In the previous section, I discussed the use of zero-rate risk-free, zero-rate dividend, and volatility term structures to infer the appropriate forward (risk-free, dividend, and volatility) rates to value path-dependent options.

TABLE 6.6 Calculation of Averaging Option Premium in the Presence of Term Structures

As was seen, the adjustments made were done to capture the presence of additional market information that took the form of forward rates. While this was not a huge step up in terms of complexity, as compared to a flat-term structure of rates (i.e., constant risk-free, dividend, and volatility rates), trying to do this for a collection of spot-volatility rate term structures across varying in-the-moneyness of the option (i.e., the spot-volatility rate surface) becomes much trickier. This is simply due to the fact that in addition to the term of the option maturity, the in-the-moneyness of the option needs to be captured (which intuitively is equivalent to saying that the implied volatility is a function of both the time and prevailing stock price). The problem of trying to use the implied volatility surface to value path-dependent options was tackled in 1994, when Dupire published his paper on volatility smiles set in continuous time. Derman and Kani in the same year independently published their paper on the same topic but set in discrete time.

In this section, I discuss the implementation of the methodology proposed by Derman and Kani (1994) – which was basically motivated by the desire to construct a stock-price tree that is consistent with the term structure of zero rates and implied volatility surface in discrete time. More precisely they assumed a generalized version of equation (3.5)

(6.13)

where

S is the price of the stock.

r(t) is the annualized continuously compounded risk-free rate that is a function of time.

q(t) is the annualized continuously compounded dividend rate that is a function of time.

σ(t, S) is the annualized volatility of the stock price return that is a function of both time and stock price level.

dz is the random variable drawn from a standard normal probability density function.

dS is the small change in the stock price over a small time interval dt.

They constructed a binomial tree of stock prices and probabilities that was consistent with above diffusion equations – a more complex version of the binomial tree that was discussed in Chapter 3. Derman and Kani showed that equations (3.14a) to (3.14c) can be generalized to accommodate the presence of more market information and written out to be

(6.14a)

(6.14b)

where

[a] is the floor function of a.^{[1]}

is the Arrow-Debreu price associated with the stock price (where is defined to be 1).

is the jth node of the stock-price tree at time(where is defined to be ).

FIGURE 6.3 Implied Volatility Surface Used for Table 6.7, A to C

is the forward price associated with the stock price and applied to time

is the implied volatility of the option when the stock price is and the strike price is

is the probabilty of moving from stock price to (where pt,1 is defined to be probability of upward movement at time t).

To illustrate the use of the above algorithm to calibrate a tree that is consistent with the term structures of zero rates and volatility rates, I will reproduce the example contained in their paper, using their assumptions that the current stock price is $100, the term structure of zero dividend rates is 0 percent, the term structure of zero risk-free rates is 3 percent, and the implied volatility surface is 10 percent for at-the-money options increasing (decreasing) linearly by 0.5 percent per every $10 drop (increase) in the strike rate of the option.^{[2]} The assumed implied volatility surface is shown in Figure 6.3.

Using the above information (and assuming that one is interested in generating a binomial tree with annual time steps), it is of interest to find the values of the following implied stock-price tree (shown in Figure 6.4a) and the implied probability tree (shown in Figure 6.4b).

FIGURE 6.4A Implied Stock-Price Tree

Given the assumptions (including one in which the authors assume that the price tree is centered around the initial stock price ), it readily follows that.

Getting the Implied Stock Prices When i = 0

When i = 0, first note that j = 0,1. To get(see Figure 6.4a), one would need to use the equation (6.14a) by noting that when

FIGURE 6.4R Implied Probability Tree

i = 0. Given this observation, it is straightforward to see thatcan be obtained by using the middle of equation 6.14a (which is given by case 1A when j = 1). Similarly, it can be seen thatcan be obtained by using the last of equation 6.14a (which is given by case IB when j = 0).

Case 1A: When j = 1

(6.15)

where

Since, one can valueusing the 1-step binomial tree to obtain a value of 6.40.^{[3]}

Making the appropriate substitutions in equation (6.15) yield 110.52.

Case 1B: When j = 0

(6.16)

where

FIGURE 6.5A Implied Stock Price Tree with One Year of Information

As before, since , one can value using the one-step binomial tree to obtain a value of 3.44.^{[4]}

Making the appropriate substitutions into equation (6.16) yields

Getting the Implied Probabilities When i = 0

To get the value of(see Figure 6.4b), one can use equation (6.14b) and obtain the expression

Since,(from case IB) and (from case 1A), it readily follows that Putting all these values together, one can arrive at Figures (6.5a) and (6.5b) – which are updates of Figures (6.4a) and (6.4b).

Getting the Implied Stock Prices When i = 1

When i= 1, first note that j=0,1,2 from equation (6.14a). From Figure 6.5a, it can be seen that sinceis known, one only needs to find and. To get, one would need to use the equation

FIGURE 6.SB Implied Probability Tree with One Year of Information

(6.14a) by noting that(and hence). Given this observation, it is straightforward to see thatcan be obtained by using the top of equation 6.14a (which is given by case 2A when j = 2). Similarly, it can be seen thatcan be obtained by using the last of equation 6.14a (which is given by case 2B when j = 0).

Case 2A: When / = 2

(6.17)

where

^{[5]}

Since^{[6]}, one can value spot(van) using the 2-step binomial tree to obtain a value of 3.951.^{[7]}

Making the appropriate substitution in equation (6.17) yields 120.30.

Case 2B: When j = 0

(6.18)

where

As before, since^{[8]}, one can value using the 2-step binomial tree to obtain a value of 1.283.^{[9]} Making the appropriate substitutions in equation (6.18) yield 79.28.

Getting the Implied Probabilities When i = 1

To get the values ofand(see Figure 6.5b), one can use equation (6.5b) and obtain the expressions

Since,, (from case 2B),and(from case 2A), it readily follows that and

Putting all these values together, one arrives at Figure 6.6a and Figure 6.6b – an update of Figure 6.5a and Figure 6.5b.

One can go through similar steps and arrive at the implied price tree and an implied probability tree given in the Derman and Kani paper (reproduced in Figures 6.7a and 6.7b).

With the implied stock-price trees and the corresponding probabilities, it is straightforward for one to use the methodology outlined in Chapters 3 and 5 to value path-dependent options. Furthermore, although the

FIGURE 6.6A Implied Stock-Price Tree with Two Years of Information

FIGURE 6.6B Implied Probability Tree with Two Years of Information

FIGURE 6.7A Implied Stock-Price Tree with Four Years of Information

FIGURE 6.7B Implied Probability Tree with Four Years of Information

example discussed the use of a flat term structure of zero risk-free and dividend rates, the above methodology can be easily extended to incorporate a nonflat, deterministic zero risk-free and dividend rate term structure.

As an alternative to the above approach, one can also use stochastic volatility models (in which the volatility itself is a random variable). Two widely used stochastic volatility models are Heston (1993) and Hagan et al. (2002) (which is sometimes better known as the SABR model – Stochastic Alpha Beta Rho model). The reader is referred to the excellent treatment of implied volatility surface and stochastic volatility models by Gatheral (2006) for more details.

[1] The floor function is defined such that [a] is the largest integer that is no greater than a. For example, [1+2] = 1, [2] = 2.

[2] When the stock price is $100, the implied volatility is 10 percent. When the stock price goes to $110, $120, and $130, the implied volatility decreases to 9.5, 9, and 8.5 percent respectively. When the stock prices go to $90, $80, and $70 the implied volatility increases to 10.5, 11, 11.5 percent respectively.

[3] Using equations (3.14a) to (3.14c) for the inputs given, it can be shown that ,,, and , which simplifies to 6.40.

[4] Using equations (3.14a) to (3.14c), for the inputs given, it can be shown that , , , and which simplifies to 3.44.

[5] This is the Arrow-Debreu value,, corresponding to the stock price node This value can be obtained by calculating the probability of present value of $1 obtained in this node which is, where p the probability of an upward move from a stock price level ofto a level ofthat is consistent to the implied volatility and is given by theexpression in equation (6.14b) where. As a consequence

[6] Since the implied volatility when the stock prices are at $110 and $115 are 9.5 percent and 9 percent respectively, using linear interpolation, it can be shown the implied volatility would be 9.47 percent when the stock price is $ 110.52.

[7] Using equations (3.14a) to (3.14c), for the inputs given, it can be shown that u _ eo.o947тД _ 1.0993, d = -= 0.9096, p = – = 0.6368, and C100д10 52;0;2;o.oз, 0,0.0947Αρολναη) = e~om^p212 * *max(100u2 – 110.52,0)], which simplifies to 3.95.

[8] Since the implied volatility when the stock prices are at $110 and $90 are 10 percent and 10.5 percent respectively, using linear interpolation, it can be shown the implied volatility would be 10.47 percent when the stock price is $90.48.

[9] Using equations (3.14a) to (3.14c), for the inputs given, it can be shown that * = c009WT = 1.0993, d = i = 0.9096, p = = 0.6368, and P100,110.52,0,2,0.03, ooo947^po((шя) = <r003(2>[(l -p)2*ntax(-100d2 + 110.52,0)], which simplifies to 1.283.

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