
BreedenLitzenberger formula Imagine that, for a given underlying stock S and maturity T, we may...
BreedenLitzenberger formula Imagine that, for a given underlying stock S and maturity T, we may trade an infinite number of calls whose strikes K > 0 form a continuum. Assume that S T > 0 has a distribution density f(x), which may or may not be...
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yesterday

Derivation of the closedform formulas for European options Let D T = max(0, S T – K) be the payoff.
Derivation of the closedform formulas for European options Let D T = max(0, S T – K) be the payoff of a European call with strike K and maturity T. The following questions provide a stepbystep derivation of the closedform formulas for c 0...
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21 hours ago

Simulating a normal distribution Let X be a random variable with uniform distribution over the...
Simulating a normal distribution Let X be a random variable with uniform distribution over the interval [0, 1], and let is the inverse function of the standard normal cumulative distribution. Show that Y follows a standard normal distribution....
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yesterday

Logcontract The payoff of the logcontract on an underlying asset S is DT = – ln(ST / F0), where F0
Logcontract The payoff of the logcontract on an underlying asset S is DT = – ln(ST / F0), where F0 is the forward price of S for maturity T. (a) Draw the payoff function of the logcontract. (b) Find the fair value D0 of the logcontract in...
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yesterday

Quadratic option The payoff of a quadratic option with strike K and maturity T is: DT = (ST – K) 2..
Quadratic option The payoff of a quadratic option with strike K and maturity T is: DT = (ST – K) 2. (a) Draw the payoff function of a quadratic option with strike 100. (b) Find a closedform formula for the fair value of a quadratic option in...
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yesterday

Lognormal distribution Let X be a normally distributed random variable with mean μ and standard...
Lognormal distribution Let X be a normally distributed random variable with mean μ and standard deviation σ, and let (a) Show that Y follows a lognormal distribution. (b) Show that the expected value of Y is (c) What is the...
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22 hours ago

Digital option The payoff of a digital option with strike K and maturity T is: (a) Draw the payoff..
Digital option The payoff of a digital option with strike K and maturity T is: (a) Draw the payoff function of a digital option with strike 100. (b) Show that the fair value of the digital option in the...
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yesterday

Rule of thumb for the value of an ‘atthemoneyforward’ call Consider an ‘atthemoneyforward’...
Rule of thumb for the value of an ‘atthemoneyforward’ call Consider an ‘atthemoneyforward’ European call, i.e. a plain vanilla call whose strike K is equal to the underlying forward price F 0 . (a) What does the...
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yesterday

Butterflies and probabilities Dull Inc.’s stock currently trades at $100 in New York, and the...
Butterflies and probabilities Dull Inc.’s stock currently trades at $100 in New York, and the interest rate curve is flat at zero. The fair values of three 1year calls on Dull Inc. are shown below: (a) Draw the payoff function of a butterfly...
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yesterday

Negative time value and arbitrage (a) Based on the lognormal model, what is the value of a 3year,..
Negative time value and arbitrage (a) Based on the lognormal model, what is the value of a 3year, European inthemoney put struck at €100 on a stock trading at €60 with a 10% volatility? The price of the 3year zerocoupon is €80...
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yesterday

MonteCarlo pricer In a spreadsheet, build a pricer which computes the value of the following...
MonteCarlo pricer In a spreadsheet, build a pricer which computes the value of the following derivative payoffs using the MonteCarlo method with 5,000 simulations. The user must be able to input the following parameters: forward price F, strike K,...
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yesterday

Plain vanilla option pricer In a spreadsheet, build a pricer which computes the value of European...
Plain vanilla option pricer In a spreadsheet, build a pricer which computes the value of European calls and puts using the closedform formulas The user must be able to input the following parameters: forward price F, strike K, maturity T, riskfree...
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yesterday

Other distributions for ST Discuss the pros and cons of the following distribution assumptions to...
Other distributions for ST Discuss the pros and cons of the following distribution assumptions to model the final underlying price ST of a stock: (a) Uniform: with ; (b) Normal: where (c) where X has cumulative distribution function...
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yesterday

Closedform formulas. The three questions are independent. (a) Based on the lognormal model, what is
Closedform formulas. The three questions are independent. (a) Based on the lognormal model, what is the value of a 1month European call on Kroger Co. struck at $25? Kroger Co. currently trades at $24, no dividend is scheduled, the riskfree rate is...
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yesterday

True or False? “The lognormal model builds on the idea that the log of the final underlying price ST
True or False? “The lognormal model builds on the idea that the log of the final underlying price ST has a normal distribution centered on the log of the forward price F 0 .”
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yesterday

Riskneutral probability Consider the onestep binomial model over the period [0, T]. Let ? (u)...
Riskneutral probability Consider the onestep binomial model over the period [0, T]. Let ω (u) denote the ‘up’ scenario and ω (d) the ‘down’ scenario with respective probabilities p (u) and p (d) = 1 – p (u) ....
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yesterday

Call on a bond. This problem is about the valuation of a call on a bond and goes beyond the scope of
Call on a bond. This problem is about the valuation of a call on a bond and goes beyond the scope of equity derivatives. Consider a world where, at any given point in time t, the interest rate curve is flat at rate r t which may vary. Currently,...
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yesterday

a. Binomial pricing algorithm In a spreadsheet, build a 12step binomial tree to calculate the value
a. Binomial pricing algorithm In a spreadsheet, build a 12step binomial tree to calculate the value of a European call with a strike price of 100 and a maturity of one year. Assume that the 1month interest rate is constant at 10% p.a., that the...
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yesterday

American option Compute the value of a 2year American put with strike $50 using a twostep binomial
American option Compute the value of a 2year American put with strike $50 using a twostep binomial tree. The underlying stock pays no dividends, its current $50 price may rise or fall by 20% each year, and the riskfree rate is constant at 5% in...
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yesterday

Binomial model and dividends Consider a onestep binomial model on an underlying stock S which pays.
Binomial model and dividends Consider a onestep binomial model on an underlying stock S which pays a cash dividend D on the maturity date T. Assume that the values , already take into account any predictable stock price reduction caused...
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yesterday

Binomial model and forward contracts Consider the onestep binomial model. (a) Show that the...
Binomial model and forward contracts Consider the onestep binomial model. (a) Show that the binomial model is consistent with the arbitrage price formula for forward contracts. In other words, apply the model to a forward contract with strike K and...
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22 hours ago

‘Atthemoneyforward’ options The stock of ABC Inc. currently trades at $100 and does not pay any..
‘Atthemoneyforward’ options The stock of ABC Inc. currently trades at $100 and does not pay any dividend. Analysts predict that the price may rise or fall by 10% every 6 months and that the riskfree rate will remain at 5% per annum...
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yesterday

Twostep binomial tree The stock of Schultz AG currently trades at €120 in Frankfurt. Below is a...
Twostep binomial tree The stock of Schultz AG currently trades at €120 in Frankfurt. Below is a tree of your analysts’ expectations for the evolution of the stock price over the next two years. Your analysts also predict that no dividend...
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yesterday

Forward interest rate. This problem is about forward interest rates and goes beyond the scope of...
Forward interest rate. This problem is about forward interest rates and goes beyond the scope of equity derivatives. Consider a market with no arbitrage and infinite liquidity where investors can lend and borrow money for any maturity T at the...
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yesterday

Forward exchange rate. This problem is about forward contracts in foreign exchange and goes beyond..
Forward exchange rate. This problem is about forward contracts in foreign exchange and goes beyond the scope of equity derivatives. The spot exchange rate of the euro is S dollars, i.e. to buy one euro one must pay S dollars. The euro zone yield...
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yesterday

(i) What is the value of a knockout call with barrier H = 95? What about a knockin call with...
(i) What is the value of a knockout call with barrier H = 95? What about a knockin call with barrier H = 95? (ii) What qualitative difference do you see between a knockout call with barrier H = 80 and a knockout call with barrier H = 110? (iii)...
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23 hours ago

Barrier option A ‘knockout barrier option’ is a call or put option which may only be exercised at..
Barrier option A ‘knockout barrier option’ is a call or put option which may only be exercised at maturity if the price of the underlying never hits a preagreed barrier price H throughout the life of the option. Symmetrically, a...
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yesterday

Successive dividends Find formulas for the price f0 of the forward contract when: (a) Two cash...
Successive dividends Find formulas for the price φ0 of the forward contract when: (a) Two cash dividends D1, D2 are paid out at times 0 (b) Two proportional dividends at rates d1, d2 are paid out at times 0 (c) A cash dividend D and then a...
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yesterday

The S&P 500 index is worth $1,200, the price of a 1year atthemoney American call on the S&P 500..
The S&P 500 index is worth $1,200, the price of a 1year atthemoney American call on the S&P 500 is $144, the interest rate curve is flat at 1% per annum, and the annual dividend rate is 2%. What is the maximum price you would pay for a...
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yesterday

Arbitrage arguments Establish the following relationships at an arbitrary point in time t = T....
Arbitrage arguments Establish the following relationships at an arbitrary point in time t ≤ T. Assume no arbitrage, infinite liquidity, and the ability to shortsell. All option characteristics are the same (underlying S, maturity T, strike K)....
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yesterday

Forward price and price of the forward contract at an arbitrary time t Generalize the formulas for..
Forward price and price of the forward contract at an arbitrary time t Generalize the formulas for the forward price and the arbitrage price of the forward contract (Equations (51) p.51 and (52) p.53) at an arbitrary point in time 0 ≤ t...
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22 hours ago

Option strategies. The two questions are independent. (a) Consider a 3month atthemoney straddle..
Option strategies. The two questions are independent. (a) Consider a 3month atthemoney straddle on Kroger Co.’s stock which currently trades at $24. The cost of the straddle is $1.92 and the 3month interest rate is 3%. At what stock price...
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22 hours ago

Option payoffs. The three questions are independent. (a) In each of the three examples in identify..
Option payoffs. The three questions are independent. (a) In each of the three examples in identify the underlying assets, the maturity date, and the payoff formula. (b) Find a portfolio of European options on an underlying asset S with maturity T...
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yesterday

Forward price and price of the forward contract Microsona Corp.’s stock currently trades at 5,200 in
Forward price and price of the forward contract Microsona Corp.’s stock currently trades at 5,200 in Tokyo. The 1year riskfree interest rate is 1% p.a. Calculate the 1year forward price of Microsona Corp. and the price of a 6month forward...
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yesterday

True or False? The three questions are independent. (a) “The value of a derivative security is...
True or False? The three questions are independent. (a) “The value of a derivative security is always positive.” (b) “With options you can never lose any money.” (c) “Assuming no dividends, the time value of a European...
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21 hours ago

Portfolio optimization on n uncorrelated assets Consider n uncorrelated assets with volatilities s 1
Portfolio optimization on n uncorrelated assets Consider n uncorrelated assets with volatilities σ 1 , σ 2 , ... , σn. Let P be a portfolio with weights w 1 , w 2 , ... , wn. Show that the minimum variance of the ...
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yesterday

Portfolio optimization on n correlated assets Consider n assets all having the same volatility of...
Portfolio optimization on n correlated assets Consider n assets all having the same volatility of 100% and pairwise correlation ρ. Let P be a portfolio with weights w 1 , w 2 , ... , w n . (a) Show...
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yesterday

General portfolio optimization on 2 assets Consider two assets A and B with returns R A and R B ,...
General portfolio optimization on 2 assets Consider two assets A and B with returns R A and R B , volatilities σ A and σ B, and correlation ρ. Let P be a portfolio of A and B with weights w and 1 – w respectively. (a) Can the...
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yesterday

Currency portfolio. It is recommended that you solve this problem using a spreadsheet. You are a...
Currency portfolio. It is recommended that you solve this problem using a spreadsheet. You are a eurozone investor with 1 billion euros to be invested in dollars (USD), yen (JPY), or pounds sterling (GBP). You are given the following market data and...
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yesterday

Volatility and Sharpe ratio (a) Based on Table 41 p.35, calculate the pairwise correlation...
Volatility and Sharpe ratio (a) Based on Table 41 p.35, calculate the pairwise correlation coefficients ρ1,2, ρ1,3 and ρ2,3 for Kroger Co., the TBond and Coast Value LP. (b) Verify that Equation (41) p.41 matches the level of...
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yesterday

Risk premium and CAPM Following the CAPM, the riskfree rate is 3% and the expected market return is
Risk premium and CAPM Following the CAPM, the riskfree rate is 3% and the expected market return is 7%. Calculate the risk premium and Sharpe ratio of the following assets: (a) Pschitzer Pharmaceuticals (stock): 15% volatility and 1.5 beta; (b)...
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yesterday

a.Riskfree rate and Sharpe ratio Using the data for the Treasury bond in determine the theoretical.
a.Riskfree rate and Sharpe ratio Using the data for the Treasury bond in determine the theoretical riskfree rate rf so that the Sharpe ratio of the TBond be equal to 1. b: Risk and return of Richky Corp. The table below gives the stock price of...
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yesterday

True or False? The three questions are independent. (a) “The average monthly return of Kroger Co. in
True or False? The three questions are independent. (a) “The average monthly return of Kroger Co. in 200910 was 0.28% (including dividends). Therefore, its annual return was (1 + 0.28%) 12 − 1 ≈...
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yesterday

Zerocoupon rate curve and expectations. The shortterm zerocoupon rate curve of the euro zone is..
Zerocoupon rate curve and expectations. The shortterm zerocoupon rate curve of the euro zone is given as: The current refinancing rate of the European Central Bank (ECB) is at 2.75%. This is the rate at which banks can borrow from the ECB for 2...
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yesterday

Price sensitivity and convexity. It is recommended to use a spreadsheet to solve this problem....
Price sensitivity and convexity. It is recommended to use a spreadsheet to solve this problem. Suppose the US zerocoupon rate curve is given as: Consider the...
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yesterday

Arbitrage price formula Let A be a financial security that pays 3 annual cash flows F 1 , F 2 , F 3.
Arbitrage price formula Let A be a financial security that pays 3 annual cash flows F 1 , F 2 , F 3 , and let X, Y, Z be the 1, 2, 3year zerocoupon bonds with face value 1 and prices P X , P Y , P Z respectively. (a) Show that the arbitrage price...
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yesterday

Zerocoupon rate curve On the German bond market investors can buy and sell: ? Bond A – maturity: 1.
Zerocoupon rate curve On the German bond market investors can buy and sell: • Bond A – maturity: 1 year, zero coupon, price: €97 • Bond B – maturity: 2 years, fixed annual coupon, yield: 4%, price: €100 •...
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yesterday

a. Arbitrage price Using the zerocoupon rate curve in Figure 33 p.28, calculate the arbitrage...
a. Arbitrage price Using the zerocoupon rate curve in Figure 33 p.28, calculate the arbitrage price of a 10year bond with $500 face value and 6% annual coupon. b: Zerocoupon bond portfolio Using the following...
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yesterday

Dividend announcement At the annual stockholders’ meeting of Tankai Corp., a hightech company...
Dividend announcement At the annual stockholders’ meeting of Tankai Corp., a hightech company headquartered in Tokyo, a dividend distribution of 400 per share was voted. The market reacted positively to the news and the stock price is about to...
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yesterday

Bond arbitrage On the UK bond market investors can buy and sell: ? Bond A – maturity: 1 year, zero..
Bond arbitrage On the UK bond market investors can buy and sell: • Bond A – maturity: 1 year, zero coupon, face value: £100, price: £90 • Bond B – maturity: 2 years, face value: £1,000, annual coupon: £50,...
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yesterday