Sample of Ph.D. Advisory Exam For MathFinance

Transcription

1 Sample of Ph.D. Advisory Exam For MathFinance Students who wish to enter the Ph.D. program of Mathematics of Finance are required to take the advisory exam. This exam consists of three major parts. The first part (Math- Finance I) addresses mainely some of the probability tools used in discrete-time financial models. The second part of this exam deals with some problems in discrete-time financial models. These questions reveals to the applicant how part I of this exam enters financial modeling and analysis. The third part treats a discrete-time incomplete financial market model. This illustrates how MathFinance leads to an interplay between different areas (optimization, analysis and probability for instance). We recommand you the following books: 1. Probability, A.N. Shiryaev 2. Introduction to Stochastic Calculus Applied to Finance, D. Lamberton and B. Lapeyre (1991) The applicant is invited to consult the topics related to the proposed problems (conditional expatation/probability, dicrete-time martingales,..., discrete-time financial models for instance).

2 1 MathFinance I In this part, we consider a probability space (Ω, F, P ) on which we consider a sequence of integrable random variables η 0,...η n. Definition 1.1 Let (M k ) k=0,...,n be a process and (F k ) k=0,...,n a nondecreasing sequence of σ-algebra F 0 F 1... F n = F defined on the probability space above. The couple (M k, F k ) k=0,...,n is said to be a martingale if the following statements are fullfilled: a. M k is F k -measurable and E ( M k ) <. b. For each k = 0,..., n 1, we have E (M k+1 F k ) = M k P a.s. k = 0,..., n, we simply say that (M k ) k=0,...,n is a martin- In the case when F k = σ{m 0, M 1,..., M k }, gale. Question 1. Let D 0 D 1... D n be a sequence of σ-algebra with D 0 = {, Ω} and D n = F such that η k is D k -measurale, k = 0,.., n. Show that (ξ k, D k ) k=0,...,n : ξ k := (η i E (η i D i 1 )) is a martingale. Question 2. Suppose that E (η i η 1,..., η i 1 ) = 0. Show that the sequence ξ = (ξ k ) k=1,...,n where ξ 1 = η 1 and ξ k+1 = η i+1 f i (η 1,..., η i ) and f i are given bounded functions, is a martingale. Question 3. Let (ξ k, D k ) k=0,...,n and (η k, D k ) k=0,...,n be two square integrable martingales such that ξ 1 = η 1 = 0. show that (m k, D k ) k=0,...,n, m k := ξ k η k (η i η i 1 )(ξ i ξ i 1 ) is a martingale. In particular, when ξ k = η k, deduce that the process (η 2 k ) k=0,...,n can written as a sum of a martingale and a non-negative and non-decreasing process. 2

3 Question 4. Let ξ and η be two independent random variables indenticaly diostributed such that E(ξ) is defined. Show that E (ξ ξ + η) = E (η ξ + η) = ξ + η. 2 3

4 2 MathFinance II Problem I. We consider one risky asset whose price process is denoted by S n, n = 0,..., N (N is a positive integer). The mouvement of this process S is given as follows. In each step in time the process can take only two values: going up or down as follows. S n+1 = { Sn (1 + a) S n (1 + b) where 1 < a < b and the initial capital, S 0, is supposed known. We consider also a non-risky asset ( bank account) whose price is denoted by Sn 0 = (1 + r)n. Put T n = Sn S n 1, n = 1,..., N. 1. Show that the discounted price process Sn is a martingale if and only if Sn 0 E (T n+1 F n ) = 1 + r, n = 0,..., T Let Q be a probability measure such that Q (T n+1 = 1 + a) = q [0, 1] for each n = 0,..., T 1. Prove that Sn S is a martingale under Q and 0 < q < 1 if and only if a < r < b. Determine q in terms n 0 of a, b and r in this context. 3. Suppoe that the condition a < r < b is not satisfied. Provide an investement policy (borrowing money from the bank and investing in the stock S n, n = 0,..., N) that leads to a profit free from risk. 4. Suppose that the sequence of random variable T 1,..., T N are independent identically distributed such that P (T i = 1 + a) b r b a = 1 P (T i = 1 + b). Prove that Sn S is a martingale. n 0 5. Conversely, suppose that Sn S is a martingale, prove that the random variables T n 0 1,..., T N independent with the same distribution. are Economic interpretations: The condition 0 < q < 1 permits Q to be an equivalent martingale measure for the market described by S n and S 0 n. This mathematical formulation represents the financial concept of absence of arbitrage opportunities in this framework. An arbitrage opportunity is an investement strategy that leads to a profit with no risk taken. 4

5 3 MathFinance III On a probability space (Ω, F, P ), we consider a non-decreasing sequence of σ-algebra (F k, k = 0,..., T ) and a process X := (X k, k = 0,..., T ) such that X k is a square integrable and F k -measurable random variable. Hereafter, we denote X k := X k X k 1, k = 1,..., T. Definitions We say that a process θ = (θ k ) k=1,...,t is predictable if θ k is F k 1 -measurable. 2. X is said to satisfy the nondegeneracy condition, denoted hereafter by (ND), if there exists δ (0, 1) such that ) E ( X k F k 1 ) 2 δe (( X k ) 2 F k 1, P a.s. For a predictable process θ, we consider the associated gain process given by G k (θ) = θ j X j, j=1 k = 0,..., T The following set respresents the set of predictable processes allowed to construct gain processes. Θ := {θ predictable such that θ k X k L 2 (P ), k = 1,.., T } Notations: (b) Consider the following processes α k := A k E (( X k ) 2 F k 1 ), k = 1,.., T K k := Ẑ k := k j=1 1 α j X j 1 α j A j, k = 0,.., T (E [ X i F i 1 ]) 2, k = 0,.., T. V ar ( X i F i 1 ) The former process is called the mean-variance tradeoff process. Conventions: Hereafter, we will take into account the following conventions: a. A sum on a empty set is 0. b. A product on a empty set is 1. c. 0/0 = 0 Question 1. Prove that X can be written in unique sense as follows X = X 0 + M + A 5

6 where M is a square integrable martingale and A is a predictable process. (Hint. Consider A k = E ( X k F k 1 ) and use Question 1 of Part I) Question 2. Prove that the following statements are equivalent 2.a. X satisfies (ND) 2.b. α k A k δ, P a.s k = 1,..., T Question 3. Prove that in case X satisfies (ND), the process Ẑ is a square integrable martingale. (Hint: prove that ) 1 E (Ẑ2 k, k = 0,..., T.) (1 δ) k Question 4. Define the process β as follows ( β k := E ) X Tj=k+1 k (1 β j X j ) F k 1 E (( X k ) 2 ), k = 1,.., T. T j=k+1 (1 β j X j ) 2 F k 1 In the remaing part of MathFinance III, assume that the process β is well defined and T E (1 β j X j ) 2 F k < +, P a.s k = 0,..., T. (a) Prove that j=k+1 ( T ) ( T ) E (1 β j X j ) 2 F k 1 = E (1 β j X j ) F k 1 1, k = 1,.., T i=k (b) Prove that i=k ( ) E Z T X k F k 1 = 0, k = 1,..., T. Question 5. Suppose that the mean-variance tradeoff process K is deterministic (does not depend on the randomness, it depends on time only). Show that the processes β and α coincide (use backward induction). Question 6. Let ξ Θ, H L 2 (P ) and c IR. Prove that the following statement are equivalent: (6.a) For each θ Θ E{[H c G T (ξ)] G T (θ)} = 0, θ Θ (6.b) For each k = 1,..., T E{[H c G T (ξ)] X k F k 1 } = 0, P a.s. 6

7 Question 7. Example1: Suppose that X 0 = 0 and X 1 takes the values 1, 0 and 1 with probability 1/3 for each value. Given X 1 1, the distribution of X 2 takes 1 and 1 with probability of 1/2. The distribution of X 2 given X 1 = 1 is given by ν, a measure such that x 2 ν(dx) <, ν({0}) < 1 a. Calculate the processes α and β in this case. b. Prove that the mean-variance tradeoff process is bounded in this case. c. Provide a necessary and sufficient condition in terms of the measure ν for the two processes α and β to differ. d. Prove that the processes α and β are different if and only if X is a martingale. Optional Question. Let H be a square integrable random variable. Suppose that there exist ξ Θ, c a constant and L a square integrable martingale such that and Show that for each θ Θ, we have Or equivalently that ξ is a solution to: E ( L k M k F k 1 ) = 0, k = 1,..., T, P a.s. E H = H 0 + G T (ξ) + L T, P a.s. (3.1) [ (H c G T (ξ)) 2] E [ (H c G T (θ)) 2]. [ min E (H c G T (θ)) 2]. (3.2) θ Θ Economic Interpretations: The process X denotes a discounted risky asset price process. The set Θ models the set of admissible strategies for trade in the market. The process G k (θ) determine the aggregate gain resulting from the investment policy θ up to time k. The optional question proposes a methodology for reducing the risk issued from a claim in incomplete market. Indeed this method consists of splitting the claim H -in a unique sense- into two part: the first part is exactely replicated (G T (ξ)) and the second part is totally non-replicated (L T ) by trading in the market model under consideration. We can prove (in a some delicate manner) that the condition (ND) implies that the decomposition (3.1) exists for any H L 2 (P ). Hence the minimization problem (3.2) admits a solution for any square integrable claim H. We can also prove that the problem (3.2) admits a solution for any square integrable claim H if and only if the subspace G T (Θ) is closed in L 2 (P ). 7