Question

In: Statistics and Probability

Problem 2: Valuation on a Multiplicative Binomial Lattice This problem reviews some of the main ideas...

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.

  1. (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.)

Solutions

Expert Solution

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 , Su(1) = S(0)*u , where "u" is up factor

Node-C = it shows down price of stock , Sd(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 = Su(2) = S(1)*u

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

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


Related Solutions

Valuation on a Multiplicative Binomial Lattice This problem reviews some of the main ideas of valuation...
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)...
Problem 2: Valuation on a Multiplicative Binomial Lattice This problem reviews some of the main ideas...
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...
This problem reviews some of the main ideas of valuation on a binomial lattice and the...
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...
Can you complete this concept map that reviews some key ideas about species and speciation?
Can you complete this concept map that reviews some key ideas about species and speciation?
Discuss some of the main ideas and theories that expanded the body of knowledge on Trade...
Discuss some of the main ideas and theories that expanded the body of knowledge on Trade Theory over the period 1920 to 1995.
This problem will relate some of the ideas of microstates, ensemble, statisticalmechanical entropy, and our ideal...
This problem will relate some of the ideas of microstates, ensemble, statisticalmechanical entropy, and our ideal model of many identical oscillators, to intuitive notions of entropy and disorder in a very complex system – a protein. A protein is basically a long chain of amino acid residues that folds into a regular shape and performs a huge variety of biological functions. Biochemistry and biology students will already know a lot about proteins, you may know a lot after having taken...
Medicare is an even bigger problem and solving it is a political minefield. Some ideas privatize...
Medicare is an even bigger problem and solving it is a political minefield. Some ideas privatize as much as possible and encourage insurance only for catastrophic health bills to encourage a more thrifty approach to health care by seniors. This one is arguably the toughest fix of all and arguably needs to make health care as a whole more efficient and not just Medicare itself. How would you solve the national debt problem? Also explain which solution you feel is...
tell me some greatest fact and main ideas about west society to speak in serach paper...
tell me some greatest fact and main ideas about west society to speak in serach paper and essay? please I started to read but the imformarion.seems too broad? does not matter I have to look for the fact or ideas you probably mention. i want to have north to follow clearly thanks
1.Identify the main research problem. 2.Analyze the main underlying causes of the existing problem. 3.Establish the...
1.Identify the main research problem. 2.Analyze the main underlying causes of the existing problem. 3.Establish the cause-and-effect relations between the various aspects. 4.Formulate the best solutions to address the problem. CASE 1: ZAPPOS FACES COMPETITIVE CHALLENGES Zappos Faces Competitive Challenges Zappos, based in Las Vegas, is an online retailer with the initial goal of trying to be the best website for buying shoes by offering a wide variety of brands, styles, colors, sizes, and widths. The Zappos.com brand has grown...
what are the some main problem U.S franchisers encounter when attempting to expands their buisness in...
what are the some main problem U.S franchisers encounter when attempting to expands their buisness in a country such as china?
ADVERTISEMENT
ADVERTISEMENT
ADVERTISEMENT