Two and more binomial periods

IC: 5

Course: M339D/M389D - Intro to Financial Math

University of Texas at Austin

In-Class Assignment 5

Two and more binomial periods

5.1. Two binomial periods. 5.1.1. Binomial asset pricing. Recall the form of our two-period binomial tree:

Page: 1 of 6

Suu

Su

S0

Sud Sdu

Sd

Sdd

With the given up factor u and down factor d, we have that

Su = uS(0), Sd = dS(0), Suu = u2S(0), Sud = Sdu = udS(0), Sdd = d2S(0).

With the length of a single time-period denoted by h, the continuously-compounded, risk-free

interest rate r, and the continuous dividend yield , the risk-neutral probability of the stock price

taking a step up has the following expression:

Solution:

p

=

e(r-)h - d u-d

.

Now, we can follow the different paths that the stock price can take throught the binomial tree

and obtain the risk-neutral probabilities that the stock-price attains any of the final stock prices

available at the leaves of the tree.

Instructor: Milica C udina

IC: 5

Course: M339D/M389D - Intro to Financial Math

Page: 2 of 6

Under the risk-neutral probability, the stock-price at time T , i.e., the final stock price is a random variable S(T ) whose distribution can be written in a table as follows:

Stock price

Suu

Sud

Sdd

Risk-neutral probability of the price Solution: (p)2 Solution: 2p(1 - p) Solution: (1 - p)2

5.1.2. Pricing derivative securities. When we price European-style options, the first step is to figure out the payoffs at the different states-of-the-world, i.e., for different final asset prices available in the model. Generally speaking, If the payoff function of the derivative security is v, then we have

Vuu = v(Suu), Vud = v(Sud), Vdd = v(Sdd).

The payoff V (T ) is a random variable with support encompassing the above three values. The risk-neutral probabilities of reaching the three different payoffs are

Payoff

Vuu

Vud

Vdd

Risk-neutral probability of the payoff Solution: (p)2 Solution: 2p(1 - p) Solution: (1 - p)2

Remember that the risk-neutral pricing can be interpreted as the discounted expected payoff with:

(1) the discounting done with respect to the risk-neutral interest rate r, and (2) the expectation calculated under the risk-neutral probability measure. So, the risk-neutral pricing formula for the price of the derivative security at time-0 is Solution:

V (0) = e-rT E[V (T )] = e-rT (p)2Vuu + 2p(1 - p)Vud + (1 - p)2Vdd .

Problem 5.1. Consider a two-period binomial model for the stock-price movement over the following year. The current stock price is S(0) = 100, the up factor is given to be u = 1.3 and the down factor is d = 0.8 The stock pays no dividends. The continuously compounded risk-free interest rate is given to be 0.05.

Calculate the price VAC(0) of an asset call with exercise date in one year which pays one share of stock in case that the stock price exceeds $100.

Instructor: Milica C udina

IC: 5

Course: M339D/M389D - Intro to Financial Math

Page: 3 of 6

Solution: In this problem,

Suu = 1.32 ? 100 = 169, Sud = 1.3 ? 0.8 ? 100 = 104, Sdd < 100.

The risk-neutral probability is

So, the asset call has the price

p

=

e0.025 - u-d

d

0.45.

VAC (0) = e-0.05[0.452 ? 169 + 2 ? 0.45 ? 0.55 ? 104] = e-0.05 ? 85.7025 = 81.52.

As we have shown in class, we can construct the two-period derivative-security tree as well as its (dynamic) replicating portfolio. The form of the tree is

Vuu

Vu

V0

Vdu Vud

Vd

Vdd

5.2. More binomial periods. The stock-price binomial tree is constructed in the same way as above if we decide to split the time horizon T into n binomial periods, each one of length h = T /n. We set up our binomial asset-pricing model by positing u and d. The initial stock price S(0) can be observed.

While it is impossible to draw the complete multiperiod derivative-security tree, Figure 5.2 shows what it would look like in broad strokes:

The indexation of the possible payoffs is deliberate. The index itself stands for the number of times the stock-price took a step down in order to reach that particular payoff. We have that given the payoff function v, the payoff is V (T ) = v(S(T )). So, for any index k, we have

Vk = v(S(0)un-kdk). Recalling the binomial distribution, we realize that the payoff Vk is reached with the risk-neutral probability

Instructor: Milica C udina

IC: 5

Course: M339D/M389D - Intro to Financial Math Figure 1. The multiperiod derivative-security tree

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V0

V1

V2 V (0)

Vn Instructor: Milica C udina

IC: 5

Course: M339D/M389D - Intro to Financial Math

Page: 5 of 6

Solution:

n k

(p)n-k(1 - p)k

for all k = 0, . . . , n. We will not prove the risk-neutral pricing formula for European-style options,

although the proof is straightforward (especially if you took discrete mathematics and can still

remember mathematical induction). We have

Solution:

V (0) = e-rT

n

Vk

n k

(p)n-k(1 - p)k.

k=0

Problem 5.2. Let the continuously compounded interest rate be r = 10%. Assume that the initial price of a non-dividend-paying stock is $100 per share.

Consider a 5-period binomial model for the evolution of the stock price over the next year. Let u = 1.04 and d = 0.96.

(i) What is the price of a one-year, 100-strike cash call on the above asset? Solution: The payoff function is

v(s) = 1 if s > 100 0 if s 100

= I(100,)(s)

So, we can get the possible payoffs of the cash call in the above tree as follows: u5S(0) = 1.045 ? 100 = 121.67 V0 = 1,

u4dS(0) = 1.044 ? 0.96 ? 100 = 112.31 V1 = 1, u3d2S(0) = 1.043 ? 0.962 ? 100 = 103.67 V2 = 1.

The remaining possible final stock prices are below the threshold of $100. So, the payoffs at those final nodes are equal to zero. In fact,

VCC (0) = e-rT P[S(T ) > 100]

where P stands for the risk-neutral probability measure consistent with p. We can calculate p as

We get

p

=

e(0.1/5)-0.96 1.04 - 0.96

= 0.7525.

VCC (0) = e-0.1 (0.7525)5 + 5(0.7525)4(1 - 0.7525) + 10(0.7525)3(1 - 0.7525)2 = 0.8135.

(ii) What is the price of a one-year, at-the-money European call on the above asset? Solution: Using the final stock prices we calculate above we get these possible payoffs at the uppermost final nodes:

V0C = 21.67, V1C = 12.31, V2C = 3.67. The remaining payoffs are all zero as the call goes unexercised. The call price is VC (0) = e-0.1[21.67 ? 0.75255 + 12.31 ? 5(0.7525)4(1 - 0.7525) + 3.67 ? 10(0.7525)3(1 - 0.7525)2] = 10.0176.

Instructor: Milica C udina

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