FM441 Exam 2018 solutions PDF

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© LSE ST 2018/FM 441Summer 2018 examinationFM 441Derivatives2017 /18 s yllabus only -- not for resit candidatesInstructions to candidatesThis paper contains four questions, each worth 25 marks. Answer all questions.Time Allowed - Reading Time: None Writing Time: 2 hoursYou are supplied with: A list ...

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SOLUTIONS Summer 2018 examination

FM441 Derivatives 2017/18 syllabus only -- not for resit candidates

Instructions to candidates This paper contains four questions, each worth 25 marks. Answer all questions.

Time Allowed - Reading Time: Writing Time:

None 2 hours

You are supplied with:

A list of facts that you may find useful, on the last page of this exam paper

You may also use:

No additional materials

Calculators:

Calculators are allowed in this examination

© LSE ST 2018/FM441

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1. Suppose the evolution of a stock over the next 12 months follows a two-step binomial process. The initial value of the stock, S0 , is £20. The value of the stock in 6 months, S1/2 , can be either £25 or £16, with equal probability. Conditional on the price going up in the first six months, the value S1 of the stock in 12 months can be either £28 or £22 with equal probability, whereas if the stock price falls in the first six months, S1 can take value £20 or £12 with equal probability. Suppose the interest rate is constant at 4% per annum with semi-annual compounding. (a) (13 marks) Consider a binary cash call on the stock expiring in 12 months, struck at £17, with cash payment £100. Value this option without calculating a replicating strategy (you will be asked to do that in the next part).

The effective riskfree rate of return over 6 months is 2%. The risk-neutral probabilities are given by: 25q0 + 16(1 − q0 ) = 1.02 × 20 28qu + 22(1 − qu ) = 1.02 × 25 20qd + 12(1 − qd ) = 1.02 × 16

⇒ ⇒ ⇒

q0 = 0.489 qu = 0.583 qd = 0.540

The payof f of the option is 100 at all nodes except the node at which the terminal stock price is 12. Value of the option at node u: Vu =

100 1.02

Value of the option at node d: Vd =

1 [0.54 1.02

= 98.039. × 100] = 52.941.

Value of the option initially: 1 [q0 Vu + (1 − q0 )Vd] 1.02 1 = [0.489 × 98.039 + (1 − 0.489) × 52.941] 1.02 = 73.52.

V0 =

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(b) (12 marks) Describe in detail the dynamic self-financing strategy that replicates the binary cash call.

At node u, the replicating portfolio has to be invested in the riskfree asset to yield 100 = 98.039. 100, at a cost of 1.02 At node d, we need to hold δd units of stock and save βd in the riskfree asset so that 20δd + 1.02βd = 100, 12δd + 1.02βd = 0. This gives us δd = 12.5 and βd = −147.059 (the replicating portfolio is very leveraged). The cost of this portfolio is 16δd + βd = 52.941. Finally at node 0, we need to hold δ0 units of stock and save β0 in the riskfree asset so that 25δ0 + 1.02βd = 98.039, 16δ0 + 1.02β0 = 52.941. This gives us δ0 = 5.011 and β0 = −26.699 (the replicating portfolio is leveraged). The cost of this portfolio is 20δ0 + β0 = 73.52, which is equal to V0 calculated in part (b).

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2. In this question you are asked to show that the price of an option is convex in the strike price. (a) (8 marks) Let C(K) be the Black-Scholes price of a European call option with strike price K. Calculate ∂ 2 C/∂K 2 and show that it is positive, i.e. the Black-Scholes call price is convex in the strike price. Hint: Use the following fact: Sφ(d1 ) − e−r(T −t) Kφ(d2 ) = 0, where φ(·) is the standard normal pdf, and d1 and d2 are as defined in the Black-Scholes formula in the formula sheet. Differentiating, we get: ! " ∂d1 ∂C ∂d2 ′ −r(T −t) ′ = SΦ (d1 ) KΦ(d2 ) +Φ (d2 ) −e ∂K ∂K ∂K √ Since d2 = d1 − σ T − t, we have ∂d1 /∂K = ∂d2 /∂K. Also ′Φ(x) = φ(x). Therefore, # $ ∂d1 ∂C = Sφ(d1) − e−r(T −t) Kφ(d2 ) − e−r(T −t) Φ (d2 ). ∂K ∂K Using the fact given in the hint: ∂C = −e−r(T −t) Φ (d2 ). ∂K Differentiating again, ∂2C ∂d2 = −e−r(T −t) ′Φ(d2 ) 2 ∂K ∂K e−r(T −t) φ(d2 ) √ = , Kσ T − t which is positive.

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(b) (12 marks) Let C(K) be the price of a European call option with strike price K, but not necessarily in the Black-Scholes setting. Consider strike prices K1 , K2 (with K1 < K2 ) and Kλ defined as follows: Kλ = λK1 + (1 − λ)K2 , where 0 < λ < 1. Show that C(Kλ ) ≤ λC(K1 ) + (1 − λ)C (K2 ), i.e the call price is convex in the strike price. It suffices to show that the payof f of the option with strike Kλ is lower than the payof f of the portfolio combining λ options with strike K1 and (1 − λ) options with strike K2 , i.e. (ST − Kλ )+ ≤ λ(ST − K1 )+ + (1 − λ)(ST − K2 )+ . For ST ≤ Kλ , the Kλ option is OTM, so that the LHS = 0 and the inequality is trivially satisfied. For Kλ < ST < K 2 , the K2 option is OTM while the other two are ITM. The inequality can be written as ST − Kλ ≤ λ(ST − K1 ) which is satisfied if

f

(1 − λ)ST ≤ Kλ − λK1 = (1 − λ)K2 ,

which is satisfied (with strict inequality). Finally, if ST ≥ K2 , none of the options is OTM, and the inequality becomes (ST − Kλ ) ≤ λ(ST − K1 ) + (1 − λ)(ST − K2 ), which is satisfied with equality.

(c) (5 marks) Using the result in part (b), establish the analogous result for European puts. (You should be able to answer this question even if you were unable to answer part (b)). Using put-call parity, we have # $ P (Kλ ) + S − PV(Kλ ) ≤ λ P (K1) + S − PV(K1 ) # $ + (1 − λ) P (K2 ) + S − PV(K2 ) ⇒ P (Kλ ) ≤ λP (K1 ) + (1 − λ)P (K2). Thus the European put price is also convex in the strike price.

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3. (a) For this question, allow the instantaneous riskfree rate to follow an arbitrary stochastic process. Consider a forward contract, entered into at date 0, to buy a non-dividend paying underlying at date T . i. (8 marks) Find the marked-to-market value Vt of this contract at time t < T in terms of the spot price at date t, St , and the forward price at date 0, F0 . At t, consider the trading strategy of buying the underlying and selling it forward at T . This involves an investment of St and a single riskless payof f of Ft . Hence it must earn the same as the yield on a zero-coupon maturing at T : Ft = ey(t,T )(T −t) St . The marked-to-market value of the forward contract is given by Vt = e−y(t,T )(T −t) (Ft − F0 ) % & = e−y(t,T )(T −t) ey(t,T )(T −t) St − F0 = St − e−y(t,T )(T −t) F0 .

ii. (7 marks) Now suppose that St = S0 . Find a condition on zero-coupon prices under which Vt = 0. We have Vt = St − e−y(t,T )(T −t) ey(0,T )T S0 If St = S0 , Vt = 0 if and only if y(0, T )T = y(t, T )(T − t), or, equivalently, P (0, T ) = P (t, T ).

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(b) Consider a futures contract in a discrete-time setting with a deterministic (not necessarily constant) riskfree rate. i. (5 marks) Show that the futures price is a martingale under the risk-neutral measure. Consider the trading strategy wherein you buy the futures contract at time t and sell it after settlement at time t + 1. Your payof f is 0 at time t, (ft+1 − ft ) at time t + 1, and 0 thereafter. Therefore, 0 = e−rt EQt (ft+1 − ft ), where rt is the interest rate for the period t to t + 1, and we take the interval between the two settlement dates to be the time unit. It follows that EQt (ft+1 − ft ) = 0 ⇒

ft = EQ t (ft+1).

Using the law of iterated expectations, ft = EQ t (ft+s ),

∀s ≥ 0.

Hence f is a martingale under Q. ii. (5 marks) Show that the futures price is equal to the forward price. Using the martingale property of the futures price, ft = EtQ(fT ) = EQt (ST ). For a forward contract: 0 = e−(rt +rt+1+...+rT −1 ) EQt (ST − Ft )

⇒

Ft = EQ t (ST ).

Therefore, ft = Ft .

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4. (a) (8 marks) What is a payer swaption? Show that it can be viewed (and valued) as a coupon-bond option. A payer swaption with strike sK and maturity date T is an option to enter into a swap at T , paying the fixed swap rate sK . Let N be the notional of the underlying swap. The swaption allows you to enter a swap at T whereby you receive floating and pay the fixed rate sK . Exercising the option effectively means exchanging a fixed-coupon bond (with coupon rate sK ) for a floating-coupon bond (which is worth N), giving the former and receiving the latter. Hence the swaption is akin to an option to sell the fixed-coupon bond at a price of N (i.e. at par). In other words, holding a payer swaption is equivalent to holding a put, struck at N, on a fixed-coupon bond with face value N and coupon rate sK .

(b) You have bought a swaption giving you the right to pay a fixed swap rate sK = 2% over a 2-year period (rates are expressed as annual rates with semi-annual compounding). The swaption expires today. The notional of the underlying swap is £1m and the swap has four payment dates – one payment every six months starting in six months. The prices of zero-coupon bonds with face value £100 and maturities 6, 12, 18 and 24 months are respectively £98, £96, £94 and £92. Before the expiration date, of course, the swaption payof f is unknown. Its realization depends on the yield curve at expiration. In the present case, it is not the entire shape of the yield curve at expiration that matters, but only four points on the curve − at maturities 6, 12, 18 and 24 months. This information determines the swap rate prevailing at maturity (part a) and it also determines the annuity value (part b).

i. (8 marks) What is the fair swap rate today on a two-year swap with semi-annual payments? The fair swap rate sT is such that the value of the fixed-rate bond embedded in the swap is equal to the notional (since the value of the floating-rate bond embedded in the swap is equal to the notional), i.e. 0.5sT N(0.98 + 0.96 + 0.94 + 0.92) + (N × 0.92) = N. Therefore, sT =

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1 − 0.92 = 4.21%. 0.5 × (0.98 + 0.96 + 0.94 + 0.92) Page 8 of 10

ii. (9 marks) What is the value of the swaption today? The swaption has positive value, corresponding to the value of an annuity with semi-annual payments of the amount 0.5 × (0.0421 − 0.02) × 1, 000, 000 = £11, 050 four times starting in 6 months. Indeed, we can obtain this stream of cash f lows costlessly by exercising the (payer) swaption and at the same time entering a swap position, receiving fixed and paying floating, at current market conditions. Given the prices of zero-coupon bonds of all relevant maturities, the value of the swaption is (0.98 + 0.96 + 0.94 + 0.92) × 11, 050 = £41, 990.

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Useful Facts • Put-call parity (non-dividend paying stock): Ct = Pt + St − e−r(T −t) K. • Lognormal property of GBM: Suppose dSt = µSt dt + σSt dWt . Then the conditional distribution of log ST at time t is normal: # % & $ log ST ∼ N log St + µ − σ 2 /2 (T − t), σ 2 (T − t) .

• Black-Scholes formula:

The price of a European call option at t < T is given by Ct = St Φ (d1 ) − e−r(T −t) KΦ (d2 ), where Φ (·) is the standard normal cdf, and d1 =

log(St /K) + (r + σ 2 /2)(T − t) √ , σ T −t

√ d2 = d1 − σ T − t.

• Black-Scholes PDE: Consider a derivative on a stock with payof f g(ST ) at time T , whose price at time t ≤ T is given by F (S, t). The Black-Scholes PDE is 1 Ft (S, t) + rSFS (S, t) + σ 2 S 2 FSS (S, t) − rF (S, t) = 0. 2 • Standard normal density:

1 1 2 φ(x) = √ e− 2 x . 2π

• Expectation of the exponential of a normal random variable: If X normally distributed, then " ! 1 E[exp(X)] = exp E(X) + Var(X) . 2

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