-Rubenstein (Binomial) Mode!
The Cox-Ross-Rubenstein Model makes the assumption that over any period of time theunderlying contract can move up (u) or down (d) by a given amount. The probability ofan up move is given by p, and the probability of a down move is given by 1 ~p. Forexample, suppose a certain contract is trading at 100, and over the next time period thecontract will move up to 105 (u=5) or down to 95 (d=-5), and that there is a 50 percentchance (p=.5) of either outcome. If this is the only time period remaining to expiration,and there are no interest considerations, we can calculate the value of a 100 call as itsexpected return at the end the period. The call is worth intrinsic value if the underlyingcontract is above the exercise price, and zero if the underlying contract is below theexercise price. Its expected return is:
.9 X (105 - 100) + 0 = 2.50
4. Wilcox, Darren, “Spread Options in Energy Markets”, Research Paper, Goldman, Sachs & Co., March1990.
5. Cox, John C.; Ross, Stephen A.; and Rubenstein, Mark; “Option Pricing: A Simplified Approach”;Journal of Financial Economics, No. 7, 1979; pages 229-263For a good discussion of the computation required, see: Meisner, James E. and Labuszewski, John W.;“The Cox-Ross-Rubenstein Model for Alternative Underlying Instruments”; Advances in Futuresand Options Research, Vol. 2, 1987; pages 263-278The Mathematics of Option Pricing 435
In the same way we can calculate the value of a 90 call as:5 x (105 - 90) + .5 x (95 - 90) = 10
We can extend this approach by dividing the time to expiration into many smallperiods, and assume that over each period the underlying contract will always move up(u) or down (d). The result at expiration will be a binomial tree, with many possibleprices at expiration for the underlying contract. An example of a three-period binomialtree is shown in Figure B-1.
If we assume that the probability, p, of an up move and the probability, 1-p, of adown move, is the same at each node of the binomial tree, it is a simple matter tocalculate the probability of each price outcome at expiration. The expected return forthe option is the sum, for each outcome where the option is in-the-money, of thedifference between the option’s exercise price and the underlying price, multiplied bythe probability of that outcome. All outcomes where the option is out-of-the-money areset to zero.
In order to create a binomial tree which will approximate a lognormal distributionwe can define the following:
us etnd= 1/uwhere n = __ thenumber of periods to expiration (the number of
branches on the binomial tree)
Figure B-1A3 Period Binomial Treeu=+5 115 (100 call = 15)d= 5a m105 105 (100 call = 5)100 10095 95 (100 call = 0)90
85 (100 call = 0)436 Appendix B
vt
the annual volatility of the underlying contractthe time to expiration in years
Not
As n becomes very large, the resulting terminal prices of the binomial tree converge toa lognormal distribution.
The probability, p, of an up move is determined by the requirement that theunderlying market be arbitrage-free, ie., that no profit can be made trading theunderlying contract. If there are no interest considerations, which is the case with afutures contract, we can ensure that the underlying futures market is arbitrage-free bydefining the probability, p, of an up move as:
p = (1-d)/(u-d)
In an arbitrage-free stock market, the stock price must increase by the amount ofcarrying costs over each period. If r is the risk-free rate, the rate, rr, by which the stockprice must increase over each period is:
rr= 1 + (rt/n}We can ensure that the underlying stock market is arbitrage-free by defining theprobability, p, as:p = (rr-d)/(u-d)
Finally, we must discount the expected return for the option by the carrying costs overthe life of the option by multiplying the expected return by 1/(rr)".The basic form of the Cox-Ross-Rubenstein Model is:
n
1 no= {—p)"-k, kqn-kUECc crt » ki(n-k)! Pp ( Dp) max(0,u! d )
 
 
P= ! > a pk(1—p)*-*max(0,E~ukd"+U)
~ (rr) o ki(n-k)!
 
 
In order to use the Cox-Ross-Rubenstein Model to evaluate an American option,we need to check whether at the beginning of each time period the value of the optionis such that early exercise is warranted. For example, in our two-period example withthe underlying at 100, we saw that the expected return for the 90 call was 10. If theoption were settled in cash, the theoretical value of the option would be 10 less thecarrying costs over the holding period. If the carrying costs were .25, the theoreticalvalue of the option would be 9.75. If, however, the option were American, everyonewould exercise the option in order to obtain 10 points right now. In other words, thefact that the option is American would require us to set the theoretical value at 10.
If we define U(i,j) as the j* underlying price at the end of the i® time period (seeFigure B-2), and C(i,j) or P(i,j) as the call or put value at each U(i,j), then we must checkto see whether each C(i,j) < Ufi,j)-E, if we are evaluating an American call, or whetherThe Mathematics of Option Pricing 437
Figure B-2~ U4,0)U(,j) is the jth underlying price UG.0)at the end of the ith period ,U(2,0) U(4,1)UG,1)
u(1,0)a Ul2,1) (4,2)
U(1,1) UG,2)U(2,2) U(4,3)(0,0) is the current price u@3)of the underlying contract ,u(4,4)
each P(i,j) < E-Uli,j), if we are evaluating an American put. If either condition is truethe option becomes an early exercise canditate, and we set the value of C(i,j) equal toU(i,j)-E, or the value of P(i,j) equal to E-U(i,j).
We begin by calculating each terminal value U(n,j} for j=0,...,.n. From these valueswe can calculate the value of the option at each previous node, C(n-1 ,j) or P(n—I ,j) forj=0,...,n-1. If C(n-1,j) turns out to be less than U(n-1,j)-E, or if P(n-1,j) turns out to beless than E-U(n-1,j), the option value is set to parity, and the iterative procedurecontinues. We continue to work backwards to C(0,0) or P(0,0) (the current value of thecall or put), always setting the value at each node, U(i,j), to max|C(i,j), U(i,j)-E] for a call,or max[P(i,j), E-U(i,)] for a put:
pci+1,j) + d-pyCda+i,j+
CG, ) = max IT , UG, i) ~ F
pPd+1,j) + d-p)PG+1,j+)rr
 
PCi, )) = max ? E- Uti, 0Since the delta of an option is its change in value fora given change in the price ofthe underlying contract, we can also calculate call and put deltas from a binomial model:
_ cqd,4) = C(1,0)
call A = Ud,1) - UCL,0)438 Appendix B
P(1,1) — P(1,0)Ud,1)—Ud,0)
The larger n is chosen, the more accurate the option value generated by the mode!will be. Unfortunately, as n increases, the number of calculations required to evaluatean American option increases geometrically. Most traders who use the Cox-Ross-Ruben-stein model choose n somewhere between 25 and 50. This represents a reasonabletrade-off between computer time and accuracy.
put A =
Ill. The Whaley (Quadratic) Model®The Whaley Model uses an approximation technique to estimate the critical price, U",at which an American option should optimally be exercised early. This critical price isthen used to determine the value of the option.In order to find U* for a call, we must solve the equation:
E - Ut = e@9YU*N(hHEN(h-vvi)] + ((1-e©'N(h-wit))(U*/a,)1(For convenience we will refer to the two sides of this equation as LHS (the left-handside] and RHS {the right-hand side].)
where b = carrying costs on the underlying instrument (for futuresb = 0; for stock b = r) and all other abbreviations are thoseused in the Black-Scholes Model
Ut venf = | + [>+F}
>ll
wtGd = (-(N-1) + YQ(N-1)? + 4M/k)|/2M = 2r/v2N = 2b/v?k =~ Let
To solve for U* we pick a desired degree of accuracy, €, satisfying:(LHS - RHS) /E<eWe then perform progressive iterations, each time replacing U* , With:U*u1 = (E+ RHS —bU? |] / (1-)where b, = e®-9tN(hU*)(1-1/q,) + [1-26 N'(hU*/(wv t/a,
Once U" has been calculated ta the desired degree of accuracy, the value of anAmerican call, C, and its delta, A, can be found by solving:
C=c + A(U/U)%
 
6.  Barone-Adesi, Giovanni and Whaley, Robert E., “Efficient Analytic Approximation of American OptionValues,” Journal of Finance, Vol. 42, No. 2, June 1987, pages 301-320.The Mathematics of Option Pricing 439
A= 8 + Ag(U/UNY/U
where Cc = the value of a European call5 = the delta of a European callA, = (U/q,)fl-e'N(hU")]
In order to find U* for a put, we must solve the equationE - Ut = e@*(EN(h- wt) - U*N(h)] — [(1 - et N(h)) (U/q))](We again refer to the two sides of this equation as LHS [the left-hand side] and RHS [theright-hand side.)
where b = carrying costs on the underlying instrument (for futuresb = 0; for stock b = r) and al! other abbreviations are thoseused in the Black-Scholes Model
us ve«(=| + +5}
h wt
q, =  KXN-l) - VON-1D? + 4M/k)|/2M = 2t/y?
N = 9 2b/y2
k = 1e™
To solve for U* we pick a desired degree of accuracy, e, satisfying:(LHS — RHS) /E<eWe then perform progressive iterations, each time replacing vy with:U",,, = [E- RHS - b,U*] / (1-D)where b, = e®*N(hU")(1~1/q,) + [I-e@ N‘(hU")/(vVt)|/q,
Once U™ has been calculated to the desired degree of accuracy, the value of anAmerican put, P, and its delta, A, can be found by solving:
P=p+ A,U/U*)%A= 8 + Ayq,(U/U)2/U
where p
the value of a European putthe delta of a European put— (U*/q,}[1-e®*N(hU")]
anol440 Appendix B
NORMAL DISTRIBUTIONS
Since the assumption that price changes are normally distributed plays such an impor.tant role in many theoretical pricing models, it may be useful to know how to computesome of the numbers associated with a normal distribution.
The Mean (m)
The mean (m) of a distribution of n occurrences is the average outcome of all theoccurrences (x,}:
n{m=7 x,1=1
The Standard Deviation (co)The standard deviation (o) of a distribution of n occurrences is defined as:”
o-V
n1i my1=1
If a distribution is normal, approximately 68.3 percent of the occurrences fallwithin one standard deviation of the mean, approximately 95.5 percent of the occur-rences fall within two standard deviations of the mean, and approximately 99.7 percentof the occurrences fall within three standard deviations of the mean.
Skewness (Sk) and Kurtosis (Ku)
Ifa distribution is approximately normal, it may be useful to know how the distributiondeviates from a true normal distribution. We can determine this by computing theskewness and kurtosis of the distribution.
Many of the measures associated with a distribution are derived from a group ofnumbers called moments. In general, the j moment of a distribution about the meanis:
Bl
m, =
¥ @-m)1=1
To determine the skewness and kurtosis of a distribution, we need the second, third,and fourth moments:
n1M, = 5 - my?1=1
7. Because we are dividing by (n-1) this is technically the sample standard deviation. This is the formwhich is most commonly used for volatility calculations.The Mathematics of Option Pricing 441
n1m; = >> @,-m)*1=1_ly ‘my = nv -M)1=1
The skewness of a distribution is defined as:
 
ske nswness =myvM2A perfectly normal distribution has a skewness of zero. If the distribution is positivelyskewed {Sk > 0), the right tail is tonger than the left. If the distribution is negativelyskewed (SK < 0}, the left tail is longer than the right.
The kurtosis of a distribution is defined as:
. Mgkurtosis = —5 - 3M,A perfectly normal distribution has kurtosis of zero (mesokurtic). If a distribution has apositive kurtosis (Ku > 0), more occurrences fall in the middle and at the extreme tailsof the distribution (leptokurtic). If a distribution has a negative kurtosis (Ku < 0), feweroccurrences fall in the middle and at the extreme tails of the distribution (platykurtic).
While almost all spreadsheets include the mean and standard deviation functions,and many also include skewness and kurtosis, it may be worthwhile to do a samplecalculation.
Consider, for example, the pinball distribution in Figure 4-2 (reproduced in FigureB-3}. In order to compute the mean, we must multiply the number of balls in eachtrough by the number of the trough, add up the total (563), and divide by the totalnumber of balls (75):
mean = 563/75 = 7.507
To compute the standard deviation, for each ball we square the difference betweenits trough and the mean, add up all 75 results, divide by the number of occurrences lessone (74), and take the square root of this number.
standard deviation = V1
Sy hs my?nl
2 ¥ (x, ~ 7.507)
1=1Appendix B
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 The Mathematics of Option Pricing 443
- Y 660.747
74= ¥8.929= 2.988
 
While a detailed calculation of the skewness and kurtosis would require aninordinate amount of space, the relevant values are:
skewness
kurtosis
Wott ou
Il Ul
8.810185213.455
M,MVM,1858.810 V8.810185
26.149+.007 (the right tail is slightly longer than the left)
 
213.4558.81022.750 - 3 ,~ .250 (the distribution has a slightly flatter peak
than one would expect)
 
- 3
VOLATILITY CALCULATIONS
Historical Volatility
The historical volatility is defined as the standard deviation of the logarithmic pricechanges measured at regular intervals of time. Since settlement prices are usuallyconsidered the most reliable, the most common method of computing volatility involvesusing settlement-to-settlement price changes. We define each price change (x,) as:
I fiae Pit
where P, is the price of the underlying contract at the end of the i® time interval
P,/P,, is sometimes referred to as the price relative.As an example, consider the volatility of the price changes in Figure 5-1.444 Appendix B
Underlying Deviation DeviationWeek Price In(P,/P,-1) Mean from Mean Squared0 101.35
1 102.26 +.008939 007771 .0000602 99.07 —.031692 —.032859 — .0010803 100.39 +.013236 ~ _ 012069 .0001464 100.76 +.003679 2 .002512 .0000065 103.59 +.027699 3 026532  .0007046 99.26 ~.042698 + —.043865 0019247 98.28 ~.009922 —.011089 = .0001238 99.98 +.017150 015982 0002559 103.78 +.037303 036136 00130610 102.54 ~.012020 ~.013188 —§ .000174+.011674 005778
We first calulate the standard deviation of the logarithmic price changes:
standard deviation = V(.005778/9)= V.000642= .025338
We then calculate the annual volatility by multiplying the standard deviation by thesquare root of the time interval between price changes. Since we looked at price changesevery week, the time interval is 365/7:
025338 x V(365/7)025338 x ¥52.14025338 x 7.221829 (18.29%)
We can calculate the volatility of a stock, such as in Figure 5-2, in the same wayexcept that we must remember that the volatility is based on the forward price. Thisrequires two modifications. Over any interval of time, in order to maintain a theoreti-cally arbitrage-free market, the stock price is expected to increase by the carrying costson the stock over that period. If we record stock price changes at weekly intervals, thestock price is expected to rise by r/52, where ris the annualized, risk-free interest rate.The price changes are therefore expressed as:
1 miSN /5P,_,
When the stock gives up its dividend, this does not really cause a change in the stockprice. We therefore include the dividend, D, in the price of the stock when calculatingprice change over the period when the stock goes ex-dividend:
annualized volatility
i
M
UI
ilThe Mathematics of Option Pricing 445
=| P+DHe IP /520P,)
Over short periods of time, or in a low-interest-rate environment, the impact ofinterest rates on the forward price is small, and can usually be ignored when calculatinghistorical stock price volatility.
The Extreme Value Method®
When reliable settlement prices are unavailable, the extreme value method is anacceptable alternative method of calculating historical volatility. The method uses highand low prices during a period, and each data point, x,, is equal to:
Xx, = .601 - In(H/L,)where H, = __ the highest price over time period tL the lowest price over time period t
The volatility is the standard deviation of all X,, annualized by multiplying by thesquare root of the number of time periods, t, in a year. If t is a daily interval, we must
Figure B-4: Using a Newton-Raphson Search to Find the Implied Volatility
 
theoretical valueat first guess
             
 
theoretical valueat second guess
Theoretical Value —______—_________ Jj
theoretical valueat third guessOptionPrice
vega at; second guessi
 
 
 
ImpliedVolatility
Volatility >
 
 
8. Parkinson, Michael, “The Extreme Value Method for Estimating the Variance of the Rate of Return’,Journal of Business, Vol. 53, No.1, 1980, pages 61-65.446 Appendix B
multiply by the square root of the number of trading days in a year (approximately 253).if t is a weekly interval, we must multiply by the square root of the number of tradingweeks in a year (approximately 52).
Calculation of Implied Volatility
While it is not possible to invert the Black-Scholes Model to arrive at an implied volatilityfrom a known option price, a Newton-Raphson search can be used to converge on theimplied volatility quite quickly. We first make a guess as to the option’s impliedvolatility, and then use the option’s vega (sensitivity to a change in volatility) to comeprogressively closer to the true implied volatility. This method is illustrated in FigureB-4.
Because an option’s vega is relatively linear, the method converges very quickly,usually with no more than four iterations, even with a poor first guess. The iterativeprocess is:
x =x?i+1 i v,where p = __ the option’s pricex = the volatilityy, = the option’s theoretical value at volatility x,vy, = the option’s vega at volatility x,
We choose a desired degree of accuracy, €, and continue with the process until ly, — pl <€, at which point x, is the desired implied volatility.
THE EXPONENTIAL AND NATURALLOGARITHM FUNCTIONSSince the natural logarithm (In(x)} and exponential function (e*) play such an important
role in most theoretical pricing models, a brief discussion of their use may be worth-while.
Suppose r = an annual interest rate,expressed as a decimal fractionI = the amount of an investmentt = the time period over which the investment is made,
expressed in years
When r is continuously compounded, the value, V, of I at the end of the timeperiod, t, is given by:
Ve=et.]
When r is continuously compounded, the initial investment, I, required to yield Vat the end of the time period is given by:The Mathematics of Option Pricing 447
T=et.v
Lis referred to as the present value of V, i.e., V discounted by its carrying costs.The yield, y, resulting from the continuous compounding of the interest rate, r, overthe time period is given by:
y = et—]
The annualized yield is y/t.Example: If r=10% (.10), the value of a $2,000 investment in three months(t=.25) if interest is compounded continuously will be:
V = et. $2,000 = 210-25 . $2,000= 2925. $7,000 = 1.0253 - $2,000 = $2,050.63
Example: If r=6% (.06) and is compounded continuously, the amount of moneyone would need to invest to have $5,000 after eight months (t=.667) would be:
1 = ett. $5,000 = 2-967 . $5,000e~04. $5,000 = .9608 - $5,000 = $4,803.95
Example: If r=15% (.15} and is compounded continuously, the total yield on asix-month (t=.5) investment will be:
y=et—j] =e15-5—] = e075_ | = 0779 (7.79%)The total annualized yield is:y/t = .0779/.5 = 1558 (15.58%)
If 1 is the amount initially invested, the continuously compounded rate of return,r,, needed to yield V over some period of time, t, is given by:r, = In(V/I)
If V is greater than I, the resulting rate of return will be a positive number, If V is lessthan I, the resulting rate of return will be a negative number. The annualized rate ofreturn is:
Hi
t=r/t=In(V/)/t
Example: Assuming continuous compounding, the annual rate of return requiredfor an initia! investment of $3,000 to yield $3,200 in nine months (t=.75) is:
In(3,200/3,000)/.75 = In(1.0667)/.750645/75 = .0861(8.61%)
Note that the exponential and logarithmic functions are inverses:
In(e*) = eh) = x
T448 Appendix B
Because volatility is also a rate of return, and is assumed to be continuouslycompounded, the exponential and logarithmic functions can be used to calculateexpected price changes of an underlying contract.
Example: Suppose a certain futures contract is trading at a price, P, of 50 and hasan annual volatility, v, of 12%. A one standard deviation upward price change is
ev-P=e!2.50 = 1.1275 - 50 = 56.37A one standard deviation downward price change isev. P=en!2. 50 = 8869 - 50 = 44.35Since we know that one standard deviation takes in about 68 percent of all outcomes,we know that if the volatility figure of 12% is correct, there is a 68 percent chance thatone year from now the same futures contract will be trading between 44.35 and 56.37.
What about a two standard deviation price change? If the move is upward, wecalculate:
e122. 50 =e24- 50 = 1.2712 - 50 = 63.56if the move is downward, we calculate:e122. 50 = e-44 . 50 = 7866 - 50 = 39.33Since we know that two standard deviations takes in about 95% of all outcomes, weknow that if the volatility figure of 12% is correct, there is a 95% chance that one yearfrom now the same futures contract will be trading between 63.56 and 39.33.
For time periods other than one year we must also take into consideration thesquare root relationship between time and volatility. If a one standard deviation pricechange over a period is given by v, then a one standard deviation price change over aperiod which is twice as long is v V2. Similarly, a one standard deviation price changeover a period only half as long is v V.5. The generalized rule is:
volatility (standard deviation) over the period t=v-Vt
where v is the annualized volatility and t is the time period in years. This enables us toexpress a move of n standard deviations over t as either:
env. P (an upward move]orenv. P (a downward move)
where P is the current price of the contract.
Example: With an underlying contract at 84.00 and an annual volatility of 16%,a one and two standard deviation price change over a three-month period (t = .25) is:e}6V.25 . 84.00 = e165. 84.00 = &8. 84.00= 1.0833 - 84.00 = 91.00 (up 1 standard deviation)The Mathematics of Option Pricing 449
2.16425. 84.00 = e2-16-5. 84.00 = e!S. 84.00= 1.1735. 84.00 = 98.57 (up 2 standard deviations)
e165. 84.00 = 8 . 84.00.9231- 84.00 = 77.54 (down | standard deviation)
e-2-16V-25 . 84.00 = e2-165. 84.00 = e-!6- 84.00= 8521.84.00 = 71.58 (down 2 standard deviations)
e125 . 84.00
u
Given a volatility and time period we can always calculate the number of standarddeviations required to reach an outcome. If we have a table of standard deviations andtheir associated probabilities, we can then find the probability associated with thatoutcome.
In the case of options, we might often be interested in the probability of an optionwith a certain exercise price being in-the-money at expiration. The movement, instandard deviations, required for an underlying contract with price, P, to go through anexercise price, E, at expiration is given by:
number of standard deviations = In(E/P)wt
Example: If conditions are the same as in the last example (v=.16, P=84.00), themovement, in standard deviations, required for