What volatility should be used to price 20-, 40-, and 60-day options?
. Suppose that there is an event that increases the current volatility by 0.5% to 2% per day.Estimate the effect on the volatility in 20, 40, and 60 days.
e. By how much does the event increase the volatilities used to price 20-, 40-, and 60-dayoptions?
anoOYrT>U_|ma
NUMERICAL PROCEDURES
This chapter discusses three numerical procedures that can be used to value derivatives when exactformuias are not available. These invoive the use of binomial trees, Monte Carlo simulation, andfinite difference methods, respectively. Monte Carlo simulation is used primarily for derivativeswhere the payoff is dependent on the history of the underlying variable or where there are severalunderlying variables. Binomial trees and finite difference methods are particularly useful when theholder has early exercise decisions to make prior to maturity. In addition to valuing a derivative, allthe procedures can be used to calculate Greek letters such as delta, gamma, and vega.
18.1 BINOMIAL TREES
 
In Chapter 10, we introduced one- and two-step binomial trees for non-dividend-paying stocks andshowed how they lead to valuations for European and American options. These trees are veryimprecise models of reality. A more realistic model is one that assumes stock price movements arecomposed of a much larger number of small binomial movements. Thijs is the assumption thatunderlies a widely used numerical procedure first proposed by Cox, Ross, and Rubinstein.'Consider the evaluation of an option on a non-dividend-paying stock. We start by dividing thelife of the option into a large number of small time intervals of length 51. We assume that in eachtime interval the stock price moves from its initial value of Sp to one of two new values, Souand Sod. This model is illustrated in Figure 18.1. In general, u > 1 and d < 1. The movement
Figure 18.1 Stock
Telit]rigur
' See J. C. Cox, S. A. Ross, and M. Rubinstein, “Option Pricing: A Simplified Approach,” Journal of FinancialEconomics, 7 (October 1979), 229-63.
392Numerical Procedures 393
 
from Sy to Sou is therefore an “up” movement, and the movement from Sy to Sod is a “down”movement. The probability of an up movement will be denoted by p. The probability of a downmovement is | — p.
Risk-Neutral Valuation
The risk-neutral valuation principle, introduced in Chapters 10 and 12, states that an option (orother derivative) can be valued on the assumption that the world is risk neutral. This means thatfor valuation purposes we can assume the following:
1. The expected return from all traded securities is the risk-free interest rate.
2. Future cash flows can be valued by discounting their expected values at the risk-free interestrate.
We make use of this result when using binomial trees. The tree we will build for a non-dividend-paying stock represents stock price movements in a risk-neutral world.
Determination of p, u, and d
The parameters p, u, and d must give correct values for the mean and variance of stock pricechanges during a time interva! of length 5t. Because we are working in a risk-neutral world, theexpected return from a stock is the risk-free interest rate, r. Hence the expected vaiue of the stockprice at the end of a time interval of length dis Se", where S is the stock price at the beginning ofthe time interval. It follows that
Sel = pSu + (i — p)Sd (18.1)or
e”* = pu +(1— pid (18.2)
The stochastic process we developed for stock prices in Chapter |1 implies that the variance of thepercentage change in the stock price in a small time interval of length dt is 0” 5t. Because theyariance of a variable Q is defined as E(Q°) —[E(Q)P, it follows that
pu? + (1 — p)d? — [pu + (1 — p)dP = 07 8tSubstituting for p from equation (18.2) reduces this toeu +d) —ud — "= 07 bt (18.3)
Equations (18.2) and (18.3) impose two conditions on p, u, and d. A third condition used by Cox,Ross, and Rubinstein is
 
dThese three conditions implya-d= 18.p=. (18.4)u = ove (18.5)
d=e ve (18.6)394 CHAPTER 18
wherea=e* (18.7)
and terms of order higher than dr are ignored.” The variable a is sometimes referred to as thegrowth factor. Equations (18.4) to (18.7) are consistent with the results in Section 10.7.
Tree of Stock Prices
Figure 18.2 illustrates the complete tree of stock prices that is considered when the binomial modelis used. At time zero, the stock price, Sy, is known. At time ét, there are two possible stock prices,Sou and Sod; at time 2 t, there are three possible stock prices, Sou’, So, and Sod’; aud so on. Ingeneral, at time 7ét, i+ 1 stock prices are considered. These are
Syd i, 7=0, 1,..., i
Note that the relationship wu = 1/d is used in computing the stock price at each node of the tree inFigure 18.2. For example, Spu’d = Spu. Note also that the tree recombines in the sense that an upmovement followed by a down movement leads to the same stock price as a down movementfollowed by an up movement.
  
Spd*
Figure 18.2 Tree used to value a stock option
2 To see this, we note that equations (18.4) and (18.7) satisfy the condition in equation (18.2) exactly. From Taylorseries expansions, when terms of order higher than 4¢ are ignored, equation (18.5) implies that w = 1 + o/8t+ ko? btand equation (18.6) implies that d= 1 —oVJ8t + 40°41, with e = 1+r6t and e7 = 1 +4 2rdt. It follows thatequations (18.5) and (18.6) satisfy equation (18.3) when terms of order higher than é¢ are ignored.Numerical Procedures 395
Working Backward through the Tree
Options are evaluated by starting at the end of the tree (time 7) and working backward. The valueof the option is known at time 7. For example, a put option is worth max(K — Sy, 0) and a calloption is worth max(S; — K, 0), where S; is the stock price at time T and K is the strike price.Because a risk-neutral world is being assumed, the value at each node at time T — dt can becalculated as the expected value at time T discounted at rate r for a time period $f. Similarly, thevalue at each node at time T—26t can be calculated as the expected value at time T — dtdiscounted for a time period 6t at rate r, and so on. If the option is American, it is necessary tocheck at each node to see whether early exercise is preferable to holding the option for a furthertime period 6t. Eventually, by working back through all the nodes, we are able to obtain the valueof the option at Gime zero,
Example 18.1 Consider a five-month American put option on a non-dividend-paying stock whenthe stock price is $50, the strike price is $50, the risk-free interest rate is 10% per annum, and thevolatility is 40% per annum. With our usual notation, this means that Sy = 50, K = 50, r = 0.10,o = 0.40, and T = 0.4167. Suppose that we divide the life of the option into five intervals of lengthone month (= 0.0833 year) for the purposes of constructing a binomial tree. Then $t = 0.0833 and,using equations (18.4) to (18.7),
uae = 14204, d= e °% — 0.8909a=e*=1,0084, p=2—4~0.5073u—d
1— p=0.4927
Figure 18.3 shows the binomial tree produced by DerivaGem. At each node there are two numbers.The top one shows the stock price at the node; the lower one shows the value of the option at thenode. The probability of an up movement is always 0.5073; the probability of a down movement isalways 0.4927.
The stock price at the jth node (j=0,1,...,4) at time iét (= 0,1,...,5) is calculated asSywid'/. For example, the stock price at node A (i= 4, j = 1) (i.c., the second node up at theend of the fourth time step) is 50 x 1.1224 x 0.89093 = $39.69,
The option prices at the final nodes are calculated as max(K — Sy, 0). For example, the optionprice at node G is 50.00 — 35.36 = 14.64. The option prices at the penultimate nodes are calculatedfrom the option prices at the final nodes. First, we assume no exercise of the option at the nodes. Thismeans that the option price is calculated as the present value of the expected option price one timestep later. For example, at node E, the option price is calculated as
(0.5073 x 0 + 0.4927 x 5,45)¢70.10%0-0833 _ 9.66whereas at node A it is calculated as(0.5073 x 5.45 + 0.4927 x 14.64)e 010%0.9833 _ 9 99
We then check to see if early exercise 1s preferable to waiting. At node E, early exercise would give avalue for the option of zero because both the stock price and strike price are $50. Clearly it is bestto wait. The correct value for the option at node E is therefore $2.66. At node A it is a differentstory. If the option is exercised, it is worth $50,00 — $39.69, or $10.31. This is more than $9.90. Ifnode A is reached, the option should therefore be exercised and the correct value for the option atnode A is $10.31.396 CHAPTER 18
 
At each node:
Upper value = Underlying Asset PriceLower value = Option Price
Shading indicates where option is exercised
Strike price = 50Discount factor per step = 0.9917
Time step, dt = 0.0833 years, 30.42 daysGrowth factor per step, a = 1.0084Probability of up move, p = 0.5073Up step size, u = 1.1224Down step size, d = 0.8909
       
Node Time:0.0000 0.0833 0.1667 0.2500 0.3333 0.4167
Figure18.3 Binomial tree from DerivaGem for American put on non-dividend-paying stock(Example 18.1)
Option prices at earlier nodes are calculated in a similar way. Note that it is not always best toexercise an option early when it is in the money. Consider node B. If the option is exercised, it isworth $50.00 — $39.69, or $10.31. However, if it is held, it is worth
(0.5073 x 6.38 + 0.4927 x 14,64)}e7010%00833 _ 19 36
The option should therefore not be exercised at this nodc, and the correct option value at the nodeis $10.36.
Working back through the tree, the value of the option at the initial node to be $4.49. This is ournumerical estimate for the option’s current value. In practice, a smaller value of St, and many morenodes, would be used. DerivaGem shows that with 30, 50, 100, and 500 time steps we get values forthe option of 4.263, 4.272, 4.278, and 4.283.Numerical Procedures 397
Expressing the Approach Algebraically
Suppose that the life of an American put option on a non-dividend-paying stock is divided intoN subintervals of length dt. We will refer to the jth node at time ir as the (i, j) node, where0<i< Nand0< j <i. Define f, ; as the value of the option at the (i, 7) node. The stock price atthe é. i) node is Spu/d’-/, Because the value of an American put at its expiration date ismax(K — S;, 0), we know that
fj = max(K — Suid’ 4, 0), f=0,1,...,N
There is a probability p of moving from the (i, /) node at time i 6¢ to the (i+ 1, 7 + 1) node at time(i+ 1)6t, and a probability 1 — p of moving from the (i, /) node at time / dt to the (i + 1, j/) node attime (i + 1) ét. If we assume no early exercise, risk-neutral valuation gives
Fig =6 Ee hint, jt $C Py fiat fl
for0 <i < N-—1land0 < j <i. When early exercise is taken into account, this value for f;, ; mustbe compared with the option’ § intrinsic value, and we obtain
fi,j = max {K — Souid', eo Ep fiat ja +(1— p) fis, 1}
Note that, because the calculations start at time 7 and work backward, the value at time iétcaptures not only the effect of early exercise possibilities at time 7 dt but also the effect of earlyexercise at subsequent times.
In the limit as df tends to zero, an exact value for the American put is obtained. In practice,N —30 usually gives reasonable results. Vigure 1 18.4 shows the convergence of the option price inthe example we have been considering. This figure was calculated using the Application Builder
functions provided with the DerivaGem software (see Sample Application A).
4 Optionvalue5.00
4.804.60 |4.40
4204
4.00 7
3.80 4
 
No. of steps3.60 p} ag pp5 10 15 20 25 30, 35 Ab 45 50
 
Figure 18.4 Convergence of the price of the option in Example 18.1 calculated from the DerivaGemApplication Builder functions398 CHAPTER 18
 
Estimating Delta and Other Greek Letters
It will be recalled that the delta, A, of an option is the rate of change of its price with respect to achange in the underlying asset price. It can be estimated as
bfA=6S
where 5S is a small change in the stock price and 3f is the corresponding small change in theoption price. At time dt, we have an estimate f,, for the option price when the stock price is Sgu,and an estimate f,) for the option price when the stock price is Sod. This means that, when5S = Sou — Sod, the value of Sf is f\; — fio. An estimate of A at time 4 is therefore
a — Suz fo foSou — Sod
osany00boSo
To determine gamma, T, note that we have two estimates of A at time 2 8t. When S = L(Squ? + So)(halfway between the second and third node), delta is (fy - far)/(Sou? — So) when S=4S + Spd”) (halfway between the first and second node), delta is (f:) — fao)/(Sp — Syd”). Thedifference between the two values of S is h, where
h = 0.5(Sgu™ — Sod”)
Gamma is the change in delta divided by h:
 
[2 = Fadi Sou? = So) = fay = fa0)/(So = Sod”)]
r=h’
(18.9)These procedures provide estimates of delta at time 6¢ and of gamma at time 2 6r. In practice, theyare usually used as estimates of delta and gamma at time zero as well.*
A further hedge parameter that can be obtained directly from the tree is theta, ©. This is the rateof change of the option price with time when all else is kept constant. If the tree starts at time zero,an estimate of theta is
_ fa — Sooo= 2m (18.10)
Vega can be calculated by making a small change do in the volatility and constructing a new tree toobtain a new value of the option. (The time step 3¢ should be kept the same.) The estimate of vega is
_f-f
ba
 
where f and f* are the estimates of the option price from the original and the new tree,respectively. Rho can be calculated similarly.
3 If slightly more accuracy is required for delta and gamma, we can start the binomial tree at time —2 6¢ and assumethat the stock price is Sp at this time. This leads to the option price being calculated for three different stock prices attime zero.Numerical Procedures 399
Example 18.2 Consider again Example 18.1. From Figure 18.3, fio = 6.96 and f,, = 2.16.Equation (18.8) gives an estimate of delta of2.16 — 6.9656.12—4455 7°"From equation (18.9), an estimate of the gamma of the option can be obtained from the values atnodes B, C, and F as[(0.64 — 3.77)/(62.99 — 50.00)] — [(3.77 — 10.36)/(50.00 — 39.69)]11.65 ~
From equation (18.10), an estimate of the theta of the option can be obtained from the values atnodes D and C as
 
0.03
3.77 — 4,490.1667or —0.012 per calendar day. These are, of course, only rough estimates. They become progressively
better as the number of time steps on the tree is increased. Using 50 time steps, DerivaGem providesestimates of —0.415, 0.034, and —0.0117 for delta, gamma, and theta, respectively.
= —4.3 per year
18.2 USING THE BINOMIAL TREE FOR OPTIONS ON INDICES,CURRENCIES, AND FUTURES CONTRACTS
 
 
As shown in Scction 13.2, the binomial tree approach to valuing options on non-dividend-payingstocks can be adapted to valuing American calls and puts on a stock providing a dividend yield attate q.
Because the dividends provide a return of g, the stock price itself must, on average, in a risk-neutral world provide a return of r— q. Hence equation (18.1) becomes
Se = Su + (1 — p)Sdso thatef = pu + (1 — p)dEquation (18.3) becomese949 + d) — ud — ee" = & bt
It can be shown (as in footnote 2) that equations (18.4), (18.5), and (18.6) are still correct (whenterms of order higher than 8t are ignored) but with
(r—g) 8taz=ee
(8.11)Therefore, with this interpretation of g, the binomial tree numerical procedure can be used exactlyas before except that the old value of a is replaced with this new one.
Recall from Chapter 13 that stock indices, currencies, and futures contracts can, for thepurposes of option evaluation, be considered as assets providing known yields. In the case of astock index, the relevant yield is the dividend yield on the stock portfolio underlying the index; inthe case of a currency, it is the foreign risk-free interest rate; in the case of a futures contract, it isthe domestic risk-free interest rate. The binomial tree approach can therefore be used to valueoptions on stock indices, currencies, and futures contracts providing g in equation (18.11) isinterpreted appropriately.400 CHAPTER 18
Example 18.3 Consider a four-month American call option on index futures where the currentfutures price is 300, the exercise price is 300, the risk-free interest rate is 8% per annum, and thevolatility of the index is 30% per annum. We divide the life of the option into four one-monthperiods for the purposes of constructing the tree. In this case, / = 300, K = 300, r = 0.08, o = 0.3,T = 0.3333, and dt = 0.0833. Because a futures contract is analogous to a stock paying dividends at arate r, q should be set equal to r in equation (18.11). This gives a = 1. The other parameters necessaryto construct the tree are
w= 0% — 1.9905, d=t= 0.9170
a “ =04784, 1 — p= 0.5216
 
p~~_The tree, as produced by DerivaGem, is shown in Figure 18.5 (The upper number is the futures price;the lower number is the option price.) The estimated value of the option is 19.16. More accuracy isobtained using more steps. With 50 time steps DerivaGem gives a value of 20.18; with 100 time stepsit gives 20.22.
At each node:
Upper value = Underlying Asset PriceLower value = Option Price
Shading indicates where option is exercised
Strike price = 300Discount factor per step = 0.9934
Time step, dt = 0.0833 years, 30.42 daysGrowth factor per step, a = 1.0000 424.19Probability of up move, p = 0.4784 i SsUp step size, u = 1.0905Down step size, d = 0.9170
     
 
 
Node Time:0.0000 0.0833 0.1667 0.2500 0.3333
Figure 18.5 Binomial tree produced by DerivaGem for American call optionon an index futures contract (Example 18.3)Numerical Procedures 401
Example 18.4 Consider a one-year American put option on the British pound. The currentexchange rate is 1.6100, the strike price is 1.6000, the U.S. risk-free interest rate is 8% per annum,the sterling risk-free interest rate is 9% per annum, and the volatility of the sterling exchange rate is12% per annum. In this case, Sy = 1.61, K = 1.60, r = 0.08, ry = 0.09, o = 0.12, and T = 1.0. Wedivide the life of the option into four three-month periods for the purposes of constructing the tree,so that 6t = 0.25. In this case, q = rz and equation (18.11) gives
 
g = ell.08-0.00.25 _ g 9975
The other parameters necessary to construct the tree are
u=e™™ = 1.0618, d=t= 0.9418
_a-dPad
 
= 0.4642, 1 — p =0.5358The tree, as produced by DerivaGem, is shown in Figure 18.6. (The upper number is the exchange
At each node:
Upper value = Underlying Asset PriceLower value = Option Price
Shading indicates where option is exercised
Strike price = 1.6Discount factor per step = 0.9802
Time step, dt = 0.2500 years, 91.25 daysGrowth factor per step, a = 0.9975Probability of up move, p = 0.4642Up step size, u = 1.0618Down step size, d = 0.9418
      
 
Node Time:0.0000 0.2500 0.5000 0.7500 1.0000
Figure 18.6 Binomial tree produced by DerivaGem for American put option on a currency(Example 18.4)402 CHAPTER 18
rate; the lower number is the option price.) The estimated value of the option is $0.0710. (Using50 time steps, DerivaGem gives the value of the option as 0.0738; with 100 time steps, it alsogives 0.0738.)
18.3. BINOMIAL MODEL FOR A DIVIDEND-PAYING STOCK
 
We now move on to the more tricky issue of how the binomial model can be used for a dividend-paying stock. As in Chapter 12, the word dividend will, for the purposes of our discussion, be usedto refer to the reduction in the stock price on the ex-dividend date as a result of the dividend.Known Dividend Yield
If it is assumed that there is a single dividend, and the dividend yield (i.e., the dividend as apercentage of the stock price) is known, the tree takes the form shown in Figure 18.7 and can beanalyzed in a manner similar to that just described. If the time i8t is prior to the stock goingex-dividend, the nodes on the tree correspond to stock prices
Suid /, jf =0,1,...,iwhere u and d are defined as in equations (18.5) and (18.6). If the time i4t is after the stock goes
Soul - 8)
Spu3l — 3)
Sout — 8)
So
So(1 — 8)
        
Sgd(1 - 8)
Sod? — 8) Sod? - 8)
Spd3(1 — 8)Ex-dividend date
Sod4(l - 8)
Figure 18.7 Tree when stock pays a known dividend yield at one particular timeNumerical Procedures 403
ex-dividend, the nodes correspond to stock pricesSoi— suid, jf=O li
where 4 is the dividend yield. Several known dividend yields during the life of an option can bedealt with similarly. If 8; is the total dividend yield associated with all ex-dividend dates betweentime zero and time 7 5t, the nodes at time i ét correspond to stock prices
Known Dollar Dividend
In some situations, the most realistic assumption is that the dollar amount of the dividend ratherthan the dividend yield is known in advance. If the volatility of the stock, o, is assumed constant, thetree then takes the form shown in Figure 18.8. It does not recombine, which means that the numberof nodes that have to be evaluated, particularly if there are several dividends, is liable to become verylarge. Suppose that there is only one dividend, that the ex-dividend date, t, is between & 6 and(k + 1) ét, and that the dollar amount of the dividend is D. When i < &, the nodes on the tree at timeidt correspond to stock pricesSod i, f=0,1,...,3
as before. When i = k + 1, the nodes on the tree correspond to stock prices
Sid /—D, j=0,1,...5i
Sy-D
_<
Ex-dividend date
AWWA
Figure18.8 Tree when dollar amount of dividend is assumed known and volatility is assumed constant404 CHAPTER 18
When i =k + 2, the nodes on the tree correspond to stock prices(Souid''-i — Dyw and (Syuid'~'-/ — Dyd
for /=0,1,...,i—1, so that there are 2i rather than i+1 nodes. When i=k-+™m, there arem(k + 2) rather than k +m 4 1 nodes.
The problem can be simplified by assuming, as in the analysis of European options inSection 12.12, that the stock price has two components: a part that is uncertain and a part thatis the present value of all future dividends during the life of the option. Suppose, as before, thatthere is only one ex-dividend date, r, during the life of the option and that két < 1 < (k + I)ét.The valuc of the uncertain component, S$”, at time /8f is given by
S*=8S, wheniét>tand byS*=S$— De") when idt <
where D is the dividend. Define o* as the volatility of S* and assume that o* is constant.* Theparameters p, u, and d can be calculated from equations (18.4) to (18.7) with o replaced by o* and atree can be constructed in the usual way to model S*. By adding to the stock price at each node thepresent value of future dividends (if any), the tree can be converted into another tree that models S.Suppose that So is the value of S$” at time zero. At time i St, the nodes on this tree correspond to thestock pricesStudi + De, 5 =0,1,...,3when iét < t andSguwd'!, 7 =0,1,....3
when i ét > t. This approach, which has the advantage of being consistent with the approach forEuropean options in Section 12.12, succeeds in achieving a situation where the tree recombines sothat there are i + 1 nodes at time i dt. It can be generalized in a straightforward way to deal withthe situation where there are several dividends.
Example 18.5 Consider a five-month put option on a stock that is expected to pay a singledividend of $2.06 during the life of the option. The initial stock price is $52, the strike price is $50,the risk-free interest rate is 10% per annum, the volatility is 40% per annum, and the ex-dividenddate is in 34 months.
We first construct a tree to model S*, the stock price less the present value of future dividendsduring the life of the option. At time zero, the present value of the dividend is
2.06e70 2917104 = 2.00
The initial value of S*, therefore, is 50.00. Assuming that the 40% per annum volatility refers to S*,we find that Figure 18.3 provides a binomial tree for S*. (This is because $* has the same initial valueand volatility as the stock price that Figure 18.3 was based upon.) Adding the present value of thedividend at each node leads to Figure 18.9, which is a binomial model for §. The prohabilities at eachnode are, as in Figure 18.3, 0.5073 for an up movement and 0.4927 for a down movement. Workingback through the tree in the usual way gives the option price as $4.44. (Using 50 time steps,DerivaGem gives a value for the option of 4.202; using 100 steps, it