USING IMPLIED BLACK-SCHOLES VOLATILITY SURFACE AND ZERO RATE TERM STRUCTURE TO VALUE OPTIONS
In this section, I discuss how an implied Black-Scholes volatility surface together with the term structure of zero risk-free and dividend rates can be used to value path dependent options. To do this, I start with the special case when the volatility surface is assumed to be parallel to the in-the-money axis (i.e., no in-the-moneyness effects) – a consequence of which is that the implied volatility surface collapses to an implied volatility term structure. Following this, I revisit the problem of pricing path-dependent options in the presence of in-the-moneyness effects, so as to help the reader better appreciate the nuances and complexity associated with extraction of information that is inherent in the volatility surface.
Using Volatility Term Structure
I will first assume that the zero risk-free rate, zero-dividend rate, and spot- volatility term structures are the only sources of market information available. In this instance, the valuation of vanilla European-style options is an easy exercise simply because to infer the inputs rt,T, qt,T, and σt,T one just has to read them off the appropriate term structures. Given this backdrop, I will focus my discussion on path-dependent option valuation.
To understand how term structures can be used to price path-dependent options, one needs to first observe that when valuing these options, one major underlying assumption (given by equation (3.5)) is that lnST is normally distributed with a mean of and a variance of . The consequence of this assumption is that ST does not depend on any information prior to time t. Thus, if Su represents some time in the future (where t <u< T), then the pdf of lnSj given InSt can be easily decomposed as a sum of the following two pdfs:
1. is normally distributed with a mean and a variance of
2. lnST is normally distributed with a mean and a variance
Thus, a conditional pdf of lnST on lnSu is dependent on the forward rates ru,T, qu,T, and σu,T – among other things. Hence it is important for one to be able extract these forward rates from the given sources of market term structures.
As stated in equation (2.7), for any two given zero risk-free rates rt,u and (where T > u> t), the forward risk-free rate rt,u,T (a rate that is applicable from time и to T using the zero rates at time t) is given by the expression
Since the zero-dividend rates work the same way as the zero risk-free rates, one can go through similar arithmetic to show that for any two given zero-dividend rates qt,u and qt,T (where T > u> t), the forward dividend rate qt,u,T (a rate that is applicable from time и to T using the zero rates at time t) is given by the expression
Unlike forward risk-free and dividend rates, forward volatility rates cannot be obtained using a formula that is similar to equations (2.7) and (6.6). The reason for this stems from my earlier comment on the ability to decompose the conditional pdf of InST (given InSt) as a sum of the conditional pdfs of lnSu (given InSt) and InST (given lnSu). Given this decomposition, it is easy to see that
As a consequence, it readily follows that if σt,u and σt,Τ are any two spot-volatility rates (where T > u> t), the forward volatility rate σt,u,T (a rate that is applicable from time и to T using the spot volatility rates at time t) is given by the expression
Table 6.4 shows the implementation of equations (2.7), (6.6), and (6.7).
TABLE 6.4 Calculation of Forward Rates ru,T, qu,T, σu,T
Now that I have illustrated how to imply forward rates from the appropriate zero (risk-free and dividend) and spot-volatility rate term structures, I will show how forward rates can be used to value path-dependent options. To do this, I discuss one example involving the derivation of an accurate analytical solution and another involving the use of the Monte Carlo method.
Installment Options In Chapter 5, I discussed the valuation of a call-on- call option and showed for a flat term structure of a zero risk-free rate, zero- dividend rate, and spot-volatility rate (i.e., rt,T = rt,u,qt,T = qt,u, σt,T = σt,u for all T > u> t), the formula for the first installment premium is given by equations (5.12a), (5.12b), and (5.12c) which can be rewritten for ease of reading as follows:
and, S* is the solution to
To consider the presence of the term structure of rates and obtain an analogous expression to value an installment option, I will as before assume that the:
■ Current time is t.
■ Second optional premium of X is paid at time и.
■ Underlying option expiry upon payment of the second optional premium is time T (i.e., option life of T – u).
Given the above assumptions, it is straightforward to see that the first installment premium (associated with the installment option), in the presence of term structures, is given by
Since the values of ru,T, qu,T, and σu,T are unknown and assumed to be deterministic at time t, one needs to find an estimate for these variables using the forward zero rate (rt>u>T), forward dividend rate (qt,u,T), and forward volatility rate (σt,u,T) – all of which have been defined in equations (2.7), (6.6), and (6.7).
Putting all these together with (6.8), one gets
and S* is the solution of the equation
To evaluate equation (6.9a), first observe that the third integral and the terms accompanying it (after expansion and substitution of equation (6.9b)) becomes
Substituting equations (6.9c) and (6.9d), one gets
Now observe that the second integral and the terms accompanying it in equation (6.9a) (after expansion and substitution of equation (6.9b)) becomes
Letting the above simplifies to
Making the substitution with equations (6.9c) and (6.9d) and letting
Finally observe that the first integral and the terms accompanying it in equation (6.9a) (after expansion and substitution of equation (6.9b))
Letting , it is easy to see that
Letting and then substituting with equations (6.9c) and (6.9d),
Putting all the pieces together, equation (6.9a) can be simplified with the aid of equations (6.10a), (6.10b), and (6.10c) to arrive at the formula,
and, S* is the solution to
Table 6.5 shows the implementation of a call-on-call installment option (equations (6.11a), (6.11b), and (6.11c)) in the presence of term structures.
TABLE 6.5 Calculation of Installment Option Premium in the Presence of Term Structures
As can be seen from Table 6.5, the calculations are quite similar to Table 5.4 except that one has to factor in the concepts of forward risk-free rate, dividend rate, and volatility rate.
Averaging Options In the implementation of the Monte Carlo method to value path-dependent options described in Chapter 5, one had to recursively generate a stock price using the immediately generated stock price. More precisely, I used the recursive relationship derived using the idea in footnote 21 of Chapter 4:
where z represents the random standard normal variate and At represents time over which the simulation is done.
Equation (6.12a) can be more generally written as
the ith random standard normal variate.
, andrepresent the forward risk-free rate, dividend rate, and volatility rate respectively.
Table 6.6 shows the revaluation of the averaging option illustrated in Table 5.6.
As can be seen from Table 6.6,1 have only demonstrated the formulae involving the computation of the forward rates and the generation of the stock price since the rest of the formulae is no different from those produced in Table 5.6.