two and more binomial periods · the stock-price binomial tree is constructed in the same way as...

IC: 5 Course: M339D/M389D - Intro to Financial Math Page: 1 of 6

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:

SH0L

Sd

Su

Suu

Sud=Sdu

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-freeinterest rate r, and the continuous dividend yield δ, the risk-neutral probability of the stock pricetaking a step up has the following expression:Solution:

p∗ =e(r−δ)h − du− d

.

Now, we can follow the different paths that the stock price can take throught the binomial treeand obtain the risk-neutral probabilities that the stock-price attains any of the final stock pricesavailable at the leaves of the tree.

Instructor: Milica Cudina


Under the risk-neutral probability, the stock-price at time T , i.e., the final stock price is a randomvariable 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 figureout the payoffs at the different states-of-the-world, i.e., for different final asset prices available inthe 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. Therisk-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 payoffwith:

(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 isSolution:

V (0) = e−rTE∗[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 follow-ing year. The current stock price is S(0) = 100, the up factor is given to be u = 1.3 and the downfactor is d = 0.8 The stock pays no dividends. The continuously compounded risk-free interest rateis given to be 0.05.

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



Solution: In this problem,

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

The risk-neutral probability is

p∗ =e0.025 − du− d

≈ 0.45.

So, the asset call has the price

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

VH0L

Vd

Vu

Vdd

Vdu=Vud

Vuu

5.2. More binomial periods. The stock-price binomial tree is constructed in the same way asabove 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) canbe observed.

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

The indexation of the possible payoffs is deliberate. The index itself stands for the number oftimes the stock-price took a step down in order to reach that particular payoff. We have that giventhe 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-neutralprobability



Figure 1. The multiperiod derivative-security tree

V (0)

V0

V1

V2

Vn



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 stillremember mathematical induction). We haveSolution:

V (0) = e−rTn∑k=0

Vk

(n

k

)(p∗)n−k(1− p∗)k.

Problem 5.2. Let the continuously compounded interest rate be r = 10%. Assume that the initialprice 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. Letu = 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 payoffsat those final nodes are equal to zero. In fact,

VCC(0) = e−rTP∗[S(T ) > 100]

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

p∗ =e(0.1/5)−0.96

1.04− 0.96= 0.7525.

We get

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 atthe uppermost final nodes:

V C0 = 21.67, V C

1 = 12.31, V C2 = 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.



Problem 5.3. Let the continuously compounded interest rate be r = 11% per annum. Assumethat the initial price of a non-dividend-paying stock is $100 per share.

Consider an 11−period binomial model for the evolution of the stock price over the next year.Let u = 1.04 and d = 0.96.

What is the price of a one-year, 130-strike cash call on the above asset?

Solution: Calculating the possible final stock prices is inefficient. What we do is find the “critical”value of the possible down-steps that the stock price can take as it moves through the tree. At thisnumber of down-steps, the payoff is still one, if there is even one more step down in the stock’spath through the tree, the payoff is zero. Symbolically, we are looking for the largest k = 0, . . . , 11such that

un−kdkS(0) > K ⇔(d

u

)k>

K

S(0)un⇔ k <

ln(K/S(0))− n ln(u)

ln(d/u).

Taking into account the given data, the above inequality becomes k < 2.11. So, only the highestthree stock prices yield a $1 payoff. We need to calculate the risk-neutral probability of the stockprice going up in a single step:

p∗ =e0.11/11 − 0.96

1.04− 0.96= 0.6256.

Finally, the cash-call price is

VCC(0) = e−0.11[(p∗)11 + 11(p∗)10(1− p∗) + 55(p∗)9(1− p∗)2] = 0.1404.


two and more binomial periods · the stock-price binomial tree is constructed in the same way as...

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