Exam Quantitative Finance (35V5A1) Part I: Discrete-time finance Exercise 1 (20 points) a. Provide the definition of the pricing kernel k q. Relate this pricing kernel to the set of discount factors D p. [3 pts.] b. Provide the definition of the expectation kernel k e, and show how it can be found given a set of basic assets x j M (j = 1,..., J). [3 pts.] c. Suppose there are 3 states and 2 securities with payoffs x 1 = (1, 1, 0), and x 2 = (0, 1, 1). The probabilities related to each state are 1/3. Find the expectations kernel k e. [4 pts.] d. Let X be the J S matrix with payoffs on J assets and p a vector with associated prices. S is the number of scenarios. Define Π = diag(π 1,..., π S ), i.e. Π is a diagonal matrix with the probabilities π i (i = 1,..., S) on the diagonal. Show and explain carefully that the pricing kernel equals X h, with h = (X ΠX) 1 p. [5 pts.] e. Take a z M, with M the payoff space. Consider the projection ( ( )) z = αk e + βk q + ɛ, E ɛ k e = 0. k q Proof that z is on the mean-variance frontier if it satisfies z = αk e + βk q for some α R, and β R. [5 pts.] 1
Exercise 2 (15 points) Suppose we are given some payoff space M with some corresponding stochastic discount factor m. Consider the performance measure l m : l m : R S r l m (r) = k E ( m ( r r ref)) R. a. Determine the performance of a portfolio that has return r a M. [3 pts.] b. Show the performance implications if we evaluate a portfolio that has return r a / M. [3 pts.] c. Provide a proof that shows that the choice of reference return r ref does not matter numerically for the performance measure l m. [3 pts.] d. If we have a stochastic discount factor m = E(m) [1 + b (f E(f))], and a constant k = 1 E(m) = r, show that the performance measure l m can be interpreted as the constant term in a regression of the excess returns r r on the factors f. [6 pts.] 2
Exercise 3 (15 points) Following the discussions in the Pennacchi book, the CAPM can be expressed as E[R i ] = R f + β i γ, where E[ ] is the expectation operator, and R i is the realized return on asset i. R f is the risk-free return, β i is asset i s beta, and γ is a positive market risk premium. Now, consider a stochastic discount factor of the form m = a + br m, where a and b are constants and R m is the realized return on the market portfolio. Also, denote the variance of the return on the market portfolio as σ 2 m. In answering the following questions, show clearly the steps you make in your derivations. a Derive an expression for γ as a function of a, b, E[R m ], and σm. 2 [7 pts.] b Note that the equation 1 = E[mR i ] holds for all traded assets. Consider the case of the risk-free asset and the case of the market portfolio, and solve for a and b as a function of R f, E[R m ], and σm. 2 [5 pts.] c Show that γ = E[R m ] R f. [3 pts.] 3
Part II: Continuous-time finance Put your answers to the remaining exercises on a separate set of papers! Exercise 4 (20 points) (ai) [2pts.] When is a continuous-time stochastic process said to be a Markov process? (aii) [2pts.] When is a continuous-time stochastic process said to be a martingale? (aiii) [2pts.] Give the definition of the quadratic variation process related to a continuoustime stochastic process X. (b) Let W a Brownian motion with variance σ 2 > 0 per unit of time independent of a Poisson process N with intensity λ > 0. Introduce, for c R, a new process X by X t = W 2 t + N t + ct, t 0. (i) [3pts.] Write down a SDE for the process Z t = Wt 2 stochastic integral notation. and present this SDE also in (ii) [3pts.] Is X a Markov process? (iii) [3pts.] Determine c such that X is a martingale. (iv) [3pts.] Determine the quadratic variation of the process (W 2 t ) t 0 (hint: exploit (i)) and the quadratic variation of N. (v) [2pts.] Determine the quadratic covariation between the processes (W 2 t ) t 0 and N and determine the quadratic variation of X. 4
Exercise 5 (30 points) Consider the standard Black-Scholes model. Recall that the dynamics of the stock price and money market account are given by ds t = µs t dt + σs t dw t, (1) and db t = rb t dt, (2) where W is a standard Brownian motion and B 0 = 1 and S 0 = s 0. (a) [2pts.] Give the definition of a self-financing portfolio. (b) [5pts.] State the risk-neutral version of the First Fundamental Theorem of Finance. (c) [6pts.] There is a process W Q of the form W Q t = γt + W t such that, under the risk-neutral (equivalent martingale) measure corresponding to taking the asset B as a numéraire, the process W Q is a standard Brownian motion. Determine γ. (d) Notice that we have ds t = rs t dt + σs t dw Q t. Assume that the price process of a European option, with maturity T > 0, is given by p t = C(t, S t ), where C is a smooth function. (i) [6pts.] Use Itô s lemma to obtain a SDE for dp t = dc(t, S t ) in terms of W Q. (ii) [6pts.] Use (i) and Part (b) to show that C is a solution to the Black-Scholes PDE: F t (t, s)+ 1 2 σ2 s 2 F ss (t, s)+rsf s (t, s) rf (s, t) = 0, for (almost) all 0 t T, s > 0. (e) [5pts.] What is the precise (financial) interpretation of a function F that is a solution to the Black-Scholes PDE? 5
Answer sketches to questions 1-3 (discrete-time finance) Note that the answers provided below are sketches. material. No rights can be claimed from this Question 1 a. See LN, page 29. b. See LN, page 39. c. (1) Write down the expected payoffs E(k e x i ), i = 1, 2; (2) k e lies in the asset span: k e = h 1 x 1 + h 2 x 2 for some portfolio (h 1, h 2 ); (3) Substitue in (1) and solve for (h 1, h 2 ). d. m D p, so Xy = p, and m = Π 1 y m is also a pricing kernel, i.e. m = X h. Premultiply this with XΠ on both sides and realize that XΠm = E[mX] = p! Solve h from p = XΠX h. e. See LN, page 44. Question 2 a. LN, p.48: l m = 0! b. LN, p.48. c. LN, p.48. d. LN, p.49. Question 3 This question is directly taken from the Pennacchi book. See the accompanying document. 6