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derivative pricing

Derivative Pricing 1st Edition Ambrose Lo - Solutions

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 be the time-0 price of the following 4-year European asset-or-nothing option on the stock. The option
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) Determine all values of λ for which(b) Determine all values of λ for which Explain your reasons. C(K)
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 the above options are denoted by P(K1) and P(K2), respectively.Use no-arbitrage arguments to show
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. Because the British pound may lose value relative to the US dollar, Jeff decides to buy appropriate
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 dividends continuously at a rate proportional to its price. The dividend yield is 6%.(iv) The
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 of a 1-year euro-denominated European put option on dollars with a strike price of €0.70 is
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 price of a 3-year dollar-denominated European call option on euros with a strike price of $1.50 is
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 Pb(K, T)), respectively. (ii) The current ask and bid prices for the stock are Sa(0) and Sb(0),
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 the put option.(iii) Both the call option and put option will expire in 4 years.(iv) Both the call
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 Put A, which is a 6-month at-the-money European put option on the stock. Although the market price
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 price per period if the price goes up.(iii) d = 0.9, where d is one plus the percentage change in the
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 can enter into to exploit an arbitrage opportunity (if one exists). Strike
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 each strategy:• Identify the principle upon which your strategy is based.• Describe clearly the
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. Option I II III IV V European/American European European American American American Strike Price Time to Expiration 1 years 2 years 1
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 that there are no transaction costs.The risk-free annual force of interest is a constant r. Let S
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 the stock if the price goes up.(iv) d = 0.756, where d is one plus the rate of capital loss on the
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 compounded expected rate of return on the stock. Consider a one-period put option on the stock with strike
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 share of stock j.You are given:(i) S1(0) = S2(0) = 200.(ii) Stock 1 and Stock 2 share the same
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 the compound option is 3.• The compound option expires in 6 months.• The underlying option
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 price at each month’s end.Based on the above stock prices, calculate the payoff of the option.(A)
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, S(0), is 150.The index price when the option expires, S(3), is 200.The maximum index price over the
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 Asian call option (the average is calculated based on monthly closing stock prices) with a strike of
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 price is 100.(iii) The stock’s volatility is 20%.(iv) The continuously compounded risk-free
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, each of which will expire at time 3 with a strike price of $100.The chooser option price is $20 at
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 of K2. Prove thatIdentify K∗. Cgap (K₁, K₂) - pgap (K₁, K₂) = FT(S) - K*e-T.
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 European exchange option that provides the right to obtain S1(0)/S2(0) shares of stock 2 in exchange
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 risk-free interest rate is 5%.(iv) The following table shows the prices of 3-month 4-strike European
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 price is 80.(iii) The stock’s volatility is 30%.(iv) The continuously compounded risk-free
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 = 1.0594, where u is one plus the percentage change in the stock price per period if the price goes up.(v)
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 and a strike of 65(2) Down-and-out call option with a barrier of 58 and a strike of 65(3) Up-and-in
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 time 3. The payoff of the claim isYou are given:(i) S1(0) = $100.(ii) S2(0) = $200.(iii) Stock 1 pays
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 call option is on a stock that pays no dividends.(ii) The stock’s volatility is 30%.(iii) The
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) $35(E) $40 Type of Option down-and-out up-and-out down-and-in up-and-in down rebate up
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 numbers with a < b. Determine an expression for the price of such a spread using common symbols.
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 European put option on the stock every three months, with the first one being bought immediately.You
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) = 50.Consider a special 6-month 60-strike European “knock-in, partial knock-out” call option that knocks in
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 1’s volatility is 0.18.(iii) Stock 2’s volatility is 0.25.(iv) The correlation between the
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 compounded expected rate of return on the stock is 8%.(iv) The continuously compounded risk-free
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 options cannot be priced with the Black-Scholes formula.(D) Compound options can be priced with the
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 continuously at a rate proportional to its price; the dividend yield is 2%. It has a volatility of 20%.The
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 than $100, the payoff is S(1) – 90;Otherwise, the payoff is zero. You are given: (i) S(0) =
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 the percentage change in the stock price per period if the price goes up.(iv) d = 0.890947, where d
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 time T is greater than $100, the payoff is S(T) − 90; otherwise, the payoff is zero.You are
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) The current price of the stock is 100.(iii) The stock’s volatility is 100%.(iv) Each gap call
Assume the Black-Scholes framework. For a European put option and a European gap call option on a stock, you are given:(i) The expiry date for both options is T.(ii) The put option has a strike price of 40.(iii) The gap call option has strike price 45 and payment trigger 40.(iv) The time-0 gamma of
You are given the following:• The current price to buy one share of XYZ stock is 500.• The stock does not pay dividends.• The annual risk-free interest rate, compounded continuously, is 6%.• A European call option on one share of XYZ stock with a strike price of K that expires in one year
You are given:(i) The following 1-year European put option prices on the same stock:(ii) The continuously compounded risk-free interest rate is 6%. You take advantage of any possible mispricing by means of an appropriate spread position. Calculate your 1-year profit when the 1-year stock price is
Explain intuitively when you will expect that CE ≈ FP0,T(S).
Consider European and American options on a nondividend-paying stock.You are given:(i) All options have the same strike price of 100.(ii) All options expire in six months.(iii) The continuously compounded risk-free interest rate is 10%.You are interested in the graph for the price of an option as a
On April 30, 2007, a common stock is priced at $52.00. You are given the following:(i) Dividends of equal amounts will be paid on June 30, 2007 and September 30, 2007.(ii) A European call option on the stock with strike price of $50.00 expiring in six months sells for $4.50.(iii) A European put
You are given:• C(K, T) denotes the current price of a K-strike T-year European call option on a nondividend-paying stock.• P(K, T) denotes the current price of a K-strike T-year European put option on the same stock.• S denotes the current price of the stock.• The continuously compounded
For a stock, you are given:(i) The current stock price is $50.00.(ii) δ = 0.08.(iii) The continuously compounded risk-free interest rate is r = 0.04.(iv) The prices for one-year European calls (C) under various strike prices (K) are shown below:You own four special put options, each with one of
You are given:(i) The following prices of 3-year European call options on the same stock:(ii) The continuously compounded risk-free interest rate is 2%. To earn arbitrage profit, you buy two 100-strike call options, two 120-strike call options, sell some 110-strike call options, and invest the
A nine-month dollar-denominated call option on euros with a strikeprice of $1.30 is valued at $0.06. A nine-month dollar-denominated put option on euroswith the same strike price is valued at 0.18. The current exchange rate is $1.2/euro andthe continuously compounded risk-free rate on dollars is
Near market closing time on a given day, you lose access to stock prices, but some European call and put prices for a stock are available as follows:All six options have the same expiration date.After reviewing the information above, John tells Mary and Peter that no arbitrage opportunities can
You are given:(i) The current exchange rate is 0.011$/¥.(ii) A four-year dollar-denominated European put option on yen with a strike price of $0.008 sells for $0.0005.(iii) The continuously compounded risk-free interest rate on dollars is 3%.(iv) The continuously compounded risk-free interest rate
Given the following chart about call options on a particular dividend-paying stock, which option has the highest value?(A) Option A(B) Option B(C) Option C(D) Option D(E) Option E Option A B с D E Option Style European American European American American Time Until Expiration 1 year 1 year 2
You are given the following information on a stock:Assume that the log returns are normally distributed. (a) Calculate the 95% prediction interval for the stock price in one year. (b) Calculate the expected stock price and the standard deviation of the stock price in one year. Initial
Assume the Black-Scholes framework. You are given:(i) The stock, whose current price is 100, pays dividends continuously at a rate proportional to its price.(ii) The stock’s volatility is 0.35.(iii) The continuously compounded expected rate of stock-price appreciation is 15%.(iv) The
Assume the BlackScholes framework. Let S(t) denote the time-t price of a stock, which pays dividends continuously at a rate proportional to its price.You are given:(i) S(0) = 8 (ii) The 90% lognormal prediction interval for S(2) is (13.10, 41.93).Calculate the width of the 95% lognormal
Assume the Black-Scholes framework. Determine an expression for E[S(T)|a < S(T) < b], where a and b are positive constants with a < b.
Assume the Black-Scholes framework. For t ≥ 0, let S(t) be the time-t price of a stock. You are given:(i) The stock pays dividends continuously at a rate proportional to its price.(ii) The 90% lognormal prediction interval for S(2) is (13.1072, 41.9448).(iii) The 95% lognormal prediction interval
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 2-year at-the-money European put option on the stock will be exercised is 0.5279.(ii) The 99% lognormal prediction interval for the 3-year
Assume the Black-Scholes framework. You are given:(i) The current price of a stock is 80.(ii) The stock’s volatility is 25%.(iii) The stock pays dividends continuously at a rate proportional to its price.(iv) The continuously compounded risk-free interest rate is 6%.(v) The expected 1-year stock
Assume the Black-Scholes framework. You are given:(i) The stock pays dividends continuously at a rate proportional to its price. The dividend yield is identical to the continuously compounded (total) expected rate of return on the stock.(ii) The expected 6-month stock price is 150.(iii) The
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 an 8-month European put option on the stock will be exercised is 0.512.(ii) The expected 8-month stock price is 10.134.(iii) The expected
Which 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 lognormally distributed.(D) All transaction costs are included in stock returns.(E) The risk-free
For a six-month European put option on a stock, you are given:(i) The strike price is $50.00.(ii) The current stock price is $50.00.(iii) The only dividend during this time period is $1.50 to be paid in four months.(iv) σ = 0.30.(v) The continuously compounded risk-free interest rate is 5%.Under
Your company has just written a one-year European putoption on an equity index fund.The equity index fund is currently trading at 1000. It pays dividends continuously at arate proportional to its price; the dividend yield is 2%. It has a volatility of 20%.The strike price of the put option is set
Assume the Black-Scholes framework. You are given:(i) The stock price is 100.(ii) The stock pays dividends continuously at a rate proportional to its price. The dividend yield is 2%.(iii) The continuously compounded risk-free interest rate is 6%.(iv) The stock’s volatility is 40%.Calculate the
You are considering the purchase of 100 units of a 3-month 25-strike European call option on stock.You are given:(i) The Black-Scholes framework holds.(ii) The stock is currently selling for 20.(iii) The stock’s volatility is 24%.(iv) The stock pays dividends continuously at a rate proportional
You are considering the purchase of a three-month 41.5-strike American call option on a nondividend-paying stock.You are given:(i) The Black-Scholes framework holds.(ii) The stock is currently selling for 40.(iii) The stock’s volatility is 30%.(iv) The current call option delta is 0.5.Determine
Consider a one-year 45-strike European put option on a stock S. You are given:(i) The current stock price, S(0), is 50.00.(ii) The only dividend is 5.00 to be paid in nine months.(iii) Var[ln FPt,1(S)] = 0.01 × t, 0 ≤ t ≤ 1.(iv) The continuously compounded risk-free interest rate is 12%.Under
Assume the Black-Scholes framework. For an at-the-money, 8-month European put option on a stock, you are given:(i) The stock pays dividends continuously at a rate proportional to its price. The dividend yield is 2%.(ii) The continuously compounded risk-free interest rate is 5%.(iii) Var[ln S(t)] =
On January 1st, 2007, the following currency information is given:• Spot exchange rate = $0.82/euro• Dollar interest rate = 5.0% compounded continuously• Euro interest rate = 2.5% compounded continuously• Exchange rate volatility = 0.10What is the price of 850 dollar-denominated euro call
You are asked to determine the price of a European put option on a stock.Assuming the Black-Scholes framework holds, you are given:(i) The stock price is $100.(ii) The put option will expire in 6 months.(iii) The strike price is $98.(iv) The continuously compounded risk-free interest rate is r =
Assume the Black-Scholes framework. Consider a 3-year European contingent claim on a stock. For t ≥ 0, let S(t) be the time-t price of the stock.You are given:(i) S(0) = 45.(ii) The stock’s volatility is 20%.(iii) The stock pays dividends continuously at a rate proportional to its price. The
Assume the Black-Scholes framework. You are given:(i) The stock price is 100.(ii) The stock pays dividends continuously at a rate proportional to its price. The dividend yield is 2%.(iii) The continuously compounded risk-free interest rate is 6%.(iv) The stock’s volatility is 40%.Calculate the
Company A is a US international company, and Company B is a Japanese local company. Company A is negotiating with Company B to sell its operation in Tokyo to Company B. The deal will be settled in Japanese yen. To avoid a loss at the time when the deal is closed due to a sudden devaluation of yen
Assume the Black-Scholes framework. You are given:(i) The current price of a stock is 80.(ii) The stock’s volatility is 25%.(iii) The stock pays dividends continuously at a rate proportional to its price.(iv) The continuously compounded risk-free interest rate is 5%.(v) The continuously
Which of the following otherwise identical European options has the highest gamma?(A) 1-day deep out-of-the-money call option(B) 10-day deep in-the-money call option(C) 1-year at-the-money put option(D) 10-year at-the-money put option(E) There is not enough information to determine the answer
Assume the Black-Scholes framework. Consider a 9-month at-the-money European put option on a futures contract. You are given:(i) The continuously compounded risk-free interest rate is 10%.(ii) The strike price of the option is 20.(iii) The price of the put option is 1.625.If three months later the
Consider a stock with current price $50. You are given:(i) There will be only one dividend; $2 will be paid in three months.(ii) σ = 0.30.(iii) The continuously compounded risk-free interest rate is 5%.Use the Black-Scholes methodology to price a nine-month at-the-money European put option on the
You compute the current delta for a 50-60 bull spread with the following information:(i) The continuously compounded risk-free 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 expiration is 3
A call option is modeled using the Black-Scholes formula with the following parameters.• S = 25• K = 24• r = 4%• δ = 0%• σ = 20%• T = 1Calculate the call option elasticity, Ω.(A) Less than 5(B) At least 5, but less than 6(C) At least 6, but less than 7(D) At least 7, but less than
Assume the Black-Scholes framework. For a 9-month 45-55 put bear spread on a stock, you are given:(i) The current stock price is 50.(ii) The only dividends during this time period are 2.50 to be paid in two months and five months.(iii)(iv) The continuously compounded risk-free interest rate is
Assume the Black-Scholes framework. Consider a 1-year European contingent claim on a stock. You are given:(i) The time-0 stock price is 45.(ii) The stock’s volatility is 25%.(iii) The stock pays dividends continuously at a rate proportional to its price. The dividend yield is 3%.(iv) The
Assume the Black-Scholesframework. You are given:(i) The current stock price is $82.(ii) The stock’s volatility is 30%.(iii) The stock pays no dividends.(iv) The continuously compounded risk-free interest rate is 8%.Using the above information, you calculate the price of a 3-month 80-strike
Assume the Black-Scholes framework. Consider a stock, and a European call option and a European put option on the stock. The current stock price, call price, and put price are 45.00, 4.45, and 1.90, respectively.Investor A purchases two calls and one put. Investor B purchases two calls and writes
Assume the Black-Scholes framework. For a 3-month at-the-money European put option on a stock, you are given:(i) The stock is currently selling for 50.(ii) The stock will pay a single dividend of 1.5 in two months.(iii) Var[ln FPt,0.25(S)] = 0.09t, for 0 ≤ t ≤ 0.25.(iv) The continuously
You are given the following information about 50-strike and 60-strike European put options with the same stock and time to expiration:Calculate the elasticity of a 50-60 European put bull spread. Strike price 50 60 Elasticity -4.9953 -3.4267 Put premium 3.7295 9.5865
Assume the Black-Scholes framework. Consider a 3-month European contingent claim on a stock.You are given:(i) The stock is currently selling for 50.(ii) The stock will pay a single dividend of 1.5 in two months.(iii) Var[ln FPt,0.25(S)] = 0.09t, for 0 ≤ t ≤ 0.25.(iv) The continuously compounded
You are given:(i) The current dollar-euro exchange rate is 1.50$/AC.(ii) The volatility of the exchange rate is 20%.(iii) The continuously compounded risk-free interest rate on dollars is 3%.(iv) The continuously compounded risk-free interest rate on euros is 4%.Consider a 6-month at-the-money
You have ordered a Rolls Royce car for the price of 200,000 British pounds, which you will pay when the car is delivered to you in three months. The current exchange rate is 1.60 US dollars per British pound, and your insanely rich mom will give you US $320,000 three months from now. Because the US
To settle an urgent debt of US $300,000 payable in three months, you have decided to (reluctantly!) sell your favorite Rolls Royce car for the price of 200,000 British pounds, which you will receive when the car is delivered to the buyer in three months. Because the British pound may lose value
Assume the Black-Scholes framework. You are given:(i) The current dollar/euro exchange rate is 1.50$/€.(ii) The volatility of the exchange rate is 20%.(iii) The continuously compounded risk-free interest rate on dollars is 4%.(iv) The continuously compounded risk-free interest rate on euros is
Assume the Black-Scholes framework.You are given:(i) The current price of the P&K 777 index is 500.(ii) The P&K 777 index pays dividends continuously at a rate proportional to its price. The dividend yield is 2%.(iii) The continuously compounded risk-free interest rate is 6%.(iv) The

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