Question

Problem 2: Valuation on a Multiplicative Binomial Lattice

This problem reviews some of the main ideas of valuation on a binomial lattice and the properties of put and call options. You may wish to review the relevant lecture material and readings.

Suppose that the price of a share of KAF stock is S(0) = £120 in period 0. At the beginning of period 1, the price of a share can either move upward to S(1) = u S(0) or downward to S(1) = d S(0). Suppose that u = 4/3 = 1.333 and d = 3/4 = .75, so that S(1) = u S(0) = £160 after an up move and S(1) = d S(0) = £90 after a down move. Suppose that the probability of an up move is p = 0.5.

Similarly, suppose that, at the beginning of period 2, the share price either moves up or down by the same multiplicative factors and with the same probability (0.5) of an up move. (If the probability of an up move in a period is 0.5, then the probability of a down move in a period is also 0.5.) Hence, if the share price in period 1 is S(1), then the share price at the beginning of period 2 is either S(2) = u S(1) = 4/3 S(1) or S(2) = d S(1) = 3/4 S(1).

For simplicity, suppose that a period is a year, and let the riskless interest rate be r = .12, that is, 12% per period.

2. (i) Briefly explain what is meant by an Arrow-Debreu 1-period-ahead down security and the 1-period-ahead down state price.

(ii) Write down two equations that describe a replicating portfolio of KAF shares and riskless bonds that has a payoff of 1 in period 1 if the price of a KAF share moves downward at the beginning of period 1 and has a payoff of 0 otherwise. Briefly explain your reasoning.

Calculate the number of shares of KAF stock and the amount of riskless bonds in the replicating portfolio.

(Recall that buying riskless bonds is equivalent to lending money at the riskless rate of interest and selling riskless bonds short is equivalent to borrowing money at the riskless rate. A negative number of shares in a portfolio corresponds to selling those shares short.)

Answer 1

2 Stock Price (SO 3 up factor 4 down factor 120 1.3333 0.75 0.5 0.5 ul 6 p(d Binomial lattice for Share Price 10 Period-0 Period-1 Period-2 12 213.3333333 16 17 18 19 20 21 160 120 120 67.5 23 24

Formula reference -

2 Stock Price (SO 3 up factor 4 down factor 120 -4/3 3/4 0.5 6 p(d 1-B5 Binomial lattice for Share Price 9 10 Period-0 Period-1 Period-2 12 B16"ВЗ 16 17 18B2 19 20 21 -A18"ВЗ B20"ВЗ A18 B4 B20 B4 23

• Binomial option Pricing Model

In simple words, as binomial means consisting of two terms, binomial pricing model suggest for each period a stock can move only two direction either up or down and accordingly it determines the stock prices at each subsequent period.

For determination of stock prices at each period under binomial pricing model, we must have following factors

• Up factor (u) = factor for rise in price
• Down factor(d) = factor for fall in price

For determination of option value under binomial pricing model, we must have following factors -

• Probability (P) = Probability for rise in price
• (1-P) = Probability for fall in price
• r = risk free interest rate

Please refer to below diagram to understand how stock price calculated at each node.

ご2 S(0)

at time =0

Node A = it shows current stock Price S(0)

from this node Stock Price may move to up or down at time=1

At time = 1

Node-B = it shows up price of stock , S_u(1) = S(0)*u , where "u" is up factor

Node-C = it shows down price of stock , S_d(1) = S(0)*d , where "d" is down factor

Similarly, at time=2 , we can calculate the price of stock at Node - D,E & F.

Node-D = S_u(2) = S(1)*u

Node-E = Sd(2) = Su(1)*d

Node-F = Sd(2) = Sd(1)*d