Exams 2012-2015, questions and answers PDF

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King’s College London University Of London This paper is part of an examination of the College counting towards the award of a degree. Examinations are governed by the College Regulations under the authority of the Academic Board. ATTACH this paper to your script USING THE STRING PROVIDED

Candidate No: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Desk No: . . . . . . . . . . . . . . . . . . . . . . .

BSc and MSci Examination 6CCM388A (CM388Z) Mathematical Finance II: Continuous Time MOCK EXAM Summer 2012 Time Allowed: Two Hours All questions carry equal marks. Full marks will be awarded for complete answers to FOUR questions. Only the best FOUR questions will count towards grades A or B, but credit will be given for all work done for lower grades. Within a given question, the relative weights of the different parts are indicated by a percentage figure.

You are permitted to use a Calculator. ONLY CALCULATORS APPROVED BY THE COLLEGE MAY BE USED.

DO NOT REMOVE THIS PAPER FROM THE EXAMINATION ROOM TURN OVER WHEN INSTRUCTED c 2012 King’s College London

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1.

6CCM388A (CM388Z)

Consider the standard one-period binomial model where the stock price goes from S to Su or Sd with d < 1 < u, and consider an option which pays fu or fd in each case, and assume the interest rate is r and time to maturity is T . a. Derive the formula for the Delta ∆ of the option, i.e. the amount of stock we need to buy to hedge against selling the option. Solution: Buy ∆ shares and sell 1 option. Riskless portfolio is worth the same amount if S goes up or down, thus ∆Su − fu = ∆Sd − fd . Re-arrange to obtain ∆ =

fu − fd . Su − Sd [25%]

b. Compute the no-arbitrage price of the option. Solution: Equate discounted final value and the initial value of the riskless portfolio ∆S − f0 = e−rT [∆Su − fu ] . where f is the initial option price. Re-arrange to get f0 = ∆S − e−rT (∆Su − fu ) . [25%] c. The risk-neutral probability of S going up is p=

erT − d . u−d

if d ≤ erT ≤ u. Write down the expression for the initial option price in terms of p (proof not required) The risk-neutral probability of S going up is p=

erT − d . u−d See Next Page

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6CCM388A (CM388Z)

if d ≤ erT ≤ u. Write down the expression for the initial option price in terms of p (proof not required) [25%] Solution: f = e−rT [pfu + (1 − p)fd ] .

(1)

d. Price the option for the following numbers S = 1.05, u = 1.1, d = .9, fu = .1, fd = 0, r = .04, T = 1. [25%] Solution:

.

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2.

6CCM388A (CM388Z)

Let W denote a standard one-dimensional Brownian motion. a. Let St = ebt+σWt for σ > 0. Use Ito’s lemma write a stochastic differential equation (SDE) for dSt in terms St . Solution: Let St = f (t, Wt ) where f (x, t) = ebt+σx . Then ft = bf , fx = σf , fxx = σ 2 f and 1 dSt = ft dt + fx dWt + fxxdt 2 1 = bSt dt + σSt dWt + St σ 2 dt 2 1 2 = St ((b + σ )dt + σdWt ) . 2 [25%] b. Let St be a solution to the SDE dSt = (βSt + 1 − β)dWt .

(2)

Apply the general version of Ito’s lemma to Xt = βS t + 1 − β, and write dXt as an SDE in terms of Xt

(3) [25%] c.

(4) [25%] See Next Page

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6CCM388A (CM388Z)

d. Consider a process Xt satisfying the SDE dXt = X t2 dWt Compute the SDE for Rt = 1/Xt in terms of Rt .

(5) [25%]

Solution:

(6)

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3.

6CCM388A (CM388Z)

Let St denote a stock price process governed by the Black-Scholes model dSt = St [µdt + σdWt ]

(7)

for σ > 0. a. Derive the Black-Scholes PDE for the price of a call option under the Black-Scholes model. [20%] Solution: Let f (S, t) denote the call option price and we assume that f is twice differentiable in S and once in t. Then, by Ito’s lemma we have 1 df = (ft + µSt fS + σ 2 fSS St2 )dt + fS St σdWt . 2

(8)

We now wish to hedge this option by selling ∆ = fS units of stock. Thus the total portfolio value Π = f − ∆S satisfies dΠ = df − fS dSt

1 = (ft + µSt fS + σ 2 fSS St2)dt + fS St σdWt − fS St (µdt + σdWt ) 2 1 2 = (ft + σ fSS St2 )dt . (9) 2

But the change in the portfolio has no dWt term, i.e. is riskless. Thus the portfolio should earn the risk free rate, i.e. 1 ft + σ 2 fSS St2 = rΠ . 2

(10)

Substituting for Π and re-arranging, we obtain the Black-Scholes PDE 1 ft + rSfS + σ 2 fSS S 2 = rf . 2

(11)

b. State the Feynman-Kac formula for a solution to the PDE 1 ft + µ(x)fx + σ(x)2 fxx = V (x)f , 2 subject to the terminal condition f (x, T ) = φ(x), with µ(x), σ(x), V (x) bounded. Solution:

See Next Page

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6CCM388A (CM388Z)

The Feynman-Kac formula states that f (t, x) has the stochastic representation f (x, t) = E(e−

∫T t

V (Xs )ds

φ(XT ) | Xt = x)

where Xt is a stochastic process which satisfies the SDE dXt = µ(Xt )dt + σ(Xt )dWt . [20%] c. Using the Feynman Kac formula and the Black Scholes PDE, write down the probabilistic representation for the price of the call option? [20%] Solution:

C = e−r(T −t) EQ (max(ST − K, 0)).

(12)

dSt = St [rdt + σdWt ]

(13)

where St follows

under Q. d. Price a call option under the Black-Scholes model with S0 = 1, K = 1.05, σ = .2, T = .2, r = 0, q = 0 (q is the dividend rate) using the BlackScholes formula and the Normal tables provided: C(S, τ ) = SΦ(d1 ) − Ke−rτ Φ(d2 ) , S √ +(r+21σ 2 )τ log K √ , d2 = d1 − σ τ where Φ(x) = where τ = T − t and d1 = σ τ ∫ x e−z 2 /2 √ is the standard cumulative Normal distribution function. −∞ 2π

Solution:

d1 = −0.500769259031127

d2 = −0.590211978131119 C = 0.016866321627

(14) [20%] See Next Page

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6CCM388A (CM388Z)

e. Compute the Gamma of the option using the formula in the notes, where 1 −z 2 /2 . Describe the meaning of Gamma. n(z) = √2π e Solution: Γ = 3.934695160204

(15)

Gamma is the rate of change of the delta of the option wrt to the underlying stock price. [20%]

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4.

6CCM388A (CM388Z)

Let Wt denote standard Brownian motion, and let Mt = max0≤s≤t Ws . a. Use the reflection principle to derive an expression for P(Mt > b, Wt < x). Solution: W has no memory and thus starts afresh after it hits the level b. More precisely Wτb +t − Wτb ∼ N (0, t) where τb is the first time that W hits the level b. This means that if W hits b for the first time at time τb < t, then it starts afresh at τb , and is then equally likely to go up or down by a further amount b − x, i.e. is equally likely to end up below x or above b + b − x = 2b − x. Thus P(Mt > b, Wt < x) = P(Mt > b, Wt > 2b − x)

(16)

for x ≤ b. But if Wt > 2b − x then W must have exceeded b at some point, so P(Mt > b, Wt > 2b − x) = P(Wt > 2b − x). Thus P(Mt > b, Wt < x) = P(Wt > 2b − x) .

(17) [20%]

b. Use this to show that P(Mt > b) = 2P(Wt > B). Solution: Setting b = x, we obtain P(Mt > b, Wt < b) = P(Wt > b) = P(Mt > b, Wt > b) . But the events {Mt > b, Wt < b} and {Mt > b, Wt > b} are disjoint, and both have probability equal to P(Wt > b) . Thus we can add both sides to obtain P(Mt > b) = P(Mt > b, Wt < b ∪ Mt > b, Wt > b)

= P(Mt > b, Wt < b) + P(Mt > b, Wt > b) = 2P(Wt > b) . [20%]

c. Using part (b), derive the density of the hitting time τb to the level b for b > 0. [20%]

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6CCM388A (CM388Z)

Solution: THE EVENT THAT WE HIT b BEFORE TIME t IS THE SAME AS THE EVENT Mt ≥ b: P(τb < t) = P(Mt ≥ b) .

(18)

The left hand side is the distribution function of τb . To obtain its density, we just differentiate wrt t using the chain rule d c b d P(Mt ≥ b) = 2Φ ( √ ) dt dt t = = √

2 b e−b /2t . 2πt3

d. Let St follow the Black-Scholes model dSt = St σdWt with µ = 0 and assume r = q = 0. Derive the fair price of a One Touch contract with infinite maturity, which pays 2 dollars if S hits the barrier level B at the moment when S hits? (you may use that St → 0 almost surely as t → ∞. [20%] Solution: Consider buying 1 unit of stock at time zero at price S0 . If the barrier hits B, then we can sell the stock for B. Otherwise, if we never hit B, the stock price will eventually tend to zero. Thus the fair price of a contract which pays B if we hit the barrier is S0 , hence the fair price of a contract that pays 2 at the barrier is B2 S0 . e. Prove that the Black-Scholes SDE dSt = St σdWt

(19) 2

with µ = 0 is a martingale. You may use that E(St2) = S02e2σ t < ∞.

Solution: Integrating the SDE from s to t, we obtain ∫ t Su σdWu . St = Ss +

(20)

s

Thus E(St | Fs ) = Ss + E(

∫

t

Su σdWu ) .

(21)

s

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6CCM388A (CM388Z)

The rightermost term is a stochastic integral, so we expect its expectation to be zero. To prove this, we have to check ∫ t ∫ t 2 2 E( Su σ du) = E( Su2σ 2 du) s

s

=

< ∞.

(22)

See Next Page

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5.

6CCM388A (CM388Z)

Let ˜ t = Wt − γt W for γ constant, where W is standard Brownian motion under the measure P. ˜ under which ˜ in terms of the measure P State Girsanov’s theorem for W ˜ is standard Brownian motion. W Solution Let ˜ t = Wt − γt W for γ constant, where W is standard Brownian motion under the measure P. Theorem 0.1. (Girsanov). Let ∫t

Zt = e

0

γdWs − 12

∫t 0

γ 2 ds

˜ as follows and define the new probability measure P ˜P(A) = EP (1A ZT )

(23)

˜ is standard driftless Brownian for any event A, and some T > 0. Then W motion under the new measure ˜P. [25%] ˜ t < a) where b. a. Using Girsanov’s theorem, write down an expression for P(M Mt = max0≤s≤t Ws , as an expectation under the measure P, and then rewrite this a double integral in terms of f (x, b), where f (x, b) is the joint density of (Wt , Mt ) for standard driftless Brownian motion. Solution: ˜P(Mt < a) = EP (eγWt −21γ 2 t 1M a is a Borel set. [20%]

See Next Page

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2.

6CCM338A (CM338Z)

Let W denote a standard one-dimensional Brownian motion. 1 2

a. Let St = e−2 σ t+σWt for r, σ > 0. Use Ito’s lemma to write a stochastic differential equation (SDE) for dSt in terms of St . [25%] b. Let Yt = S1 t . Apply the general version of Ito’s lemma to Yt , and write dYt as an SDE in terms of Yt [25%] c. Write down an SDE for Zt = sin(Wt ) when Wt ∈ (− 21 π, 21 π) (you may use the trigonometric identity cos(θ)2 + sin(θ)2 = 1). [25%] d. Consider a process Xt satisfying the SDE dXt = X t2 dWt Compute the SDE for Rt = 1/Xt in terms of Rt .

[25%]

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3.

6CCM338A (CM338Z)

Let St denote a stock price process governed by the Black-Scholes model dSt = St [µdt + σdWt ] for σ > 0. a. Derive the Black-Scholes PDE for the price of a call option under the Black-Scholes model using a self-financing portfolio (φt , ψt ) in the stock and the risk free Bond which evolves as dBt = rBt dt . [30%] b. State the alternative probabilistic representation for the price of a put option with strike K and maturity T which pays max(K − ST , 0) at time T , i.e. how do we price the put option using risk-neutral valuation? [10%] c. State the Feynman-Kac formula for a solution to the PDE 1 ft + µ(x)fx + σ(x)2 fxx = V (x)f , 2 subject to the terminal condition f (x, T ) = φ(x), with µ(x), σ(x), V (x) bounded. [20%] d. Price a call option under the Black-Scholes model with S0 = 1, K = 1.1, σ = .2, T = 1, r = .05, q = 0 (q is the dividend rate) using the BlackScholes formula: C(S, τ ) = SΦ(d1 ) − Ke−rτ Φ(d2 ) ,

S √ +(r+21σ 2 )τ log K √ where τ = T − t and d1 = , d2 = d1 − σ τ where Φ(x) = σ τ ∫ x e−z 2 /2 √ dz is the standard cumulative Normal distribution function (leave −∞ 2π the answer expressed in terms of the Φ function, no need to use tables or calculator).

[20%] √

∂C = Se−qτ n(d1 ) τ numerically, e. Compute the vega of the option Λ = ∂σ 1 −z 2 /2 where n(z) = √2π e . Describe the meaning of vega.

[20%] See Next Page

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4.

6CCM338A (CM338Z)

Let Wt denote standard Brownian motion, and let Mt = max0≤s≤t Ws . a. Use the reflection principle to derive an expression for P(Mt > b, Wt < x) for x ≤ b, b > 0. [25%] b. From part a) we can show that P(Mt > b) = 2P(Wt > b) (Proof of this not required). Using this equality, compute the density of τb = inf {t : Wt = b}, where τb is the first hitting time to b. [25%] c. Let St follow the Black-Scholes model dSt = St σdWt with µ = 0 and assume r = q = 0. Derive the fair price of a No-Touch contract with infinite maturity, which pays 1 if S does not exceed the barrier level B for all time? (you may use that St → 0 almost surely as t → ∞). [25%] d. Now assume St is governed by a general continuous non-negative martingale dSt = St σt dWt and interest rates are zero and σt > 0, and consider a One-Touch option which pays 1 at time T if S hits B before time T , where B > S0 . By considering both scenarios: i.e. S hits B before time T or S does not hit B before time T , show that Price of One Touch ≤

CK,T B−K

(1)

where CK,T is the price of a K-strike call option with maturity T and strike K ≤ B at time zero (you may use the fact that the value of a call option is greater than or equal to its intrinsic value max(S0 − K, 0)). How do we make the bound in (1) as tight as possible? [25%]

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5.

6CCM338A (CM338Z)

a. Let ˜ t = Wt − γt W for γ constant, where W is standard Brownian motion under the measure ˜ in terms of the measure P ˜ under which P. State Girsanov’s theorem for W ˜ is standard Brownian motion. W [25%] b. Using Girsanov’s theorem, write down an expression for ˜P(Mt > a) where Mt = max0≤s≤t Ws , as an expectation under the measure P, and then re-write this as a double integral in terms of f (x, b), where f (x, b) =

)2 2(2b − x) − (2b−x 2t √ e 2πt3

is the joint density of (Wt , Mt ) for standard driftless Brownian motion. You do not have to evaluate the integral explicitly. [25%] c. Now let Xt = − 12 σ 2 t + σWt for σ > 0. Using Girsanov’s theorem or your answer to part (b), write down an expression for P(X¯t > a) where X¯t = max0≤s≤t Xs , as a double integral in terms f (x, b). From this, derive an expression for the price of a No-Touch option under the Black-Scholes model dSt = St [(r − q)dt + σdWt ] as a double integral in terms of f (x, b). The No-Touch option pays 1 at time T if S does NOT go above the level B > S0 before time T . [25%] d. Let Wt be standard Brownian motion. Compute the mean and variance of ∫ 1 t At = Ws ds t 0 [25%]

Final Page

King’s College London University Of London This paper is part of an examination of the College counting towards the award of a degree. Examinations are governed by the College Regulations under the authority of the Academic Board. ATTACH this paper to your script USING THE STRING PROVIDED

Candidate No: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Desk No: . . . . . . . . . . . . . . . . . . . . . . .

BSc and MSci Examination 6CCM338A (CM338Z) Mathematical Finance II: Continuous Time. May 2014

Time Allowed: Two Hours All questions carry equal marks. Full marks will be awarded for complete answers to FOUR questions. Only the best FOUR questions will count towards grades A or B, but credit will be given for all work done for lower grades. Within a given question, the relative weights of the different parts are indicated by a percentage figure.

You are permitted to use a Calculator. ONLY CALCULATORS APPROVED BY THE COLLEGE MAY BE USED.

DO NOT REMOVE THIS PAPER FROM THE EXAMINATION ROOM TURN OVER WHEN INSTRUCTED c 2014 King’s College London

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1.

6CCM338A (CM338Z)

a. Let A be an algebra and let A1 , A2 , . . . , An , . . . ∈ F . Is ∩∞ n=1 An ∈ A ? (explain). [25%] b. Prove that {1} ∪ {2} is a Borel set. [25%] c. Let Ω = R, F = B(R), and X : Ω → R be a function such that X(ω) = 0 if ω < 0 and X(ω) = 1 otherwise. Prove that X is a random variable (you may use the fact that [0, ∞) is a Borel set). [25%]

d. Let U be a uniform random variable on [0, 1]. Using U , describe how we can generate a random variable X such that X ∼ N (0, 1), i.e. X has a standard Normal distribution with mean 0, variance 1. [25%]

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2.

6CCM338A (CM338Z)

Let W denote a standard one-dimensional Brownian motion. 1

2

a. Let St = e− 2 σ t+σWt for σ > 0. Use Ito’s lemma to write a stochastic differential equation (SDE) for dSt in terms of St . [25%] b. Apply Ito’s lemma to W t4 − 6Wt2t + 3t2 .

[25%]

c. Consider the following SDE dXt = σ(Xt )dWt . ∫x 1 Compute the SDE for g(Xt ), where g(x) = 0 σ(u) du. [25%] d. Consider the following SDE dRt =

2δ − 1 dt + 2dWt Rt

(1)

for δ ≥ 0, R0 > 0. Compute the SDE for Zt = Rt2 in terms of Zt (you may assume that Rt ≥ 0). [25%]

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3.

6CCM338A (CM338Z)

Let St denote a stock price process governed by the Black-Scholes model dSt = St [µ dt + σdWt ] for σ > 0. a. State the Black-Scholes PDE and boundary condition for the price of a put option under the Black-Scholes model. [25%] b. Derive a formula for the price of an option under the Black-Scholes model which pays 2 dollars at time T if ST > K and zero otherwise [25%] c. Price a European call option under the Black-Scholes model at time zero with S = 1, K = 1.1, σ = .1, T = 1, r = 0.05, using the Black-Scholes formula: C(S, τ ) = SΦ(d1 ) − Ke−rτ Φ(d2 ) , S √ +(r+21σ 2 )τ log K √ , d2 = d1 − σ τ where Φ(x) = where τ = T − t and d1 = σ τ ∫ x e−z 2 /2 √ is the standard cumulative Normal distribution function (leave −∞ 2π the answer expressed in terms of the Φ function, no need to use tables or calculator). ). How do we price a European put option with the same strike without re-computing the Black-Scholes formula?

[25%] 2

d. The volga of an option under the Black-Scholes model is defined as ∂∂σC2 , i.e. the second derivative of the call price with respect to the volatility σ. Describe how one would approximate volga using finite differences (no numerical calculations required). [25%]

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4.

6CCM338A (CM338Z)

Let Wt denote standard Brownian motion, and let Mt = max0≤s≤t Ws . a. Use the reflection principle to derive an expression for P(Mt > b, Wt < x) for b > 0, x ≤ b.

[25%]

b. Show that P(Mt > b) = 2P(Wt > b) for b > 0. [25%] c. Using that P(Mt > b) = 2P(Wt > b), compute P(τb > t), where τb = min{s ≥ 0 : Ws = b} is the first hitting time to b > 0, [25%]

d. Now assume for simplicity that we have a stock price process which is just equal to standard Brownian motion St = W t

(2)

and suppose we wish to hedge a No-Touch option with barrier B > 0, which pays 1 ...