1. Please provide brief explanations for the following

(a) [5] The fundamental theorem of asset pricing in terms of a numeraire asset.

(b) [5] An interest rate Floor.

2.  (a) [5] Sketch the Price, Delta, and Gamma of a Call option, when t = 0 and t = T = 1yr , indicating any important points and/or features. Price, Delta, and Gamma on di↵erent plots, but two curves in each plot: one for t = T = 1yr and one for t = 0.

(b) [2.5]  Sketch how realized  volatility  evolves  (in  time) in real markets.   Describe any important features.

(c) [2.5] Sketch how implied volatility varies as a function of strike.  Describe any important features.

3. [10] Please indicate true or false (no explanations required ).

+2 for correct answer; 0 for no answer or wrong answer.

(a)  [T]    [F]

The following assets are traded in an economy. This economy is arbitrage free.

(b)  [T]      [F]

You have bought a call option struck at $100 on XYZ shares and you are simultaneously delta-hedging the position. Suppose that important (unexpected) good news arrives declar- ing exceptional sales of XYZ products resulting in an increase in share value from S = $100 to S = $110. You must sell shares of XYZ to maintain your hedge.

(c) [T]    [F]

Suppose that an equity price is modeled as in Black-Scholes but interest rates are model as in the Vasicek model. The distribution, under the risk-neutral measure, of the asset price at a fixed point in time is log-normal.

(d) [T]    [F]

Consider the compound option: a call, maturing at T1  with strike K1 , on a put, maturing at T2  > T1  and strike K2 , on a stock. As the spot price increases, the compound option price increases.

(e) [T]    [F]

In the Vasicek model, the forward rate of interest process `t   = T2T1   ⇣  − 1⌘ is log-

normally distributed under the risk-neutral measure.

4. In this question, W = (Wt )t≥0  is a P-Brownian motion and the risk-free interest rate is assumed to be constant and equal to 0.

(a) [5] Suppose that the price process of a tradable asset X = (Xt )t≥0  satisfies the SDE (i.e.,

the Black-Scholes model):

dXt  = µXt dt + σ Xt dWt .

Using numeraire techniques, and without integrating or using the density, derive the for- mula for the price process f = (ft )t≥0 of an asset or nothing option which pays fT  = XT 1XT>K .

(b) [5] Suppose that F = (Ft )t≥0  denotes a futures price process and F satisfies the SDE dFt  = Ft (µ(t,Ft )dt + σ(t,Ft )dWt )

where W = (Wt )t≥0 is a P-Brownian motion (and P denotes the historical measure), and µ(· , · ) and σ(· , · ) are two given functions. You may write µt(F)  = µ(t,Ft ) and σt(F)  = σ(t,Ft ) to simplify notation.

Let g = (gt )t2 [0,T]  denote the price process of a European option on the futures price that pays G(FT ) at time T. Assume there exists a function g(t,F) such that gt  = g(t,Ft ) with enough di↵erentiability to apply Ito’s Lemma.

Using the dynamic hedging argument, where you use a self-financing strategy (↵ , β , −1) in the bank account, the futures price, and the option, derive a PDE for g(t,F).

5. Let r = (rt )t≥0  denote the stochastic interest rate, and let (x,y) = (xt ,yt )t≥0  be two stochastic factors such that rt  = φt + xt + yt , with φt  deterministic. Suppose that x and y satisfy the SDEs

dxt  = −axt dt + σt dWtx ,

dyt  = −byt dt + ⌘t dWty ,

where (Wx ,Wy ) = (Wtx ,Wty )t≥0  are correlated risk-neutral Brownian motions with correlation p, and σt , ⌘t  are deterministic functions of time.

(a) [3] Show that the solution to the SDE for x is (y is similar, and you may use this result in

the other parts.)

xu  = e−a (u−t) xt + Zt u e−a(u−s) σs dWs(x) ,    Au ≥ t ≥ 0 .

(b)  [3] Demonstrate that a bond of maturity T has the affine form Pt (T) = eAt(T)−Bt(T) xt−Ct(T) yt   for

some deterministic functions At(T) , Bt(T), and Ct(T) . Do not compute At(T) , Bt(T), and Ct(T)   explicitly – an integral or ODE representation suffices.

(c) [4] Derive an expression for the price of a caplet with tenure T1  and T2 . Recall that a caplet pays (` − K)+ , where ` =  ⇣  − 1⌘ , at T2   (here, ∆T = T2  − T1 ).   You may leave various expressions in terms of integrals and you may assume that the affine functions from part (b) are known, so that Pt (T) = eAt(T)−Bt(T) 北t−Ct(T) yt   for some known deterministic functions At(T) , Bt(T), and Ct(T) .

6. Consider a domestic and foreign market each with bank accounts and risky assets denoted by Bd  = (Bt(d))t≥0  and Sd  = (St(d))t≥0  (domestic) and Bf  = (Bt(f))t≥0  and Sf  = (St(f))t≥0  (foreign) where the short rate of interest in each economy is constant and equal to rd  and rf , respectively. Further, let X = (Xt )t≥0  denote the exchange rate from the foreign to domestic economy (1 unit of the foreign currency is worth Xt  units of domestic currency). Assume that X, Sd  and Sf  satisfy the SDEs:

dXt  = µXt dt + σ Xt dWt北

dSt(d)  = v St(d) dt + ⌘ St(d) dWtd

dSt(f)  = ↵ St(f) dt + β St(f) dWtf

where (W北 ,Wd ,Wf ) = (Wt北 ,Wtd ,Wtf )t≥0  are correlated P-Brownian motions with correlation pi,j for all i,j 2 {X,d,f} (where P represents the real-world/historical probability measure).

(a) [5] Let Q⇤ denote the equivalent martingale measure induced by Sd . Show, using martingale conditions, that the SDEs for X, Sd , and Sf  may be written

dXt  = Xt ((rd − rf + p北d ⌘σ) dt + σ dW⇤ )

dSt(d)  = St(d)  ⇣(rd + ⌘2 )dt + ⌘ dW⇤ ⌘

dSt(f)  = St(f)  ⇣(rf  − p北f σβ + pdf ⌘β)dt + β dW⇤ ⌘

where (W北,⇤,Wd, ⇤ ,Wf, ⇤ ) = (W⇤ ,W⇤ ,W⇤ )t≥0  are correlated Q⇤-Brownian motions with d[Wi ,Wj ]t  = pij dt for all i,j 2 {X,d,f}.

(b) [5] Let f = (ft )t≥0  denote the price process of a foreign-domestic exchange option that pays fT  = (ST(d) − aST(f))+  where a is a constant (forward) FX rate so that f is in domestic currency. Derive a valuation formula for this option.

7. Here we address a stochastic volatility model.  Let S = (St )t≥0  and v = (vt )t≥0  denote a tradable asset price process and its variance process, and we assume a time varying Heston model (with r = 0) so that they satisfy the SDEs

dSt  = St ^vt dWts ,

dvt  = K(✓ − vt )dt + ⌘t ^vt dWtv ,

where (Ws ,Wv ) = (Wts ,Wtv )t≥0  are correlated Q-Brownian motions with correlation p 2 (−1, 1), K, ✓ > 0 and constant, and ⌘t  a deterministic function of time.

(a) [3] Derive the moment generating function of log(ST /St ) at all points in time, i.e, derive an

expression for   t (z) := Et [exp{z log(ST /St )}].  You may leave the answer in terms of solutions to ODEs or integral representations.

(b) [7] You are to value a forward exchange option with price process f = (ft )t≥0  with fT2   = (ST0   − aST1)+

and T2  > T1  > T0  > 0, a > 0. Note all three dates in the above are distinct.

Conditioned on the sigma-algebra G  := σ ((vu )u<T1), the asset price process is a time varying Geometric Brownian motion of some form. Determine what the conditional SDE is, and use it to develop a Monte Carlo estimate of the price (at t = 0) in terms of averages of Black-Scholes like formulae. To obtain full marks, you must also specify how to simulate the system to obtain the estimate.