Question

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.

3. (i) Using the replicating portfolio you determined in question 2, calculate the numerical value of πd, the 1-period-ahead down state price for the state where the price of a share of KAF stock makes a down move from period 0 to period 1. Also calculate the numerical value of πu, the 1-period-ahead up state price for the state where the price of a share of KAF stock makes an up move from period 0 to period 1. Explain your reasoning.

ii) Replicating portfolios, risk neutral probabilities, and state prices can each be used to calculate the market value in period 0 of any asset or portfolio of assets with payoffs in period 1 that are determined by the movement in the price of a share of KAF stock. Write down the formulas that connect the relevant risk neutral probabilities for period 1, pu and pd, and the 1-periodahead state prices, πu and πd. What are the numerical values of the risk neutral probabilities, pu and pd ?

iii)

Consider a portfolio that consists of the following three assets: two identical European call options each for one share of KAF stock that expire in period 1 and each have an exercise price of K = 120, and one European put option for one share of KAF stock that expires in period 1 and has an exercise price of K = 140.

What are the possible payoffs of this portfolio in period 1? Explain your reasoning.

Using the 1-period-ahead up and down state prices, πu and πd, write down a formula for the market value in period 0 of this portfolio. Also, write down a formula for the market value in period 0 of the portfolio in terms of the risk neutral probabilities, pu and pd. Using your formulas, calculate a numerical value for the market value of the portfolio in period 0. Briefly explain your answers.

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