In Chapters 3, 4, and 5,1 discussed the valuation of options when the variability entered only through future price (value or rate) movements. As a consequence, the bulk of my discussion in this chapter will focus on the estimation of model parameters when there is only price stochasticity. As in Chapter 5, I will continue to assume that the underlying is an index that pays continuous dividends.

I showed in Chapter 3 how equations (3.7a) and (3.7b) – were used to value European-style options on a stock that pays a continuously compounded dividend. For the ease of reading, I have re-stated these equations

again:

(3.7a)

(3.7b)

where

When illustrating the implementation of equations (3.7a) and (3.7b) in Table 3.4, I assumed that all the inputs going into the model (e.g., St,K,t,T,rt,T,qt,T,σt,T) are known – which is far from the truth in practice. In reality, the only inputs that are known with 100 percent objectivity are St, K, t, T. In fact, while St represents the value of the observable index at the time of valuation, the other three inputs (i.e., K, t, T) are terms associated with the European-style vanilla option contract. Furthermore, the input rt,T, which used to be thought of as known, transparent, and objective in the past has been shown to be none of those for the following reasons:

■ As seen in Figure 2.5, there can be a big discrepancy in the zero rates arising from how the zero-rate curve is constructed (i.e., linear function versus cubic polynomial). In the event that the option maturity lies in the region where this discrepancy is great, the value of rt,T that is used for both growing the stock and present valuing the option payoff can be drastically different – leading to differing option values.

■ Since the decision to implement Dodd-Frank, the market has started to move away from the use of the uncollateralized FIBOR as a floating rate to the use of a collateralized overnight indexed swap (OIS) rate as a floating rate – in the process redefining the term risk-free rate. See also footnote 20 of Chapter 2.

Despite this, relative to both qt,T and σt,T, there is a lesser degree of uncertainty associated with the estimation of rt,T.

Given the above backdrop, I will now discuss methods used by practitioners to estimate the unknown parameters rt,T, qt,T and at,T. Before doing this, it would be useful for the reader to recognize that since an option premium is supposed to reflect the risks embedded in the option, it is imperative for the parameters rt,T, qt,T, and to accurately represent the average value of the parameters during the life of the option, as it would be these factors (among other things) that determine the value of ST. Clearly the longer the option maturity, the greater the difficulty to accurately guesstimate these values.

To help with the estimation of the parameters rt,T, qt,T, and σt,T, one typically resorts to using at least one of the two following methods in practice:

1. Implied:^{[1]} Refers to the use of currently available liquid market information (e.g., option price, zero risk-free rate curve, and so on) to determine the implied values of the parameters, as this would be the market's perception of the realized values.

2. Historical: Refers to the use of historical data to statistically estimate the values of the parameters, as this would be the practitioner's view on the realized values.

Starting with a discussion on how parameter values can be implied from market data, I conclude this chapter with a discussion on various statistical methods that can be deployed to estimate the parameter values using historical data.

[1] This is sometimes referred to as calibrating the model to market information (as the inputs in the model are adjusted so that the resulting output matches the market value of the option) – although there are some exceptions to this when the model is an all-encompassing model with a lesser number of estimable parameters relative to the amount of available liquid market data. See also the section in this chapter on Calibration of Interest Rate Option Model Parameters.

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