MAS452/MAS6052. MAS452/MAS Turn Over SCHOOL OF MATHEMATICS AND STATISTICS. Stochastic Processes and Financial Mathematics

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1 t r t r2 r t SCHOOL OF MATHEMATICS AND STATISTICS Stochastic Processes and Financial Mathematics Spring Semester hours t s s tt t q st s 1 r s r t r s rts t q st s r t r r t Please leave this exam paper on your desk Do not remove it from the hall str t r r r ts t t 2 st t 1 Turn Over

2 Blank 2 Continued

3 t Ω = {1,2,3,4} s r t s t A = {,Ω,{1},{2,3} } t t A s t σ st t ts σ(a) t σ r t 2 A r s X : Ω R 2 X(ω) = ω s t t X σ(a) s r st 2 2 r s r r s t X r r 1 2 Y = e X = n=0 X n n! s r r r s 2 s st r r s ts t s r t2 s s r ts ts r r s r t 2 r r 2 st t t X 1 r r t str t P[X 1 = 1] = P[X 1 = 1] = 1 2 s q r r s (X n ) 2 s 2 t t r n N t Y = X 1 X n+1 = X n. t st t ts r tr r s r 2 st 2 2 r s r s X n d Y s n X n P Y s n X n a.s. Y s n r s 3 Turn Over

4 s q st s r s t s r t t t t ss ts s st r s r2 t ss t t t t s t r2 r s t t h t = (x t,y t ) r t = 1,...,T rt str t 2 t V h t t ts r ss t s t r (h t ) t r t t t t Φ(S T ) r t t t t r t rs d,r u s q t t t r r tr r s T = 2 t t r t rs t p u = p d = 0.5 u = 1.5 d = 0.75 r = 0 s = 2 s r t t t t 1 r s t T = 2 Φ(S T ) = max(s T 10) r r tr t st r r ss t t s t = 0,1,2 t t 2 r tr t s t r tr r r r Φ(S T ) t t rt str t 2 t t r t s Φ(S T ) r s t (X n ) n N s q t r r s t t str t X n 2 P[X n = 1] = P[X n = 1] = 1 2n 2, P[X n = 0] = 1 1 n 2. t r S n = n X i r S 0 = 0 i=1 t t S n s rt t r s t t tr t F n r t 2 X n t E[ X n ] s t t r n E[ S n ] n i=1 1 i 2 r s t t t r 1 sts r r S s t t S n a.s. S s n 1 r 2 2 t s s t t st s r 2 t r S n t t 2 2 s r s t T = inf{n 0 ; r m n,s m = S }. s T st t r st t r 2 r s r t t E[S T ] = E[S ] = 0 r s 4 Continued

5 t t t t r t t B t r t t F t ts r t tr t t 0 s t s t t r t t s t t E[B t F s ] = B s t t E[Bt 2 Bs 2 F s ] = t s r s t t r s r t2 t r t r s r s t B t st r r t t Z t = (sinb t ) 2 t t Z t s s t t st st r t q t dz t = (1 2Z t )dt+2sinb t cosb t db t. t t f(t) = E[Z t ] s t s s t r r2 r t q t f (t)+2f(t) = 1 1 r ss f(t) s 1 t t t s t t f(t) 1 2 s t r s 5 Turn Over

6 t B t st r r t t α R σ > 0 t X t t r ss s t s 2 X 0 > 0 dx t = αx t dt+σx t db t. t st st r t dz t r Z t = logx t t Φ(S T ) = log(s T ) t t t t T > 0 t t s s t t t t s t t t t t [0,T] s 2 e r(t t)( logs t +(r 1 2 σ2 )(T t) ). r s r s r2 t s ss t t t t s t r2 r s t s r rt t V(t,S t ) t t t t t t t s r t s rt t t tr t t t r st r s 2 t t t s t t tr rt r t r = 1,S 0 = 1 σ 2 = 2 s t t t t 0 r rt s sts tr t t t t Φ(S T ) = log(s T ) s t r r r t t x st t t t t 2 t r rt r r t r rt t tr t t 0 r s 6 Continued

7 s r t t t t t r A C E B D F t t r t s η j = 1 1+j r s r2 t ss t t t t s t r2 r s t s t t A s ts ts s t t r t2 t t t r s t s ts s s r2 t t t r t st t r s ts t t t s t r st r s End of Question Paper 7

8 MAS352/452/6052 Formula Sheet Part One Where not explicitly specified, the notation used matches that within the typed lecture notes. Modes of convergence X n d X for any x R, limn P[X n x] P[X x]. X n P X for any a > 0, limn P[ X n X > a] = 0. X n a.s. X P[X n X as n ] = 1. X n L p X E[ X n X p ] 0 as n. The binomial model and the one-period model The binomial model is parametrized by the deterministic constants r (discrete interest rate), p u and p d (probabilities of stock price increase/decrease), u and d (factors of stock price increase/decrease), and s (initial stock price). The value of x in cash, held at time t, will become x(1+r) at time t+1. The value of a unit of stock S t, at time t, satisfies S t+1 = Z n S n, where P[Z t = u] = p u and P[Z t = d] = p d, with initial value S 0 = s. When d < 1+r < u, the risk-neutral probabilities are given by q u = (1+r) d u d, q d = u (1+r). u d The binomial model has discrete time t = 0,1,2,...,T. The case T = 1 is known as the one-period model. Conditions for the optional stopping theorem (MAS452/6052 only) The optional stopping theorem, for a martingale M n and a stopping time T, holds if any one of the following conditions is fulfilled: (a) T is bounded. (b) M n is bounded and P[T < ] = 1. (c) E[T] < and there exists c R such that M n M n 1 c for all n.

9 MAS352/452/6052 Formula Sheet Part Two Where not explicitly specified, the notation used matches that within the typed lecture notes. The normal distribution Z N(µ,σ 2 ) has probability density function f Z (z) = 1 2πσ e (z µ)2 2σ 2. Moments: E[Z] = µ, E[Z 2 ] = σ 2 +µ 2, E[e Z ] = e µ+1 2 σ2. Ito s formula For an Ito process X t with stochastic differential dx t = F t dt + G t db t, and a suitably differentiable function f(t,x), it holds that { f dz t = t (t,x f t)+f t x (t,x t)+ 1 2 } f f 2 G2 t x 2(t,X t) dt+g t x (t,x t)db t where Z t = f(t,x t ). Geometric Brownian motion For deterministic constants α,σ R, and u [t,t] the solution to the stochastic differential equation dx u = αx u dt+σx u db u satisfies X T = X t e (α 1 2 σ2 )(T t)+σ(b T B t). The Feyman-Kac formula Suppose that F(t,x), for t [0,T] and x R, satisfies F t (t,x)+α(t,x) F x (t,x)+ 1 2 β(t,x)2 2 F x 2 (t,x) = 0 F(T,x) = Φ(x). Then F(t,x) = E t,x [Φ(X T )], where X u satisfies dx u = α(u,x u )dt+β(u,x u )db u.

10 The Black-Scholes model The Black-Scholes model is parametrized by the deterministic constants r (continuous interest rate), µ (stock price drift) and σ (stock price volatility). The value of a unit of cash C t satisfies dc t = rc t dt, with initial value C 0 = 1. The value of a unit of stock S t satisfies ds t = µs t dt+σs t db t, with initial value S 0. At timet [0,T], the price F(t,S t ) ofacontingent claim Φ(S T ) (satisfyinge Q [Φ(S T )] < ) withexercise date T > 0 satisfies the Black-Scholes PDE: The unique solution F satisfies F t (t,s)+rs F s (t,s)+ 1 2 s2 σ 2 2 F (t,s) rf(t,s) = 0, s2 F(T,s) = Φ(s). F(t,S t ) = e r(t t) E Q [Φ(S T ) F t ] for all t [0,T]. Here, the risk-neutral world Q is the probability measure under which S t satisfies ds t = rs t dt+σs t db t. The Gai-Kapadia model of debt contagion (MAS452/6052 only) A financial network consists of banks and loans, represented respectively as the vertices V and (directed) edges E of a graph G. An edge from vertex X to vertex Y represents a loan owed by bank X to bank Y. Each loan has two possible states: healthy, or defaulted. Each bank has two possible states: healthy, or failed. Initially, all banks are assumed to be healthy, and all loans between all banks are assumed to be healthy. Given a sequence of contagion probabilities η j [0,1], we define a model of debt contagion by assuming that: ( ) For any bank X, with in-degree j if, at any point, X is healthy and one of the loans owed to X becomes defaulted, then with probability η j the bank X fails, independently of all else. All loans owed by bank X then become defaulted. Starting from some set of newly defaulted loans, the assumption ( ) is applied iteratively until no more loans default.