- Assume the Black-Scholes framework. You are given:(i) The current stock price is 95.(ii) The stock’s volatility is 10%.(iii) The stock pays dividends continuously at a rate proportional to its
- Assume the Black-Scholes framework. Let S(t) denote the price at time t of a stock, which will pay a dividend of $1 after 3 months.Consider a European gap option which matures in 9 months. If the
- Michael has ordered a Rolls Royce car for 200,000 British pounds, which he will pay when the car is delivered to him in three months. Because Michael has got an A+ in ACTS:4830 from the devilish
- Consider a “sad” 1-year European contingent claim on a stock. You are given:(i) The time-0 stock price is 70.(ii) The stock pays dividends continuously at a rate proportional to its price. The
- Assume the Black-Scholes framework. For t ≥ 0, let S(t) denote the time-t price of a stock. Consider a 1-year European contingent claim. If the 1-year stock price is less than $60, the payoff of
- Assume the Black-Scholes framework. For a stock which pays dividends continuously at a rate proportional to its price, you are given:(i) The continuously compounded expected rate of stock-price
- Assume the Black-Scholes framework. For a stock which pays dividends continuously at a rate proportional to its price, you are given:(i) The probability that a 3-month 70-strike 75-trigger European
- Assume the Black-Scholes framework. For t ≥ 0, let S(t) be the time-t price of a stock. You are given:(i) S(0) = 65.(ii) The stock pays dividends continuously at a rate proportional to its price.
- Assume the Black-Scholes framework. For t ≥ 0, let S(t) be the time-t price of a stock. You are given:(i) S(0) = $48.(ii) The stock pays dividends continuously at a rate proportional to its price.
- Assume the Black-Scholes framework. For t ≥ 0, let S(t) be the time-t price of a stock that pays dividends continuously at a rate proportional to its price.Consider a 1-year European gap option. If
- You are given the following generic Black-Scholes-type pricing function:and all variables are positive. Your boss, who knows nothing about option pricing, has asked you to analyze the following
- Assume the Black-Scholes framework. You are given:(i) The current stock price is 100.(ii) The stock pays dividends continuously at a rate proportional to its price. The dividend yield is 2%.(iii) The
- Assume the Black-Scholes framework. You are given the following information about two stocks:(i)(ii) The correlation between the continuously compounded returns on the two stocks is −0.3. (iii)
- Assume the Black-Scholes framework. For a 5-month European gap put option on a nondividend-paying stock, you are given:(i) The current price of the stock is 120.(ii) The stock’s volatility is
- Assume the Black-Scholes framework. Two actuaries, A and B, are computing the prices of a European call and a European put using different parameters.You are given:Describe the relationship between
- Consider two nondividend-paying stocks whose time-t prices are denoted by S1(t) and S2(t), respectively.You are given:(i) S1(0) = S2(0) = 10.(ii) Stock 1’s volatility is 25%.(iii) Stock 2’s
- Consider a European option to exchange Stock 2 for Stock 1 at a certain future date. Each stock pays dividends continuously at a rate proportional to its price.Determine whether each of the
- For j = 1, 2, and t ≥ 0, let Sj(t) denote the price of one share of stock j at time t (in years). Both stocks pay no dividends.Let π be the current price of a 4-year European exchange option that
- Assume the Black-Scholes framework. For j = 1, 2 and t ≥ 0, let Sj(t) denote the time-t price of Stock j.(a) Consider a T-year European contingent claim whose payoff is the maximum of the two
- You are given:(i) The current prices of Stock 1 and Stock 2 are 100 and 200, respectively.(ii) Stocks 1 and 2 pay dividends continuously at a rate proportional to their prices. The dividend yield of
- Consider a model with two nondividend-paying stocks, Stock 1 and Stock 2, and a special 5-year European straddle on Stock 1, with a strike price given by the 5-year price of Stock 2.You are given:(i)
- You are given:(i) Stock XYZ pays no dividends.(ii) Derivative A gives its holder the right, but not the obligation, to buy an at-the-money European call option for $6 at the end of 6 months. The call
- Assume the Black-Scholes framework. Consider two nondividend-paying stocks whose time-t prices are denoted by S1(t) and S2(t), respectively.You are given:(i) S1(0) = $100 and S2(0) = $150.(ii) Stock
- Assume the Black-Scholes framework. For t ≥ 0, let S1(t) and S2(t) be the time-t prices of Stock 1 and Stock 2, respectively. You are given:(i) S1(0) = $100 and S2(0) = $120.(ii) The volatility of
- Which of the following statements about exotic call options is/are correct?(A) The gamma of a European cash-or-nothing call option must be positive.(B) The vega of a European cash-or-nothing call
- For a threeperiod binomial stock price model, you are given:(i) The length of each period is one year.(ii) The current price of a nondividend-paying stock is 100.(iii) u = 1.1, where u is one plus
- You use the following information to construct a binomial forward tree for modeling the price movements of a stock:(i) The length of each period is 6 months.(ii) The current stock price is 100.(iii)
- For t ≥ 0, let S(t) be the time-t price of a stock. You are given:(i) S(0) = 15.(ii) The stock pays dividends continuously at a rate proportional to its price. The dividend yield is 3%.(iii) The
- You use the following information to construct a binomial forward tree for modeling the price movements of a nondividend-paying stock:(i) The length of each period is 6 months.(ii) The current stock
- You use the following information to construct a binomial forward tree for modeling the price movements of a stock:(i) The length of each period is 6 months.(ii) The current stock price is 100.(iii)
- You use the following information to construct a binomial forward tree for modeling the movements of the dollar/euro exchange rate:(i) The length of each period is 3 months.(ii) The current
- Consider the following European options having the same strike price, time to maturity, and underlying stock:I. A plain vanilla call option II. A gap call option III. An extrema lookback call
- For a binomial model for the price of a nondividend-paying stock, you are given:(i) The length of each period is 1 year.(ii) The current price of the stock is 120.(iii) u = 1.15, where u is one plus
- For a binomial tree modeling the price movements of a nondividend-paying stock, you are given:(i) The length of each period is 4 months.(ii) The current stock price is 120.(iii) u = 1.2212, where u
- Consider the following up-and-in 60-strike European call options on the same underlying stock with the same time to expiration:The current price of the stock is 50.Rank the four barrier options, from
- Assume the Black-Scholes framework. You are given:(i) The current price of a nondividend-paying stock is 80.(ii) The stock’s volatility is 30%.(iii) The continuously compounded risk-free interest
- Assume the Black-Scholes framework. You are given:(i) The current stock price is 40.(ii) The stock pays no dividends.(iii) The stock’s volatility is 30%.(iv) The continuously compounded risk-free
- For a binomial stock price model, you are given:(i) The length of each period is 1 year.(ii) The current price of a nondividend-paying stock is 100.(iii) u = 1.15, where u is one plus the percentage
- For a four-period binomial stock price model, you are given:(i) The length of each period is 3 months.(ii) The current price of a nondividend-paying stock is 100.(iii) u = 1.11.(iv) d = 0.90.(v) The
- You use the following information to construct a binomial forward tree for modeling the price movements of a stock:(i) The length of each period is 1 year.(ii) The current stock price is 200.(iii)
- Consider a chooser option (also known as an as-you-like-it option) on stock ABC. At time τ (in years) with 0 You are given:(i) The current price of stock ABC is 32.(ii) Dividends of 1.5 are paid at
- Assume the Black-Scholes framework. Consider a special chooser option (also known as an as-you-like-it option) on a nondividend-paying stock. One year from now, its holder will choose whether it
- Consider a chooser option (also known as an as-you-like-it option) on a nondividend-paying stock.At time 1, its holder will choose whether it becomes a European call option or a European put option,
- Consider a chooser option (also known as an as-you-like-it option) on two stocks. At time 1, its holder will choose whether it becomes a European option to exchange two units of Stock B for one unit
- The current time is t = 0 (in years). Assume the Black-Scholes framework. You are given:(i) The current stock price is 40.(ii) The stock’s volatility is 40%.(iii) The stock pays dividends
- Assume the Black-Scholes framework. Let S(t) be the time-t price of a stock. Consider a special 3-year European contingent claim which pays a certain amount three years from now, provided that S(3)
- Assume the Black-Scholes framework. Consider a special forward start option which, 1 year from today, will give its owner a 1-year European put option with a strike price equal to the one-year stock
- Assume the Black-Scholes framework. Consider a special forward start option which, 2 years from today, will give its owner a 1-year 100-strike European gap call option whose payment trigger is equal
- You are given:(i) The current price of a stock is 100.(ii) The stock pays dividends continuously at a rate proportional to its price. The dividend yield is 1%.(iii) The continuously compounded
- Assume the Black-Scholes frame-work. For t ≥ 0, let S(t) be the time-t price of a stock.You are given:(i) S(0) = 20.(ii) The stock’s volatility is 25%.(iii) The stock pays dividends continuously
- Assume the Black-Scholes framework. Three months ago, Embryo (Ambrose’s twin brother) bought a 1-year 50-strike European cash-or-nothing call option of $1,000 on a nondividend-paying stock. He
- Assume the Black-Scholes framework. For t ≥ 0, let S(t) be the time-t price of a stock. You are given:(i) S(0) = 100.(ii) The stock’s volatility is 25%.(iii) The stock pays dividends continuously
- This problem shows how the pricing formula for a T-year K-strike European cash-or-nothing call option of $1 can be retrieved from that of a plain vanilla T-year K-strike European call option.(a)
- Assume the Black-Scholes framework. For t ≥ 0, let S(t) denote the time-t price of a stock that pays dividends continuously at a rate proportional to its price. The dividend yield δ is 2%.Let π
- For K ≥ 0, let C(K) denote the price of a K-strike European call option on a stock. Let 0 1 2. Define Kλ = λK1 + (1 − λ)K2, where λ is a real number, not necessarily between 0 and 1.(a)
- Suppose the current time is 0. Consider two European put options on the same underlying stock and the same maturity date T, but with different strike prices K1 and K2, where K1 ≤ K2. The prices of
- To settle an urgent debt payable in US dollars in one year, Jeff has decided to (reluctantly!) sell his favorite Rolls Royce car in exchange for a fixed sum payable in British pounds in one year.
- Consider European and American call options on the same underlying stock.You are given:(i) Both options have the same strike price of 100.(ii) Both options expire in six months.(iii) The stock pays
- Justify PE ≤ FP0,T (K) by means of:(a) Intuitive explanations (b) A no-arbitrage proof (c) An algebraic proof
- You are given:(i) The current euro/dollar exchange rate is 0.72.(ii) The price of a 1-year euro-denominated European call option on dollars with a strike price of €0.70 is €0.09.(iii) The price
- You are given:(i) The current dollar-euro exchange rate is $1.25/€.(ii) The price of a 3-year dollar-denominated European call option on euros with a strike price of $1.20 is $0.06545.(iii) The
- You are given: (i) The quoted ask (resp. bid) prices of a K-strike T-year call option and a K-strike T-year put option on the same stock are denoted by Ca(K, T) and Pa(K, T) (resp. Cb(K, T) and
- Consider a European call option and a European put option on a nondividend-paying stock. You are given(i) The current price of the stock is 60.(ii) The call option currently sells for 0.15 more than
- Assume the Black-Scholes framework. The current price of a nondividend-paying stock is $60 and the continuously compounded risk-free interest rate is 5%.Your boss has asked you to quote a price for
- The current price of a stock is 60. A dividend of 2 will be paid 6 months from now. The one-year forward price is 61.80. Calculate the continuously compounded risk-free annual rate of interest.
- You are given the following regarding the stock of Iowa Actuarial Association (IAA):(i) The stock is currently selling for $100.(ii) u = 1.1, where u is one plus the percentage change in the stock
- You are given:(i) The following prices of 1-year European call options on the same stock for various strikes:(ii) The continuously compounded risk-free interest rate is 6%. Describe transactions you
- You are given the following European call and put prices on the same stock: All six options have the same expiration date.Propose two sets of arbitrage strategies based on different principles. For
- You are given the following European and American call options written on the same stock:Rank, as far as possible, the prices of these five options.
- An investor purchases a nondividend-paying stock and writes a t-year, European call option for this stock, with call premium C. The stock price at time of purchase and strike price are both K.Assume
- For a one-period binomial model for the price of a stock, you are given:(i) The period is one year.(ii) The stock pays no dividends.(iii) u = 1.433, where u is one plus the rate of capital gain on
- Consider a one-period binomial tree. The length of the period is h years. The stock price moves from S to S × u or to S × d, 0 < d < u.The stock pays no dividends. Let α be the continuously
- Assume the Black-Scholes framework. Consider two stocks, each of which pays dividends continuously at a rate proportional to its price. For j = 1, 2 and t ≥ 0, let Sj (t) be the time-t price of one
- You are given the following information on a compound CallOnPut option:• The continuously compounded risk-free rate is 5%.• The strike price of the underlying option is 43.• The strike price of
- For a fixed strike price K1, determine the value of the payment trigger that maximizes the payoff of a European gap option.
- At the beginning of the year, a speculator purchases a six-month geometric average price call option on a company’s stock. The strike price is 3.5. The payoff is based on an evaluation of the stock
- Several lookback options are written on the same underlying index. They all expire in 3 years.Let S(t) denote the value at time t of the index on which the option is written.The initial index price,
- You have observed the following monthly closing prices for stock XYZ:The following are one-year European options on stock XYZ. The options were issued on December 31, 2007.(i) An arithmetic average
- You use the following information to construct a binomial forward tree for modeling the price movements of a nondividend-paying stock:(i) The length of each period is 6 months.(ii) The current stock
- Consider a chooser option (also known as an as-you-like-it option) on a nondividend-paying stock. At time 1, its holder will choose whether it becomes a European call option or a European put option,
- Let Cgap(K1, K2) and Pgap(K1, K2) be the current prices of a T-year European gap call option and a T-year European gap put option, respectively, both with a strike price of K1 and a payment trigger
- Consider two nondividend-paying stocks. For j = 1, 2, and t ≥ 0, let Sj (t) denote the price of one share of stock j at time t (in years). Under the Black-Scholes framework, you price a four-month
- You are given:(i) The current price of a stock is 42.(ii) The stock pays dividends continuously at a rate proportional to its price. The dividend yield is 3%.(iii) The continuously compounded
- You use the following information to construct a binomial forward tree for modeling the price movements of a nondividend-paying stock:(i) The length of each period is 6 months.(ii) The current stock
- You are given the following information about a binomial stock price model:(i) The length of each period is 1 month.(ii) The current stock price is 800.(iii) The stock pays no dividends.(iv) u =
- The current stock price is 60. The price of a 6-month 65-strike European call has a price of 5.Consider the following 6-month European barrier options:(1) Down-and-in call option with a barrier of 58
- Consider a model with two stocks. Each stock pays dividends continuously at a rate proportional to its price. Sj(t) denotes the price of one share of stock j at time t.Consider a claim maturing at
- Consider a forward start option which, 1 year from today, will give its owner a 1-year European call option with a strike price equal to the stock price at that time.You are given:(i) The European
- Barrier call option prices are shown in the table below. Each option has the same underlying asset and the same strike price.Calculate $X, the price of the up-and-in option.(A) $20(B) $25(C) $30(D)
- Assume the Black-Scholes framework. A European cash-or-nothing spread on a stock pays $1 at time T if and only if the stock price at time T lies in the interval [a, b], where a and b are positive
- You own one share of a nondividend-paying stock. Because you worry that its price may drop over the next year, you decide to employ a rolling insurance strategy, which entails obtaining one 3-month
- Prices for 6-month 60-strike European up-and-out call options on a stock S are available. Below is a table of option prices with respect to various B, the level of the barrier. Here, S(0) =
- Assume the Black-Scholes framework. Consider two nondividend-paying stocks whose time-t prices are denoted by S1(t) and S2(t), respectively.You are given:(i) S1(0) = 10 and S2(0) = 20.(ii) Stock
- 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) = 180.(ii) The stock’s volatility is 20%.(iii) The continuously
- Which one of the following statements is true about exotic options?(A) Asian options are worth more than European options.(B) Barrier options have a lower premium than standard options.(C) Gap
- Your company has just written one million units of a one-year European asset-or-nothing put option on an equity index fund.The equity index fund is currently trading at 1000. It pays dividends
- Assume the BlackScholes framework. Let S(t) denote the price at time t of a nondividend-paying stock.Consider a European gap option which matures in one year. If the one-year stock price is greater
- For a binomial model for the price of a nondividend-paying stock, you are given:(i) The length of each period is one month.(ii) The current stock price is 50.(iii) u = 1.122401, where u is one plus
- Let S(t) denote the price at time t of a stock that pays dividends continuously at a rate proportional to its price. Consider a European gap option with expiration date T, T > 0.If the stock price at
- A market-maker sells 1,000 1-year European gap call options, and delta-hedges the position with shares.You are given:(i) Each gap call option is written on 1 share of a nondividend-paying stock.(ii)

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