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1 Million+ Step-by-step solutions * Q:Write out a balanced chemical equation for the reaction thatWrite out a balanced chemical equation for the reaction that occurs in step 15 of the procedure. Q:Butter contains esters of glycerol with several different carboxylic acids.Butter contains esters of glycerol with several different carboxylic acids. We refer below to butyric (butanoic) acid (R = C3) for simplicity, but many of the acids have R= C15-C17. Q:Name some other household products that contain fats that couldName some other household products that contain fats that could be used in this experiment instead of butter. Q:Why is the soap layer the upper layer and theWhy is the soap layer the upper layer and the aqueous layer the lower layer? Q:What is the colour of the universal indicator paper? WhatWhat is the colour of the universal indicator paper? What is the pH? Explain your observations. Q:What did you observe when you added the soap toWhat did you observe when you added the soap to the hard water and soft water ? Explain your observations. Q:Note what you observe when you add salt to theNote what you observe when you add salt to the soap solution, and explain it. Q:What is the cause of the cleansing properties of soap?What is the cause of the cleansing properties of soap? (Hint: look up “micelles” in an organic chemistry textbook). Q:Predict the product of the following reaction:Predict the product of the following reaction: Q:In the context of Example 4.6, use dynamic programming toIn the context of Example 4.6, use dynamic programming to compute the optimal portfolio for the general power utility, U(x) = x γ/γ . Q:Find the optimal portfolio for the exponential utility from terminalFind the optimal portfolio for the exponential utility from terminal wealth in continuous time by solving the HJB PDE. Q:Using the martingale approach in the single-period binomial model, findUsing the martingale approach in the single-period binomial model, find the optimal portfolio strategy for maximizing E[log(X(1)] and E[Xγ|(1)= γ], for γ Q:You have a choice between two investment opportunities. One paysYou have a choice between two investment opportunities. One pays $20,000 with certainty, while the other pays $30,000 with probability 0.2, $6,000 with probability 0.4, and $1,000 with probability 0.4. Your utility is of the type U(x) = xγ, 0 Q:Using the martingale approach in the two-period binomial model, findUsing the martingale approach in the two-period binomial model, find the optimal portfolio strategy for maximizing E[1−expf−X(1)g]. Q:In the context of Example 4.10, find the optimal portfolioIn the context of Example 4.10, find the optimal portfolio for the exponential utility using the duality approach. Q:Find the optimal portfolio and consumption strategies for the logFind the optimal portfolio and consumption strategies for the log utility in continuous-time, by solving the HJB PDE (4.43). try to find a solution of the HJB PDE of the form V (t, x) = f(t) + g(t) log x.) Q:Given a random variable C whose value is known byGiven a random variable C whose value is known by time T, and such that E[|C|] is finite, show that the process M(t) := Et[C] is a martingale on the time interval [0, T]. Q:Let A(x) denote the absolute risk aversion of utility functionLet A(x) denote the absolute risk aversion of utility function U(x). What is the absolute risk aversion of utility function V (x) = a + bU(x)? Q:Suppose your utility function is U(x) = log(x). You areSuppose your utility function is U(x) = log(x). You are considering leasing a machine that would produce an annual profit of $10; 000 with probability p = 0:4 or a profit of $8; 000 with probability p = 0:6. What is the certainty equivalent for this random return? Q:Consider a single-period binomial model: The price of the stockConsider a single-period binomial model: The price of the stock at time 0 is S(0) = 100. At time 1 it can move up to 110 with probability 1/3, and down to 90 with probability 2/3. There is also a bank account that pays interest r = 5% per period. The agent has exponential utility U(X(1)) = −e−0.03X(1). If the agent has $100 as initial capital, how much should she invest in the stock, in order to maximize her expected utility? Q:You can invest in asset 1 with μ1 = 0.1,You can invest in asset 1 with μ1 = 0.1, σ1 = 0.3 and asset 2 with μ2 = 0.2, σ2 = 0.5, with correlation p = 0.2. You can also invest in the risk-free asset with return R = 0.05. Find the optimal mean-variance portfolio for the given mean return μ = 0.2. Q:Suppose that the risk-free rate is 5%. There are threeSuppose that the risk-free rate is 5%. There are three risky portfolios A, B and C with expected returns 15%, 20%, and 25%, respectively, and standard deviations 5%, 10%, and 14%. You can invest in the risk-free security and only one of the risky portfolios. Which one of them would you choose? What if the risk-free rate is 10%? Q:Compute the historical daily 90% VaR of a portfolio whoseCompute the historical daily 90% VaR of a portfolio whose daily losses in the last 10 days were, in millions of dollars (minus sign indicates a profit). 1,−0.5,−0.1, 0.7, 0.2, 0.1,−0.2,−0.8,−0.3, 0.5 . Q:Compute the daily 99% and 95% VaR of a portfolioCompute the daily 99% and 95% VaR of a portfolio whose daily return is normally distributed with a mean of 1% and a standard deviation of 0.5%. The current value of the portfolio is $1 million. Q:Small investor Taf has 70% of his portfolio invested inSmall investor Taf has 70% of his portfolio invested in a major market-index fund, and 30% in a small-stocks fund. The mean monthly return rate of the market-index fund is 1.5%, with standard deviation 0.9%. The small-stocks fund has the mean monthly return rate of 2.2% with standard deviation of 1.2%. The correlation between the two funds is 0.13. Assume normal distribution for the return rates. What is the monthly VaR at 99% level for the Taf’s portfolio if the portfolio value today is $100, 000? Q:Consider a mutual fund F that invests 50% in theConsider a mutual fund F that invests 50% in the risk-free security and 50% in stock A, which has expected return and standard deviation of 10% and 12%, respectively. The risk-free rate is 5%. You borrow the risk-free asset and invest in F so as to get an expected return of 15%. What is the standard deviation of your investment? Q:Consider two European call options on the same underlying andConsider two European call options on the same underlying and with the same maturity, but with different strike prices, K1 and K2 respectively. Suppose that K1 > K2. Prove that the option prices c(Ki) satisfy K1 − K2 > c(K1) − c(K2). Q:Provide no-arbitrage arguments for equation (6.9).Provide no-arbitrage arguments for equation (6.9). Q:Argue equation (6.13) for forward contracts.Argue equation (6.13) for forward contracts. Q:Provide detailed no-arbitrage arguments for expression (6.5).Provide detailed no-arbitrage arguments for expression (6.5). Q:Consider a single-period binomial model of Example 6.3. Suppose youConsider a single-period binomial model of Example 6.3. Suppose you have written an option that pays the value of the squared difference between the stock price at maturity and $100.00; that is, it pays [S(1)−100]2. What is the cost C(0) of the replicating portfolio?. Construct arbitrage strategies in the case that the option price is less than C(0) and in the case that it is larger than C(0). Compute the option price as a risk-neutral expected value. Q:Given a random variable C whose value is known byGiven a random variable C whose value is known by time T, and such that E[|C|] is infinite, show that the process M(t) : = Et[C] is a martingale on the time interval [0, T]. Q:Assume that the future dividends on a given stock SAssume that the future dividends on a given stock S are known, and denote their discounted value at the present time t by (t). Argue the following:Q:Why is an American option always worth more than itsWhy is an American option always worth more than its intrinsic value? (As an example, recall that the intrinsic value at time t for the call option is max(S(t) − K; 0).) Q:Given stock trades at $95, and the European calls andGiven stock trades at $95, and the European calls and puts on the stock with strike price 100 and maturity three months are trading at $1.97 and $6 respectively. In one month the stock will pay a dividend of $1. The prices of one-month and three-month T-bills are $99.60 and $98.60; respectively. Construct an arbitrage strategy, if possible. Q:In order to avoid the problem of implied volatilities beingIn order to avoid the problem of implied volatilities being different for different strike prices and maturities, a student of the Black-Scholes theory suggests making the stock’s volatility σ a function of K and T, σ(K, T). What is wrong with this suggestion, at least from the theoretical/modeling point of view? (In practice, though, traders might use different volatilities for pricing options with different maturities and strike prices.) Q:In a two-period CRR model with τ = 1% perIn a two-period CRR model with τ = 1% per period, S(0) = 100, u = 1.02 and d = 0.98, consider an option that expires after two periods, and pays the value of the squared stock price, S2(t), if the stock price S(t) is higher than $100.00 when the option is exercised. Otherwise (when S(t) is less or equal to 100), the option pays zero. Find the price of the American version of this option. Q:Find the price of a 3-month European call option withFind the price of a 3-month European call option with K = 100, r = 0.05, S(0) = 100, u = 1.1 and d = 0.9 in the binomial model, if a dividend amount of D = $5 is to be paid at time τ = 1.5 months. Use the binomial tree with time step Δt = 1=12 years to model the process SG(t) = S(t) − e−r(τ −t)D for t Q:Consider the following two-period setting. the price of a stockConsider the following two-period setting. the price of a stock is $50. Interest rate per period is 2%. After one period the price of the stock can go up to $55 or drop to $47 and it will pay (in both cases) a dividend of $3. If it goes up the first period, the second period it can go up to $57 or down to $48. If it goes down the first period, the second period it can go up to $48 or down to $41. Compute the price of an American put option with strike price K = 45 that matures at the end of the second period. Q:In a two-period CRR model with r = 1% perIn a two-period CRR model with r = 1% per period, S(0) = 100, u = 1.02, and d = 0.98, consider an option that expires after two periods, and pays the value of the squared stock price, S2(t), if the stock price S(t) is higher than $100.00 when the option is exercised. Otherwise (when S(t) is less or equal to 100), the option pays zero. Find the price of the European version of this option. Q:Consider a Merton-Black-Scholes model with r = 0.07, σ =Consider a Merton-Black-Scholes model with r = 0.07, σ = 0.3, T = 0.5 years, S(0) = 100, and a call option with the strike price K = 100. Using the normal distribution table (or an appropriate software program), find the price of the call option, when there are no dividends. Repeat this exercise when (a) the dividend rate is 3%, (b) the dividend of $3.00 is paid after three months. Q:In the context of the previous two problems, with noIn the context of the previous two problems, with no dividends, compute the price of the chooser option, for which the holder can choose at time t1 = 0.25 years whether to hold the call or the put option. Q:Let S(0) = $100:00, K1 = $92:00, K2 = $125:00,Let S(0) = $100:00, K1 = $92:00, K2 = $125:00, r = 5%. Find the Black-Scholes formula for the option paying in three months $10:00 if S(T) ≤ K1 or if S(T) ≥ K2, and zero otherwise, in the Black-Scholes continuous-time model. Q:Show that, if S is modeled by the Merton-Black-Scholes model,Show that, if S is modeled by the Merton-Black-Scholes model, then S and its futures price have the same volatility. Q:Compute the price of a European call on the yen.Compute the price of a European call on the yen. The current exchange rate is 108, the strike price is 110, maturity is three months and t he price of a three-month T-bill is $98.45. We estimate the annual volatility of the yen-dollar exchange rate to be 15%. A three-month pure-discount yen-denominated risk-free bond trades at 993 yen (nominal 1,000) Q:Consider a single-period binomial model with two periods where theConsider a single-period binomial model with two periods where the stock has an initial price of $100 and can go up 15% or down 5% in each period. The price of the European call option on this stock with strike price $115 and maturity in two periods is $5.424. What should be the price of the risk-free security that pays $1 after one period regardless of what happens? We assume, as usual, that the interest rate r per period is constant. Q:Suppose that the stock price today is S(t) = 2:00,Suppose that the stock price today is S(t) = 2:00, the interest rate is r = 0%, and the time to maturity is 3 months. Consider an option whose Black-Scholes price is given by the function V (t, s) = s2e2(T−t) , where the time is in annual terms. What is the option price today? What is the volatility of the stock equal to? Q:Show that the interest rate r(t) in the Vasicek modelShow that the interest rate r(t) in the Vasicek model has a normal distribution. Q:In the previous problem, show thatIn the previous problem, show thatQ:Let σ(t) be a deterministic function such that Consider the process UseLet Ï(t) be a deterministic function such that*

Consider the process

Use Ito’s rule to show that this process satisfies

dZ = ÏZdW.

Deduce that this process is a martingale process. Use this fact to find the moment-generating

Function

f(y) = E[eyX]

of the random variable

Finally, argue that X is normally distributed, with mean zero and variance

Q:Do you think that the put-call parity holds in theDo you think that the put-call parity holds in the presence of default risk? Why?

Q:The price of three-month and nine-month T-bills are $98.788 andThe price of three-month and nine-month T-bills are $98.788 and $96.270, respectively. In our model of the term structure, three months from today the six-month interest rate will be either 5.5% or 5% (in equivalent annual terms). Compute the price of a three-month European put written on the nine-month pure discount bond, with strike price $97.5.

Q:You are a party to a swap deal with aYou are a party to a swap deal with a notional principal of $100 that has 4 months left to maturity. The payments take place every three months. As a part of the swap deal you have to pay the three-month LIBOR rate, and in exchange you receive the fixed 8% rate (total annually) on the notional principal. The prices of the one-month and four-month risk-free pure discount bonds (nominal $100) are $99:6 and $98:2, respectively. At the last payment date the three-month LIBOR was 7%. Compute the value of the swap.

Q:Show that the value of the swaption S+(T) is equalShow that the value of the swaption S+(T) is equal to the value of the cash flow of call options ΔT[R(T)− ]+ paid at times t = T1,. . . , Tn, where R(T) is the swap rate at time T, for the swap starting at t = T and maturing at t = Tn.

Q:Consider a floating-rate coupon bond which pays a coupon ciConsider a floating-rate coupon bond which pays a coupon ci at time Ti, i = 1, . . , n, where the coupons are given by

ci = (Ti − Ti−1)L(Ti−1, Ti)

and Ti – Ti-1 = ΔT is constant. Show that the value of this bond at time t Q:In Example 8.3 find the price of the at-the-money callIn Example 8.3 find the price of the at-the-money call option on the three-year bond, with option maturity equal to two years.

Q:Consider a binomial model with a stock with starting priceConsider a binomial model with a stock with starting price of $100. Each period the stock can go up 5% or drop 3%. An investment bank sells for $0.80 a European call option on the stock that matures after five periods and has a strike price of $120. Interest rate per period is 2%. Describe the steps to be taken by the investment bank in order to start hedging this short position at the moment the option is sold.

Q:In the previous problem suppose that another option with theIn the previous problem suppose that another option with the same maturity is available with the Black-Scholes price given by the function

c(t; s) = s3e6(T−t) .

If you still hold 10 units of the first option, how many options of the second type and how many shares of the stock would you buy or sell to make a portfolio both delta neutral and gamma-neutral (gamma equal to zero)?

Q:Show that the payoff given by equation (9.6) is, indeed,Show that the payoff given by equation (9.6) is, indeed, equal to the payoff of the butterfly spread. Also show that the butterfly spread can be created by buying a put option with a low strike price, buying another put option with a high strike price, and selling two put options with the strike price in the middle.

Q:The Black-Scholes price of a three-month European call with strikeThe Black-Scholes price of a three-month European call with strike price 100 on a stock that trades at 95 is 1.33, and its delta is 0.3. The price of a three-month pure discount risk-free bond (nominal 100) is 99. You sell the option for 1:50 and hedge your position. One month later (the hedge has not been adjusted), the price of the stock is 97, the market price of the call is 1:41, and its delta is 0.36. You liquidate the portfolio (buy the call and undo the hedge). Assume a constant risk-free interest rate and compute the net profit or loss resulting from the trade.

Q:The stock of the pharmaceutical company “Pills Galore” is tradingThe stock of the pharmaceutical company “Pills Galore” is trading at $103. The European calls and puts with strike price $100 and maturity in one month trade at $5.60 and $2.20, respectively. In the next three weeks the FDA will announce its decision about an important new drug the company would like to commercialize. You estimate that if the decision is positive the stock will jump to above $110, and if the decision is negative it will drop below $95. Is it possible to construct a strategy that will yield a profit if your estimates are correct? Explain.

Q:Consider a two-period binomial model with a stock that tradesConsider a two-period binomial model with a stock that trades at $100. Each period the stock can go up 25% or down 20%. The interest rate is 10%. Your portfolio consists of one share of the stock. You want to trade so that the value of your modified portfolio will not drop below $90 at the end of the second period. Describe the steps to be taken in order to achieve this goal. Only the stock and the bank account are available for trading.

Q:Suppose that the annual interest rate is 4%. You haveSuppose that the annual interest rate is 4%. You have a liability with a nominal value of $300, and the payment will take place in two years. Construct the durationimmunizing portfolio that trades in two pure discount bonds: a one-year pure-discount bond with nominal value $100 and a three-year pure-discount bond with nominal value $100.

a. How many units of each bond should the portfolio hold?

b. If the rate drops to 3:5% after one year, is the value of all your positions at that time positive or negative? How much is it exactly?

Q:Consider a zero-coupon bond with nominal $100 and annual yieldConsider a zero-coupon bond with nominal $100 and annual yield of 5%, with one year to maturity. You believe that after one week the yield will change from 5% to 5:5%. Find the expected change in the bond price in three ways:

a. Exactly, computing the new price

b. Approximately, using the initial duration

c. Approximately, using the initial duration and convexity

Q:Explain why expression (11.14) is the continuous-time version of expressionExplain why expression (11.14) is the continuous-time version of expression (11.13), and find the Black-Scholes formula for the call option on the continuous geometric mean (11.14).

Q:Use the retrieval of volatility method to find the initialUse the retrieval of volatility method to find the initial value of the optimal portfolio for maximizing the expected log-utility of terminal wealth E[log(Xx,π(T))], in the Black-Scholes model, with any parameters you wish to choose. Compare to the exact analytic solution derived in chapter 4.

Q:The one-year spot rate is 6%: According to your modelThe one-year spot rate is 6%: According to your model of the term structure you simulate the values of the one-year spot interest rate that could prevail in the market one year from now. You do only five simulations and get 6%, 6:52%, 6:32%, 5:93%, and 6:41%: Compute, using Monte Carlo, the rough approximation of the price of a one year European call on the two-year pure discount bond with strike price 94.

Q:Why do you think it is not easy to applyWhy do you think it is not easy to apply the Monte Carlo method to compute prices of American options?

Q:Recompute all the examples in section 12.1 with the logRecompute all the examples in section 12.1 with the log utility replaced by the exponential utility U(x) = 1 − e−ax.

Q:The risk-free rate, average return of portfolio P and averageThe risk-free rate, average return of portfolio P and average return of the market portfolio are, respectively, 4%, 8%, and 8%. The estimated standard deviation of the market portfolio is 12%, and the estimated nonsystematic risk of portfolio P is 15%. The Jensen index for portfolio P is 1%. What can you say about the performance of portfolio P?

Q:In a CAPM market, the expected return of the marketIn a CAPM market, the expected return of the market portfolio is 20%, and the risk-free rate is 7%. The market standard deviation is 40%. If you wish to have an expected return of 30%, what standard deviation should you be willing to tolerate? How would you attempt to achieve this if you had $100.00 to invest?

Q:Suppose that we estimate the standard deviation of a portfolioSuppose that we estimate the standard deviation of a portfolio P to be 10%, the covariance between P and the market portfolio to be 0.00576, and the standard deviation of the market portfolio to be 8%. Find the idiosyncratic risk of P.

Q:The expected return and standard deviation of the market portfolioThe expected return and standard deviation of the market portfolio are 8% and 12%, respectively. The expected return of security A is 6%. The standard deviation of security B is 18%, and its specific risk is (10%)2. A portfolio that invests 1/3 of its value in A and 2/3 in B has a beta of 1. What are the risk-free rate and the expected return of B according to the CAPM?

Q:Using electronegativity values from Table 1-2 (in Section 1-3), identifyUsing electronegativity values from Table 1-2 (in Section 1-3), identify polar covalent bonds in several of the structures in Problem 25 and label the atoms δ+ and δ- , as appropriate.

Q:Draw a Lewis structure for each of the following species.Draw a Lewis structure for each of the following species. Again, assign charges where appropriate.

(a) H-

(b) CH3-

(c) CH3+

(d) CH3

(e) CH3NH3+

(f) CH3O-

(g) CH2

(h) HC2-(HCC)

(i) H2O2 (HOOH)

Q:For each of the following species, add charges wherever requiredFor each of the following species, add charges wherever required to give a complete, correct Lewis structure. All bonds and nonbonded valence electrons are shown.

(a)

(b)

(c)

(d)

(e)

(f)

Q:(a) The structure of the bicarbonate (hydrogen carbonate) ion, HCO3-.(a) The structure of the bicarbonate (hydrogen carbonate) ion, HCO3-. is best described as a hybrid of several contributing resonance forms, two of which are shown here.

(i) Draw at least one additional resonance form,

(ii) Using curved “electron-pushing” arrows, show how these Lewis structures may be interconvert by movement of electron pairs,

(iii) Determine which form or forms will be the major contributors) to the real structure of bicarbonate, explaining your answer on the basis of the criteria in Section 1-5.

(b) Draw two resonance forms for formaldehyde oxime, H2CNOH. As in parts (ii) and (iii) of (a), use curved arrows to interconvert the resonance forms and determine which form is the major contributor.

(c) Repeat the exercises in (b) for formaldehyde oximate ion, [H2CNO]-.

Q:Several of the compounds in Problems 25 and 28 canSeveral of the compounds in Problems 25 and 28 can have resonance forms. Identify these molecles and write an additional resonance Lewis structure for each. Use electron-pushing arrows to illustrate how the resonance forms for each species are derived from one another, and in each case indicate the major contributor to the resonance hybrid.

Q:Draw two or three resonance forms for each of theDraw two or three resonance forms for each of the following species. Indicate the major contributor or contributors to the hybrid in each case.

(a) OCN-

(b) CH2CHNH-

(c) HCONH2 (HCNH2)

(d) O3 (OOO)

(e) CH2CHCH2-

(f) C1O2- (OC1O)

(g) HOCHNH2+

(h) CH3CNO

Q:Compare and contrast the Lewis structures of nitromethane, CH3NO2, andCompare and contrast the Lewis structures of nitromethane, CH3NO2, and methyl nitrite, CH3ONO. Write at least two resonance forms for each molecule. Based on your examination of the resonance forms, what can you say about the polarity and bond order of the two NO bonds in each substance?

Q:Write a Lewis structure for each substance. Within each group,Write a Lewis structure for each substance. Within each group, compare

(i) Number of electrons,

(ii) Charges on atoms, if any,

(iii) Nature of all bonds, and

(iv) Geometry.

(a) Chlorine atom, CI, and chloride ion, CI-

(b) Borane, BH3, and phosphine, PH3

(c) CF4 and BrF4- (C and Br are in the middle)

(d) Nitrogen dioxide, NO2, and nitrite ion, NO2- (nitrogen is in the middle)

(e) NO2, SO2, and C1O2 (N, S, and CI are in the middle)

Q:Use a molecular-orbital analysis to predict which species in eachUse a molecular-orbital analysis to predict which species in each of the following pairs has the stronger bonding between atoms.

(a) H2 or H2+

(b) He2 or He2+

(c) O2 or O2+

(d) N2 or N2+

Q:For each molecule below, predict the approximate geometry about eachFor each molecule below, predict the approximate geometry about each indicated atom. Give the hybridization that explains each geometry.

(a)

(b)

(c)

(d)

(e)

(f)

Q:For each molecule in Problem 35, describe the orbitals thatFor each molecule in Problem 35, describe the orbitals that are used to form every bond to each of the indicated atoms (atomic s, p, hybrid sp, sp2, or sp3).

Q:Draw and show the overlap of the orbitals involved inDraw and show the overlap of the orbitals involved in the bonds discussed in Problem 36.

Q:Describe the hybridization of each carbon atom in each ofDescribe the hybridization of each carbon atom in each of the following structures. Base your answer on the geometry about the carbon atom.

(a) CH3CI

(b) CH3OH

(c) CH3CH2CH3

(d) CH2 == CH2 (trigonal carbons)

(e) HC === CH (linear structure)

(f)

(g)

Q:Depict the following condensed formulas in Kekule (straight-line) notation. (SeeDepict the following condensed formulas in Kekule (straight-line) notation. (See also Problem 42.)

(a) CH3CN

(b)

(c)