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CALIBRATION OF PARAMETERS IN THE BLACK – SCHOLES MODEL

As mentioned earlier, there are three input parameters (namely rt,T, qt,T, and σt,T) that need to be estimated for use in equations (3.7a) and (3.7b). However, before one can decide how to best estimate these parameters, it is important to first understand the function of these parameters. To do this, I will provide a quick description for each of these parameters – the reader is referred to Hull (2012) for more details:

rt,T: The annualized continuously compounded risk-free rate that represents the fully collateralized risk-free rate (that is obtained for the time period (t,T) using the zero risk-free rate curve at time t) is the rate at which money can be borrowed or lent for the life of the option (i.e., T – t).

qt,T: The annualized continuously compounded dividend rate that represents the rate (that is obtained for the time period (t, T) using the zero dividend rate curve at time t), is the rate at which dividends are paid out to shareholders of the underlying shares for the life of the option (i.e., T – t) (where it is assumed that the dividends are not reinvested).

at,T: The annualized rate that represents the spot volatility rate (that is obtained for the time period (t, T) using the spot volatility rate curve at time t) is the volatility experienced by the underlying stock for the life of the option (i.e., Tt).

When I discussed the construction of a zero risk-free rate curve in Chapter 2, the reader will recall that for the constructed curve to be deemed effective, the curve should be able to

■ Serve as a platform that can be used to compare the relative value of

one interest-bearing instrument over the other.

■ Reproduce the market prices of actively trading financial instruments.

■ Infer (deduce) prices of nonliquid and/or complex financial instruments (so as to assess how much premium is attached to such instruments – relative to where liquid instruments trade).

The reason for my drawing reference to the zero risk-free rate curve construction is to show the striking similarity between what was discussed in Chapter 2 and what will be discussed in this chapter.

To infer rt,T,qt,T, and σt,T, from market information, it is important for the reader to first understand that for a given maturity T, these parameters can be estimated by going through the following steps:

Step 1: Estimate rt,T using the zero risk-free rate curve.

Step 2: Estimate qt,T using the value of rt,T (in step 1) and the value of forward/futures contract with the same maturity.

Step 3: Estimate σt,T using the values of rt,T and qt,T (in steps 1 and 2) and the value of an options contract with the same maturity.

Since the implementation of step 1 is straightforward (given the backdrop of Chapter 2), I will henceforth only focus on the inference of qt,T and σt,T.

 
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