IFM

Let's review the assumptions for the Black-Scholes formula.

The Black-Scholes formula makes several assumptions about how the underlying stock is distributed: Continuously compounded returns on the stock are normally distributed and independent over time. There are no sudden jumps in the stock price. Volatility is known and constant. Future dividends are known. In addition, the Black-Scholes formula makes several assumptions about the economic environment: The risk-free rate is known and constant. There are no taxes or transaction costs. Short-selling is allowed at no cost. Investors can borrow and lend at the risk-free rate.

Which of the following is an assumption of the Black-Scholes framework? A. Continuously compounded returns on the stock are lognormally distributed and independent over time. B. The borrowing rate is known but not a constant. C. The yield curve is upward sloping. D. Short-selling is allowed at a cost that is known and constant. E. The market is frictionless.

Statement A is false.Continuously compounded returns on the stock are normally distributed and independent over time. It follows that stock prices are lognormally distributed. Statement B is false.Investors can borrow or lend at the risk-free rate, and the risk-free rate is known and constant. Thus, the borrowing rate is known and constant. Statement C is false.The risk-free rate is known and constant. Thus, the yield curve is flat; it is neither upward sloping nor downward sloping. Statement D is false.Short-selling is allowed at no cost. Statement E is true.A frictionless market is a market with no transaction costs. The Black-Scholes framework assumes the market has no transaction costs.

Which one of the following is an assumption of the Black-Scholes option pricing model? A. Stock prices are normally distributed. B. Stock price volatility is a constant. C. Changes in stock price are log-normally distributed. D. All transaction costs are included in stock returns. E. The risk-free interest rate is a random variable.

Statement A is false.Under the Black-Scholes framework, stock returns (continuously compounded), not stock prices, are normally distributed. Statement B is true.The stock price volatility is constant, and calculated as the standard deviation of the continuously compounded returns. Statement C is false.The ratios, not changes, of stock prices are lognormally distributed. Mathematically, S(t)S(0)S(t)S(0) is lognormally distributed, which also means lnS(t)S(0)ln⁡S(t)S(0) is normally distributed. Statement D is false.The Black-Scholes option pricing model assumes that there are no transaction costs. Statement E is false.Like the stock price volatility, the risk-free interest rate is a constant, not a random variable.

Assume all options are of the European or American type. Determine which of the following statements about options is true. A. Naked writing is the practice of selling options and taking an offsetting position in the underlying asset. B. A covered put involves taking a short position in an asset together with a long put on the same asset. C. An American-style put option must have a longer time to expiration than an otherwise equivalent European put option. D. A short position in a call or put option that is out-of-the-money at expiration always has a positive profit. E. A long position in a call or put option that is in-the-money at expiration always has a positive profit.

(A) is false. Naked writing is the practice of selling options and and not taking an offsetting position in the underlying asset. (B) is false. A covered put involves writing a put while taking a short position in the underlying asset. (C) is false. The time to expiration of an American-style put option is not bound by the time to expiration of an otherwise equivalent European put option. (E) is false. A long position in an in-the-money call or put option at expiration may have a negative profit at expiration if the accumulated value of the premium is greater than the payoff of the option. (D) is true. An out-of-the-money call or put option has a payoff of zero. Call and put option writers receive premiums at time 0 for writing these options; as a result, the accumulated value of the premium plus the option payoff of 00 will be positive. Thus, the correct answer choice is (D).

Determine which of the following statements about options is true. A. Naked writing is the practice of buying options without taking an offsetting position in the underlying asset. B. A covered call involves taking a long position in an asset together with a written call on the same asset. C. An American style option can only be exercised during specified periods, but not for the entire life of the option. D. A Bermudan style option allows the buyer the right to exercise at any time during the life of the option. E. An in-the-money option is one which would have a positive profit if exercised immediately.

A) is false.Naked writing involves selling, not buying options.(B) is true.When an option writer sells a call and "covers" it by longing the underlying asset, the investor has created a covered call. Short call + Long asset=Write a covered callShort call + Long asset=Write a covered call (C) is false.An American option can be exercised at any time.(D) is false.A Bermudan option can only be exercised during specific periods.(E) is false.Being in-the-money means there is a positive payoff, not necessarily a positive profit.Thus, the correct answer choice is (B).

Determine which of the following statements regarding comparing options with respect to style and maturity is TRUE: 1. An American call option's price cannot exceed the prepaid forward price of the underlying asset. 2. A European put option's price must be at least as great as the present value of the strike price. 3. An American put option on a nondividend-paying stock should never be exercised prior to expiration. 4. A European call with more time to expiration is at least as valuable as an otherwise identical call with less time to expiration. 5. An American put with more time to expiration is at least as valuable as an otherwise identical put with less time to expiration. Grade & Review

An American put with more time to expiration is at least as valuable as an otherwise identical put with less time to expiration.

Professor Merton wishes to use the Black-Scholes formula to price a derivative, but would like to relax a few of its restrictions. Determine which of the following statements is most likely FALSE. A. He could adjust the risk-free rate to a predetermined rate. B. He could adjust the volatility according the economic environment. C. He could adjust the present value of the dividends to reflect decreasing future cash flows. D. He could adjust the stock price in a discrete manner to account for any earning announcements. E. He could adjust the volatility and interest rate to vary over time in a known way.

As Derivatives Markets states, "with a small change in the [Black-Scholes] formula, we can permit the volatility and interest to vary over time in a known way" (McDonald, p. 353, 3rd Ed.). Thus, (A), (B), and (E) are true. Answer choice (C) is true because we can always adjust the present value of the dividends and then recalculate FP(S)FP(S) and finally the option price. (D) is false. The Black-Scholes model assumes that continuously compounded returns are normally distributed with no "jumps" in the stock price.

A binomial tree is used to model stock prices. As the number of periods in the tree increases, which distribution of stock price will the binomial tree approximate? A. Uniform B. Normal C. Lognormal D. Exponential E. None of (A), (B), (C), and (D) are correct.

As the number of periods in the tree increases, the time interval for each period decreases. Then, the continuously compounded returns are (approximately) normally distributed, and the stock price is lognormally distributed. So the answer is C.

Which of the following statements is/are true about exotic options? I. Average price Asian options are worth less than or equal to the otherwise equivalent standard European options. II. Barrier options are path-dependent. III. The payoff of a gap option must be non-negative. IV. A compound "put-on-call" option gives the option holder the option to buy a put. A. I and II only B. II and III only C. II and IV only D. I, II and III only E. The correct answer is not given by (A), (B), (C), or (D).

Correct answer is A Statement I is true.The prices of average price Asian options are subject to the averaging effect which reduced volatility. A lower volatility leads to a lower premium. Statement II is true.Barrier options have payoffs that depend on whether or not the price of the underlying asset reaches a pre-specified barrier. This is determined by the path of the underlying asset price. Statement III is false.Exercising a gap option is not optional. A gap option must be exercised when the trigger price is hit, even if it results in a negative payoff. Negative payoffs are possible for a gap call option when the trigger price is less than the strike price. Negative payoffs are also possible for a gap put option when the trigger price is greater than the strike price. Statement IV is false.A compound "put-on-call" option is an option to sell a call.

A binomial tree is used to model stock prices. As the number of periods in the tree increases, which distribution of stock price will the binomial tree approximate? A. Uniform B. Normal C. Lognormal D. Exponential E. None of (A), (B), (C), and (D) are correct.

Correct answer is C. As the number of periods in the tree increases, the time interval for each period decreases. Then, the continuously compounded returns are (approximately) normally distributed, and the stock price is lognormally distributed.

Which of the following effects are correct on the price of a stock option? (Choose one, none, or a combo) I. The premiums would not decrease if the options were American rather than European. II. For European put, the premiums increase when the stock price increases. III. For American call, the premiums increase when the strike price increases.

I only

Determine which one of the following is an assumption of the Black-Scholes option pricing model? A. Continuously compounded returns on the stock are dependent over time. B. The volatility of continuously compounded returns is an unknown constant. C. The cost to short-sell is greater than 0. D. The yield curve is upward sloping. E. It is possible to borrow at the risk-free interest rate.

Statement A is false. Continuously compounded returns on the stock are independent over time. Statement B is false. The volatility of continuously compounded returns is known and constant. Statement C is false. It is possible to short-sell costlessly. Statement D is false. The risk-free rate is known and constant, implying a flat yield curve. Statement E is correct.

Determine which of the following statements is TRUE. A. The value of a European call can never exceed the strike price. B. The value of a European put can never exceed the stock price. C. The value of a European call on a stock with dividends (with time-to-maturity T2T2) is always at least as great as the value of an otherwise equivalent European call (with time-to-maturity T1T1), where T1≤T2T1≤T2. D. The value of a European call on a nondividend-paying stock (with time-to-maturity T2T2) is always at least as great as the value of an otherwise equivalent European call (with time-to-maturity T1T1), where T1≤T2T1≤T2. E. The value of a European put option (with time-to-maturity T2T2) is always at least as great as the value of an otherwise equivalent European put option (with time-to-maturity T1T1), where T1≤T2T1≤T2.

Statement A is false. The value of a European call can never exceed the stock price. In general: S≥CAmer(S,K,T)≥CEur(S,K,T)≥max[0,FP0,T(S)−Ke−rT]S≥CAmer(S,K,T)≥CEur(S,K,T)≥max[0,F0,TP(S)−Ke−rT] Statement B is false. The value of a European put can never exceed the strike price. In general: K≥PAmer(S,K,T)≥PEur(S,K,T)≥max[0,Ke−rT−FP0,T(S)]K≥PAmer(S,K,T)≥PEur(S,K,T)≥max[0,Ke−rT−F0,TP(S)] Statements C and E are false. Statement D is true. A European option with a longer time to expiration on a nondividend-paying stock must be at least as valuable as a European option with a shorter time to expiration. Although this property is also generally true for European call options on dividend-paying stocks and European puts, there are two exceptions: Exception 1: Consider a dividend-paying stock that will pay a liquidating dividend in 99 months. A liquidating dividend means that the company will sell all of its assets and close its business. After the liquidating dividend is paid, the stock price falls to $0$0. Consider a 6-month and a 1-year European call on the stock. For the 1-year European call option, since it will never have a positive payoff after the liquidating dividend is paid, the option will expire worthless. For the 6-month European call option, the option can have a positive payoff before it expires, and thus it can expire in-the-money. Thus, a European call option with a longer time to expiration on a dividend-paying stock may not be as valuable as a European option with a shorter time to expiration. Exception 2: When a company declares bankruptcy, its stock becomes worthless, and a European put option on the stock will have a payoff equal to the strike price, KK. The value of the European put option is then the present value of the strike price, Ke−rTKe−rT. The shorter the time to expiration, the less we discount, and thus the more the option is worth.

Which of the following statements regarding Asian options is/are true? The value of a geometric average price Asian call option is always less than or equal to the value of an otherwise equivalent arithmetic average price Asian call option. As the number of averaging periods increases, the price of an average strike Asian call option increases. As the number of averaging periods increases, the price of an average price Asian call option increases. I only II only III only I and II only I and III only

Statement I is true. The geometric average is always less than or equal to the arithmetic average. Thus, the value of a geometric average price Asian call option is always less than the value of an otherwise equivalent arithmetic average price Asian call option. Statement II is true. For average strike options, as the frequency of sampling (N) increases, the option's value increases. As NN increases, it is more likely that the average stock price, S¯S¯, will be further away from the final stock price, S(T)S(T). As a result, the value of an average strike Asian option increases as NN increases. Statement III is false. For average price options, as the frequency of sampling (N)(N) increases, the option's value decreases. As NN increases, the average stock price, S¯S¯, has less volatility, and this makes it less likely to have a large payoff. As a result, the value of an average price Asian option decreases as NN increases.

Determine which of the following statements is TRUE regarding the forward price of a stock. A. As the dividend yield decreases, the forward price increases. B. As the risk-free interest rate decreases, the forward price increases. C. As the current stock price decreases, the forward price increases. D. Assuming the risk-free interest rate is greater than the dividend yield, as the time until maturity decreases, the forward price increases. E. The expected stock price is lower than the forward price.

The forward price for stock that pays continuous dividends is: F0,T=S0e(r−δ)TF0,T=S0e(r−δ)T Note that: As δδ decreases, F0,TF0,T increases. Thus, answer choice (A) is true. As rr decreases, F0,TF0,T decreases. Thus, answer choice (B) is false. As S0S0 decreases, F0,TF0,T decreases. Thus, answer choice (C) is false. If r>δr>δ, as TT decreases, F0,TF0,T decreases. Thus, answer choice (D) is false. The expected stock price is E[ST]=S0e(α−δ)TE[ST]=S0e(α−δ)T. The forward price is F0,T=S0e(r−δ)TF0,T=S0e(r−δ)T. αα is the expected return on the stock. rr is the risk-free return. Rational investors would require a higher return than the risk-free rate to compensate them for taking risk participating in a risky investment such as stock. For that reason, α>rα>r. Thus, the expected stock price should be higher than the forward price. Answer choice (E) is false.

Consider a European call option and a European put option on a nondividend-paying stock. You are given: The continuously compounded risk-free interest rate is 8%. Both the call option and the put option will expire in 2 years. Both the call option and the put option have a strike price of 100. There are 365 days in a year. An investor creates a portfolio by purchasing one share of the stock, writing one call option, and longing one put option. Calculate the value of the investor's portfolio 14 days from now. 85.21 85.48 92.17 100.00 It cannot be determined from the information given above.

The investor has created a synthetic treasury, as shown by rearranging put-call parity for a nondividend-paying stock: C(K)−P(K)=S(0)−Ke−rT ⇒Ke−rT=S(0)−C(K)+P(K) Thus, the portfolio will grow at the risk-free rate of 8% for 14 days: [Ke−rT]ert=[100e−0.08(2)]e0.08(14/365)=85.48