CALIBRATION OF INTEREST RATE OPTION MODEL PARAMETERS

In the previous section, I discussed how and why practitioners use market prices to imply the volatility, risk-free, and dividend inputs for the Black- Scholes model. Although it was easy to incorporate information from a term structure of implied spot-volatility rates to value early exercise and path dependent options, to do so using a surface of implied spot-volatility rates, trying to value path-dependent options was a challenge. Regardless, one was still able to find a way to solve the problem using the concept of local volatilities – while trying to keep the stochastic component contained to stock-price movements. Although I concluded the previous section by commenting on the use of stochastic volatility models, throughout my discussion I have primarily kept to the use of a lognormal pdf for price-rate movements. While this is, in general, true for the valuation of options on equities and currencies, practitioners use a plethora of models when valuing interest rate options, ranging from equilibrium models to market-based models (models that describe the movements of the entire term structure of interest rates). The reader is referred to Hull (2012) for an overview of some of the models.

In using models that are all encompassing (that describe all term structure movements), the practitioner needs to balance between the ability to reproduce liquid market prices and the need to value path-dependent options. The reason for this stems from the fact that many of these models have only a finite number of parameters that need to be inferred vis-a-vis the abundance of available market information. As a consequence, the practitioner is faced with the problem of a possible overfill, simply because after fitting, the user is not able to reproduce all the market prices that were used to estimate or infer the parameters of this model. In this section, I will present one example in the context of the widely used Hull-White one-factor model that is used to model short interest rates (e.g., overnight rate).

Unlike equation (3.5) where the underlying index was assumed to follow the diffusion process, the Hull-White one-factor model for the short rates is assumed to follow the diffusion process

(6.19)

where

r is the annualized continuously compounded short rate.

is a time-related function that is chosen so as to ensure that the model fits the term structure.

a is the mean reversion rate associated with the short rates.

σ is the annualized standard deviation of the short rate.

dz is the random variable drawn from a standard normal probability

density function.

dr is the small change in the short rate over a small time interval dt.

See Hull (2012) for further details of this model and the use of this model to value options.

Using the short-rate process assumption in equation (6.19), it can be shown that the formulae to price European-style call and put options on zero coupon bonds are given by the equations (6.20a) and (6.20b) respectively.

(6.20a)

(6.20b)

where

K is the strike price of the option.

L is the face value of the zero coupon bond.

T – t is the option life.

s – f is the zero coupon bond life (where s > T).

P(t,s) is the s-t-year discount factor using the yield curve at time t. P( 0, T) is the T-year discount factor using the yield curve at time

Table 6.7 shows the implementation of equations (6.20a) and (6.20b) on a spreadsheet.

TABLE 6.7 Valuing Options on Zero Coupon Bonds

Now that I have illustrated how the Hull-White one-factor model can be applied to value options on zero coupon bonds, I will show how a practitioner can imply the parameters of the model using market information.

In the Black-Scholes framework, I discussed a practitioner's dilemma in trying to estimate values of rt,T, qt,T, and σt,T. Similarly, as can be seen from the description of the Hull-White one-factor model in equation (6.8), there are essentially three unknowns associated with the model. These are

a: The mean reversion rate which the short rate reverts back to its longterm mean(identifying one of the characteristics in the interest rate market in that interest rates cannot keep increasing or decreasing indefinitely).

σ: The standard deviation associated with the short rates (which is different from the definition of the volatility in the Black-Scholes framework).

: The long-term, time-dependent mean rate.

Whileis calibrated to be consistent with the zero curve (i.e., the term structure of zero rates),^{[1]} the other two parameters of the Hull-White model are calibrated using option prices on the underlying bonds/rates. Furthermore, as can be seen by the option-pricing formulae on zero coupon bonds given in this section, unlike the explicit presence of a and σ, the only way values ofenter the formulae is through the presence of discount factors P(0,T) and P(0,s). As a consequence, to illustrate the computation of the implied values of a and σ from a list of option premiums, I will assume the discount factors take the form in Table 6.8 and the option prices for various maturities in Table 6.9.

Given the market information in Tables 6.8 and 6.9, to find the best estimates for both a and σ using the seven option premiums, I have used equation (6.20b) to compute the premiums for each of the seven options (whererepresents the calculated premium for put option i for i= 1,2,3,...,7). Lettingrepresent the observed market premium for put option i and defining the error function

TABLE 6.8 Curve of Discount Factors

Time

Discount Factors

0

1.00000

1

0.97045

1.5

0.95238

2

0.92302

2.5

0.89131

3

0.85544

3.5

0.81621

4

0.78650

4.5

0.64317

5

0.59139

5.5

0.54859

6

0.50645

6.5

0.46932

7

0.42908

7.5

0.38644

one would be interested in finding the values of a and σ that minimize the value of the function f(a,o).

From Table 6.10, it can be seen that f (a, σ) is minimized when a = 10 percent and σ = 1.6 percent. It should be pointed out that sometimes these values can be quite unstable (i.e., there could be several local mini- mums) and the global minimum would be achieved at values of a and σ that may be unrealistic or impractical. As a consequence, it is imperative for the user to ensure common sense checks are applied to whatever combinations of a and σ are produced as optimal values. Furthermore, when the number of market-data points increases, it is important for the reader to note

TABLE 6.9 Option Premiums by Term

Option Life (Yrs)

Bond Life (Yrs)

Put Option Premium

1

1

6

0.578

2

1.5

6.5

2.423

3

2

7

4.150

4

2.5

7.5

5.958

5

3

8

7.165

6

3.5

8.5

8.609

7

4

9

10.929

TABLE 6.10 Relative Errors from Fitting

that despite finding the optimal values of a and σ (which minimize the sum of squareds of the relative error), it would not be possible for the user to reproduce the market prices – thereby not achieving one of the objectives of this exercise. Hence, unlike the Black-Scholes model where the parameters in the model can be uniquely inferred from market information, practitioners using more powerful models are faced with the problem of not being able to uniquely infer model parameters (hence the inability to reproduce market prices accurately). As a consequence, it should not be a surprise to see traders keep a variety of models in their arsenal and use what, in their judgment, seems to be best suited for their objectives.

[1] Like the bootstrapping of the yield curve discussed in Chapter 2 or the calculation of implied volatility that was discussed earlier in this chapter, 0(t) can be uniquely implied from the zero risk-free rate curve.

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