BIRKBECK (University of London) MSc EXAMINATION FOR INTERNAL STUDENTS MSc FINANCIAL ENGINEERING DEPARTMENT OF ECONOMICS, MATHEMATICS AND STATIS- TICS

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1 BIRKBECK (University of London) MSc EXAMINATION FOR INTERNAL STUDENTS MSc FINANCIAL ENGINEERING DEPARTMENT OF ECONOMICS, MATHEMATICS AND STATIS- TICS PRICING EMMS014S7 Tuesday, May , 10:00am-13.15pm (includes 15 minutes reading time). The paper is divided into four sections. There are four questions in each section. Students who attended lectures this academic year should answer TWO questions from section A, and TWO questions from section B. All questions carry the same weight. Students who attended lectures before this academic year should answer TWO questions from section C, and TWO questions from section D. All questions carry the same weight. page 1 of 22 Please turn over

2 SECTION A ( Students: Answer TWO questions from this section.) 1. We want to price European derivatives written on a stock whose price S t is modeled by a Geometric Brownian motion: ds t = µs t dt + σs t dw t. (1) Here (W t ) t 0 is a standard Brownian motion and we assume that µ and σ are constants. Let r be the risk-free rate, assumed to be constant also. We suppose that the stock pays out a continuous dividend at a positive constant rate of q. Let V (S, t) be the price of a European derivative, as function of the stock price and of time. We assume that the derivative s pay-off is a function F (S T ) of stock-price only; here T is the derivative s maturity date. (a) Derive the Black and Scholes equation for V (S, t). (b) By using the Feynman-Kac theorem, or otherwise, show that the price of the derivative is given by ( ) V (S, t) = e rτ F Se (r q 1 2 σ2 )τ+σ τ z e z2 /2 dz, (2) 2π where τ = T t is time-to-maturity. (c) (i) State the definition of the derivative s, and explain the financial importance of this quantity for derivative trading. [1 point] (ii) Starting from (2), derive an integral formula for the derivative s. [3 points] (d) Compute the of a European call with strike K and maturity T. (e) (i) Explain the concept of implied volatility and its relevance. (ii) It is sometimes market practice to quote implied volatilities as a function of the of a call or put, instead of as a function of the moneyness. Explain why this is possible. page 2 of 22

3 2. We consider American options written on futures contract f t whose pricedynamics is modeled by df t = µf t dt + σf t dw t, (3) with (W t ) t 0 a Brownian motion, and µ and σ constants. Let V (f, t) be the price of an American option written on this futures contract. We will specify different pay-off functions later on. (a) (i) How much does it cost to enter a futures contract? Suppose you enter a futures contract at time t 0. How much will you receive net at a later time t 1 > t 0? (ii) Show that the price V (f, t) of a derivative written on the future satisfies prior to exercise the following PDE: V t σ2 f 2 2 V = rv. (4) f 2 (b) Suppose now that the American derivative is perpetual, with a pay-off which does not explicitly depend on time, but only on the price of the underlying. Its price will then be a function V (f) of f only. Show that V (f) is of the form V (f) = C + f λ + + C f λ, (5) where λ ± = σ2 ± σ 4 + 8σ 2 r 2σ 2, (6) and where C +, C are two constants. Also show that λ < 0 while λ + > 1. (c) Determine the price of a perpetual American call with strike K. (d) Consider now a perpetual American straddle with strike K, whose pay-off upon exercise is given by f t K. Assume the straddle will be exercised as soon as the futures contract s price goes above a certain upper limit f or below a certain lower limit f. [3 points] (i) Assuming the perpetual straddle s price is of the form (5), set up a system of four equations in four unknowns for C +, C, f, f. Please note that you are not asked to solve this system. (ii) The pay-off of a straddle is identical to that of being long a put and a call with identical strike K. Will the price of the American straddle therefore be the sum of that of of an American call and an American put? Explain your answer. page 3 of 22 Please turn over

4 3. One of the defects of the Black Scholes model is the assumption of constant volatility for the underlying price process. Assume instead that the stock price is given by ds t = µs t dt + σ t S t dw t, with stochastic volatility σ t evolving according to: dσ t = a(σ t )dt + b(σ t )dz t. Here, W t and Z t are two correlated Brownian motions with constant correlation ρ, and a(σ) and b(σ) are two given functions of σ. Let V = V (S, σ, t) represent the price of a European derivative (such as a call) written on the stock, as function of stock price, volatility and time. (a) Explain how you can continuously hedge the derivative V (S t, σ t, t) by taking a suitable position in the derivative, in the underlying and in a second derivative, V 1 (S t, σ t, t) (such as a call with a different strike and/or maturity) ). Specify the number of stock and derivatives V 1 you need for hedging. What should the return of the resulting portfolio be equal to? Explain your answer. (b) (i) Show, as a consequence, that t V + L(V ) + rs S V rv σ V = tv 1 + L(V 1 ) + rs S V 1 rv 1 σ V 1 where L(V ) is given by L(V )(S, σ, t) = 1 2 σ2 S 2 2 SV + ρσb(σ)s 2 SρV b(σ)2 2 σv, (7) and where r is the risk-free rate, assumed to be constant. [6 points] (ii) From this deduce a pricing equation for V = V (S, σ, t), introducing a new auxiliary function q(s, σ, t). (c) Suppose we write the pricing equation in the form V t + L(V ) + rs S V + (a λb) σ V = rv, for some suitable function λ = λ(s, σ, t). Let Π t = V (S t, σ t, t) ( S V )S t, (8) be the usual hedging portfolio of the derivative which is -hedged with respect the underlying only. page 4 of 22

5 (i) Show that dπ t rπ t dt = λb σ V dt + b σ V dz t. (9) (ii) Explain why λ can be interpreted as the market price of volatility risk. (a) Suppose that σ t follows a mean-reverting Ornstein - Uhlenbeck process: dσ t = α(θ σ t )dt + βdz t. (10) Indicate a natural modeling choice for λ(s, σ, t) in this situation. page 5 of 22 Please turn over

6 4. In practice, the volatility of a stock is not a constant, but stochastic. Volatility derivatives allow investors to take view on future realized values of the volatility or, equivalently, of its square, the variance. Suppose the stock-price S t evolves according to a general stochastic volatility model: ds t = µs t dt + σ t S t dw t. (11) We make no specific modeling assumptions on the evolution of σ t. The quantity T 0 σ 2 t dt, (12) is called the realized variance at the (future) time T and is used as underlying for a range of volatility derivatives, notably the so-called variance swaps, whose pay-off at T is given by T here F is called the swap rate of the volatility swap. 0 σ 2 t dt F ; (13) (a) (i) Show that 1 2 σ2 t dt = ds t S t d (log S t ). (14) (ii) By integrating the relationship found in (i), show that the following trading strategy will give a pay-off at T which is identical to that of a volatility swap: At any time t up till maturity, holding an amount of 2/S t in stock. Shorting a European option with pay-off at T equal to 2 log(s T /S 0 ) and borrowing F e rt at 0. (b) We would like to replicate the European option with pay-off log(s T /S 0 ) with a (continuous) combination of European digital puts and calls. Recall that a European digital put (call) with strike K has pay-off H(K S T ) (resp. H(S T K)) at maturity T, where H(x) is the Heaviside function: H(x) = { 1, x > 0 0, x < 0. (15) (i) Show that 1 0 H(K S) dk K = { log S if S < 1 0 if S > 1. (16) page 6 of 22

7 (ii) Similarly, 1 H(S K) dk K = { log S if S > 1 0 if S < 1. (17) (c) Show that log(s T /S 0 ) = log S 0 + H(S T K) dk 1 1 K H(K S T ) dk 0 K, (18) and use this relation to express the price at time 0 of the European option with pay-off log(s T /S 0 ) at T in terms of prices of digital puts and calls. (d) Determine the value at time 0 of a variance swap with swap rate F. You may assume that the fair value of a future pay-off is obtained by taking discounted expectations, assuming that the evolution of S t is risk-neutral: µ = r, the risk-free rate. (e) How should the swap rate be set to make the variance swap fairly priced at time 0? page 7 of 22 Please turn over

8 SECTION B ( Students: Answer TWO questions from this section.) 5. (a) Consider a Black and Scholes market model with one risky asset S t and one risk-free asset B t, with price-dynamics ds t = µs t dt + σs t dw t db t = rb t dt, B 0 = 1, where W t is a Brownian motion with respect to the probability measure P, and where r, µ and σ are constants. We let F t be the Brownian filtration at time t. (i) Consider a self-financing portfolio V t = φ t S t + ψ t B t, where φ t and ψ t are the units of S t and B t respectively in the portfolio at time t. Prove that the discounted portfolio Ṽt = V t /B t has dynamics dṽt = φ t d S t [3 points] (ii) Prove that, if V T is a self financing portfolio which replicates an F T - measurable claim X, then: 1) there exists a probability measure Q such that S t = S t /B t is a martingale with respect to Q and 2) the time-0 price of X is given by X 0 = E Q (X)/B T. (iii) Consider a claim, with final payoff X at time T given by [6 points] X = { ST if S T S t1 0 elsewhere, where t 1 is an intermediate fixed date with 0 < t 1 < T. Use martingale pricing (risk neutral valuation) to find the time t-price of the claim with t < t 1. [6 points] (b) Consider a random variable z N(0, 1) with respect to a probability P. Suppose that you want to estimate P(z < a) (that is, the probability of z < a with respect to P) by simulation, so that with n simulations you can write n P(z < a) = E P 1 {z< a} 1 {zi < a}/n, where z 1,..., z n are the outcomes of the n simulations. i=1 page 8 of 22

9 (i) Show how to calculate P(z < a) when simulations are sampled from a normal distribution N( a, 1) with mean a instead of 0. [You will have to change probability measure and find the Girsanov weight (Radon-Nykodim derivative).] (ii) Briefly explain what is the advantage of changing probability measure as suggested in point (b)-(i) for estimating P(z < a). [1 point] page 9 of 22 Please turn over

10 6. Consider a market with two risky assets S 1,t and S 2,t, with price-evolution with respect to a probability measure P given by ds 1,t = µ 1 S 1,t dt + σ 1 S 1,t dw t ds 2,t = µ 2 S 2,t dt + σ 2 S 2,t dw t db t = rb t dt, B 0 = 1 with r, µ 1, µ 2, σ 1 and σ 2 constants, and W t a Brownian motion. (a) Taking S 1,t as a numéraire, put B t := S 1 1,t B t and S 2,t := S 1 1,t S 2,t. (i) Show that the P-dynamics of B t and S 2,t are given by: and d B t B t = (r µ 1 + σ 2 1)dt σ 1 dw t, (19) d S 2,t S 2,t = (µ 2 µ 1 + σ 2 1 σ 1 σ 2 )dt + (σ 2 σ 1 )dw t. (20) [You can use the following result without proof d(1/s 1 ) = (σ 2 1 µ 1 )dt/s 1 σ 1 dw/s 1.] [3 points] (ii) Show that if there exists an equivalent probability measure R, with, dr = e γwt γ2 /2t dp, such that B t and S 2,t are martingales with respect to R, then we must have that γ = µ 1 r σ 1 σ 1, (21) and µ 2 r σ 2 = µ 1 r σ 1. (22) [3 points] (b) Assume the market is arbitrage free and focus on the asset S 1 only, which for brevity will simply be denoted as S, with ds t = µs t dt + σs t dw t (that is, µ 1 = µ and σ 1 = σ). Consider a claim with final payoff at time T equal to: { ST if S X = T K 0 elsewhere, where K is a positive constant. (i) Let X t denote the price of the claim at time t prior to T. By using S t as numéraire, show that X t can be written as: X t = S t E R ( 1{ST K} F t ), (23) (where F t is the Brownian filtration) and compute the price. [Reformulate {S T > K} as a condition on B T. ] [6 points] page 10 of 22

11 (ii) Let Q denote the risk-neutral probability measure, so that S t /B t is a Q-martingale and dw Q t = dw t + γ 1 dt is a Brownian motion with respect to Q with γ 1 = (µ r)/σ. Show that dw R t = dw Q t [From part a(ii), we know that dwt R σ.] + γ 2 dt with γ 2 = σ. = dw t +γdt with γ = (µ r)/σ [3 points] (iii) By using Girsanov theorem (and changing probability measure) show that equation (23) can be rewritten as: ( ) ST X t = B t E Q 1 {ST >K} F t, B T [After changing probability measure in equation (23), notice that e 1 2 σ2 (T t)+σ(w Q T W Q t ) = S T St B t B T ] page 11 of 22 Please turn over

12 7. In this question, p(t, T ) denotes the price of a 0-coupon bond maturing at T, and L(T i 1, T i ) the LIBOR-rate set at T i for the period [T i 1, T i ], while L(t, T i 1, T i ) is the corresponding LIBOR forward rate for t T i 1. (a) Suppose we have deterministic interest rates. Show that the relation p(t, T 2 ) < p(t, T 1 ) p(t 1, T 2 ) (24) for some t < T 1 < T 2, leads to an arbitrage opportunity. (b) Consider a payer swap with reset and payment dates T α, T α+1,..., T β, a fixed year fraction τ between T i 1 and T i, and fixed payments KτN, with K > 0 the fixed rate and N the nominal. Show that at time t < T α the value of the swap equals π S (t) = Nτ (S α,β (t) K) β i=α+1 p(t, T i ), (25) where S α,β (t) = p(t, T α) p(t, T β ) τ β i=α+1 p(t, T i). (26) (c) The following cap, floor and swap contracts share the reset and payment dates T α, T α+1,..., T β, a fixed year fraction τ between T i 1 and T i, fixed rate K > 0 and the nominal N. Assume t T α. We will write x + for max(x, 0). (i) Show that the cash flow of the i-th caplet τ (L(T i 1, T i ) K) + at time T i is equivalent to the cash flow ( 1 (1 + Kτ) 1 + τk p(t i 1, T i ) at maturity T i 1 of (1 + τk) times a put option on the T i -bond with strike (1 + Kτ) 1. (ii) Let π C (t) and π F (t) be the values of the cap and the floor at time t T α. Show the parity relation π C (t) π F (t) = τ β i=α+1 ) + p(t, T i )(L(t, T i 1, T i ) K). [You may use that x + ( x) + = x for any real number x. ] page 12 of 22

13 (iii) Show that a payer swaption price is always dominated by the corresponding cap price. [You may use the inequality (x + y) + x + + y +, x, y R, as well as the fact that for ay real-valued random variable X, we have that E(X) + E(X + ), where E(X) + = max(e(x), 0). ] page 13 of 22 Please turn over

14 8. (a) A CDS maturing at T pays out a continuous premium at a rate of s, while the holder of the CDS will have to pay out the loss upon default of some reference entity. Assume an intensity-based credit risk model for the reference entity, with a constant intensity of default λ and a constant of recovery rate R. The risk-free rate is allowed to be stochastic. (i) Derive an integral expression for the premium leg of the CDS in terms of the 0-coupon bond prices and the survival probabilities. (ii) Similarly, derive an integral expression for the protection leg of the CDS. (iii) Determine the fair value of the CDS spread s at time 0 in terms of λ and R. (b) Consider an intensity-based credit model with constant intensity and recovery rate 0.2. Assume a CDS for which the premium is paid continuously is quoted with spread 400 basis points. (i) Find the intensity. (ii) Calculate the probability of default within 2 years and the probability of surviving 4 years. (iii) What is the probability of surviving an infinite time? [1 points] (c) Discuss the sensitivity of the probability of default with respect to the recovery and the CDS spread. [3 points] page 14 of 22

15 SECTION C (Pre Students: Answer TWO questions from this section.) 9. Consider a European derivative written on an asset with price S t that follows the dynamics ds t = µs t dt + σs t dw t, (27) where µ and σ are constants, and W t (t 0) is a Brownian motion. Let the derivative s pay-off at maturity T, as function of the underlying s price S, be given by the function F (S). The asset does not pay any dividends. Let V (S, t) be the value of the derivative at time t T if S t = S. (a) Using a hedging argument, derive the Black and Scholes equation for V (S, t), valid for t < T : V t σ2 S 2 2 V V + rs = rv, (28) S2 S with boundary condition V (S, T ) = F (S) at time T. (b) By using the Feynman - Kac theorem, or otherwise, show that the solution to the boundary value problem of part (a) is given by V (S, t) = e rτ F ( ) Se (r 1 2 σ2 )τ+σ τ z e z2 /2 dz, (29) 2π where r is the risk-free rate (assumed constant) and τ = T t is the timeto-maturity. (c) Derive integral expressions for the derivative s ρ and υ (vega). (d) We now specialize to a call, with pay-off F (S) = max(s K, 0), where K is the call s strike price. Compute the υ of the call. page 15 of 22 Please turn over

16 10. Let C(S, t) be the price at time t of a European call option with strike K and maturity T written on a stock S, and let C A (S, t) be the price of its American counterpart. We assume that S does not pay out any dividends. (a) (i) By comparing the value at T of the underlying with that of a portfolio consisting of the European call and a suitable amount of money in a savings account, show that C(S, t) S Ke r(t t). (ii) Show that if S does not pay any dividends, the American call will never be exercised before maturity. We now suppose that the stock-price S t evolves according to ds t S t = µdt + σdw t, where W t is a Brownian motion, and µ and σ are constants. (b) (a) Show that the price V = V (S, t) of any derivative on S before exercise has to satisfy the Black and Scholes equation V t σ2 S 2 V SS + rsv S = rv. (b) State the free boundary problem for an American put with strike K. (c) Consider a perpetual American power put, with pay-off upon exercise given by max(k S, 0) n, where n is a given power (for n = 1 this becomes an ordinary put). Assuming we would exercise when the price of the underlying falls below the level of S, for given S, determine the price of the perpetual power put. (d) By maximizing the value for the put holder, determine the optimal value of the exercise boundary, S. page 16 of 22

17 11. An investor wishes to protect his investment in a non-dividend paying risky asset S t by simultaneously buying an European put option. The investors s initial capital is normalized to 1. At time 0 he decides to buy α units of the risky asset, where 0 < α < 1 as well as the same number α European puts on the asset, with maturity of T and with a strike of K/α. We assume that the stock price is modelled by a geometric Brownian motion with volatility parameter σ, and that the usual modelling assumptions for the Black and Scholes formula apply. We let r be the risk-free rate. (a) Show that the investor s portfolio of put plus assets is worth max(αs T, K). (b) Letting p(s, t; K, T ) be the price at time t < T of a put with strike K and maturity T, when the value of the underlying is S, show that α and K have to satisfy the equation αs 0 + αp(s 0, 0; K/α, T ) = 1. (30) [3 points] (c) Recall that the Black and Scholes price at time 0 of a European call with strike K and maturity T is given by where S 0 Φ(d + ) Ke rt Φ(d ), (31) d ± = log(s 0/K) + (r ± 1 2 σ2 )T σ. T Use the put-call parity relation to derive the price of the corresponding put. (d) Show that the equation (30) cannot have a solution in α if K > e rt. (Hint: You can either study the equation directly, or use an absence of arbitrage argument by comparing the investment in the portfolio of stock plus put with an investment in some other financial instrument.) (e) Show that a solution α can be found if S 0 + p(s 0, 0; K, T ) 1. (Hint: What is the right hand side of (30) equal to when α = 0? What happens if we let α increase?) page 17 of 22 Please turn over

18 12. Merton s jump difusion model assumes that the stock price S t evolves according to ds t = µs t dt + σs t dw t + (J t 1)S t dn t, (32) where W t is a Brownian motion, N t is a Poisson process with intensity λ, and J t is the jump-size if a jump takes place at time t. It is assumed that all these processes, as well as as jump-sizes at different points in time, are all independent, and that the jump sizes are identically distributed. (a) Let k := E(J t 1). Show that E(dS t ) = (µ + λk)dt. For which value of µ will the price-process be risk-neutral (that is, on average earn the risk-free rate r)? (b) Show that the solution of (32) is given by [3 points] ( Nt ) S t = S 0 J i e (µ 1 2 σ2 )t+σw t, (33) i=0 where N t is the number of Poisson-jumps between 0 and t, and J i is the jump-size at successive jump-times t i, i N t. (c) We now consider a derivative written on S t with value V (S t, t). We hedge the derivative at time t by shorting t of the underlying, where t is to be determined. Let Π t = V (S t, t) t S t be the value of the resulting portfolio. Show that dπ t is equal to ( ) V L(V )dt + σs t S t dw t where + (V (JS t, t) V (S t, t) t (J 1)S t ) dn t, L(V ) = V ( ) V t + µs t S t σ2 St 2 2 V S. 2 Can the jump risk be completely hedged away? Explain your answer. [6 points] (d) Assuming that investors do not require to be rewarded for the jump-risk, derive Merton s PIDE for the option price V (S, t) as function of the price S of the underlying and of t. [6 points] page 18 of 22

19 SECTION D (Pre Students: Answer TWO questions from this section.) 13. Consider a Black Scholes market model having one risky asset S t and one risk-free asset B t, with price-dynamics ds t = µs t dt + σs t dw t, (34) db t = rb t dt, B 0 = 1, where W t is a Brownian motion with respect to the objective probability measure P, and r, µ and σ are constants. (a) (i) By using Girsanov s theorem, show that there exists a new process Ŵt and a new probability measure Q such that Ŵt is a Brownian motion with respect to Q and such that ds t = rs t dt + σs t dŵt. (35) (ii) Give the solution of the SDE (35) with initial value S 0. [3 points] (b) Consider a self-financing portfolio V t = φ t S t +ψ t B t, with φ t and ψ t adapted to the Brownian filtration. Show that the discounted portfolio has dynamics dṽt = φd S t. (*) (c) Let X be the pay-off of a replicable European claim with maturity T. By using parts (a) and (b) of this question, show that the price of the claim at time 0 is given by π 0 (X) = e rt E Q (X), the discounted expectation of X with respect to Q. (d) Now specialize to a digital European put, with pay-off X given by { 1 if ST K X = 0 if S T K. Show that the price of this digital put at time 0 is equal to ( π 0 (X) = e rt log(k/s0 ) (r 1 ) 2 Φ σ2 )T σ, T where Φ is the standard normal distribution. (Hint: π 0 (X) = e rt Q(S T K).) page 19 of 22 Please turn over

20 14. Consider a market with two risky assets S 1,t and S 2,t with price dynamics: ds 1,t = S 1,t (µ 1 dt + σ 1 dw 1,t ), ds 2,t = S 2,t (µ 2 dt + σ 2 dw 2,t ), where W 1,t and W 2,t are two correlated Brownian motions, with constant correlation ρ, 1 < ρ < 1, and σ 1, σ 2 assumed to be constant also. We will use S 1,t as numéraire to price options. Let Q be the new probability measure with respect to which asset prices discounted by S 1,t become martingales. Let F t, t 0, be the filtration generated by the two Brownian motions. (a) Show that with respect to Q, the relative price P t = S 2,t /S 1,t solves the SDE dp t = P t ( σ2 dw 2,t σ 1 dw 1,t), where W 1,t and W 2,t are two new Brownian motions which again have correlation ρ. (You may use without proof that correlations remain unchanged after a Girsanov-type change of measure.) (b) Show that P T = P t e Z, where Z is normally distributed with variance equal to Σ 2 := (σ 2 1 2ρσ 1 σ 2 + σ 2 2)τ, where τ := T t, and mean 1 2 Σ2. (c) Consider a European exchange which gives the right to exchange S 2,T for S 1,T at time T. Explain why the price at time 0 of this option is given by π exch 0 = S 1,0 E Q (max(p T 1, 0)). (36) where P t is as in part (b). Show that this equals with Z as in part (b). π exch (S 0,1, S 0,2 ) = E ( S 2,0 e Z S 1,0 ), (37) (d) To hedge the option at time 0, we have to compute its s with respect to the two assets, that is, the derivatives of its price with respect to S 1,0 and S 2,0. Compute these s. You may use that if U N(a, b 2 ), K > 0 and if H is the Heaviside function (H(x) = 1 for x 0 and 0 otherwise), then E ( H(e U K) ) ( ) a K = Φ, b and E ( H(e U K)e U) ( a + b 2 K = Φ b ). page 20 of 22

21 15. Consider a general short rate model of the type where r t is the short rate. dr t = a(r t )dt + b(r t )dw t, (38) (a) Specify the coefficients a(r t ) and b(r t ) in case we are dealing with (i) The Vasicek model. (ii) The Cox-Ingersoll-Ross model. Discuss the similarities and differences between the two models. (b) Explain why the price of a 0-coupon bond with maturity T is given by an expectation ( E Q e ) T t r sds F t, (39) where F t is the Brownian filtration, and Q is a (choice for the) risk-neutral measure. How would the SDE for r t under Q look like? (c) It was shown in the Lectures that in the Vasicek model the price at time t of a 0-coupon bond maturing at T is given by p(t, T ) = e A(t,T ) B(t,T ) rt, where ) A(t, T ) = (θ σ2 (B(t, T ) T + t) σ2 2κ 2 4κ B2 (t, T ) B(t, T ) = 1 ( ) 1 e κ(t t), κ with θ, σ and κ the parameters under the risk-neutral measure, and where r t is the prevailing interest rate at t. Suppose that after calibration of the model to market date we found that κ = 0.5, θ = 0.04 and σ = 0.2, while today s prevailing interest rate is Use these data to price a bond with a nominal of 1000, 000 and a maturity of 2 years, which pays a yearly coupon of 4%. (d) Recall that yield R(t, T ) of a 0-coupon bond with maturity T is defined by p(t, T ) = e R(t,T ) (T t). (i) Compute R(t, T ) for the Vasicek model. (ii) Still for the Vasicek model, compute the correlation between R(t, T 1 ) and R(t, T 2 ) for T 1 different from T 2. Is this realistic for an actual interest rate market? (iii) Would you expect the same, or a different, answer for other short rate models such as the Cox-Ingersoll-Ross. (iv) Give an example of another type interest rate model in which we can let these correlations vary with T 1 and T 2. page 21 of 22 Please turn over

22 16. Consider the Merton model of defaultable debt, in which a firm is financed by equity and a single 0-coupon bond with face value F. We assume that the risk-neutral evolution of the firm s value V t is given by dv t = (r δ)v t dt + σv t dw t, (40) where W t is Brownian motion. Here, δ is the dividend rate, r the risk-free interest rate and σ is the volatility of the (return on the) firm value. (a) By examining the pay-offs to equity holders and debt holders at maturity T, show that equity is modelled by a European call option on the firm s value with strike F, while debt holders hold the difference of a risk-free investment with face-value F and a put. (b) Let S t be the value of equity at time t < T, and p t,t the value of the firm s debt. By using risk-neutral pricing, show that S t + p t,t = e δ(t t) V t. (Hint: Compute S T + p T,T, the value of the right hand side at maturity; you will also need the solution to (40).) (c) Compute the risk-neutral probability of default in Merton s model. at time 0, given that the firm s value at that time is V 0. (d) (i) How does the risk-neutral probability of default change with increasing F? (ii) And with increasing σ? You may assume for simplicity that r = δ σ2? [3 points] page 22 of 22