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Chapter 8: Problem 11

Compare the prices of an American call and a European call with strike price \(X=120\) dollars expiring at time 2 on a stock with initial price \(S(0)=120\) dollars in a binomial model with \(u=0.2, d=-0.1\) and \(r=0.1\).

Short Answer

Expert verified
European call: 19.02 dollars, American call: 19.65 dollars.

Step by step solution

01

Define the Binomial Model Parameters

Identify the parameters provided in the exercise:- Initial stock price, S(0) = 120 dollars- Strike price, X = 120 dollars- Up factor, u = 0.2- Down factor, d = -0.1- Risk-free rate, r = 0.1- Expiry time, T = 2 periods
02

Calculate the Stock Prices at Each Node

Calculate the possible stock prices at each node in the binomial tree:At time 1:- Up move: \(S_1^u = S(0) \times (1+u) = 120 \times 1.2 = 144 \) dollars.- Down move: \(S_1^d = S(0) \times (1+d) = 120 \times 0.9 = 108\) dollars.At time 2: - Up鈥揢p move: \(S_2^{uu} = S_1^u \times (1+u) = 144 \times 1.2 = 172.8\) dollars.- Up鈥揇own move: \(S_2^{ud} = S_1^u \times (1+d) = 144 \times 0.9 = 129.6\) dollars.- Down鈥揇own move: \(S_2^{dd} = S_1^d \times (1+d) = 108 \times 0.9 = 97.2\) dollars.- Down鈥揢p move: \(S_2^{du} = S_1^d \times (1+u) = 108 \times 1.2 = 129.6\) dollars.
03

Calculate the Payoff at Maturity

Determine the payoff of the call options at maturity (time 2):For European call option:- Up-Up: \( \text{max}(S_2^{uu} - X, 0) = \text{max}(172.8 - 120, 0) = 52.8 \) dollars.- Up-Down or Down-Up: \( \text{max}(S_2^{ud} - X, 0) = \text{max}(129.6 - 120, 0) = 9.6 \) dollars.- Down-Down: \( \text{max}(S_2^{dd} - X, 0) = \text{max}(97.2 - 120, 0) = 0 \) dollars.
04

Calculate the Risk-neutral Probabilities

Calculate the risk-neutral probabilities using the formula: \(p = \frac{e^r - (1 + d)}{(1 + u) - (1 + d)} = \frac{e^{0.1} - 0.9}{1.2 - 0.9} \approx 0.578\) \(1 - p \approx 0.422\)
05

Calculate the Expected Payoff and Discount Backwards for European Call

Calculate the expected payoff at time 1 using risk-neutral probabilities and discount back to time 0:Expected payoff at time 1: \(E(C_2^u) = 0.578 \times 52.8 + 0.422 \times 9.6 = 32.16 \) dollars. \(E(C_2^d) = 0.578 \times 9.6 + 0.422 \times 0 = 5.55 \) dollars.Discounted back to time 0: \(C_1 = \frac{e^{-2r}}{2}\times\left(0.578\cdot52.8+0.422\cdot0.33\right)\times(1+p)\approx19.02\) dollars.
06

Determine the Early Exercise for American Call

Compare the value of holding the option vs the value of exercising early at each node:At time 1, the value to hold and exercise:\(C_1^u = \text{max}(32.16, 144 - 120) = 32.16 \) dollars.\(C_1^d = \text{max}(5.55, 108 - 120) = 5.55 \) dollars.At early-exercise, calculate the expected payoff:Discounted at time 0:\(C_0 = e^{-0.2r} \times \left(0.578 \times 32.16 + 0.422 \times 5.55 \right) \approx 19.65\).
07

Compare Prices

The price of the European call is 19.02 dollars, and the price of the American call is 19.65 dollars.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Binomial Model
The binomial model is a popular method for valuing options, providing a simple yet powerful framework that captures the changing dynamics of asset prices over time. It represents the possible future outcomes of the stock price upon each successive time period using a binomial tree structure. Here, each node in the tree shows a possible price of the stock, and each branch represents a potential move up or down in price.

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Most popular questions from this chapter

Compute the prices of European and American puts with exercise and strike price \(X=14\) dollars expiring at time 2 on a stock with \(S(0)=12\) dollars in a binomial model with \(u=0.1, d=-0.05\) and \(r=0.02\), assuming that a dividend of 2 dollars is paid at time 1 .

Find the initial value of the portfolio replicating a call option if proportional transaction costs are incurred whenever the underlying stock is sold. (No transaction costs apply when the stock is bought.) Compare this value with the case free of such costs. Assume that \(S(0)=X=100\) dollars, \(u=0.1, d=-0.1\) and \(r=0.05\), admitting transaction costs at \(c=2 \%\) (the seller receiving \(98 \%\) of the stock value).
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