black model to options on T-bond futures. The Black model also applies directly to European options on a bond. Earlier in this section, we discussed forward rates and the market for FRAs. In this section, we also evaluate options on FRAs, which are known as call and puts on LIBOR. All of these applications are well suited to the Black model.

### Options on T-bond Futures [viewable here in Excel]

As discussed above, there is a robust market for options in interest rates futures. T-bond futures and options on T-bond futures trade on the CBOT. The T-bond futures contract has \$100,000 of bond principal as the underlying good, with the specification that the deliverable bonds are of a maturity greater than or equal to 15 years. Prices are quoted as a percentage of the par value of the underlying bonds. T-bond futures options trade with expirations tied to each futures expiration and have prices expressed as a percentage of par. Futures options generally expire in the latter part of the month before the corresponding futures expires.

As an example, assume that today is February 20, and consider an option on the September T-bond futures. The option will expire in six months on August 20. (As we are not interested presently in day count conventions, we treat the expiration as 0.5 years or = 6/12.) The current T-bond futures price is 115-11 (that is, 115 + 11/32 percent of par = 115.34375). The standard deviation of the futures price is 0.15. The current short-term interest rate of 6.05 percent. The yield curve is slightly upward sloping, so we assume that the six-month zero-coupon factor is Z0,6 = 1.031746. We will price a call and a put, each with a strike price of 110.

d1F = ln (Ft/X) + 0.5σ22)*(6/12))/(0.15*SQRT(0.5))

σ√T-t =

d2F = d1F - σ√T-t = 0.500268 - 0.106066 = 0.394202

Finding the cumulative normal values for the put and call, we have N(d1F) = .691557, N(d2F) = 0.653294, N(-d1F) = 0.308443, and N(-d2F) = 0.346717.

Assuming continuous discounting at the short-term rate of 6.05 percent we have the following:

cFt = exp-r(T-t)[Ft x N(d1F) - X N(d2F)]

= exp-0.0605 x 0.5 *(115.34375 x 0.691557 - 110 x .0653284) = 7.669976

pFt = exp-r(T-t)[XN(-d2F) - FtN(-d2F)]

= exp(.0605*0.5)*(110 x 0.346716 - 115.34375 x 0.308443) = 2.4855

If we wanted to take account of the shape of the term structure in pricing these options, we would use our zero-coupon factor, Z0,6 = 1.031746

cFt = 1/Zt,T[Ft x N(d1F) - X N(d2F)]

= 1/1.031746 *(115.34375 x 0.691557 - 110 x .0653284) = 7.66229064

pFt = 1/Zt,T [XN(-d2F) - FtN(-d2F)]

=1/1.031746*(110 x 0.346716 - 115.34375 x 0.308443) = 2.4855

Because the term structure is upward sloping, the prices reflecting the shape of the term structure are slightly lower than the prices based on the short-term rate. In our example, the term structure was only modestly upward sloping, so the price difference was quite small - less than one cent on each option. This outcome illustrates that, in general, option prices are not very sensitive to the discounting of the payoff back to the present.

### European Bond Options

We now explore the application of the Black model to European bond options. A bond option is simply an option with a bond as the underlying good. For example, the underlying good might be a Treasury bond or note.

Applying the Black model to value an option on a bond present three complications. First, the input price t the model should be the forward price of the bond. Second, the application of the model must account for accrued interest and any coupon payments between the valuation date and the expiration of the option. Third, the input volatility should be the volatility of the forward bond price. We consider each of these issues in turn.

### The Forward Price of the Bond

In our discussion of the Black model, we noted that the input price for the underlying good is the forward price of that good for the expiration date of the option. In applying the Black model to futures, this was not a particular problem, because the futures price is immediately observable and is closely analogous to a forward price anyway. For typical bonds, there is no quoted forward price. Therefore, application of the Black model to value options on bonds must address the issue of finding the forward price of the underlying bond. We could find the forward price of the bond by looking to the term structure of interest rates. There are two ways of dealing with this problem. First, we could use the zero-coupon yield curve to value the bond as of the expiration date of the option. Second, we could take the term structure of interest rates between the present date and the expiration date of the option using the term structure of interest rates between the present date and the expiration date of the option. If the bond pays any coupon payments between the valuation date of the option and the options' expiration, we will have to take these into account.

### Accrued Interest and Coupon Payments

Bond prices are typically quoted without accrued interest. To apply the Black model successfully, we need to focus on the actual cash flows that are at stake. Therefore, if the bond price is quoted without accrued interest, we will have to take the accruals into account. When the underlying bond is a coupon bond, there will likely be coupon payments between the time the option is being valued and the option expiration. This will affect the estimation of the forward price of the underlying good. This situation is really analogous to pricing an option on a dividend paying stock. Payments of intervening dividends on a stock or intervening coupons on a bond both represent a "leakage of value" from the underlying good that must be considered. If we use the term structure to value the bond as of the expiration date of the option, we can ignore the intervening coupon payments. However, if we use the present price of the bond and compound it forward to the expiration date of the option to find the forward bond price, we will need to reduce the present price of the bond by the present value of the intervening coupons. This adjusted bond price -- the current cash price of the bond minus the present value of any intervening coupon payments -- can be compounded to the expiration date of the option to find the forward price of the bond. These two techniques should give exactly the same forward price.

### The Input Volatility

The correct volatility for the model is the volatility of the forward bond price. Like the forward price of the bond, the volatility of the forward price is not immediately observable. It will require estimation. Usually, data on bond yields are available, and we can use bond yield volatility to estimate the volatility of the forward bond price by considering how the forward bond price would respond to a change in yields. That bond yield can be used to estimate the volatility of the forward bond price.

### A Comprehensive European Bond Option Example [viewable here in Excel]

All of the complications associated with applying the Black model to value a European bond option can be illustrated by working through an extended example. Consider a European bond option that expires in seven months on a T-note that matures in 28 months and has a semi-annual coupon rate of 8 percent. The exercise price on the option is 1,020.00. This note has two months of accrued interest and the following remaining cash flows, assuming a par value of \$1,000:

 Months Cash flows (assuming par value of 1,000) 4 \$40 10 \$40 16 \$40 22 \$40 28 \$1,040

Table 19.6 presents information on two term structure environments for a 48-month horizon. For this example, we will use the upward-sloping yield curve, presented in column 2 of Table 19.6. The third column of the table shows the appropriate zero-coupon factor for each month, and we will use monthly compounding for all calculations in this example. Based on the cash flows stated above and yield curve data of Table 19.6, the actual value of the bond is as follows:

Po = 40/1.019 + 40/1.0493 + 40/1.0818 + 40/1.1166 + 10140/1.1538 = \$1,051.54

With two months having elapsed since the last coupon payment, the accrued interest on the bond is 0.3333 x \$40 = \$13.33. The corresponding quoted price of the bond should be \$1,038.21, the present value of the future cash flows minus the accrued interest of \$13.33.

 Table 19.6 Sample upward and downward yield curve data (monthly compounding) Upward-sloping curve Downward-sloping curve Terms to Maturity Annualized par yield (Monthly YTM) Zero-coupon discount factor Annualized par yield Zero-coupon discount factor 1 0.055200 1.004600 0.078000 1.006500 2 0.055670 1.009300 0.078343 1.013100 3 0.056528 1.014200 0.077895 1.019600 4 0.056594 1.019000 0.077842 1.026200 5 0.056814 1.023900 0.077711 1.032800 6 0.057108 1.028900 0.077542 1.039400 7 0.057278 1.033900 0.077516 1.046100 8 0.057513 1.039000 0.077295 1.052700 9 0.057665 1.044100 0.077195 1.059400 10 0.05787 1.049300 0.077069 1.066100 11 0.058113 1.054600 0.076923 1.072800 12 0.058291 1.059900 0.076854 1.079600 13 0.058504 1.065300 0.076678 1.086301 14 0.058742 1.070800 0.076571 1.093101 15 0.058856 1.076199 0.076378 1.099800 16 0.059072 1.081800 0.076248 1.106600 17 0.059244 1.087400 0.076108 1.113401 18 0.059439 1.093100 0.076016 1.120300 19 0.059652 1.098901 0.075858 1.127100 20 0.059827 1.104700 0.075745 1.134000 21 0.06002 1.110600 0.075623 1.140900 22 0.060227 1.116600 0.075493 1.147799 23 0.060402 1.122601 0.075357 1.154700 24 0.06059 1.128700 0.075256 1.161700 25 0.06075 1.134799 0.075108 1.168600 26 0.060962 1.141101 0.074993 1.175600 27 0.061144 1.147399 0.074872 1.182599 28 0.061337 1.153801 0.074746 1.189600 29 0.061537 1.160299 0.074615 1.196599 30 0.061713 1.166801 0.074481 1.203601 31 0.061896 1.173400 0.074343 1.210601 32 0.062116 1.180201 0.07423 1.217700 33 0.062311 1.187001 0.074113 1.224800 34 0.062511 1.193901 0.073966 1.231801 35 0.062663 1.200702 0.073842 1.238899 36 0.062872 1.207801 0.073716 1.246002 37 0.063035 1.214800 0.073586 1.253099 38 0.063227 1.222000 0.073477 1.260299 39 0.063399 1.229199 0.073343 1.267400 40 0.063576 1.236502 0.073207 1.274500 41 0.063755 1.243900 0.07309 1.281701 42 0.063937 1.251399 0.07297 1.288898 43 0.064122 1.259001 0.072849 1.296102 44 0.064308 1.266700 0.07275 1.303435 45 0.064496 1.274502 0.0726 1.310501 46 0.064666 1.282298 0.072473 1.317700 47 0.064856 1.290298 0.072362 1.325001 48 0.06503 1.298302 0.072232 1.332199

### The Forward Price of the Bond

We need the forward price of the bond seven months from now. During that seven months, the bond will pay a coupon. This is directly analogous to a dividend on the stock. Both the coupon payment on the bond and the dividend payment on the stock reduces the future value of both instruments from what they would have been without the payment. Therefore, in projecting the forward price of the bond, we need to take that coupon payment into account. We can use the current cash price to find this forward price by subtracting the present value of the intervening coupons and compounding this result forward for seven months.

Forward bond price = (current cash price - PV of intervening coupons) x Z0,7 = 40/1.019

= (1,051.54 - 39.25) x 1.0339 = (1,051.54 - 40/1.019) x 1.0339 = 1,046.61

As an alternative, we can find the forward bond price for the expiration date of the option by using the term structure to compute the bond price as of that date. Seven months from now, the bond price will be between coupon dates, with three months having elapsed since the last coupon payment. The forward bond price will be the present value of the remaining cash flows valued at that seven months from now:

Forward bond price = C10/(Z10/Z7) + C16/(Z16/Z7) + C22/(Z22/Z7) + C28/(Z28/Z7) [Use Table 19.6]

= 40/(1.0493/1.0339) + 40/(1.0818/1.0339) + 40/(1.1166/1.0339) +10 40/(1.1538/1.0339) = 1,046.61

Either of these equivalent methods may be used to find the appropriate forward bond price to serve as the price input to the model.

The volatility of the forward bond price We also need the volatility of the forward bond price. As mentioned above, the volatility of the forward bond price is not immediately available. Furthermore, there is not even a series of forward bond prices that we could use to compute an estimated volatility from historical data. (This contrasts with the futures market, in which futures price quotations are available each day and can be used to estimate the volatility of the futures price. However, there is a variety of time series for yields on bonds of various types, such as Treasury issues of various maturities and AAA bond yields, as well as many others. At the expiration of the option, the underlying T-bond will have a maturity of 21 months. This is fairly close to a two-year maturity. Let us assume that the annualized standard deviation of the two-year Treasury bond yield is 0.20. To emphasize, this is the volatility of the yield, not the bond itself. We can use this yield volatility to estimate the volatility of the bond price to serve as the volatility impute for the Black model.

Macaulay's duration (D) and Macaulay's modified duration (MD) of a bond are as follows:

D = {t=1ΣM t [Ct/(1 = YTM)t]}/(bond price)

Equation 19.7 MD = {t=1ΣM t [Ct/(1 = YTM)t]}/(bond price/ 1 + r)

where Ctr is the cash flow from the bond at time t and YTM is the bond's yield to maturity expressed in the same time unit as t. (Note that the modified duration is just duration divided by 1+r). Then, the resulting duration measure will be expressed in the same time units as t. The duration price change formula is as follows.

ΔP = -MDP*ΔAYTM

where AYTM in Equation 19.8 is the yield to maturity expressed in annual terms. This holds for a parallel shift in the yield curve, and results in an appropriate price change that is quite close to the actual price change. From the price change formula, it follows that:

Equation (19.8)

ΔP/P = -MDP x AYTM x Δ AYTM/AYTM

Equation (19.9)

σ = MD x AYTM x σAYTM

Thus, given the annual yield to maturity of the forward bond (AYTM), we can compute the modified duration (MD). Then, given the standard deviation of the forward bond yield (σAYTM), we can approximate the standard deviation of the forward bond price (σP).

For our forward bond, the monthly yield to maturity is 0.005231, which satisfies the forward pricing equation (notice the months):

Forward bond price = 40/(1 + YTM)3 + 40/(1 + YTM)9 + 40/(1 + YTM)15 + 1040/(1 + YTM)21

YTM = 0.005231 (Use Trial and Error)= 40/(1 + .005231)3 + 40/(1 + .005231)9 + 40/(1 + .005231)15 + 1040/(1 + .005231)21

and the annualized yield to maturity, assuming monthly compounding, is (1 + .005231)12-1 = 0.064607.

Next we find the MD:

MD = 3 x 40/(1 + YTM)3 + 9 x 40/(1 + YTM)9 + 15 x 40/(1 + YTM)15 + 21 x 1040/(1 + YTM)21 / forward bond price/ 1 + YTM = 19.78.

The MD measured in months is 19.78, so the annualized MD = 1.6483 (= 19.78/12). Recalling that we are assuming a standard deviation of 0.20 for the two year Treasury yields, we have, in annual terms,

σ = MD x AYTM x σAYTM = 1.6483 x 0.064607 x 0.20 = 0.021298

We now have all of the required inputs for the Black model. The forward price is 1,046.61; the time to expiration is seven months, or 0.5833 (=7/12) years, the exercise price is 1,020.00; the standard deviation of the forward price is 0.021298; and the seven-month factor is 1.0339 (Table 19.6). We begin by computing the d1F and the d2F terms:

d1F = ln (Ft/X) + 0.5σ2 / σ√T-t

d1F = (ln(1046.61/1020.00) + (0.5 x 0.0212982) x 0.5833)/(0.021298 x σ√0.5833) = 1.591359623

d2F = d1F - σ√T-t = 1.5914 - 0.016266 = 1.575072

Finding the cumulative normal values for the put and call, we have:

N(d1F) = N(1.591339) = .944233
N(d2F) = N(1.575072) = .94380087
N(-d1F) = N(-1.591339) = 0.05576652
N(-d2F) = N(-1.57072) = 0.057619913

The values of the call and put are as follows:

cFt = 1/Zt,T[Ft x N(d1F) - X N(d2F)]

= 1/1.0339 *(1,046.61 x .944233348 - 1020 x .942380087) = 26.1305

pFt = 1/Zt,T [XN(-d2F) - FtN(-d2F)]

= 1/1.0339 *(1,046.61 x .05576652 - 1020 x .05576652) = .3931847

Therefore, the call is worth \$26.13 and the put is worth \$0.39.

### Calls and Puts on Libor

Earlier in this section, we discussed the active market in forward interest rates based on LIBOR. Market makers are active in FRAs with one-month, three-month, six-month, and one-year maturities, and will provide quotations on other maturities as well. A parallel options market exists in which calls on LIBOR and puts on LIBOR trade. For example, there is a market for calls on three-month LIBOR and for puts on one-month LIBOR. These options are generally European in form.

### Payoffs for Options on Libor

The payoff on calls and puts on LIBOR depends on the maturity of LIBOR being quoted and the notional principal. Consider first a call on three-month LIBOR with a notional principal of \$1 million and an exercise price, or strike rate, of 9 percent. The call pays off if the observed three-month LIBOR rate on the expiration date of the option exceeds the strike rate. In that case, the payoff on this call equals the difference between the observed rate and the strike rate multiplied by the quarter of a year maturity of the rate, times the notional principal. For our example call option, if the observed rate is 10.5 percent, the payoff will be:

(0.1050 - 0.0900) x 0.25 x \$1,000,000 = \$3,750

Let FRAC be the fraction of the year covered by the maturity of the underlying LIBOR instrument, let NP be the notional principal, and let SR be the strike rate. A general expression for the payoff on a call on LIBOR will be:

Equation (19.10) MAX {0, (Observed LIBOR - SR) x FRAC x NP}

where Observed LIBOR is the rate observed at the expiration of the option for LIBOR of the appropriate maturity. A put on LIBOR pays off when the observed rate at expiration is less than the strike rate. The payoff for a put on LIBOR is:

Equation (19.11) MAX {0, (SR -Observed LIBOR) x FRAC x NP}

For example, consider a put on one-month LIBOR with a strike rate of 8.5 percent and a notional principal of \$25 million. At expiration of the option, assume that the one-month LIBOR rate stands at 7.03 percent and Observed LIBOR = 0.073. The payoff on this put would be:

(0.085 - 0.0703) x (1/12) x \$25,000,000 = \$30,625

### Determination in Advance: Settlement in Arrears

In our discussion of FRAs, we noted that they are generally "determined in advance and settled in arrears." Thus, the payment on a six-month FRA generally occurs six months after the settled in arrears." Thus, the payment on six-month FRA generally occurs six months after the determination date. Calls and puts on LIBOR are generally structured in a parallel manner. The payoffs are determined at the expiration date, but the payment is made at a date that lags the determination date by the maturity of the LIBOR quotation. (When the lag between determination and payment equals the maturity of the LIBOR quotation, the lag is said to be the natural time lag.) Thus, the actual payoffs on our sample call and put would be three months and one month after the determination dates, respectively. In pricing calls and puts on LIBOR, we need to take this settlement in arrears into account.

### The Black Model and Option on LIBOR [viewable here in Excel]

The Black model applies quite directly to calls and puts on LIBOR. The currently observed forward LIBOR (FLIBOR) plays the role of the futures price. Specifically, FLIBOR, is observed at time t, when the option is being valued. It is the forward LIBOR with a maturity corresponding to the maturity of the option and it is the forward rate with a time horizon that matches the expiration date of the option. Notice that FLIBOR, should be identical to the FRA rates for the same period in the future.

Using FLIBOR, in the Black model is tantamount to assuming that the forward LIBOR is log-normally distributed. The strike rate plays the role of the exercise price, while the standard deviation of the LIBOR forward rate (FLIBOR) of the requisite maturity is the volatility measure for the model:

Equation (19.12)

ctFLIBOR = NP x FRAC x exp-r(T + FRAC - t)[FLIBOR1 x N(d1FLIBOR) - SR N(d2FLIBOR)]

ptFLIBOR = NP x FRAC x exp-r(T + FRAC - t)[SR x N(-d2FLIBOR) - FLIBORt N(-d1FLIBOR)]

dFLIBOR1 = ln (FLIBORt/SR) + 0.5σ2

d2FLIBOR = d2FLIBOR - σ√T-t

In this equation we have the Black model with the following substitutions, FLIBOR, takes the role of the futures price and SR substitutes for the exercise price. The payoff on each option is discounted for a longer period, the time to expiration plus FRAC to account for the delayed payment, and the values of each option are multiplied by NP(FRAC) to convert the values to dollar amounts.

The following equation is identical to Equation 19.12, except that it uses the zero-coupon factor instead of the continuously compounded rate to discount the payoff on the option from the pay off date (the expiration of the option plus FRAC) to the present:

Equation (19.13)

ctFLIBOR = NP x FRAC x 1/Zt,T+FRAC x [FLIBOR1 x N(d1FLIBOR) - SR N(d2FLIBOR)]

ptFLIBOR = NP x FRAC x 1/Zt,T+FRAC x [SR x N(-d2FLIBOR) - FLIBORt N(-d1FLIBOR)]

d1FLIBOR = ln (FLIBORt/SR) + 0.5σ2

d1FLIBOR = d1FLIBOR - σ√T-t

### Applying the Black model to Option on LIBOR

As an example, consider a call and put on one-month LIBOR that both expires in eight months. The strike rate on the call and put is 7 percent. The notional principal is \$10 million. From the FRA market, the historical standard deviation of the one-month LIBOR rate has been 0.23, which is our estimate of the standard deviation of LIBOR. The yield curve is downward sloping, as illustrated in the last two columns of Table 19.6, which assumes monthly compounding. The one-month forward rate for a period from month eight to month nine is the value we need for FLIBOR. From Table 19.6, we see that the eight- and nine-month zero-coupon factors are 1.0527 and 1.0594. The one-month forward rate is, therefore, 1.0594/1.0527 - 1 = 0.006365 . With monthly compounding, the annualized rate is (1 + .006365)12 - 1 = 0.079111.

The one-month maturity corresponds to 0.0833 years (= 1/12). Using our zero-coupon factor of 1.0594 to cover the nine months until payment on the options would be received, the option values according to Equation 19.13 are as follows:

We first find the d1FLIBOR and d2FLIBOR values:

1/Zt,T+FRAC = 9-month zero-coupon rate = 1.0594

√ = √8/12 = .6666667

FRAC = 1/12 = .0833333

d1FLIBOR = ln (FLIBORt/SR) + 0.5σ2

=( ln (0.079111/0.0700) + 0.5 x (0.23)2 x 0.6667) / (0.23 x √1/12) = .745448434

d1FLIBOR = d1FLIBOR - σ√T-t

=0.7454484-(SQRT(8/12)*0.23) = .5576541

Finding the cumulative normal values for the put and call, we have,

N(d1F) = N(0.745104) = 0.77189
N(d2F) = N(0.55731) = .711342178
N(-d1F) = N(-0.745104) = 0.228104426
N(-d2F) = N(-0.55731) = 0.288657822

The cost of the call and put are as follows:

ctFLIBOR = NP x FRAC x 1/Zt,T+FRAC x [FLIBOR1 x N(d1FLIBOR) - SR N(d2FLIBOR)]

= 10,000,000 x .0833 1/1.0594 *(.07910 x .771896 - .07 x .711342) = \$8,859.70

ptFLIBOR = NP x FRAC x 1/Zt,T+FRAC x [SR x N(-d2FLIBOR) - FLIBORt N(-d1FLIBOR)]

= 10,000,000 x .0833 x 1/1.0594 *(.07 x .288658 - .079106 x .228104) = \$1,699.70

Therefore, the call is worth \$8,859.69 and the put is worth \$1,699.70.

As we will see shortly, these calls and puts on LIBOR are important elements in controlling interest rate risk on loans and investments.

### Forward Put-Call Parity [viewable here in Excel]

We will now consider the put-call parity relationship for European options. For a put and call with the same expiration date and a common exercise price equal to the price of the common underlying stock, the price of the call equals the price of the put. As we saw there the purchase of a call, the sale of a put, and the purchase of a risk-free bond paying the exercise price at the expiration of the options gives a portfolio that has exactly the same value as the underlying stock.

Put-call parity for European options, where X = St

Equation (19.14) ct - pt + exp-r(T - t) = S

In this section, we explore forward put-call parity to see how the purchase of a call and the sale of a put on an interest rate can exactly replicate a forward contract on the interest rate.

As we have observed, there is an explicit forward rate market for interest rate contracts based on LIBOR. This is the market for forward rate agreements, or FRAs, that we have already considered. We also have noted the equivalence between forward LIBOR rates (FLIBOR) and the rate for an FRA to cover the same future period. Consider now a call and a put on LIBOR with a common strike rate, SR, and underlying instrument, and the same notional principal. For a long call/short put portfolio, the payoff will be as follows:

MAX {0, (Observed LIBOR - SR) x FRAC x NP} -

MAX {0, (SR -Observed LIBOR) x FRAC x NP}

= (Observed LIBOR - SR) x FRAC x NP

where Observed LIBOR is the rate observed at the expiration of the option for the appropriate maturity, FRAC is the fraction of the year for the underlying instrument, and NP is the notional principal.

For an FRA to cover the time period and with the same notional principal, the payoff will be:

= (Observed LIBOR - FRA) x FRAC x NP

where FRA is the rate on the forward rate agreement, and Observed LIBOR is the observed rate on the determination date for the forward rate agreement, for LIBOR of the appropriate maturity. For identical time periods, it must be the case that

FLIBORt = FRAx,y

as a no-arbitrage condition. Consider now the special case in which the strike rate for the long call/short put portfolio is chosen such that

SR = FLIBORt = FRAx,y

where the time periods for all three measures are the same.

In this special case, when the strike rate on the options is the same as the forward LIBOR rate or the rate of an FRA, the payoff on the long call/short put will be the same as the payoff on the forward rate agreement. Therefore, the long call/short put portfolio is equivalent to the forward rate agreement. This is the principal of forward put-call parity.

Forward put-call parity

Equation (19.15) ctFLIBOR - ptFLIBOR = 0

where the common strike rate on the put and call options equals the current prevailing forward rate, SR = FRAx,y and the time periods covered by the options and the forward rate agreement are the same.

This result makes sense considering that the cost of entering an FRA, negotiated at the current forward rate prevailing in the market and reflecting the current term structure, is zero, and it will pay off at the expiration date based on the observed LIBOR at that date relative to the contract rate.

To illustrate forward put-call parity, recall the previous section, where we saw that the one-month LIBOR rate eight months forward was 0.079106, based on the downward-sloping yield curve of Table 19.6. Consider now a FRA entered for that forward rate. We also consider a put and call entered on the same terms: eight month expiration, one month underlying LIBOR, and a strike rate on the options equal to the forward LIBOR rate of .079106. Thus, this example meets the following condition that

SR = FLIBORt = FRAx,y

Figure 19.5 shows the payoffs for a long call plus a short put on LIBOR for a common strike rate of 0.079106 in the upper panel. The lower panel shows the payoffs for a long position in the corresponding FRA. They are identical. Therefore, the long call/short put portfolio exactly replicates the FRA payoffs, and the option portfolio must have the same price as the FRA. Since the FRA is costless, the option portfolio also must be costless. The option portfolio can be costless only when the call and put have the same price, since the portfolio is constructed by buying the call and selling the put. To see this in yet another way, let us return to Equation 19.12 and 19.13, the equations for calls and puts on LIBOR. In this special case we are considering, the strike rate on the options equals the forward LIBOR rate. In terms of our notation, the special case is SR = FLIBORt. We first evaluate the d1FLIBOR terms in this special circumstance. When SR = FLIBORt, replace FLIBORt with SR:

d1FLIBOR = ln (SR/SR) + 0.5σ2 (T - t) / σ√T-t

= 0 + 0.5σ2 σ√T-t √T-t /σ√T-t

= 0.5 σ√T-t

d2FLIBOR = d1FLIBOR - σ√T-t

= 0.5σ√T-t - σ√T-t

= 0.5 σ√T-t

d2FLIBOR = -d1FLIBOR

In this special circumstance, when the call and put have the same underlying instrument, expiration, and strike rate, we now see that SR = FLIBOR, and d2FLIBOR = -d1FLIBOR . Therefore, replacing FLIBOR, with SR, and replacing d2FLIBOR with -d1FLIBOR, we have the following:

ctFLIBOR = NP x FRAC x exp-r(T + FRAC - t)[SR x N(d1FLIBOR) - SR x N(-d1FLIBOR)]

ptFLIBOR = NP x FRAC x exp-r(T + FRAC - t)[SR x N(d1FLIBOR) - SR x N(-d1FLIBOR)]

ctFLIBOR - ptFLIBOR = NP x FRAC x exp-r(T + FRAC - t) x SR x N(d1FLIBOR) - SR x N(-d1FLIBOR)] - N(d1FLIBOR) - SR x N(-d1FLIBOR)= 0

because the sum of the cumulative normal probabilities in the bracketed expression is zero.

However, even though the long call/short put portfolio costs zero, the individual options may have value. They just have to have the same value. To illustrate this, we compute the value of the call and put, when the stake rate equals the forward rate of .079106. First, the d1FLIBOR and d2FLIBOR terms are as follows:

d1FLIBOR = 0.5σ√T-t = 0.5 x 0.23 x σ√0.6667 = 0.093897

d2FLIBOR = -d1FLIBOR = -0.093897

The cumulative normal probabilities are:

N(d1F) = N(.093897) = 0.537405
N(d2F) = N(-.093897) = .4625995488
N(-d1F) = N(-0.093897) = 0.4625995488
N(-d2F) = N(0.093897) = 0.537404512

The values of the call and put are as follows:

ctFLIBOR = NP x FRAC x 1/Zt,T+FRAC x [FLIBOR1 x N(d1FLIBOR) - SR N(d2FLIBOR)]

= 10,000,000 x .0833x 1/1.0594 *(.079106 x .537404512 - .079106 x .462595488)= \$4,654.16

ptFLIBOR = NP x FRAC x 1/Zt,T+FRAC x [SR x N(-d2FLIBOR) - FLIBORt N(-d1FLIBOR)]

= 10,000,000 x .0833x 1/1.0594 *(.079106 x .537404512 - .079106 x .462595488) = \$4,653.16

We now turn to applications of call and puts on LIBOR in controlling interest rate risk.

Sources: Damodaran, Hull, Basu, Kolb, Brueggeman, Fisher, Bodie, Merton, White, Case, Pratt, Agee.

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