Introductory Course On Financial Mathematics 1st Edition M V Tretyakov - Solutions

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=+4, i.e. an option which promises to pay f(ST) at the maturity time T to its holder with ST being the price of the underlying at T, can be constructed from an appropriate portfolio of a bank account and stock in advance.
=+derivative, i.e. we created the payoff of the considered option using other financial instruments (the bank account and stock in this case).In the above consideration we have not directly used the precise form of the payoff function. So, any European-type derivative with a payoff function f(s)
=+If the writer bought this (hedging) portfolio (ϕ, ψ) at t = 0 and held it, the equations (7.3) guarantee that he achieves his goal: if the stock moves up then the portfolio becomes worth fb and if the stock moves down the portfolio becomes worth fa.Thus, the fair price (premium) the writer
=+Because of our aim to find a fair price, for every possible outcome, the writer must have where f(s) is the payoff function of the option considered (e.g., f(s) = (s – K)+ for a call and f(s) = (K – s)+for a put). If at least one realisation is such that V1 < f(S1), the writer will be unable
=+=1. Consider the writer’s portfolio (ϕ, ψ), namely ϕ amount of stock S andψ amount of money on the bank account B. If we buy the portfolio at time zero, it costs At the next discrete-time moment, t = Δt, it would be worth one of two possible values as demonstrated in Fig. 7.2.Fig. 7.2
=+Table 7.1 The writer’s strategy in Example 7.1.Then the writer has a positive chance of making a profit with no risk of making a loss, i.e. there is arbitrage. The price of the option, $3, is too high though it follows from the law of large numbers.How do we price it correctly, i.e. in an
=+ Due to what we have learned in Part I, we can guess that the answer is ‘no’. The writer of the call the investor is buying could use the strategy (one of many)given in Table 7.1.
=+The investor follows the logic of the law of large numbers from Section 2.1 and calculates the expected value of the option at the maturity time:So, the investor agrees to pay $3 for the option, but is this a fair price?
=+The current price of a certain stock is S0 = $15. A European call maturing in one month (i.e. T = 1/12) has strike price K = $18. An investor believes that with probability of 1/2 the stock price will be either$24 or $12. The riskless borrowing rate is zero, i.e. r = 0. By solving the equations
=+Find the value of the payer swap Πp (t) at t ∈ [Ti–1, Ti ), i = 1, . . . , n, entered into at time s ≤ T0 .
=+(ii) If you speculate on the market, which financial derivatives would you use and how would you use them to try to profit from your prediction?
=+(i) Which financial instruments will you recommend to a pension fund owning a large amount of shares of this company to use in order to protect their business from this risk?
=+28. You expect that due to a reduction of your national government spending on new building programmes there is a large chance of a significant decrease of share prices of a big cement production company within the next two years.
=+with the same strike and on the same underlier are £1 and £2, respectively. Assume that the risk free interest rate is 1% p.a. with continuous compounding. Is there an arbitrage opportunity present?Provide a full justification of your answer.
=+Suppose the bid and ask prices for a one-year European call with strike£30 on a non-dividend paying underlier with spot price £35 are £6 and£7, respectively, and the bid and ask prices for a one-year European put
=+27. In a real market, bid and ask prices5 differ, i.e. there is a bid-ask spread.
=+26. What is the value of a European put option with strike K = 0?
=+25. Today is 10 August and the spot price of an XYZ share is £5. Also, European calls and puts on XYZ with strikes £3.5, £4, £4.5, £4.75, £5,£5.25, £5.5 and with maturities in September, November this year and January next year are now sold on the market. The XYZ’s executive board will
=+24. (Chooser option). Let time moments T and s ∈ [0, T], a strike price K >0 and a continuously compounded interest rate r be given. A chooser option is a contract sold at time t = 0 that gives its holder the right to choose at the time t = s either a call or a put which both expire at time t=
=+23. Let the interest rate be zero. Consider a European put option with exercise price K and maturity time T on the underlier with the price St at time t. Denote by Dt the value of this put at time t. Confirm by arbitrage arguments that Dt≥ (K – St)+for all t ≤ T.
=+22. If two otherwise identical European call options with values and have exercise prices K1 and K2, respectively, and K1< K2, then confirm by arbitrage arguments that for all t ≤ T.
=+21. Assuming that the interest rate r is constant, prove the following inequality for a European call price C0 at time t = 0 with strike K and maturity T:S0– Ke–rT ≤ C0, where S0 is the price of the underlier at t = 0.
=+20. Use arbitrage arguments to confirm the following inequality for European call option:Ct≤ St, for all t ≤ T.
=+19. Draw the payoff diagram at maturity of a bull spread with a long position in a call with strike £30 and a short position in a call with strike £35 (both calls are on the same underlying and with the same maturity).
=+18. At time t = 0 an investor bought (long position) one European call with strike K1=£20 and one European call with strike K2=£40, she also borrowed (short position) two European calls with strike K3=£30. All the options are on the same underlying and with the same maturity T.Such a portfolio
=+17. Show that the values of European put and call with the same strike, maturity and underlying are equal if and only if the strike is equal to the forward price.
=+16. How can a long forward contract on a stock with a particular delivery price and delivery date be created from European options?
=+synthesise this derivative (i.e. how do you recreate its payoff) using plain vanilla put options?
=+15. Let 0 < K1< K2. At its maturity T, a bear spread pays $A = K2– K1 if the spot price ST of the underlying is less than K1, it pays K2– ST if ST is between K1 and K2, otherwise it expires worthless. How do you
=+14. Give a possible reason why an investor might purchase a call option.
=+(ii) If one agrees to buy a put for $14, what arbitrage opportunities does this create? Demonstrate them via an appropriate strategy.
=+(i) What is the value of the corresponding put?
=+13. Suppose an XYZ stock is selling for $125 per share in February and an XYZ European call with the strike price $130 and expiry in August is selling for $10. Assuming a continuously compounded interest rate of 3% per year,
=+12. What is the difference between a forward and an option?
=+11. The current price of cocoa is $3,136 per tonne. Forward contracts are available to buy or sell 50 tonnes of cocoa at $170,000 with delivery in one year. Money can be borrowed at 3% p.a. What should be the arbitrageur strategy? (Neglect costs of storage and convenience yields.)
=+6.1. Compute the profit/loss of Investors X and Y at the end of 12 months.Table 6.1 The observed performance of the ABC shares and the continuously compounded interest rate rt.
=+due to these adjustments are rolled over at the three-month interest rate.The observed performance of the ABC shares and the continuously compounded interest rate rt over these 12 months are recorded in Table
=+Suppose Investor Y entered into a long futures contract also on 100 shares of ABC for 12 months, according to which margins were to be paid/received every three months in cash, i.e. the difference between the futures’ mark-to-market and book values. Her profits/losses arising
=+Suppose Investor X entered into a long forward contract on 100 shares of ABC for 12 months as OTC, with no regular margins to be paid.(Assume that there is no initial margin3 and maintenance margin4.)
=+10. Mark-to-market (also known as fair value accounting) refers to the accounting act of recording the value of an asset or liability according to its (or similar assets and liabilities) current market value rather than its book value2.
=+(c) Suppose one agrees to enter into an 18-month forward contract on 1,000 barrels of crude oil with the forward price $63,000 when the price is $60.4 per barrel and the interest rate is 2% p.a. with continuous compounding. What arbitrage opportunities does this create?Demonstrate them via an
=+(b) Six months later, the price of oil is dropped to $47.1 per barrel (the interest rate is unchanged). What is the new forward price (for the remaining 12 months) and what is the current value of the original short forward contract?
=+(a) What is the forward price for the contract at the date of entry?
=+9. An 18-month short forward contract on 1,000 barrels of crude oil is entered into when the commodity price is $60.4 per barrel1 and the interest rate is 2% p.a. with continuous compounding. (Neglect costs of storage and convenience yields.)
=+8. Explain the difference between a long and short forward.
=+7. Suppose the fixed-interest rates in the UK and Eurozone are 2% and 4% p.a., respectively, with continuous compounding. Also, assume that in the Euro market the current Sterling exchange rate is €1.200 (so that£1,000 costs €1,200). What is the fair futures price in the Eurozone for a
=+6. The risk-free fixed rate of interest is 2% p.a. with continuous compounding and the dividend yield on a share of an XYZ company is 4% p.a. with continuous compounding (i.e. the asset pays a continuously compounded interest on its instant price). The current value of the share is £100. What is
=+(d) Now suppose that costs of storage and convenience yield should be taken into account. How do these new conditions change your answer to part (a) of this question if we assume that the present value of all the storage costs per unit of this commodity that will be incurred during the life of
=+What arbitrage opportunities does this create? Demonstrate them via an appropriate strategy.
=+(c) Suppose one agrees to enter into a 12-month forward contract on 100 tonnes of feed wheat with the forward price £14,500 when the wheat price is £145.05 per tonne and the interest rate is 0.5% p.a. with continuous compounding.
=+What is the new forward price (for the remaining six months) and what is the current value of the original long forward contract?
=+(b) Six months later, there is a shortage of feed wheat on the market and its price jumps to £151.96 per tonne. The cost of borrowing also increases and the interest rate becomes 1.25%.
=+(a) What is the forward price for the contract at the date of entry?
=+5. A 12-month long forward contract on 100 tonnes of feed wheat is entered into when the commodity price is £145.05 per tonne and the interest rate is 0.5% p.a. with continuous compounding. (Neglect costs of storage and convenience yields.)
=+4. A bank quotes you an interest rate of 3.5% p.a. with quarterly compounding. What is the equivalent rate with(a) daily compounding;(b) monthly compounding;(c) continuous compounding.
=+(b) If you keep this investment for another five years, what will be its value then?
=+3. Exactly five years ago you invested £X in a company which promised to pay you a constant rate of continuously compounded interest of 8%p.a. as long as you do not withdraw the funds. Suppose that you still keep this investment and its current value is £50,000.(a) Find X.
=+An investor put £100,000 into a savings bank account with a fixed interest rate. In return she will receive £111,529 in five years. Calculate the interest rate p.a. with(a) annual compounding;(b) monthly compounding;(c) continuous compounding.
=+ Form a portfolio, which consists of the original portfolio and possibly some derivatives written on the same stock and/or the underlying stock itself, so that this new portfolio is neutral with respect to both delta and vega.
=+28. (Vega hedging). Consider a portfolio with a single underlying stock (i.e.the portfolio that can contain the stock itself and derivatives written on it).
=+Then discuss why it is difficult to hedge ATM options close to their maturity.
=+27. Consider an ATM European call or put option. What happens to its gamma when the option is close to its maturity?
=+26. Show that gamma of the European call coincides with gamma of the European put with the same strike, maturity and written on the same underlying stock.
=+(d) Plot Δ of this option as a function of the underlier’s price for different maturities and strikes. Make observations about delta behaviour with increase of strike and decrease of time to maturity.
=+(c) Give a financial explanation for the observation made in the part (b)of this question.
=+(b) Is it possible to say whether the found delta is always positive or negative?
=+(a) Compute the delta of this option.
=+25. In the Black–Scholes world the price of a European plain vanilla put on a stock that does not pay dividends is equal to where Φ(x) is the normal distribution function.
=+(f) Establish what happens to vega when T goes to zero (i.e. when an option is close to its maturity). Give a financial explanation for your answer.
=+(e) Plot the vega of this option as a function of the underlier’s price and different maturities T.
=+(d) Give a financial explanation for (c).
=+(c) Confirm that the vega you have computed in the part (b) of this question is always positive.
=+(b) Compute the vega of this option.
=+(a) Explain what S0, r, σ, K and T in this formula are.
=+24. In the Black–Scholes world the price of a European plain vanilla call on a stock that does not pay dividends is equal to C0 = v(0, S0) with v(t, x)from (18.44).
=+(b) Using your answer to part (a) of this question, find the function u(t, x).
=+(a) Write a probabilistic representation of the solution u(t, x).
=+23. Consider the Cauchy problem for a PDE where c is some constant.
=+22. Using PDE techniques (without using any probabilistic argument), find the solution of the Cauchy problem (18.42)–(18.43) with f(x) = (x − K)+and thus confirm (18.44).
=+21. The Cox–Ingersoll–Ross model for the short rate r(t) can be written in the form of the following Ito SDE:Compute the expectation Er(t) and variance V ar(r(t)). Also find limt→∞Er(t).
=+20. (Ornstein–Uhlenbeck process). Consider the linear SDE with additive noise dX = −αXdt + σdW(t), X(0) = x.Using the Ito formula, show that this SDE has the solution:
=+19. Exploiting the Ito formula, write an SDE for the process X(t) =cos(W(t)), where W(t) is a standard Wiener process, and compute EX(2).
=+is bounded. Introduce the process X(t) = ln S(t). Using the Ito formula, find an SDE for X(t).
=+with some drift μ(t) and volatility σ(t) which are Ft-adapted, continuous functions of t and Eσ2(t)S 2
=+18. Let S(t) satisfy the SDE:dS = μ(t)Sdt + σ(t)SdW (t), S(0) = S0,
=+, where n is a positive integer, and let W(t) be a standard Wiener process. Using the Ito formula, derive the equation for ξ(n)(t) :=gn(W(t)) and compute EW2n(t).
=+17. Let gn x) = x n
=+16. Using the Ito formula, show that(here W(s) is a standard Wiener process).
=+(t) be independent standard Wiener processes. Using the definition of Ito integral, show that
=+15. Let W1(t) and W2
=+(t) are independent standard Wiener processes.
=+14. Using an appropriate property of Ito integrals, compute where W1(t) and W2
=+13. Let W(t) be a standard Wiener process. Compute
=+(ii) One can prove (Shiryaev, 1996; Williams, 2001) that the rth moment of ξ is equal to the rth derivative of the moment generating function φ(λ)at λ = 0. Using this fact and your answer on (i), show that E [W4(t)] = 3t 2, where W(t) be a standard Wiener process.
=+(i) Find the moment generating function for a Gaussian random variable with zero mean and variance b 2.
=+12. Let ξ be a random variable. The function φ(λ) := E [exp(λξ)] is called the moment generating function of ξ.
=+process is also a standard Wiener process.
=+11. Let W(t) be a standard Wiener process. Fix some s ≥ 0. Show that the
=+10. Let W(t) be a standard Wiener process. Compute the covariances(a) Cov(3W(3) + W(2), W(1));(b) Cov(W(s), W(t));(c) Cov(W(t), W2(t)).
=+9. Let {Ft}t≥0 be a natural filtration for the standard Wiener process W(t).Which of the following stochastic processes are adapted to the filtration{Ft}t≥0 and which are not?(a) Y(t) := maxt≤s≤T W(s);(b) Z(t) := min0≤s≤t W(s).

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Introductory Course On Financial Mathematics 1st Edition M V Tretyakov - Solutions

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