KØBENHAVNS UNIVERSITET (Blok 2, 2011/2012) Naturvidenskabelig kandidateksamen Continuous time finance (FinKont) TIME ALLOWED : 3 hours

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1 This question paper consists of 3 printed pages FinKont KØBENHAVNS UNIVERSITET (Blok 2, 211/212) Naturvidenskabelig kandidateksamen Continuous time finance (FinKont) TIME ALLOWED : 3 hours This exam paper consists of 3 pages and 3 main questions, consisting of several subquestions. Economize with your time: Do not copy the text in the problems, and when possible refer to results from the syllabus without providing proofs. All written aids are allowed. You are allowed to write your answers in pencil. Problem 1 Let W (t) denote a Brownian motion and let T be a fixed time. Define the two processes X(t) = sin ( W (t) ) Y (t) = cos ( W (t) ). (a) Show that the process (X(t), Y (t)) solves the (2-dimensional) stochastic differential equation dx(t) = 1 X(t) dt + Y (t) dw (t), X() = 2 dy (t) = 1 Y (t) dt X(t) dw (t), Y () = 1. 2 (b) (i) Find a function f(t) such that M(t) = f(t)x(t) is a martingale. (ii) Find a constant z and a process h(t) such that sin ( W (T ) ) = z + T h(t) dw (t). (c) Let λ R be a constant. Show that E[W (T )e λw (T ) ] = λt e λ2 T/2. 1

2 Problem 2 Consider a standard Black-Scholes model, that is, a model consisting of a bank account B(t) with P-dynamics given by db(t) = rb(t) dt and a stock S(t) with P-dynamics given by ds(t) = αs(t) dt + σs(t) d W (t), S() = s > where r >, α R, σ > are constants and W (t) is a P-Brownian motion. Let T > be a given and fixed (expiry) date. As usual, let h(t) = ( h (t), h 1 (t) ) be a portfolio where h (t) is the number of units of the bank account at time t and h 1 (t) is the number of shares in the stock at time t. (a) Consider the portfolio h(t) = ( h (t), h 1 (t) ) = ( S(t)/B(t), B(t)/S(t) ). Determine whether the portfolio h is self-financing or not. Consider the square root option that at time T pays X = S(T ). Let F (t, s) be the pricing function of the option, that is, the arbitrage free price of the option at time t is given by π(t, X) = F (t, S(t)). (b) (i) Which equation does the pricing function F solve? (ii) Determine the arbitrage free price of the option at time t < T. Consider a new simple option X = Φ(S(T )) where the pay off function is given by Φ(s) = s log(s). (c) Determine the arbitrage free price of the option at time t < T. (Hint: one might use the result of Problem 1(c)). Problem 3 Consider a two-dimensional Black-Scholes model. The market model consists of three assets: A bank account B(t) and two stocks S 1 (t) and S 2 (t). For constant interest rate r, P-dynamics of B(t) are given by db(t) = rb(t) dt. The P-dynamics of S 1 (t) and S 2 (t) are given by ds 1 (t) = α 1 S 1 (t) dt + σ 1 S 1 (t) d W (t), S 1 () = s 1 > ds 2 (t) = α 2 S 2 (t) dt + σ 2 S 2 (t) d W (t), S 2 () = s 2 > where α 1, α 2 R, σ 1, σ 2 > are constants, and W (t) is a P-Brownian motion. 2

3 Let T > be a given and fixed (expiry) date. FinKont (a) Derive a condition such that the model is arbitrage free and complete. Assume that the market is arbitrage free and complete. Consider the option that at time T pays 1 if S 1 (T ) is strictly greater than S 2 (T ) and zero otherwise. (b) (i) Determine the arbitrage free price of the option at time t < T. (ii) Derive a hedge for the option. For the remainder of this problem assume that β = α 1 r σ 1 α 2 r σ 2 >. (c) Show that the model is not arbitrage free. Consider the portfolio where h 1 (t) = 1/(σ 1 S 1 (t)) is the number of shares in the stock 1 at time t, h 2 (t) = 1/(σ 2 S 2 (t)) is the number of shares in the stock 2 at time t, and choose the number of units of the bank account h (t) to make the portfolio self-financing. Let V h (t) denote the associated value process. Assume that the initial value of the portfolio is zero, that is, V h () =. (d) (i) Argue that h (t) = (V h (t) h 1 (t)s 1 (t) h 2 (t)s 2 (t))/b(t). (ii) Show that the portfolio is an arbitrage. END 3

4 This question paper consists of 3 printed pages FinKont KØBENHAVNS UNIVERSITET (Blok 2, 212/213) Naturvidenskabelig kandidateksamen Continuous time finance (FinKont) TIME ALLOWED : 3 hours This exam paper consists of 3 pages and 3 main questions, consisting of several subquestions. Use your time wisely: Do not copy the text of the problems, and when possible refer to results from the syllabus without providing proofs. All aids are allowed. You are allowed to write your answers in pencil. Problem 1 Let W (t) denote a Brownian motion and let T be a fixed time. (a) Use Feynman-Kac representation formula to solve the following boundary value problem in the domain of [, T ] (, ) (b) F t (t, x) + rxf x (t, x) + σ2 2 x2 F xx (t, x) = rf (t, x) for t < T where r R, σ > and K > are constants. F (T, x) = (x K) 2 ( T ) (i) Compute the mean value of exp t dw (t). (ii) Find a process h(t) such that ( T ) [ ( T )] exp t dw (t) = E exp t dw (t) + T h(t) dw (t). (c) Find the explicit solution to the following stochastic differential equation dx(t) = X(t) dt + e t dw (t), X() = x. 4

5 Problem 2 Consider a standard Black-Scholes model, that is, a model consisting of a bank account B(t) with P-dynamics given by db(t) = rb(t) dt and a stock S(t) with P-dynamics given by ds(t) = αs(t) dt + σs(t) d W (t), S() = s > where r R, α R and σ > are constants and W (t) is a P-Brownian motion. Let T > be a given and fixed (expiry) date. Let h (t) = (1 u) V h (t) B(t) and h 1(t) = u V h (t) S(t) be a self-financing portfolio where u is a constant and set V h () = 1. As usual, h (t) is the number of units of the bank account at time t and h 1 (t) is the number of shares in the stock at time t. (a) Find u such that the processes B(t)/V h (t) and S(t)/V h (t) both are martingales. Consider the derivative that at time T pays X = log(v h (T )). (b) Determine the arbitrage free price of the derivative at time t < T. Problem 3 Let W 1 (t) and W 2 (t) be two independent P-Brownian motions. The filtration is the one generated by the two Brownian motions, F t = σ( W 1 (s), W 2 (s) s t). Consider a market model with two assets: A bank account B(t) and a stock S 1 (t). For constant interest rate r, P-dynamics of B(t) are given by The P-dynamics of S 1 (t) is given by db(t) = rb(t) dt. ds 1 (t) = α 1 S 1 (t) dt + σ 11 S 1 (t) d W 1 (t) + σ 12 S 1 (t) d W 2 (t), S 1 () = s 1 > where α 1 R, σ 11 > and σ 12 > are constants. Let T > be a given and fixed (expiry) date. 5

6 (a) Is the model arbitrage free? Is the model complete? (b) Show that the call option X = (S 1 (T ) K) +, has a price process that does not depend on the choice of equivalent martingale measure. Moreover, determine the price process (arbitrage free price) of the call option at time t < T. (c) Show that the derivative Y = W 2 (T ), has a price process that do depend on the choice of equivalent martingale measure. For the remainder of this problem assume that we extend the market model to include a second stock with price process S 2 (t). The P-dynamics of S 2 (t) is given by ds 2 (t) = α 2 S 2 (t) dt + σ 22 S 2 (t) d W 2 (t), S 2 () = s 2 > where α 2 R and σ 22 > are constants. Thus, the new model (B, S 1, S 2 ) is a two-dimensional Black-Scholes model. (d) Is the model arbitrage free? Is the model complete? (e) (i) Determine the arbitrage free price of derivative Y at time t < T. (ii) Derive a hedge for derivative Y. END 6

7 This question paper consists of 4 printed pages FinKont KØBENHAVNS UNIVERSITET (Blok 2, 213/214) Naturvidenskabelig kandidateksamen Continuous time finance (FinKont) TIME ALLOWED : 3 hours This exam paper consists of 4 pages and 3 main questions, consisting of several subquestions. Use your time wisely: Do not copy the text of the problems, and when possible refer to results from the syllabus without providing proofs. All aids are allowed. You are allowed to write your answers in pencil. Problem 1 Let W (t) denote a Brownian motion and let F t = Ft W T be a fixed time. = σ(w (s) s t). Let Let X(t) be a geometric Brownian motion given by dx(t) = αx(t) dt + σx(t) dw (t) and X() = x > where α R and σ > are constants. Let M(t) be a process given by M(t) = a + e αt( bx(t) + cx(t) 2α/σ2 ) where a, b, c R are constants. (a) Show that M(t) is a martingale. Consider the stochastic differential equation dy (t) = ( t 2 Y (t) + α(t) ) dt + β(t) dw (t) and Y () = y where α(t) and β(t) are deterministic functions. 7

8 (b) (i) Show that the stochastic differential equation has a solution of the form ( Y (t) = e t3 /3 y + t and determine the function f(s). f(s)α(s) ds + (ii) Determine the distribution of Y (t). t ) f(s)β(s) dw (s) (c) (i) Compute the function g(t, x) such that g(t, W (t)) = E[W 4 (T ) F t ]. (ii) Find a constant z and a process h(t) such that W 4 (T ) = z+ T h(s) dw (s). Problem 2 Consider a standard Black-Scholes model, that is, a model consisting of a bank account B(t) with P-dynamics given by db(t) = rb(t) dt and a stock S(t) with P-dynamics given by ds(t) = αs(t) dt + σs(t) d W (t), S() = s > where r R, α R and σ > are constants and W (t) is a P-Brownian motion. Let T > be a given and fixed (expiry) date. Let h(t) = ( h (t), h 1 (t) ) be a portfolio where h (t) is the number of units of the bank account at time t and h 1 (t) is the number of shares in the stock at time t. Consider the derivative whose payoff at time T is X = 1 T T S(u) du and set I(t) = t S(u) du. (a) Determine the pricing function G(t, s, i) such that the arbitrage free price of the derivative at time t < T is given by Π(t, X) = G(t, S(t), I(t)). (Hint: you might use Fubini for conditional expectation: E[ b X(u)du F] = a b E[X(u) F] du). a Turn 8 over

9 Consider the self financing portfolio given by h 1 (t) = G s (t, S(t), I(t)) and h (t) = (V h (t) h 1 (t)s(t))/b(t). Assume that the initial value of the portfolio is V h () = G(, s, ). (b) Show that the derivative X can be hedged by h(t) = (h (t), h 1 (t)). Let F (t, s) be the solution to the partial differential equation F t (t, s) + rsf s (t, s) σ2 s 2 F ss (t, s) = rf (t, s) for t < T and s > F (T, s) = (s K) +. Thus, F (t, s) is the pricing function for a call option, expiry date T with strike K and with volatility σ >. Consider a portfolio given by h 1 (t) = F s (t, S(t)) and h (t) = e rt( F (t, S(t)) S(t)F s (t, S(t) ). (c) (i) Show that V h (T ) = (S(T ) K) +. (ii) Is h a hedge for the call option? Problem 3 Consider a two-dimensional Black-Scholes model. The market model consists of three assets: A bank account B(t) and two stocks S 1 (t) and S 2 (t). The P- dynamics of B(t) are given by db(t) = rb(t) dt, B() = 1 where r > is a constant. The P-dynamics of S 1 (t) and S 2 (t) are given by ds 1 (t) = α 1 S 1 (t) dt + σ 1 S 1 (t) d W 1 (t), S 1 () = s 1 > ds 2 (t) = α 2 S 2 (t) dt + σ 2 S 2 (t) ( ρ d W 1 (t) + 1 ρ 2 d W 2 (t) ), S 2 () = s 2 > where α 1, α 2 R, σ 1, σ 2 >, 1 < ρ < 1 are constants, and W 1 (t) and W 2 (t) are two independent P-Brownian motions. The filtration is the one generated by the two Brownian motions, that is, F t = σ( W 1 (s), W 2 (s) s t). Let T > be a given and fixed (expiry) date. (a) (i) Show that the model is arbitrage free and complete. (ii) What is the dynamics of S 1 (t) and S 2 (t) under the equivalent martingale measure? 9

10 Set L = e rt S 1 (T )/S 1 () and define a new probability measure d Q = LdQ. FinKont (b) What is the distribution of S 2 (T )/S 1 (T ) under the probability measure Q? Consider the derivative that at time T pays X = ( S 2 (T ) S 1 (T ) ) +. Let F (t, s1, s 2 ) be the pricing function of the derivative. (c) Compute the arbitrage free price of the derivative at time t =. (d) Which equation does the pricing function solve? END 1

11 This question paper consists of 3 printed pages FinKont KØBENHAVNS UNIVERSITET (Blok 2, 214/215) Naturvidenskabelig kandidateksamen Continuous time finance (FinKont) TIME ALLOWED : 3 hours This exam paper consists of 3 pages and 2 main questions, consisting of several subquestions. Use your time wisely: Do not copy the text of the problems, and when possible refer to results from the syllabus without providing proofs. All aids are allowed. You are allowed to write your answers in pencil. Problem 1 Let W (t) denote a Brownian motion and let F t = F W t = σ(w (s) s t). Let Z(t) be a process given by where α > is a constant. Z(t) = t (a) Determine the distribution of Z(t). u α dw (u) tα (b) (i) Is Z(t) an Ito process? (ii) Is Z(t) a martingale? Let F (t, x) be the solution to the partial differential equation F t (t, x) = α(x)f x (t, x) F xx(t, x) for t > and x R F (, x) = Φ(x) where α(x) and Φ(x) are given bounded continuous functions. 11

12 Let X(t) be the solution to the stochastic differential equation dx(t) = α ( X(t) ) dt + dw (t) X() = x. (c) (i) Show that the solution F (t, x) has the stochastic representation F (t, x) = E x [ Φ ( X(t) )]. (Hint: Consider the process ( t s, X(s) ) where t is given and fixed). (ii) Show that F (t, x) can be expressed by [ ( t F (t, x) = E x exp α ( W (u) ) dw (u) 1 2 (Hint: Girsanov theorem). t α 2( W (u) ) ) du Φ ( W (t) )]. Problem 2 Consider a standard Black-Scholes model, that is, a model consisting of a bank account B(t) with P-dynamics given by db(t) = rb(t) dt and a stock S(t) with P-dynamics given by ds(t) = αs(t) dt + σs(t) d W (t), S() = s > where r R, α R and σ > are constants and W (t) is a P-Brownian motion. Let T > be a given and fixed (expiry) date. Let h(t) = ( h (t), h 1 (t) ) be a portfolio where h (t) is the number of units of the bank account at time t and h 1 (t) is the number of shares in the stock at time t. Let u(t) = ( u (t), u 1 (t) ) be the relative portfolio. Assume that the relative portfolio u (t) =.2 and u 1 (t) =.8 is self-financing and that the initial value of the portfolio is 1. (a) Compute explicitly the corresponding portfolio h(t) = ( h (t), h 1 (t) ) as a function of W (t) and the parameters. 12

13 Consider a simple derivative X = Φ(S(T )) where the payoff function is given by K 2 s if s K 1 Φ(s) = K 2 K 1 if K 1 < s K 2 s K 1 if s > K 2 where < K 1 < K 2. (b) Find a hedging portfolio for derivative X. Consider a new derivative that at time T pays Y = Φ 1 ( S(T ) ) Φ2 ( S(T ) ) where the payoffs are given by Φ 1 (s) = ( log(s) K ) + and Φ2 (s) = ( K log(s) ) +. (c) Determine the arbitrage free price of derivative Y at time t < T. Let λ = (α r)/σ be the market price of risk and define the process Y (t) = e λ W (t) (r+λ 2 /2)t. Let V h (t) be the value process of a self-financing portfolio h(t) = ( h (t), h 1 (t) ). (d) (i) Show that Y (t) solves a stochastic differential equation. (ii) Show that Y (t)v h (t) is a martingale. (e) Show that if V h () = and V h (t) for all t then V h (t) = for all t. Consider a derivative that at time T pays Z and let π(t) denote the arbitrage free price at time t. (f) Show that π() = E P[ Y (T )Z ]. Let Q be the equivalent martingale measure and let L(t) be the associated Likelihood process. (g) Show that Y (t) = e rt L(t). END 13

14 This question paper consists of 3 printed pages FinKont KØBENHAVNS UNIVERSITET (Blok 2, 215/216) Naturvidenskabelig kandidateksamen Continuous time finance (FinKont) TIME ALLOWED : 3 hours This exam paper consists of 3 pages and 3 main questions, consisting of several subquestions. Use your time wisely: Do not copy the text of the problems, and when possible refer to results from the syllabus without providing proofs. All aids are allowed. You are allowed to write your answers in pencil. Problem 1 Let W (t) denote a Brownian motion and let F t = Ft W T > be a given and fixed time. = σ(w (s) s t). Let Let X(t) be a process given by (a) X(t) = t 1 dw (u) for t < T. T u (i) Show that the stochastic integral X(t) is well defined. That is, show that the integrand belongs to the class 2 [, t] for any t < T. (ii) Compute the mean value of X(t) and the variance of X(t). Consider the stochastic differential equation for t < T. dy (t) = Y (t) T t dt + dw (t) and Y () = (b) Show that Y (t) = (T t)x(t) solves the stochastic differential equation. (c) Show that Y (t) as t T in L 2. (Hint: Recall that X n X in L 2 if E[(X n X) 2 ] ). 14

15 Problem 2 Consider a standard Black-Scholes model, that is, a model consisting of a bank account B(t) with P-dynamics given by db(t) = rb(t) dt, B() = 1 and a stock S(t) with P-dynamics given by ds(t) = αs(t) dt + σs(t) d W (t), S() = s > where r R, α R and σ > are constants and W (t) is a P-Brownian motion. Let T > be a given and fixed (expiry) date. Let h(t) = ( h (t), h 1 (t) ) be a portfolio where h (t) is the number of units of the bank account at time t and h 1 (t) is the number of shares in the stock at time t. Consider the portfolio h(t) = ( h (t), h 1 (t) ) = (.5S(t),.5B(t) ). (a) Is the portfolio h self-financing? Consider the derivative that at time T pays X = ( log ( S(T ) )) 2 and let F (t, s) be the pricing function of the derivative. (b) (i) Determine the arbitrage free price of derivative X at time t =. (ii) Determine the equation satisfied by the pricing function F (t, s). Let C(t, s; K) denote the Black-Scholes price at time t of an European call option with strike price K and expiry date T when the current price of the underlying stock is s. (c) Show that C(t, s; K) is a convex function of K. That is, to show that C(t, s; (1 a)k 1 + ak 2 ) (1 a)c(t, s; K 1 ) + ac(t, s; K 2 ) for any a 1 and any < K 1 < K 2. 15

16 Consider the derivative that at time T pays Y = ( S(T ) cs(t ) ) + where c > is a constant and < T < T is a fixed date. (d) Determine the arbitrage free price of derivative Y at time t = T. (e) (i) Determine the arbitrage free price of derivative Y at time t < T. (ii) Find a hedging portfolio for derivative Y. Problem 3 Consider a n-dimensional Black-Scholes model. The market model consists of (n+1) assets: A bank account B(t) and n stocks S 1 (t),..., S n (t). The P-dynamics of B(t) is db(t) = rb(t) dt, B() = 1 where r R is a constant interest rate. The P-dynamics of S i (t) is ds i (t) = α i S i (t) dt + σ i S i (t) d W i (t), S i () = s i > for i = 1,..., n where α 1,..., α n R and σ 1 >,..., σ n > are constants and W 1 (t),..., W n (t) are n independent P-Brownian motions. The filtration is the one generated by the n Brownian motions, that is, F t = σ( W 1 (s),..., W n (s) s t). Let T > be a given and fixed (expiry) date. (a) (i) Is the model arbitrage free? (ii) Is the model complete? Consider the derivative that at time T pays X = n ( Si (T ) b i ) K i i=1 where b 1 >,..., b n > and K 1 >,..., K n > are constants. (b) Determine the arbitrage free price of derivative X at time t =. END 16