Department of Probability and Mathematical Statistics Faculty of Mathematics and Physics, Charles University in Prague petrasek@karlin.mff.cuni.cz Seminar in Stochastic Modelling in Economics and Finance January 10, 2011
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1 Outline 1 2 3 4
2 Economic model At each t 0 an agent owns a capital X t by investing in two assets a riskfree bond that pays interest rate r, db t = rb t dt. a risky asset with s (geometric Brownian motion + jumps) ( ) ds t = S(t ) αdt + σdw t + zñ(dt, dz). At each t 0 an agent controls the number of stocks t in his portfolio, and possibly consumption C t. 1
3 Admissible strategies Definition A strategy ( t, t 0) is admissible if 1 it is predictable, 2 the portfolio (X t, t 0) is self-financing, i.e. dx t = t ds t + (X t t S t )db t. We denote the set of admissible strategies by A(x) for a given capital X 0 = x. We can substitute for ds t and db t and obtain ( ) dx t = t S t (α r)dt + σdw t + zñ(dt, dz) + rx t dt. 1
4 Objective of investment The objective of an agent is to maximize his utility from investments by using admissible strategies t ( t, C t ). His aim is 1 to maximize his consumption over infinite horizon sup ( t,c t) A(x) 0 where β is a discount factor e βt E U(C t )dt, (1.1) 2 to maximize his terminal utility of terminal wealth in a given horizon T sup t A(x) E U(X T ), (1.2) 3 combination of the first two { sup ( t,c t) A(x) E U(X T ) + T 0 e βt E U(C t )dt }. (1.3)
5 Lévy process - definition Definition ( ) Let Ω, F, {F t } t 0, P be a filtered probability space. An adapted process L t is called a Lévy process if it is continuous in probability and has stationary, independent increments. Theorem Let L t be a Lévy process. Then L t has the decomposition L t = b + σw t + zñ(t, dz) + zn(t, dz), 0 t <. z 1 z >1 (1.4) where b R, σ 0, (Ñ) N is a (compensated) Poisson random measure with a Lévy measure ν, all adapted to filtration {F t } t 0.
6 Lévy process - Itô formula Theorem (Itô formula) Suppose L t R is an Lévy process of the form dl t = bdt + σdw t + 1 zñ(t, dz). Let f C 1,2 (R + R) and define Y t = f (t, L t ). Then Y t is again an Lévy process and dy t = f t (t, L t )dt + f x (t, L t ) (bdt + σdw t ) + 1 2 f xx(t, L t )σ 2 dt + + 1 1 (f (t, L t + z) f (t, L t )) Ñ(dt, dz) (f (t, L t + z) f (t, L t ) f x (t, L t )z) ν(dz)dt.
7 Lévy process - Generator Definition Suppose f : R 2 R. Then the generator A of process L t (from the previous theorem) is defined as where L s = x. Theorem 1 Af (s, x) = lim t 0+ t E [f (s + t, L s+t) f (s, x)], Suppose f C 1,2 (R + R). Then Af (s, x) exists and Af (s, x) = f t (s, x) + f x (s, x)b + 1 2 f xx(s, x)σ 2 + 1 (f (s, x + z) f (s, x) f x (s, x)z) ν(dz).
8 Outline 1 2 3 4
State process Y t = Y (u) t is a stochastic process (on filtered probability space) with s dy t = b(y t, u t )dt + σ t (Y t, u t )dw t + γ(y t, u t, z)ñ(dt, dz), Y 0 = y R k where b : R k U R k, σ : R k U R k m, γ : R k U R k R k l are given functions (time homogenous), W is Wiener process (on the given probability space), Ñ compensated Poisson random measure and U R p given set. u(t) = u(t, ω) : R + Ω U R is predictable control process and Y t = Y (u) t jump-diffusion. is a controlled 9
10 Performance criterion For a fixed T (possibly T = ) we define [ ] T J (u) (y) = E f (Y t, u t )dt + g(y T ), 0 where f : S U R, g : R k R are given continuous functions, S is called solvency region. Definition Control u is admissible, denote u A if the state process has a unique, strong solution for all x S and [ ] T E f (Y t, u t )dt + g(y T ) <. 0
11 Value function Our goal is to find the value function v and an optimal control u A such that v(x) = J (u ) = sup J (u) (x). u A We consider Markov controls u(t) = u(y t ), then Av(y) = + k b i (y, u(y))v xi (y) + i=1 k i=1 R k ( σσ T ) (y, u(y))v ij x i x j (y) i,j=1 [v(y + γ j (y, u(y), z j )) v(y) v(y)γ j (y, u(y), z j )] ν j (dz j )dt.
Revision If we start the state process from any t [0, T h] it holds [ ] t+h v(y t ) E f (Y s, u(y s ))ds + v(y (u) t+h ) with equality for u = u. We know that E t v(y (u) t+h ) = v(y t) + t t+h t A (u) v(y s )ds and by substitution into (2.1) we obtain [ t+h ( ) ] 0 E f (Y s, u(y s )) + A (u) v(y s ) ds or in differential t 0 f (y, u(y)) + A (u) v(y), for any u and equality holds for u = u. (2.1) 12
13 HJB for optimal control of jump diffusion Lemma (Verification lemma) Let ṽ C 1,2 satisfies the following 1 lim t T ṽ(y t ) = g(y T ) 2 For any u A(x) f (y, u(y)) + A (u) ṽ(y) 0. 3 There is ũ A(y) such that f (y, ũ(y)) + A (ũ) ṽ(y) = 0. Then ũ = u. and ṽ(y) = v(y) = J (u ) (y), for any y S.
14 Remarks Verification theorem holds also for random time T however with additional requirements. Example T = inf {t > 0, Y t / S} The Hamilton-Jacobi-Bellman equation provides only sufficient for an optimum, but not necessary, which is provided by Pontryagin Maximum.
15 Outline 1 2 3 4
16 Investor s question We refer back to the motivation example. An investor puts his money into risky S t and riskless B t asset. His portfolio X t evolves ( dx t = t S t (α r)dt + σdw t + )+rx zñ(dt, dz) t dt c t X t dt. and he wants to maximize utility from his consumption sup ( t,c t) A(x) 0 1 e βt E U(C t )dt, (3.1) Investor knows that his utility is given by the power utility function, i.e. U(x) = x 1 p, p > 0, p 1, 1 p = log(x), p = 1.
17 Change of notation New processes θ t = ts t X t time t, c t = Ct X t is the proportion of capital invested in risky asset at denotes the consumption proportion. s of investor s portfolio: dx t =θ t X t ( (α r)dt + σdw t + + rx t dt c t X t dt. with X (0) = x, θ t F t, c t F t. 1 zñ(dt, dz) ) (3.2)
18 Computation of generator We would like to apply the verificatin lemma on the controlled process Y t = (t, X t ) T, with Y 0 = (0, x) T. Generator of v(y t ) A (u) v(y) = v t + ((α r)θ + r c) xv x + 1 2 σ2 θ 2 x 2 v xx + Consumption 1 ( v(t, x + xθz) 1 p v(t, x) θzv x ) ν(dz). f (y, u(y)) = e βt U(cx).
19 PDE We guess the form of the value function, v(t, x) = Ke βt x 1 p A (u) v(y) = Ke βt x 1 p [ β + ((α r)θ + r c) (1 p) 1 2 σ2 θ 2 p(1 p) ( + (1 + θz) 1 p 1 θz(1 p) ) ] ν(dz) 1 = Ke βt x 1 p [ β + (r c) (1 p) + h(θ)]. A (u) v(y) + f (y, u(y)) [ ] = Ke βt x 1 p β + (r c)(1 p) + h(θ) + c1 p (3.3). K(1 p)
20 We apply the verification theorem. We demand { } A (u) v(y) + f (y, u(y)) = 0. sup u A We differentiate formula (3.3) with respect to c and θ. Optimal proportion 0 = Λ(θ) = (α r) σ 2 θp + Optimal consumption 0 = (1 p) + c p K 1 ( 1 (1 + θz) p ) zν(dz). c = (K(1 p)) 1/p
21 Constant K Finally we substitute θ and c into equation (3.3) and demand equality to zero 0 = A (u ) v(y) + f (y, u (y)) 0 = Ke βt x 1 p [ β + r(1 p) + h(θ ) K 1/p (1 p) 1/p+1 + (K(1 p)) 1/p] = Ke βt x 1 p [ β + r(1 p) + h(θ ) p (K(1 p)) 1/p] nontrivial solution is K = 1 1 p [β r(1 p) h(θ )p] p.
22 Theorem (Optimal Proportion and Consumption) Assume the portfolio (3.2) and the objective. Let and Then where θ is the optimal proportion, c = (K(1 p)) 1/p Λ(θ ) = 0 β r(1 p) h(θ ) > 0. v(0, x) = Kx 1 p is the value function, K = 1 1 p [β r(1 p) h(θ )p] p. (3.4)
23 Merton proportion and consumption Merton investment proportion Merton consumption proportion c 0 = A(p) = θ 0 = α r pσ 2, β r(1 p) p Let all the no-jump variables be indexed by 0. 1 (α r) 2 1 p 2 σ 2 p.
23 Merton proportion and consumption Merton investment proportion Merton consumption proportion c 0 = A(p) = θ 0 = α r pσ 2, β r(1 p) p Let all the no-jump variables be indexed by 0.??? What is the effect of jumps on optimal values? 1 (α r) 2 1 p 2 σ 2 p.
24 Optimal (jumps included) proportion and consumption Optimal proportion θ solves the equation Λ(θ ) = (α r) σ 2 θp + Optimal consumption 1 c = (K(1 p)) 1/p for a constant K given by equation (3.4). ( 1 (1 + θz) p ) zν(dz) = 0.
25 Function Λ We know that for Λ 0 (θ) solves the Merton and can see that 1 Λ(0) = α r, Λ(θ) is a decreasing function of θ. 2 Function (1 (1 + θz) p ) z is positive for z ( 1/θ, ). We conclude that Λ(θ) < Λ 0 (θ). Corollary θ θ 0, v v 0, c c 0, p > 1, c c 0, 0 < p < 1.
26 Merton cont. S x 1 (money units in S t ) the Merton line (ν = 0) (x 1 = θ0 x 2) 1 θ0 risk decreasing jumps risk increasing jumps x 2 (money units in B t ) S
27 Approximation of small jumps Let us suppose that the measure ν has light tails (jumps are small in absolute value). We can use the taylor expansion and after the substitution into Λ 1 (1 + θz) p = pzθ + o(z 2 ) θ 1 p α r σ 2 + 1 z2 ν(dz), i.e. for smaller jumps we can approximate Lévy process by a Brownian motion with volatility σ 2 + 1 z2 ν(dz).
28 Investor s question II An investor wants to maximize his utility from the terminal wealth Optimal strategy sup E U(X T ) (3.5) t A(x) It is optimal to put constant proportion θ of his money into the risky asset, same as in the previous.
29 Outline 1 2 3 4
Maximum - intro Alternative approach for solving optimal control. In the deterministic case introduced by Russian mathematician Lev Pontryagin. State process X t = X (u) t with s dx t = b(t, X t, u t )dt+σ t (t, X t, u t )dw t + Objective [ ] T J (u) = E f (t, X t, u t )dt + g(x T ), 0 R γ(t, X t, u t, z)ñ(dt, dz). for T < deterministic, f continuous, g concave. We want to find an admissible policy u A such that J (u ) = sup J (u). u A 30
31 Hamiltonian We define a function, called Hamiltonian by H : [0, T ] R U R R R R H(t, x, u, p, q, r) = b(t, X t, u t )p + σ t (t, X t, u t )q + γ(t, X t, u t, z)r(t, z)ν(dz), (4.1) R where R is the set of functions r : [0, T ] R R such that the integral (4.1) converges. p, q, r satisfies the corresponding adjoint backward stochastic differential equation dp t = H x (t, x, u, p, q, r)dt + qdw t + r(t, z)ñ(dt, dz), p T = g (X T ). R (4.2)
32 Let u, u A and let Xt = X (u ) t, X t = X (u) t be the corresponding state processes. We know that u is optimal if J (u ) J (u), u A, and after substitution [ ] T J (u ) J (u) = E (f (t, Xt, ut ) f (t, X t, u t )) dt + g(xt ) g(x T ) Assumption 0 We assume that the integrals in the following derivation are finite.
33 Terminal wealth I Since g is concave E [g(xt ) g(x T )] E [(XT X T ) g (XT )] = E [(XT X T ) p (T )] [ T = E (Xt X t ) dp t + + + T 0 T 0 T 0 0 (σ(t, X t, u t ) σ(t, X t, u t )) q t dt p t d (X t X t ) (γ(t, X t, u t, z) γ(t, X t, u t, z)) r (t, z)ν(dz)dt ].
34 Terminal wealth II We substitute into p t and X t and not rewrite m gales with zero expected value [ T = E (Xt X t ) H x (t, Xt, ut, pt, qt, r (t,.))dt + + + T 0 T 0 T 0 0 p t (b(t, X t, u t ) b(t, X t, u t )) dt (σ(t, X t, u t ) σ(t, X t, u t )) q t dt (γ(t, X t, u t, z) γ(t, X t, u t, z)) r (t, z)ν(dz)dt ].
35 Consumption By the definition of H [ ] T E (f (t, Xt, ut ) f (t, X t, u t )) dt 0 [ T = E (H(t, Xt, ut, pt, qt, r (t,.)) 0 H(t, X t, u t, p t, q t, r (t,.))) dt T 0 T 0 T 0 p t (b(t, X t, u t ) b(t, X t, u t )) dt (σ(t, X t, u t ) σ(t, X t, u t )) q t dt (γ(t, X t, u t, z) γ(t, X t, u t, z)) r (t, z)ν(dz)dt ].
36 Terminal wealth + Consumption [ T J (u ) J (u) E (H(t, Xt, ut, pt, qt, r (t,.)) 0 H(t, X t, u t, p t, q t, r (t,.))) dt T 0 (X t X t ) H x (t, X t, u t, p t, q t, r (t,.))dt and if we find such that J (u ) J (u) 0 (4.3) we know that u is the optimal control. ]
37 Theorem Theorem (Sufficient maximum ) Let u A with corresponding solution X = X (u ) and suppose there exists a solution (p t, q t, r (t, z)) of the corresponding adjoint equation. Moreover, suppose that H(t, Xt, ut, pt, qt, r (t,.)) = sup H(t, Xt, u, pt, qt, r (t,.)), t [0, T ], u U and H (x) = max u U H(t, x, u, p t, q t, r (t,.)) (4.4) exists and is a concave function of x, t [0, T ] (Arrow ). Then u is optimal control.
38 Remarks to theorem Condition (4.4) is guaranteed by concavity of the function H(t, x, u, p t, q t, r (t,.)) in (x, u), t [0, T ]. To finish the proof, denote and h(t, x, u) = H(t, x, u, p t, q t, r (t,.)) h (t, x) = max h(t, x, u) u A (4.3) holds if 0 h (t, x ) h(t, x, u) (x x)h (t, x ) h (t, x ) h (t, x) (x x)h (t, x ) 0. because h is concave in x for t [0, T ].
39 For the we define criterion [ ] T s J (u) (s, x) = E f (t + s, X t, u t )dt + g(x T s ), Theorem 0 v(s, x) = sup J (u) (s, x). u A Assume v C 1,3 and that there exists an optimal control ut and corresponding state process Xt for the maximum. Define p t = v x (t, X t ), q t = σ(t, X t, u t )v xx (t, X t ), r(t, z) = v x (t, X t + γ(t, X t, u t, z)) v x (t, X t ). Then p t, q t, r(t,.) solve the adjoint equation (4.2).
40 R. Cont and P. Tankov. Financial modelling with jump processes. Chapman & Hall/CRC Financial Mathematics Series., 2004. K. Janeček. Advanced topics in financial mathematics. Study material, MFF UK, 2008. B. Øksendal and A. Sulem. Applied stochastic control of jump diffusions. 2nd ed. Universitext. Berlin: Springer., 2007.
41... Thank you for attention