For a given maturity T and implied rt,T, to infer qt,T one needs the price of the forward/futures contract expiring at same time T. In footnote 29 of Chapter 3,1 pointed out that the value of a forward/futures contract is given by the expression

This can be solved for qt,T (since St and rt,T are observables) and shown to be:

(6.1)

Table 6.1 shows the implementation of equation (6.1) when all the other relevant information is given.^{[1]}

TABLE 6.1 Implying Dividend Rate, qtT

Inferring σt,T

For a given maturity T, to infer σt,T, one would need the inferred values of rt,T, qt,T, and the options contract value expiring at time T. Knowing all the relevant information allows one to solve equations (3.7a) and (3.7b) for σt,T. Putting all these together, it is easy to see that solving for at,T is equivalent to finding the solution of the equation

(6.2)

whererefers to the observed market price of the call option.

Thus the implied volatility, σ*, is the value of σt,T that is the root of equation (6.2).

Although the root of equation (6.2) is easily solved using Excel's Goal Seek function, this unfortunately is not a practical method for arriving at the solution. The reason for this is that in practice one is faced with the problem of finding implied volatilities for hundreds of option contracts instantaneously so as to allow the trader to make real-time trading decisions. As a consequence, one has to resort to more robust and powerful numerical methods.^{[2]}

For the purposes of illustration, I will again use Newton's method to show how to compute the implied volatility associated with a market-based, vanilla call option premium. Newton's method suggests that the root of equation (6.2) can be obtained by recursively using equation (6.3) to solve for

(6.3)

whereis the initial guessed value of the iteration.

Combining equations (6.2) and (6.3), one can arrive at

(6.4a)

where is defined by equation (3.7a) and

(6.4b)

One can similarly write analogs to equations (6.4a) and (6.4b) that can be used to obtain implied volatilities from put option premiums and arrive at

(6.5a)

where is defined by equations (3.7b),

(6.5b)

and PMarketPrice represents the market price of the put option from which one is trying to the implied value of σn.

Table 6.2 shows the implementation of equations (6.4a), (6.4b), (6.5a), and (6.5b) assuming that all the other relevant inputs to the Black-Scholes model and the option prices are given.

As the reader will note^{[3]} in Table 6.2:

■ The formulae used in cells BIO through B13 have not been displayed simply because these formulae are no different from those presented in Table 3.4.

TABLE 6.2 Implying Volatility Rate, σί Τ

■ Unlike Table 3.4,1 have put in the argument σ0 in cells B10 and B11 – just to emphasize that all these values are for the initial guess of sigma (σ0) that is given in cell B3.

■ Existence of the functional formof the differentials. In implementing equations (6.4b) and (6.5b),andexisted as functional forms. In practice, when dealing with path-dependent and early- exercise options, this is seldom the case. In such an event, it is customary to compute these differentials numerically – no different how thepremiums are computed. Thus one would, for example, compute (assuming this does not have a functional form) using the numerical approximationfor each value of n.

■ When using Monte Carlo simulations to value these options, due to the randomness associated with the simulation, there is noise associated with the computed premium. This noise can potentially lead to a large error when computing the differentials numerically. Fortunately, this problem can be easily overcome by ensuring that the same set of random numbers are used when computing the option premiums (for both the base volatility and shifted volatility cases).

■ Table 6.2 only contains the result for one iteration (whose results are given in cells B16 and B17). In practice this iteration has to be continued until the desired error bound is satisfied.

Table 6.3 shows the convergence of the implied volatility for both the call and put option.

As in the zero risk-free rate term structure where each liquid interest rate instrument is used to construct a zero risk-free rate that uniquely corresponded to the maturity of the instrument, one also constructs a zero- dividend rate term structure and spot volatility term structure based on the forward/futures and vanilla European-style option contracts. This should not come as any surprise since practitioners use prices of liquid instruments of varying maturities to infer a series of q and σs, each of which uniquely corresponds to a given maturity. However, unlike the zero risk-free rate curve construction (where one had to sometimes bootstrap using zero risk-free rates of shorter maturities to obtain those for the longer maturities), the zero- dividend rate and spot volatility rate curve construction are more straightforward. The reason for this stems from the fact that while the longer-dated instruments that were used to calibrate the zero risk-free rates were coupon bearing instruments, those used for calibrating the dividend and volatility rates seem to naturally embed the zero-rate or spot-rate structure-like features. Figure 6.1 shows an example of how an implied spot volatility term structure may look in practice.

As can be seen in Figure 6.1, to connect the data points I assumed the fitting of a linear function, hence the presence of kinks at the data points. Like the zero risk-free rate curve construction, one can use a cubic polynomial to smooth out the kinks at the data points. Whatever the methodology, the constructed term structure of zero risk-free rates, zero-dividend rates, and spot-volatility rates can be used to reproduce the price of liquid instruments (consistent with the second objective) and price complex options (e.g., early exercise and path dependent) on the same underlying in the process fulfilling the third objective.

Unlike the term structures associated with the zero risk-free rates and zero-dividend rates, the term structure of the spot-volatility rates unfortunately does not tell the whole story about the exact market sentiments in regard to how likely the price movements will be. In fact, since future movements in price/rate distribution in practice are not exactly lognormal, practitioners make adjustments^{[4]} to the volatilities of lognormal pdf (depending

TABLE 6.3 Convergence of σι

FIGURE 6.1 Implied Volatility Term Structure

on the in-the-moneyness of the option) so as to better capture the nonlogarithmic behavior of the pdf – all in the spirit of dressing the volatility inputs going into a lognormal pdf so as to ensure that market prices can be reproduced. As a consequence, it is not uncommon to see an implied volatility surface associated with options on an underlying, as shown in Figure 6.2, rather than that presented in Figure 6.1 – although in the absence of any in- the-moneyness effects, the volatility surface trivially collapses to a volatility term structure.

From Figure 6.2, it is important for the reader to make the following two observations:

■ One needs^{[5]} to use a cubic polynomial to smooth the kinks at the data points so the implied volatility surface can be more readily used with complex stochastic processes for the price or price volatility models.

■ A less obvious but a very important issue is one relating to the use of the implied volatility surface to price options. While valuing vanilla European-style options is quite easy (since one only needs to read off the implied surface for the appropriate implied volatility for a given maturity and the in-the-moneyness of the option), the same cannot be said when valuing early-exercise and/or path-dependent options. The reason for this stems from the fact in valuing early-exercise or path-dependent options, one would need to use the applicable forward risk-free rate,

FIGURE 6.2 Implied Volatility Surface

Source: mathworks.com.

forward dividend rate, and forward volatility rates – all of which are easy to do when time to maturity is the only dimension for the implied volatility (i.e., there is no in-the-money effect). The moment an in-the- money effect enters the picture, one needs to capture both the forward volatility rates and the in-the-moneyness effect created by the asset-price movements during the life of the option.

[1] An alternative method to imply qt,T involves the use of both call and put options that are struck at the same level and mature at the same time. In this case, using the put-call parity of the option, it can be shown that qt,T is the solution to the equation

[2] Examples of these methods include the Bisection algorithm, Newton method, Secant method, and a hybrid of these methods. See Burden and Fraires (2010).

[3] In addition, a few more technical observations need to be made as follows: One needs to start with the right σ0 so as to converge faster. In practice, one uses the implied volatility that was last used to obtain a new implied volatility for an updated option price. In the absence of this, one typically starts off with the historical volatility as a good proxy.

[4] These are sometimes called the volatility smiles or frowns.

[5] Clearly the need to use a cubic polynomial for smoothing the interpolation around the data points is a mute point if the data grid is fine enough to mimic a continuous surface.

Found a mistake? Please highlight the word and press Shift + Enter