The Binomial Model

This converges to the lognormal distribution for stock price movements, when the number of steps is large; the price of an option computed by the binomial model will therefore converge t

7 The Binomial Model

This is one of the most important chapters of this book, so it is worth giving a road map of where we are going

Section 7.1 introduces the binomial model based on the random walk which is discussed in the Appendix This converges to the lognormal distribution for stock price movements, when the number of steps is large; the price of an option computed by the binomial model will therefore converge to the analytical formulas based on a lognormal assumption for the stock price movements

Section 7.2 shows how to go about setting up a binomial tree while Section 7.3 gives several worked examples of the binomial model applied to specific option pricing problems

The consequences of the binomial model for the derivatives industry have been enormous

It is a powerful and flexible pricing tool for a variety of options which are too complicated for analytical solution But the impact on the industry goes further than this Very little technical skill is needed to set up a random walk model; yet these models can be shown to converge reliably to the “right answer” if the number of steps is large enough This approach has therefore opened up the arcane world of option pricing to thousands of professionals, with an intuitive yet accurate method of pricing options without recourse to advanced mathematics Without these developments, option pricing would have remained the domain of a few specialists

(i) The following results are demonstrated in Appendix A.2 for a random walk with forward and

backward step lengths U and D and probabilities p and 1 − p If x n is the distance traveled

after n steps of the random walk, then

rE [x N]= N{pU − (1 − p)D}

rvar [x N]= N p(1 − p)(U + D)2

rThe distribution of x N is a binomial distribution which approaches the normal distribution

as N → ∞

Consider now the movement of a stock price It was seen in Section 3.1 that the logarithm of

the stock price (ln S t) follows a normal distribution If we observe the stock price at discrete

intervals of time, we postulate that x i = ln S i follows a random walk The distribution of x i

then approximates to a normal distribution more and more closely if we make the number of

intervals larger and larger If x i is the logarithm of the stock price at the beginning of step i then

we have x i+l= x i + U or x i+l= x i − D (i.e S i+l= S ieU or S i+l= S ie−D) with probabilities

p and 1 − p In the limit of small time intervals (N → ∞; U and D → 0), x N is normally

distributed and S N is lognormally distributed

In the following analysis we switch our attention to the behavior of S N rather than x N This actually complicates the algebra a bit, but has the advantage of providing a more intuitive picture; in any case, it conforms with the way most of the literature is written A condition for

x N to approach a normal distribution is that U and D are constants; the corresponding condition for S N to approach the lognormal distribution is that u= eU and d= e−Dshould be constant

multiplicative factors The progress of x N is described as an arithmetic random walk, while

S N follows a geometric random walk.

t

dd t dd

f 0 S 0

0

f S d = dS 0

0

f S = uS u 0

Figure 7.1 Two possible final states

(ii) Single Step Binomial Model: A description of the

binomial model starts with the simple one-step

ex-ample of Section 4.1

Suppose the stock and derivative start with prices

S0and f0 After a time intervalδt, Sδt has one of

two possible values, S u or S d, with corresponding

derivative prices f u and f d (Figure 7.1)

If a perfect hedge can be formed between one

unit of derivative and units of stock, we saw in

Section 4.1(iv) that the no-arbitrage condition

im-poses the following condition:

f u − S u = f d − S d = (1 + rδt) ( f + S)

We find it simpler to manipulate the interest term in continuous form, although many authors develop the theory in the form just given For small time steps and to first order inδt, the

results are the same These equations can then be rewritten as

S0= e−rδt ( pS u + (1 − p)S d); f0= e−rδt ( p f u + (1 − p) f d)

= f u − f d

S u − S d

S u − S d

(7.1)

It is critically important to realize that we have not started by defining p as the probability of an up-move We have started from the no-arbitrage equations in which a parameter p appears It

happens to have the general “shape” of a probability and is interpreted as the pseudo-probability

(i.e probability in a risk-neutral world) that S0moves to S uin the periodδt.

If continuous dividends were taken into account, we would make the substitution Sδt→

Sδte−qδt in the first equation and p would then be given by

p= F0 δt − S d

S u − S d where F0δt = S0e(r−q)δt is the forward rate

(iii) Conditions on Drift and Variance: Values can be obtained for u, d and p from a knowledge

of the mean and variance of Sδt Writing S u = uS0 and S d = dS0 and interpreting p as the probability of an up-move in a risk-neutral world, the mean and variance of Sδtmay be written

E[Sδt]= {pS u + (1 − p)S d } = S0{pu + (1 − p)d}

var[Sδt]=pS u2+ (1 − p)S2

u



− E2[Sδt]= S2

0p(1 − p)(u − d)2 (7.2) Recall the following results derived in Section 3.2(ii) and applied to a risk-neutral world:

E[Sδt]= S0e(r −q)δt = F0 δt; var[Sδt]= E2[Sδt]{eσ2 δt − 1} ≈ F2

0 δt σ2δt + O[δt2]

76

7.2 THE BINOMIAL NETWORK Equating these last two sets of equations and dropping terms of higher order inδt gives

F0δt = S0{pu + (1 − p) d}; F02δt σ2δt = S2

0p(1 − p) (u − d)2 (7.3)

These are two equations in three unknowns (u, d and p), so there is leeway to choose one of

the parameters; is there any constraint in this seemingly arbitrary choice?

From the first relationship, it is clear that if S u(= uS0) and S d(= dS0) do not straddle F0δt,

then either p or (1 − p) must be negative Since we wish to interpret p as a probability (albeit

in a risk-neutral world), we must impose the condition S d < F0 δt < S u

The function p(1 − p) has a maximum at p = 1

2 The second of equations (7.3) above therefore yields the following inequality:

F0 δt σ√δt

S u − S d ≤ 1

This is really saying that if the spread S u − S d is not chosen large enough, the random walk will not be able to approximate a normal distribution with volatilityσ

(iv) Relationship with Wiener Process: Another way of looking at the analysis of the last

para-graph is to say that the Wiener process S t+ δt − S t = δS t = S t (r − q)δt + S t σ√δtz can be represented by one step in a binomial process, where z is a standard normal variate so that E[Sδt]= S0(1+ (r − q) δt) and var[Sδt]= S2

0σ2δt.

We must now choose u, d and p to match these, i.e.

E[Sδt]= S0(1+ (r − q)δt) = S0( pu + (1 − p)d) var[Sδt]= S2

0σ2δt = S2

The reader may very well object at this point since this seems to be the wrong answer; equations (7.3) and (7.5) are not quite the same But recall that the entire Ito analysis is based on rejection

of terms of order higher thanδt:

F0 δt = S0e(r−q)δt = S0{1 + (r − q) δt} + O[δt2]; F0 δt σ√δt = S0σ√δt + O[√δt3]

To within this order, the results of this and the last subparagraph are therefore equivalent

(i) The stock price movement over a single step of lengthδt is of little use in itself We need to

construct a network of successive steps covering the entire period from now to the maturity of the option; the beginning of one such network is shown in Figure 7.2

The procedure for using this model to price an option is as follows:

(A) Select parameters u, d and p which conform to equation (7.2) The most popular ways of

doing this are described in the following subparagraphs

(B) Using these values of u and d, work out the possible values for the stock price at the final nodes at t = T We could work out the stock value for each node in the tree but if the tree

is European, we only need the stock values in the last column of nodes

(C) Corresponding to each of the final nodes at time t = T , there will be a stock price S m ,T where m indicates the specific node in the final column of nodes.

77

(D) Assume the derivative depends only on the final stock price Corresponding to the stock

price at each final node, there will be a derivative payoff f m ,T (S T)

(E) Just as each node is associated with a stock price, each node has a derivative price The nodal derivative prices are related to each other by the repeated use of equations (7.1) Looking at Figure 7.2 we have

f4= e−rδt {p f7+ (1 − p) f8}

f5= e−rδt {p f8+ (1 − p) f9}

f2= e−rδt {p f4+ (1 − p) f5}

This sequence of calculations allows the present value of the option, f0, to be calculated from

the payoff values of the option, f m ,T (S T); this is commonly referred to as “rolling back through the tree”

(ii) Jarrow and Rudd: There remains the question of our choice of u, d and p The options are

examined for a simple arithmetic random walk in Appendix A.2(v); we now develop the corresponding theory for a geometric random walk

3

6

5

7

8 9

m, t

S

0

S

0

uS

0

dS

2 0

u S

0

S

0

S

2 0

d S

3 0

u S

2 0

u dS

2 0

u d S

3 0

d S

0

uS

0

dS

m, T

final stock

prices S

Figure 7.2 Binomial tree (Jarrow–Rudd)

The most popular choice is to put u = d−1, giving the same proportional move up and down.

Writing u = d−1 = e , substituting in equations (7.5) and rejecting terms higher thanδt gives

= σ√δt The pseudo-probability of an up-move is then given by

p= e(r−q)δt− e−σ

√

δt

eσ√δt− e−σ√δt ≈ 1+ (r − q) δt −



1 − σ√δt +1

2σ2δt



1 + σ√δt +1

2σ2δt−1− σ√δt +1

2σ2δt

≈ 1

2+1 2

r − q − 1

2σ2

σ

√

Apart from its simplicity of form, this choice is popular because u = d−1 The effect of this

is that in Figure 7.2, S4= udS0= S0 In other words, the center of the network remains at a

constant S0 Compare this formula for p with the corresponding result for an arithmetic random

78

7.2 THE BINOMIAL NETWORK

walk given by equation (A2.7) An extra term 12σ2has appeared in the drift, which typically happens when we move from a normal distribution to a lognormal one

Final stock prices merely take the values

S0e−Nσ√δt , S0e−(N−1)σ√δt , , 0, , S0eN σ√δt

where N is the number of steps in the model.

(iii) Cox, Ross and Rubinstein: An alternative, popular arrangement of S u and S d is to start the

other way round: specify the pseudo-probability as p= 1

2 and derive a compatible pair u and

d Putting p=1

2 in equations (7.3) gives

S0

1

2u+1

2d

= F0 δt or S0(u + d) = 2 F0 δt

1

2



1−1 2



S2(u − d)2= F2

δt σ2δt or S0(u − d) = 2 F0 δt σ√δt

The equations on the right immediately yield

u = F0 δt

S0 (1+ σ√δt); d = F0 δt

The binomial network for these values is shown in Figure 7.3 The probability of an up-move

or a down-move at each node is now 12 The center line of the network is no longer horizontal, but slopes up At node 4 in the diagram the stock price is

Scenter,2δt = S4 = udS0= S0e(r −q)2δt(1− σ√δt)(1 + σ√δt)

= S0e(r −q)2δt

1−1

2σ22δt= S0e(r −q)2δt



1−1

2σ 2 T

N /2

 There are N steps altogether so thatδt = T/N, and the center line Scenterhas equation

Scenter,T = S0e(r −q)T



1−1

2σ 2 T

N /2

N /2

→ expr − q − 1

2σ2

S center

0

1

2

3

4

5

Final Stock Prices S m, T

S m, t

Figure 7.3 Binomial tree (Cox–Ross–Rubinstein)

79

Final stock prices now take values

S0(1+ σ√δt) Ne(r−q−1σ2)T , S0(1+ σ√δt) N−1(1− σ√δt) e (r−q−1σ2)T , ,

S0(1− σ√δt) Ne(r −q−1σ2)T

(iv) For completeness, we list a third discretization occasionally used:

u = exp{(r − q)δt + σ√δt}; d = exp{(r − q)δt − σ√δt}

Substituting in equation (7.5) and retaining only terms O[δt] gives p =1

2(1−1

2σ√δt) The center line of the grid now has the equation Scenter= S0e(r −q)t, which is the equation for the

forward rate (known as the forward curve).

(i) European Call: Jarrow–Rudd Method (u = d−1= eσ√d t ): consider the tree shown in

Figure 7.4 From the specification of the option and equation (7.6), the following parame-ters can be calculated:

With three stepsδt = 0.5/3 : F t t+ δt = S te(r−q)δt = 1.01005S t; e−rδt = 0.983

u= eσ√δt = 1.0851; d = e−σ√δt = 0.9216; p= F t t+ δt − S te−σ√δt

S teσ√δT − S te−σ√δt = 0.541 Using these u and d factors, we can start filling in the stock prices on the tree (shown just above each node) The intermediate values of S tare not really necessary for a European option, since the option payoff only depends on the stock price at maturity; however, they are shown for ease of understanding

The payoff values of the option are max[(S T − 100), 0] and are shown just below the final

nodes The option values at the next column of nodes to the left can be calculated as follows:

f (117 74, 4 months) = 0.983{0.541 × 27.76 + (1 − 0.541) × 8.51} = 18.609

f (100.00, 4 months) = 0.983{0.541 × 8.51 + (1 − 0.541) × 0.00} = 4.43

f (84.93, 4 months) = 0.983{0.541 × 0.00 + (1 − 0.541) × 0.00} = 0.00

100.00

108.51

117.74

127.76

92.16

100.00

84.93

108.51

92.16

78.27

27.76

8.51

0

0

18.61

4.53

0

11.95

2.41 7.44

6 months

t =o 2 months 4 months

S 0 = 100

X = 100

r = 10%

q = 4%

s = 20%

t = 0.5 year

Figure 7.4 European call: Jarrow–Rudd discretization

80

7.3 APPLICATIONS Continuing this process back to the first node (“rolling back through the tree”) finally gives a 6-month option value of 7.44 This may be compared to the Black Scholes value (equivalent

to an infinite number of steps) of 7.01 This price error is equivalent to using a volatility of 21.6% instead of 20% in the Black Scholes formula

(ii) European Call: Cox–Ross–Rubinstein Method ( p= 1

2): For purposes of comparison, we reprice the same option as in the last section, using a different discretization procedure Once again we haveδt = 0.5/3 and e −rδt = 0.983 but now we use p = 1

2 and equation (7.7),

so that

u= F t t+ δt

S t

(1 + σ√δt) = 1.093; d = F t t+ δt

S t

(1 − σ√δt) = 0.928

The tree is shown in Figure 7.5 This time, only the final stock prices are shown The procedure for rolling back through the tree is identical to that in the last section, with the

simplify-ing feature that p = (1 − p) = 1

2 The calculation for the top right-hand step in the diagram becomes

0.983 ×1

2 × (30.40 + 10.72) = 20.220

and so on through the tree For all intents and purposes, the final answer is identical to that of the last section (more precise numbers are 7.444 previously and 7.438 now)

130.40 30.40 110.72 10.72 94.00 0 79.81 0

20.22

5.27

0

12.53

2.59 7.44

6 months

0

S = 100

X = 100

r = 10%

q = 4%

s = 20%

t = 0.5 year

Figure 7.5 European call: Cox–Ross–Rubinstein discretization

(iii) Bushy Trees and Discrete Dividends: Suppose that instead of continuous dividends, the stock

paid one fixed, discrete dividend Q For purposes of illustration, we assume that it is paid the

instant before the second nodes The tree can be adjusted at these nodes by the shift shown in

Figure 7.6 S t, whatever its value, simply drops by the amount of the dividend Unfortunately,

this dislocates the entire tree as shown The tree is said to have become bushy.

Let us recall the original random walk on which the binomial model is based This is described in Appendix A.1, where we see that the tree is recombining by construction since

the up-steps U and down-steps D are additive In such a tree, the insertion of a constant Q would

not cause a dislocation since everything to the right of this point would move down by the same

amount This would have been the case if we had constructed the tree for x i = ln S i However,

81

d(uS - Q)

0

u(dS - Q)

0

S

0

uS

0

uS - Q

0

dS Q

Q

0

dS - Q

Figure 7.6 Discrete fixed dividend

we have constructed the tree for S i directly, so that the sizes of the up- and down-moves are

determined by the multiplicative factors u and d A discrete dividend must also be multiplicative

if the tree is to remain recombining Instead of a fixed discrete dividend we therefore use a

discrete dividend whose size is proportional to the value of S tat the node in question This is

illustrated in Figure 7.7, where the dividend is ku S0at the higher node where the stock price is

u S0, and kd S0at the lower node The effect of this on the following three nodes is immediately apparent: the tree recombines

0

S

0

uS

0

dS

0

uS (1 - k)

0

Q = kuS

0

Q = kdS

0

dS (1 - k)

0

duS (1 - k)

0

udS (1 - k)

Figure 7.7 Discrete proportional dividend

This proportional dividend assumption is implicit in the continuous dividend case, where each infinitesimal dividend in a periodδt is qS t δt, i.e proportional to S t

We return to the call option and discretization procedure of subsection (ii), except that instead

of a continuous q = 4% (i.e 2% over the 6-month period), there is a dividend of 2% × S t at

the second pair of nodes The parameters are similar to those of subsection (ii) with q= 0;

F t t+ δt = 102.020; p = 0.582; u = 1.0851; d = 0.9216; e rδt = 0.983.

The terminal values are calculated, taking into account the dividend as shown Rolling back through the tree shown in Figure 7.8 is exactly the same as before and nothing different needs

to be done at the dividend point; this was entirely handled by the adjustment in the stock price The initial value of the option works out to be 7.297 compared with 7.444 for the continuous dividend case This difference gradually closes as the number of steps in the model increases

At 25 steps it is only half as big

The calculation was repeated using a fixed dividend of 2 paid at the same point, so that the tree did not recombine It is not worth giving the details of the calculations, but the option

82

7.3 APPLICATIONS

100.00

108.51

106.34 92.16

88.51

115.38

98.00

83.24

125.20 25.20 106.34 6.34 90.32

76.71 0

0

(in 2 months)

0

S = 100

X = 100

r = 10%

Q = 4%

s = 20%

t = 0.5 year

Figure 7.8 European call: discrete proportional dividend

value is found to be 7.356 The difference is negligible, justifying the use of the proportional dividend model

(iv) American Options: The European call option could of course have been priced using the Black

Scholes model Binomial trees really come into their own when pricing American options

Consider an American put with X= 110 and the remaining parameters the same as for the European call of subsection (i); the same discretization procedure is used as in that section and the results are laid out in Figure 7.9

100.00

108.51

117.74

127.76

92.16

100.00

84.93

108.51

92.16

78.27

0

1.49

17.84

31.73

.67

8.85 10.00

23.81 25.07

4.87

16.63 17.84 10.64

0

S = 100

X = 110

r = 10%

q = 4%

s = 20%

t = 0.5 year

Figure 7.9 American put: Jarrow–Rudd discretization

The procedure starts the same as in subsection (i):

(A) Set up the tree and calculate the values of each S tand the terminal values of the option This time we need to put in the intermediate stock prices for reasons which become apparent below

(B) Calculate the terminal payoff values for the put option

(C) Roll back through the tree calculating the intermediate option values Starting at the top right-hand corner, we have

0.67 = e −rδt

( p × 0 + (1 − p) × 1.49)

83

(D) The next value in this column is

8.85 = e −rδt

( p × 1.49 + (1 − p) × 17.84) But an American put option at this point (S = 100, X = 110) could be exercised to give

a payoff of 10.00 The value of 8.85 must therefore be replaced by 10.00 Similarly, at the bottom node in this column, the exercise value must be used

(E) With these replacement values, the next column to the left is derived Once again, the bottom node is calculated as

16.63 = e −rδt

( p × 10.00 + (1 − p) × 25.07)

This is less than the exercise value and must be replaced by 17.84, the exercise value of the American option

(F) Finally a price of 10.64 is obtained for the option This compares with a value of 9.29 for

a similar European put

The essence of the matter is summed up in Figure 7.10 In the next chapter we will show that a binomial tree is mathematically equivalent to a numerical solution of the Black Scholes equation We saw in Section 6.1 that the price of an American option is only a solution of the Black Scholes equation in certain regions Below the exercise boundary, the value of the American put is simply its intrinsic (exercise) value:

American put price= f (S t , t); solution to BS equation; S tabove exercise boundary

X − S t ; not solution to BS equation; S tbelow exercise boundary

Option exercised here

t = 0

t = T

Exercise boundary

Figure 7.10 American puts

(v) While the pricing given in this section is useful for illustration, such a small number of steps

would never be used for a real-life pricing So what is the minimum number of steps needed

to price an option in the market?

While the answer to this depends on the specific option being priced, solutions are typically distributed as shown in Figure 7.11 The principle features are as follows:

(A) The solid appearance of the left-hand graph comes about because the answers obtained change more sharply in going from an odd number of steps to an even number than they

do between successive odd or even numbers of steps When the option price is plotted

84