Question

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.

. There are three possible share prices in period 2: S(2) = 213.333, S(2) = 120, and S(2) = 67.5.

(i) How many price paths on your multiplicative binomial lattice lead to each of these prices in period 2?

(ii) What are the numerical values of the risk neutral probabilities associated with each of the possible values for the share price, S(2), in period 2? What are the numerical values of the Arrow-Debreu state prices associated with each possible value of S(2) in period 2? Briefly explain your answers.

(iii) What is the price in period 0 of a security that pays 80 pounds in period 2 if the share price in period 2 is either S(2) = 213.333 or S(2) = 67.5, and otherwise pays nothing? Explain your reasoning.

Answer 1

Formula reference -

• 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.

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