The price S(t) of a share follows the geometric Brownian motion S(t) = Se µt+σW(t) . The price of the share at t = 0 is S = £25 and µ = 0.1. The continuously compounded interest rate is 8%. However, the volatility σ is not known. (a) Explain the definition of implied volatility. (b) A European call option on the above share with the strike price K = £23 and expiration time of 6 months has the market price £3.1. Does the equation defining the implied volatility have a solution? [6] Hint. You are not supposed to solve the equation defining the volatility to answer this question

The price S(t) of a share follows the geometric Brownian motion S(t) = Se µt+σW(t) . The price of the share at t = 0 is S = £25 and µ = 0.1. The continuously compounded interest rate is 8%. However, the volatility σ is not known. (a) Explain the definition of implied volatility. (b) A European call option on the above share with the strike price K = £23 and expiration time of 6 months has the market price £3.1. Does the equation defining the implied volatility have a solution? [6] Hint. You are not supposed to solve the equation defining the volatility to answer this question

Intermediate Financial Management (MindTap Course List)

13th Edition

ISBN:9781337395083

Author:Eugene F. Brigham, Phillip R. Daves

Publisher:Eugene F. Brigham, Phillip R. Daves

Chapter8: Basic Stock Valuation

Section: Chapter Questions

Problem 10P

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Calculate Implied Volatility: • A European call with strike price 195 and one year to maturity is worth $17.00. The underlying asset is trading at $200 per share. The risk-free rate is 0.5%. The Black Scholes Merton Model implied volatility is _______. •A European put with strike price 195 and one year to maturity is worth $28.33. The underlying asset is trading at $200 per share. The risk-free rate is 0.5%. The Black Scholes Merton Model implied volatility is _______
Suppose a non-dividend paying stock is trading at $175 per share and has a volatility of 20%. What is the fair price of a 3-month European call option with a strike price of $190 per share using 1 binomial period? Assume the risk-free rate is 1%. Round to the nearest $0.01.
Suppose a non-dividend paying stock is trading at $175 per share and has a volatility of 45%. What is the fair price of a 9-month European put option with a strike price of $190 per share using 1 binomial period? Assume the risk-free rate is 1%. Round to the nearest $0.01.
The price of an American call on a non-dividend-paying stock is $4. The stock price is $31, the strike price is $30, and the expiration date is in 3 months. The risk-free interest rate is 8%. The upper and lower bounds for the price of an American put on the same stock with the same strike price and expiration date is $3.00 and $2.41, respectively. Explain carefully the arbitrage opportunities if the American put price is greater than the calculated upper bound.
Using put-call parity formula, derive expressions for the lower bounds for European call and put options. What is a lower bound for the price of (i) a three-month call option on a non-dividend-paying stock when the stock price is R860, the strike price is R760, and the risk-free interest rate is 10% per annum? (ii) a three-month European put option on a non-dividend-paying stock when the stock price is R500, the strike price is R610, and the discrete risk-free interest rate is 9% per annum?
In a financial market a stock is traded with a current price of 50. Next period the price of the stock can either go up with 30 per cent or go down with 25 per cent. Risk-free debt is available with an interest rate of 8 per cent. Also traded are European options on the stock with an exercise price of 45 and a time to maturity of 1, i.e. they mature next period. Does put-call parity hold? Explain.
Assume the Black-Scholes framework. Let S be a stock such that S(0) = 21, the dividend rate is δ = 0.02, the risk free rate is r = 0.05, and the volatility is σ = 0.2. (a) Calculate the expected payoff of a 6 month call with strike price 17. (b) Calculate the cost of such a call. (c) Calculate the cost of a 6 month put with strike price 17. (d) Calculate the price of a derivative that pays |S(0.5) − 17| in six months
Let there exist a non-dividend-paying stock, A, which has a current price of $80. Let there exist another stock, B, which has a current price of $100. Stock B will continuously pay dividends at a dividend yield of x. A one-year European exchange call option with underlying asset A and strike asset B is sold for $5, while a one-year European exchange put option with underlying asset A and strike asset B is sold for $15. 12% is the continuously compounded, risk-free interest rate. Calculate the value of stock B's dividend yield, x.
Consider a European put and a European call option which are both written on a non-dividend paying stock, have the same strike price K = £80 and expire in T = 2 months. These options are trading for p = £21 and c = £30.80, respectively. The underlying stock price is S0 = £90. The continuously compounded risk-free rate of interest is r = 10% per annum. What is the present value of the arbitrage profit? Please explain your answer and show your workings. In your response, please show all cash flows (both today and at expiration) and explain why this is an arbitrage (i.e. risk-less) profit.
The price of a share follows the geometric Brownian motion with parameters µ = 0.2 and σ = 0.18. Presently, the share’s price is £38. Consider a call option having one year until its expiration time and having a strike price of £40. The continuously compounded interest rate is 5%. (a) What is the risk-neutral price C of this call option?
Consider a market where the assumptions behind the Black-Scholes model hold. A non-dividend paying share is currently priced at $300. A put option is available on the share with a strike price of $320 and one year to expiry. Volatility of 15% and the risk-free rate is 5% per annum. What would be the fair price of the put option.
The price of an American call on a non-dividend paying stock is $2. The stock price is $15.50, the strike price is $15, the expiration date is in 6 months, and the risk free interest rate is 4% continuously compounded. Which of the following is correct, regarding the option premium of an American put on the stock with a strike price of $15 and an expiration date in 6 months? a. The upper bound for the price of the American put is $14.70 and the lower bound is $0 b. The lower bound for the price of the American put is 0 and the upper bound is $1.50 c. The lower bound for the price of the American put is $1.20 and the upper bound is $1.50