Derivative Pricing 1st Edition Ambrose Lo - Solutions

Assume the Black-Scholes framework.Consider a 95-strike 9-month European call option on an S&V 150 futures contract which matures one year from now. You are given:(i) The current price of the S&V 150 index is 100.(ii) The volatility of the S&V 150 index price is 30%.(iii) The volatility of the
Assume the Black-Scholes framework. Which of the following graphs best represents the relationship between the delta of a deep out-of-the-money European call option on a nondividend-paying stock and the time to maturity T? (A) Delta 1 0 (C) Delta 1 (E) Delta 1 0 THERITE T 6 T (B) Delta 1 0 (D)
Assume the Black-Scholes framework.Consider a 6-month 90-strike European put option on a futures contract. You are given:(i) The price of the underlying futures contract is 95.(ii) The delta of the put option is −0.3382.(iii) The continuously compounded risk-free interest rate is 3%.(iv) The
Assume the Black-Scholes framework. Consider European call and put options on a stock that pays dividends continuously at a rateproportional to its price.Determine the signs of the following twelve Greeks:Call delta, call gamma, call theta, call vega, call rho, call psi,put delta, put gamma, put
Determine the limiting value of gamma as we approach maturity (i.e., find limT→0Γ) for each of the following options:(a) A deep in-the-money European call option(b) A deep out-of-the-money European call option(c) An at-the-money European call option For each option, provide an intuitive
(a) If the strike price of a European call option decreases, then how will the price of the call change, holding everything else constant?(b) Verify your conclusion in part (a) by general reasoning.(c) Verify your conclusion in part (a) by the Black-Scholes formula.
Assume the Black-Scholes framework. Consider a long K-strike European straddle. For each of delta and gamma, discuss intuitively how the option Greek of the straddle varies with S, the current stock price.
Youcompute the current gamma for a 50-60 bull spread with the following information:(i) The continuously compounded risk-free interest rate is 5%.(ii) The underlying stock pays no dividends.(iii) The current stock price is $50 per share.(iv) The stock’s volatility is 20%.(v) The time to
Assume the Black-Scholes framework. You are given:(i) The current dollar/euro exchange rate is 1.2.(ii) The continuously compounded risk-free interest rate in the United States is 2%.(iii) The continuously compounded risk-free interest rate in Europe is 3%.(iv) The volatility of the dollar/euro
Assume the Black-Scholes framework. You are given:(i) The current stock price is 82.(ii) The stock’s volatility is 30%.(iii) The stock pays dividends continuously at a rate proportional to its price. The dividendyield is 3%.(iv) The continuously compounded risk-free interest rate is 8%.Calculate
Assume the Black-Scholes framework.Consider a one-year at-the-money European put option on a nondividend-paying stock.You are given:(i) The ratio of the current put option price to the current stock price is 0.073445.(ii) The current put-option elasticity is −5.941861.(iii) The continuously
Assume the Black-Scholes framework. Consider a portfolio consisting of three European options, X, Y, and Z, on the same stock. You are given:Calculate the elasticity of the portfolio. Option price Option delta Option elasticity X 6.8268 5.6496 Y ? -0.4269 -6.8755 Z 1.9299 0.3537 9.1627
Assume the Black-Scholes framework. For a 3-month 32-strike European straddle on a stock, you are given:(i) The stock currently sells for $30.(ii) The stock’s volatility is 30%.(iii) The stock pays dividends continuously at a rate proportional to its price. The dividend yield is 2%.(iv) The
Assume the Black-Scholes framework. For t ≥ 0, let S(t) be the time-t price of a stock.Consider a 9-month European contingent claim on the stock. You are given:(i) The stock’s volatility is 35%.(ii) The stock pays dividends continuously at a rate proportional to its price. The dividend yield is
Assume the Black-Scholes framework. For t ≥ 0, let S(t) be the time-t price of a nondividend-paying stock. You are given:(i) S(0) = 100.(ii) Var[ln S(t)] = 0.16t, for t ≥ 0.(iii) The continuously compounded risk-free interest rate is 6%.Calculate the current volatility of a 105-strike 1-year
For two European call options, Call-I and Call-II, on a stock, you are given:Suppose you just sold 1000 units of Call-I. Determine the numbers of units of Call-II and stock you should buy or sell in order to both delta-hedge and gamma-hedge your position in Call-I.(A) Buy 95.8 units of stock and
Your company has just sold a European put option on 10,000 USD for a premium paid in Japanese yen.You are given the following:• The time to maturity T = 180 days • Yen risk-free interest rate rj = 0.4988% per annum, continuously compounded • USD risk-free interest rate ru = 4.97% per annum,
Assume the Black-Scholes framework. Four months ago, Eric bought 100 units of a one-year 45-strike European call option on a nondividend-paying stock. He immediately delta-hedged his position with shares of the stock, but has not ever re-balanced his portfolio. He now decides to close out all
Assume that the Black-Scholes framework holds. The price of a nondividend-paying stock is $30.00. The price of a put option on this stock is $4.00.You are given:(i) ∆ = −0.28(ii) Γ = 0.10 Using the delta-gamma approximation, determine the price of the put option if the stock price changes to
Several months ago, an investor sold 100 units of a one-year European call option on a nondividend-paying stock. She immediately delta-hedged the commitment with shares of the stock, but has not ever re-balanced her portfolio. She now decides to close out all positions.You are given the following
An investor has a portfolio consisting of 100 put options on stock A, with a strike price of 40, and 5 shares of stock A. The investor can write put options on stock A with a strike price of 35. The deltas and gammas of the options are listed below:Which one of the following actions would delta and
One year ago, Jacky bought 10 units of a 2-year at-the-money European straddle on a nondividend-paying stock. He immediately delta-hedged his position with shares of the stock, but has not ever re-balanced his portfolio. He now decides to close out all positions.You are given:(i) The risk-free
Assume that the Black-Scholes framework holds. Consider an option on a stock.You are given the following information at time 0:(i) The stock price is S(0), which is greater than 80.(ii) The option price is 2.34.(iii) The option delta is −0.181.(iv) The option gamma is 0.035.The stock price
You hold a trading book consisting of many long/short positions in Tempranillo Corp. stock and options. You are analyzing two possible scenarios for Tempranillo Corp. stock and want to make money by adjusting your trading book.(a) For each of the above scenarios, determine whether a positive or
Assume the Black-Scholes framework.One month ago, Tidy bought 1,000 units of a 9-month 60-strike European put option on a stock. He immediately delta-hedged the commitment with shares of the stock, but has not ever re-balanced his portfolio. He now decides to close out all positions.You are
Assume you have purchased European put options for 100,000 shares of a nondividend-paying stock and you are given the following information.• Price of stock = $49.16• Strike price = $50.00• Continuously compounded risk-free interest rate = 5% per annum• Volatility = 20% per annum• There
Assume the Black-Scholes framework. You are given:(i) The current price of a nondividend-paying stock is 80.(ii) An investor has sold 1,000 units of a one-year at-the-money European call option on the stock. He immediately delta-hedges the commitment with 750 shares of the stock.(iii) The
Consider the Black-Scholes framework. A market-maker, who delta-hedges, sells a three-month at-the-money European call option on a nondividend-paying stock.You are given:(i) The continuously compounded risk-free interest rate is 10%.(ii) The current stock price is 50.(iii) The current call option
Assume the Black-Scholes framework. You are given:(i) The current stock price is 50.(ii) The stock pays dividends continuously at a rate proportional to its price. The dividend yield is 3%.(iii) The volatility of the stock is 16%.(iv) The prices of 1-year at-the-money European call and put options
Assume the Black-Scholes framework. You are given:(i) The current price of a nondividend-paying stock is 20.(ii) The stock’s volatility is 28%.(iii) The continuously compounded risk-free interest rate is 2%.(iv) The following information about a 3-month at-the-money European call option on the
Assume the Black-Scholes framework.Eight months ago, an investor borrowed money at the risk-free interest rate to purchase a one-year 75-strike European call option on a nondividend-paying stock. At that time, the price of the call option was 8.Today, the stock price is 85. The investor decides to
Assume the Black-Scholes framework. Consider a derivative security of a stock.You are given:(i) The continuously compounded risk-free interest rate is 0.04.(ii) The volatility of the stock is σ.(iii) The stock does not pay dividends.(iv) The derivative security also does not pay dividends.(v) S(t)
A market-maker has sold 100 call options, each covering 100 shares of a dividend-paying stock, and has delta-hedged by purchasing the underlying stock.You are given the following information about the market-maker’s investment:• The current stock price is $40.• The continuously compounded
Assume the Black-Scholes framework. Yesterday, you sold a European call option on a nondividendpaying stock. You immediately delta-hedged the commitment with shares of the stock.Today, you decide to close out all positions.Which of the following statements about your delta-hedged portfolio today
For two European call options, Call-I and Call-II, on a stock, you are given: Suppose you just sold 1,000 units of Call-I. You buy or sell appropriate units of the stock and Call-II in order to both delta-hedge and vega-hedge your position in Call-I.Calculate the gamma of your hedged portfolio.
Assume the Black-Scholes framework. You are given: (i) The current price of a stock is 60. (ii) The stock pays no dividends. (iii) The stock’s volatility is 30%. (iv) The continuously compounded risk-free interest rate is 5%. Suppose you have just bought 200 1-year 60-strike European call
Assume the Black-Scholes framework. You are given:(i) The current price of a nondividend-paying stock is 50.(ii) The stock’s volatility is 30%.(iii) The continuously compounded risk-free interest rate is 8%.(iv) The following information about two European call options on the stock:In each of the
Assume the Black-Scholes framework. You are given:(i) The current price of a stock is $50.(ii) The stock pays no dividends.(iii) The stock’s volatility is 25%.(iv) The continuously compounded risk-free interest rate is 5%.Suppose you have just sold 1,000 1-year 50-strike European call options.(a)
Assume the Black-Scholes framework. The current prices of a stock and a call option on the stock are $10 and $2, respectively.You are given:(i) ∆ = 0.6(ii) Γ = 0.2Use the delta-gamma approximation to estimate the option value if the stock price jumps to $10.50.
Assume the Black-Scholes framework. For a 3-month 80-strike European put option on a nondividend-paying stock, you are given:(i) The current price of the stock is 75.(ii) The current price of the put option is 6.168.(iii) The continuously compounded risk-free interest rate is 5%.The price of the
Assume the Black-Scholes framework. Consider a 9-month at-the-money European put option on a futures contract.The continuously compounded risk-free interest rate is 8%.The futures price instantaneously decreases by 10. You are given:(i) Using the delta approximation, you find that the option price
Assume the Black-Scholes framework. You are given:(i) The current price of a nondividend-paying stock is 70.(ii) The stock’s volatility is 25%.(iii) The continuously compounded risk-free interest rate is 5%.(iv) The following information about two European put options on the stock: Suppose you
Assume the Black-Scholes framework. For t ≥ 0, let S(t) be the time-t price of a nondividend-paying stock and V (s, t) be the time-t price of a European derivative when the price of the underlying stock at that time is s. You are given:(i) S(0) = 10 and S(2) = 12.(ii) The continuously compounded
Assume the Black-Scholes framework. Determine all value(s) of a, in terms of r, σ, γ, such that V (S(t), t) := AS(t)aeγt represents the time-t price of a derivative security on a nondividend-paying stock.
Assume the Black-Scholes framework. One year ago, Kelvin bought 1,000 units of a European call option on a nondividend-paying stock. He immediately delta-hedged his position with appropriate number of shares of the stock, but has not ever re-balanced his portfolio. He now decides to close out all
Assume the Black-Scholes framework. Three months ago, Tyler bought 100 units of an at-the-money European call option on a nondividend-paying stock.He immediately delta-hedged his position with appropriate units of an otherwise identical European put option, but has not ever re-balanced his
Assume the Black-Scholes framework. Three months ago, you sold 1,000 units of a 1-year European put option on a nondividend-paying stock. You immediately delta-hedged your position with appropriate number of shares of the stock, but have not ever re-balanced his portfolio. You now decide to close
Determine which, if any, of the following positions has or have an unlimited loss potential from adverse price movements in the underlying asset, regardless of the initial premium received.I. Short 1 forward contractII. Short 1 call optionIII. Short 1 put option(A) None(B) I and II only(C) I and
Aaron has purchased a forward contract on a stock. You are given:(i) If the stock price at expiration is S, his payo would be –$5.(ii) If the stock price at expiration is 1.1S, his payoff would be $1.Calculate Aaron's profit on the long forward if the stock price at expiration is 1.2S.
You buy a 50-strike 6-month call option on a stock at a price of 5. The continuously compounded risk-free interest rate is 5%.At the end of 6 months, the profit from the long call option is 4.Calculate the price of the stock at the end of 6 months.
The market price of Stock A is 50. A customer buys a 50-strike put contract on Stock A for 500. The put contract is for 100 shares of A. Calculate the customer's maximum possible loss.(A) 0(B) 5(C) 50(D) 500(E) 5000
You are given:(i) The current price of a 100-strike 9-month European put option is 12.(ii) A 9-month forward has a forward price of 105.(iii) The continuously compounded risk-free interest rate is 3%.Calculate the stock price after 9 months such that the long put option and the short forward
Stock XYZ has the following characteristics:• The current price is 40.• The price of a 35-strike 1-year European call option is 9.12.• The price of a 40-strike 1-year European call option is 6.22.• The price of a 45-strike 1-year European call option is 4.08.The annual effective risk-free
You are given the following information about two options, A and B:(i) Option A is a one-year European put with exercise price 45.(ii) Option B is a one-year American call with exercise price 55.(iii) Both options are based on the same underlying asset, a stock that pays no dividends.(iv) Both
Jack buys a 50-strike 6-month European call option on stock ABC at a price of 8. Rose buys a 50-strike 6-month European put option on the same stock at a price of 6. The continuously compounded risk-free interest rate is 4%. 6 months later, Jack suffers a loss while Rose realizes a profit, with
Investor A wrote a 104-strike 1-year call option whose price is 2. Investor B entered into a 1-year long forward with a forward price of 105.The continuously compounded risk-free interest rate is 5%.It turns out that Investor A and Investor B earned the same profit.Calculate the 1-year stock price.
An investor purchased Option A and Option B for a certain stock today, with strike prices 70 and 80, respectively. Both options are European one-year put options.Determine which statement is true about the moneyness of these options, based on a particular stock price.(A) If Option A is
Bob writes a two-year 100-strike European put with a premium of $10. The continuously compounded risk-free interest rate is 4%.Calculate the difference between Bob's maximum profit and his minimum profit.
Comparing the profits of three puts) You are given the following premiums of one-year European put options on stock ABC for various strike prices:The effective annual risk-free interest rate is 8%. Let S(1) be the price of the stock one year from now. Determine the range for S(1) such that the
The current price of a stock is 80. Both call and put options on this stock are available for purchase at a strike price of 65.Determine which of the following statements about these options is true.(A) Both the call and put options are at-the-money.(B) Both the call and put options are
Once upon a time, Leo entered into three separate positions involving 2-year options on the same stock.• Option I was a short American-style call with strike price 30.• Option II was a long Bermuda-style put with strike price 28, where exercise was allowed at any time following an initial
For a certain stock, Investor A purchases a 45-strike call option while Investor B purchases a 135-strike put option. Both options are European with the same expiration date. Assume that there are no transaction costs.If the nal stock price at expiration is S, Investor A's payo will be 12.Calculate
Several years ago, John bought three separate 6-month options on the same stock.• Option I was an American-style put with strike price 20.• Option II was a Bermudan-style call with strike price 25, where exercise was allowed at any time following an initial 3-month period of call protection.•
Determine which of the following statements about a long European call option and a short European put option on the same underlying asset is/are correct.I. Both are long with respect to the underlying asset.II. Both involve a possible purchase of the underlying asset in the future.III. Both give
A customer buys a 50-strike put on an index when the market price of the index is also 50. The premium for the put is 5. Assume that the option contract is for an underlying 100 units of the index. Calculate the customer's profit if the index declines to 45 at expiration.(A) –1000(B) –500(C)
An investor purchased Call X and Call Y for a certain stock today, with strike prices 50 and 60, respectively. Both options are European options with the same time to expiration.Determine which of the following statements is true about the moneyness of these options, based on a particular stock
Consider a European put option on a stock index without dividends, with 6 months to expiration and a strike price of 1,000. Suppose that the annual nominal risk-free rate is 4% convertible semiannually, and that the put costs 74.20 today.Calculate the price that the index must be in 6 months so
The price of an asset will either rise by 25% or fall by 40% in 1 year, with equal probability. A European put option on this asset matures after 1 year.Assume the following:• Price of the asset today: 100 • Strike price of the put option: 130• Put option premium: 7 • Annual effective
A nondividend-paying stock currently sells for 100. One year from now the stock sells for 110. The continuously compounded risk-free interest rate is 6%. A trader purchases the stock in the following manner:• The trader pays 100 today• The trader takes possession of the stock in one
The dividend yield on a stock and the interest rate used to discount the stock's cash flows are both continuously compounded. The dividend yield is less than the interest rate, but both are positive.The following table shows four methods to buy the stock and the total payment needed for each
A certain stock costs 40 today and will pay an annual dividend of 6 for the next 4 years. An investor wishes to purchase a 4-year prepaid forward contract for this stock. The first dividend will be paid one year from today and the last dividend will be paid just prior to delivery of the stock.
Determine which of the following is NOT a distinguishing characteristic of futures contracts, relative to forward contracts. (A) Contracts are settled daily, and marked-to-market. (B) Contracts are more liquid, as one can offset an obligation by taking the opposite position. (C) Contracts are
For t ≥ 0, let S(t) be the time-t price of Stock ABC. You are given:(i) S(0) = 100(ii) At time 0.5, a cash dividend of $10 per share will be paid.(iii) From time 0.75 to time 1, dividends are paid continuously at a rate proportional to its price. The dividend yield is 10%.(iv) The continuously
A one-year forward contract on a stock has a price of $75. The stock is expected to pay a dividend of $1.50 at two future times, six months from now and one year from now, and the annual effective risk-free interest rate is 6%. Calculate the current stock price.(A) 70.75(B) 73.63(C) 75.81(D)
Determine which of the following statements about futures and forward contracts is false.(A) Frequent marking-to-market and settlement of a futures contract can lead to pricing differences between a futures contract and an otherwise identical forward contract.(B) Over-the-counter forward contracts
An investor enters a long position in a futures contract on an index (F) with a notional value of 200 × F, expiring in one year. The index pays a continuously compounded dividend yield of 4%, and the continuously compounded risk-free interest rate is 2%. At the time of purchase, the index price
The current price of stock XYZ is $80. A one-year forward contract on stock XYZ has a price of $84. Stock XYZ is expected to pay a dividend of $2 six months from now and a dividend of $3 one year from now, immediately before the one-year forward contract expires. Calculate the effective annual
You are given the following information about Stock XYZ:(i) The current price of the stock is 35 per share.(ii) The expected continuously compounded rate of return is 8%.(iii) The stock pays semi-annual dividends of 0.32 per share, with the next dividend to be paid two months from now.The
Judy decides to take a short position in 20 contracts of S&P 500 futures. Each contract is for the delivery of 250 units of the index at a price of 1500 per unit, exactly one month from now. The initial margin is 5% of the notional value, and the maintenance margin is 90% of the initial margin.
The current price of a stock is 100. The stock pays dividends continuously at a rate proportional to its price. The dividend yields is 3%. The continuously compounded risk-free interest rate is 7%. Calculate the price of the stream of dividends to be paid in the next 5 years.
The following relates to one share of XYZ stock:• The current price is 100.• The forward price for delivery in one year is 105.• P is the expected price in one year.Determine which of the following statements about P is TRUE.(A) P < 100(B) P = 100(C) 100 < P < 105(D) P = 105(E) P > 105
The current price of a medical company's stock is 75. The expected value of the stock price in three years is 90 per share. The stock pays no dividends.You are also given:(i) The risk-free interest rate is positive.(ii) There are no transaction costs.(iii) Investors require compensation for
The current price of a stock is 200, and the continuously compounded risk-free interest rate is 4%. A dividend will be paid every quarter for the next 3 years, with the first dividend occurring 3 months from now. The amount of the first dividend is 1.50, but each subsequent dividend will be 1%
It is now January 1, 3018. You are given: (i) The current price of the stock is 1,000. (ii) The stock pays dividends continuously at a rate proportional to its price. The dividend yield changes throughout the year. In March, June, September, and December, the dividend yield is 3%. In other
Determine which of the following positions has the same cash ows as a short stock position.(A) Long forward and long zero-coupon bond (B) Long forward and short forward (C) Long forward and short zero-coupon bond (D) Long zero-coupon bond and short forward (E) Short forward and short
The current price of stock XYZ is 120. Stock XYZ pays dividends continuously at a rate proportional to its price. The dividend yield is 4%. The continuously compounded risk-free interest rate is 6%. You observe a 1-year forward price of 121 on stock XYZ.Describe, giving as many details as possible,
A market maker in stock index forward contracts observes a 6-month forward price of 112 on the index. The index spot price is 110 and the continuously compounded dividend yield on the index is 2%.The continuously compounded risk-free interest rate is 5%.Describe actions the market maker could take
The current price of a nondividend-paying stock is 100. The annual effective risk-free interest rate is 4%, and there are no transaction costs.The stock's two-year forward price is mispriced at 108, so to exploit this mispricing, an investor can short a share of the stock for 100 and simultaneously
(i) The current price of a stock is 1,000.(ii) The stock pays dividends continuously at a rate proportional to its price.(iii) The continuously compounded risk-free interest rate is 5%.(iv) A 6-month forward price of 1,020 is observed in the market.Describe actions you could take to exploit an
You short sold 100 shares of stock X on November 1, 3016 and closed your position on November 1, 3018. You are given:(i) Stock X pays dividends continuously at a rate proportional to its price. The dividend yield is 4%.(ii) The continuously compounded risk-free interest rate is 5%.(iii) The
The current stock price is 80. A dividend of 2 will be paid 6 months from now. The continuously compounded risk-free interest rate is 6%.If you observe a 1-year forward price of 82, describe actions you could take as an arbitrageur, and calculate the resulting arbitrage profit (per stock unit).
Investors in a certain stock demand to be compensated for risk. The current stock price is 100. The stock pays dividends at a rate proportional to its price. The dividend yield is 2%. The continuously compounded risk-free interest rate is 5%. Assume there are no transaction costs. Let X represent
Consider a stock which pays dividends continuously at a rate proportional to its price. The dividend yield is less than the interest rate, but both are positive and continuously compounded.Rank the following quantities in ascending order (i.e., from lowest to highest):(A) = Current stock price(B) =
You observe two prices $65.1 and $65.2 quoted in the market for the stock of Company ABC.The brokerage commission includes:(i) 0.3% of the transaction amount(ii) A fixed cost of $50 per transaction.Calculate how much you gain/loss if you purchase 200 shares and then sell them all immediately.
You are given the following information: (i) The current bid price and ask price of stock X are 50 and 51, respectively. (ii) A dividend of 3 will be paid 6 months from now. (iii) The continuously compounded risk-free interest rate is 6%. (iv) The one-year forward price on stock X is 55. (v)
The ask price for a share of ABC company is 100.50 and the bid price is 100. Suppose an investor can borrow at an annual effective rate of 3.05% and lend (i.e., save) at an annual effective rate of 3%. Assume there are no transaction costs and no dividends.Determine which of the following
You are given the following historical futures prices of the P&K 689 Index observed at different time points for various maturities:On March 1, 2018, Cyrus decides to take a long position in ten 3-month P&K 689 index futures. Each contract permits the delivery of 250 units of the index. The
You are given the following information: (i) The current bid price and ask price of stock Y are 40 and 41, respectively. (ii) Stock Y pays dividends continuously at a rate proportional to its price. The dividend yield is 3%. (iii) The continuously compounded lending and borrowing rates are 6%
The P&Q futures is currently trading at 1,629. The P&Q index pays dividends continuously at a rate proportional to its price. The dividend yield is 2%. Today Peter enters into eight 3-month P&Q long futures contracts. Each contract permits the delivery of 250 units of the index. The initial margin
In Figure 3.4.1, identify all possible asset price(s) at expiration such that the profit of a long K-strike straddle is zero.Figure 3.4.1 -FVO,T[C(K) + P(K)] K Payoff Profit S(T)

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Derivative Pricing 1st Edition Ambrose Lo - Solutions

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