Backward and Forward Preferences

Transcription

1 Backward and Forward Preferences Undergraduate scholars: Gongyi Chen, Zihe Wang Graduate mentor: Bhanu Sehgal Faculty mentor: Alfred Chong University of Illinois at Urbana-Champaign Illinois Risk Lab Illinois Geometry Lab Final Presentation December 13th, 018

2 Introduction Consider an investor who has a portfolio consisting of two assets and he wants to optimize his wealth. Select a time period [0, ], adopt the binomial model, and pre-specify the time utility function of the investor at time 0. After the exogenous triplet is pre-committed at time 0, we then solve for optimal investment strategies and derive the value functions/preferences before time. We will then use hypothetical value for the model and S&P 500 data to simulate the value functions.

3 Let (Ω, F, P) be the probability space. Let F = { F t be the filtration generated by }t=0,, ξ = { ξt } t=0,,. he two random variable ξ and ξ are given by ξ u ξ, p u = P(ξ 0, = = ξ u F 0 ) ξ d, p d = P(ξ 0, = ξ d F 0 ) ξ ξ = u, pu =, P(ξ = ξu F ) ξ d, pd =, P(ξ = ξd F )

4 Suppose there are two types of assets, a risk-free asset that offers a return of r and a risky asset. Denote the price of the risky asset to be S = { St } he arbitrage-free conditions are 0 < ξ d 0 < ξ d < er < ξ u. < e r < ξ u t=0,,. and

5 he following relationships for the price hold in the two period binomial model. S S 0 ξu, p 0, = p u S = 0, S0 ξd, p 0, = p d S = ξ u, p, = pu, S 0, ξ d, p, = pd, where S 0 F 0, S F and S F.

6 t = 0 t = t = S0 ξu ξ u S 0 p u 0, S 0 ξu p u, p d, S0 ξu ξ d S0 ξd ξ u p d 0, S 0 ξd p u, p d, S 0 ξ d ξ d

7 Consider an investor who holds α shares of risky asset S, and risk-free asset B, with initial wealth x. Let { X α;x } t be a wealth process of the investor t=0,, employing control α with initial wealth x at t = 0. herefore, e r X α;x e r X α;x = e r X α;x X α;x 0 = x, = x + α 0 (S e r S 0 ), + α (S e r S e r ).

8 For simplicity, we ll use the discount price of the risky asset. Denote it by S = { S t }t=0,, = { e rt St } t=0,,. Let the up and down factor for discounted price to be { } ξt t=, = { ξt } e r t=,. he arbitrage free conditions become 0 < ξ d 0 < ξ d < 1 < ξu. < 1 < ξ u and

9 Define a discounted wealth process to be { X α;x } t t=0,,, changing S to S. X α;x X α;x X α;x 0 = x, = x + α 0 (S = X α;x + α S 0 ), (S S ).

10 Denote another wealth process { Xs α;x,t } be a wealth s=t... process of the investor employing control α with the initial wealth x at time t. herefore, X α;x X α;x, = X α;x = x + α (S S ), + α (S S X α 0,α ;x α ;X α 0 ;x, = X. α;x α;x, ) = X,

11 he value function at time is given by he objective function is V (x, ) = U (x). E[U (X α;x )]. herefore, his value function at time t = 0 is V (x, 0) = sup E[U (X α;x )]. α And the value function at t = is given by V (x, ) = sup α E[U (X α;x, ) F ].

12 By the definition and Dynamic Programming Principle, V (x, ) = sup α V (x, ) = U (x), E[V (X α ;x;, ) F ], V (x, 0) = sup E[V (X α 0;x, α 0 )]. We can solve the value functions recursively, and this is why V = { V (ω, x, t) } are coined as backward ω Ω,x R,t=0,, preferences. V (x, 0) α 0 V (x, ) α V (x, ).

13 Assume V (x, ) = U (x) = e γx, γ > 0. V (x, ) = e γx EQ [h F ], where h = h u h u ( = q u ln q u ) ( p u + q d ln q d ), p u = P(p p d,,,, = pu ξ, = q u ln ( q u p u, ) + q d ln ( q d p d, ), p u, = P(p, = pu, ξ = ξ u ); = ξ d ), and q u, qd are the conditional probabilities of going up and down under measure Q in [, ]. ( p u ) ( p d, ), ln he optimal strategy is α q = u ln q d γs (ξ u ξd ).

14 V (x, 0) = e γx EQ [h 0 +h ], where h 0 = q u ln ( qu ) p + q d u 0, ln ( qd ) p and q u u 0,, q d are the conditional probabilities of going up and down under measure Q in [0, ]. he optimal strategy α 0 = ln ( pu 0, ( ) ln pu 0, q u ) h q d u +h d. γs 0 (ξ u ξ d )

15 First, if the stock price goes up initially at t = (ξ = ξ u ),

16 And if the price of the risky asset goes up at t = (ξ = ξ u ); Or if the price of risky asset goes down at t =, (ξ = ξ d ),

17 Second, if the stock price goes down initially at t = (ξ = ξ d ),

18 And if the price of the risky asset goes up at t = (ξ = ξ u ); Or if the price of risky asset goes down at t = (ξ = ξ d ),

19 Multi-period Model Consider there are N periods, where t = 0, N, N.... For simplicity, denote h = N so that t = 0, h, h...,. Let (Ω, F, P, F) be the filtered probability space. Suppose there are only one risk-free asset and one risky asset.

20 Multi-period Model t = 0, h,..., (N 1)h, in the interval [t, t + h], the price could go up or down with respective probabilities, { ξ u (ξ t+h F t ) = t+h, pt,t+h u = P(ξ t+h = ξt+h u F t) ξt+h d, pd t,t+h = P(ξ t+h = ξt+h d F t) where ξt+h u, ξd t+h and pu t,t+h, pd t,t+h F t.

21 Multi-period Model t = 0, h,..., (N 1)h, we used the money account as the numeraire. he arbitrage-free condition is 0 < ξ d t+h < 1 < ξu t+h. he risk-neutral probability of realizing up in [t, t + h] is q u t,t+h = Q(ξ t+h = ξ u t+h F t) = 1 ξd t+h ξ u t+h ξd t+h We assume ξ u t+h, ξd t+h, pu t,t+h, pd t,t+h, qu t,t+h and qd t,t+h F 0 to simplify the simulation process..

22 Multi-period Model Denote { α t be a stochastic control process. Let }t=0,h,...,(n 1)h X α;x = { X α;x } t be the wealth process of the t=0,h,...,(n 1)h, investor using the control α with initial wealth x at time 0. herefore, X α;x 0 = x, X α;x t+h = X α;x t + α t (S t+h S t ), t = 0, h,..., (N 1)h.

23 Multi-period Model s = 0, h,..., let X α;x,s = { X α;x,s } t, be the wealth t=s,s+h,..., process of the investor using the control α with initial wealth x at time s. herefore, X α;x,s t+h X α;x,0 = X α;x, Xs α;x,s = x, = X α;x,s t + α t (S t+h S t ), t = s, s + h,..., (N 1)h. And more importantly, X α;x,t α;x,t α;xt+h,t+h = X, t = s, s + h,..., (N 1)h.

24 Multi-period Model he value function at time is given by he objective function is V (x, ) = U (x). E[U (X α;x )]. herefore, his value function at time t = 0 is V (x, 0) = sup E[U (X α;x )]. α t = 0, h,...,, value function at time t is given by V (x, t) = sup E[U (X α;x,t ) F t ]. α

25 Multi-period Model V (x, t) = sup E[U (X α;x,t ) F t ], α = sup α t,α t+h,...,α (N 1)h E[U (X α t,α t+h,...,α (N 1)h;x,t ) F t ], = sup α t,α t+h,...,α (N 1)h E[E[U (X α t,α t+h,...,α (N 1)h;x,t ) F t+h ] F t ], α t ;x,t t+h,t+h = sup E[E[U (X α t+h,...,α (N 1)h;X ) F t+h ] F t ], α t,α t+h,...,α (N 1)h = sup E[ sup α t = sup α t α t ;x,t t+h,t+h E[U (X α t+h,...,α (N 1)h;X ) F t+h ] F t ], α t+h,...,α (N 1)h E[V (X α t ;x,t t+h, t + h) F t].

26 Multi-period Model By Dynamic Programming Principle, we can derive the value functions backwardly, V (x, 0) α 0 V (x, h) α h... α (N 1)h V (x, (N 1)h) α (N 1)h V (x, ). his is the reason why V = { V (ω, x, t) } ω Ω,x R,t=0,h,..., are coined as backward preferences.

27 Multi-period Model t = 0, h,..., (N 1)h, assume V (x, ) = U (x) = e γx, γ > 0, then [ (N 1)h V (x, t) = e γx EQ i=h h i F t ], ( ) ( ) where h t = qt,t+h u ln q u t,t+h + qt,t+h d ln q d t,t+h. he optimal strategy is α t = ln ( p u t,t+h q u t,t+h ) ln ( p d p u t,t+h t,t+h q d t,t+h ) ( E Q [ (N 1)h i=t+h p d t,t+h h i Ft+h u ] EQ [ (N 1)h i=t+h h i Ft+h d ]) γs t (ξt+h u ). ξd t+h

28 Multi-period Model

29 Future Goals Derive the mechanism of forward preference. Solve the preference and optimal investment strategy explicitly. Visualize forward preference and optimal investment strategy and compare them with backward case.

30 Reference [1] Marek Musiela and haleia Zariphopoulou (004): A valuation algorithm for indifference prices in incomplete markets. Finance and Stochastics 8,