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19 Dec 2018

Consider a stock with a current price of P = $27. Suppose that over the next 6 months the stock price will either go up by a factor of 1.41 or down by a factor of 0.71. Consider a call option on the stock with a strike price of $25 which expires in 6 months. The risk-free rate is 6%.

1. Using the binomial model, what are the ending values of the stock price? What are the payoffs of the call option?

2. Suppose you write 1 call option and buy Ns shares of stock. How many shares must you buy to create a portfolio with a riskless payoff (which is called a hedge portfolio)? What is the payoff of the portfolio?

d. 3. What is the present value of the hedge portfolio’s riskless payoff? What is the value of the call option?

d. 4. What is a replicating portfolio? What is arbitrage?

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Patrina Schowalter
Patrina SchowalterLv2
22 Dec 2018

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Related questions

The current price of a non-dividend-paying stock XYZ is $30. The annual volatility of XYZ price is 20%. Suppose that you have a six-month short call position with the strike price of $32. Assume the continuously compounding risk-free rate is 5%.

A) Using risk-less portfolio approach, what long position in the XYZ stock is necessary to hedge the short call position? (Calculate the number of shares you must purchase to hedge one short call)

Strategy (portfolio you use to replicate the option behavior), and Calculations:

B) What is the value the call option?

Calculations:

C) What long position in the stock s necessary to hedge a long put option when the strike price is $32. Calculate the number of shares you purchase.

Strategy (portfolio you use to replicate the option behavior), and Calculations:

D) What is the value of the put option?

Calculations:

Question 3

Consider a stock currently worth $80 (S = 80) that can go up or down by 25 percent per period. The exercise price is $80 (X = $80) and the risk-free rate is 5 percent (r = .05). The option will expire at the next period (t = 1). You try to find a theoretically fair value of the call using a one-period binomial option pricing model. Answer the following questions. (Hint: See Ch 16 Teaching Notes and Excel Spreadsheet)

Compute the u and d in the binomial option pricing model such that the stock prices at the next period are Su and Sd.

What are two possible stock prices at the next period?

What are the intrinsic values at expiration of a European call option if the stock goes up and down?

Compute two values at expiration of a European call option (Cu and Cd)?

Compute the binomial probability (p)?

Find the theoretical fair value of the call option today using the one-period binomial pricing model.

Compute the riskless hedge ratio (h).

In order to construct a riskless hedge position by buying stocks and selling 1,000 calls using the riskless hedge ratio (h). How many stocks should you buy at what price? At what price should you sell the calls?

Compute the initial value of the riskless hedge position (V0).

Compute the market values of the riskless hedge position at the next period if the stock goes up (Vu) and goes down (Vd).

Find the rate of return on the riskless hedge position (R).

A European call option has a strike price of $20 and an expiration date in six months. The premium for the call option is $5. The current stock price is $25. The risk-free rate is 2% per annum with continuous compounding. What is the payoff to the portfolio, short selling the stock, lending $19.80 and buying a call option? (Hint: fill in the table below.)

Value of ST

Payoff

ST ≤ 20

ST > 20

How much do you pay for (or receive with) this portfolio at date 0?

Is there an arbitrage opportunity?

If there is an arbitrage opportunity, then answer the following:

What is the minimum profit, expressed as a present value?

Will investors trade to exploit the opportunity?

If they will trade to exploit the opportunity, explain why security prices change and describe how security prices change.

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