Optimization Models in Financial Mathematics

Optimization Models in Financial Mathematics John R. Birge Northwestern University www.iems.northwestern.edu/~jrbirge Illinois Section MAA, April 3, 2004 1 Introduction Trends in financial mathematics and engineering Rapid expansion of derivative market (total now greater than global equity) Rise in successful quantitative investors (e.g., hedge funds) Applications in asset management and risk management Boom market (and meltdown) Current focus on managing risks What to answer? (3 questions) Illinois Section MAA, April 3, 2004 2 1

Big Three Questions How much should you pay? What should you buy? How much can you lose? Main application areas: Pricing: How much should you pay? Portfolio Optimization:What should you buy? Risk Management: How much can you lose? Illinois Section MAA, April 3, 2004 3 Themes Optimization is a key part of the fundamental questions Duality and optimal solution properties provide a foundation for finding the answers Illinois Section MAA, April 3, 2004 4 2

Presentation Outline How much to pay? Fundamental Theorem of Asset Pricing Pricing the American option What to buy? Dynamic portfolio optimization How much to lose? Value-at-Risk and the Moment Problem Illinois Section MAA, April 3, 2004 5 Axiom of the Market Market Axiom: There is no free lunch The market does not allow arbitrage No one can trade assets, never lose money, and sometimes make a profit How to write this mathematically? Assume prices S t (1),,S t (n) for n assets at times t Own (owe) x t =x t (1),.,x t (n) shares of each Trades at t change our position from x t-1 to x t and must satisfy conservation of funds: i=1n S t (i) x t-1 (i) = i=1n S t (i) x t (i) or S t x t-1 = S t x t Illinois Section MAA, April 3, 2004 6 3

Linear Program for No Free Lunch Prices and share decisions are random variables, some distribution on events P No losses means S T x T 0 almost surely; No positive profits without losses means: 0 max E P [S T x T ] s.t. S 0 x 0 = 0, S T x T 0 (a.s.) S t x t-1 =S t x t, t=1,.,t Illinois Section MAA, April 3, 2004 7 Linear Program Dual Primal problem: 0 max c T x s.t. A x = 0, B x 0 Dual problem: π, ρ s.t. π T A - ρ T B= c T, ρ 0 What does that mean for no-arbitrage problem? Illinois Section MAA, April 3, 2004 8 4

Discrete Scenario Tree Suppose i=1 N t outcomes at time t Probability of each outcome i at t is p(i,t) Each outcome i at t has ancestor a(i,t) at t-1 and descendants D(i,t) at t+1 Prices S t (i) and actions x t (i) depend on outcomes i a(1,t)=a(2,t)=1 1 2 N T Illinois Section MAA, April 3, 2004 9 Constraints for No Arbitrage A x = 0 corresponds to: S 0 x 0 = 0 -S t (i) x t-1 (a(i,t))+s t (i) x t (i)=0, i, t B x 0 corresponds to S T (i) x T (i) 0, i Starting structure of A: S0 0 0 0 S1(1) S1(2) 0 S (1) 1 0 S (1) 2 0 S (2) 1 0 0 0 S (1) 2 Illinois Section MAA, April 3, 2004 10 5

Dual for No Arbitrage π T A + ρ T B = c T becomes π 0 S 0 - i=1 N1 π 1 (i) S 1 (i)=0 π t (i) S t (i) - j D(i,t) π t+1 (j) S t+1 (j) = 0, t=1..t-1 π T (i) S T (i) - ρ T (i) S T (i) = p(i,t) S T (i) Suppose Asset 1 is a mattress (riskfree investment Treasury bonds), price of Asset 1is S t (1,i) = 1 for all t,i This means: π t (i) = j D(i,t) π t+1 (j), t=0..t-1 Let q t+1 (j)= j D(i,t) π t+1 (j)/π t (i) then S t (i) = j D(i,t) q t+1 (j) S t+1 (j) = E Q [S t+1 ], t=1..t-1 where Q is called a risk-neutral equivalent measure to P Illinois Section MAA, April 3, 2004 11 Fundamental Theorem of Asset Pricing The absence of arbitrage is equivalent to the presence of a risk-neutral equivalent measure such that the expected return on all assets is the same with respect to this measure Q is also called a martingale measure Can price any asset (in fact, derivative) using Q Illinois Section MAA, April 3, 2004 12 6

Results on European Options Black-Scholes-Merton formula can be found using Q Find find values of Calls and Puts for buying and selling at K at T: Call=e -rt E Q [(S T -K) + ] American options: Call K -Put Can exercise before T No parity Calls not exercised early if no dividend Puts have value of early exercise Illinois Section MAA, April 3, 2004 13 American Option Complications Price American options Decision at all t - exercise or not? Find best time to exercise Put (optimize!) K S Exercise? 1 2 3 T Time Illinois Section MAA, April 3, 2004 14 7

American Options Difficult to value because: Option can be exercised at any time Value depends on entire sample path not just state (current price) Model (stopping problem): sup 0 t T e -rt V t (S 0t ) Approaches: Linear programming, linear complementarity, dynamic programming, duality Illinois Section MAA, April 3, 2004 15 Formulating as Linear Program At each stage, can either exercise or not Suppose price goes from S to us or ds each δ time step (binomial model): V t (S) K-S and e -rδ (pv t+ δ (us)+(1-p) V t+ δ (ds)) If minimize over all V t (S) subject to these bounds, then find the optimal value. Linear program formulation (binomial model) min t kt V t, kt s. t. V t,kt K-S t,kt, t=0,δ,2δ,,t; V T,kT 0 V t,kt e -rδ (pv t+δ,u(kt) +(1-p) V t+ δ,d(kt) ) t=0,δ,2δ,,t-1; kt=1,,t+1;s t+δ (U(kt))=uS(kt); S t+δ (D(kt))=dS(kt); S 0,1 =S(0). Result: can find the value in a single linear program Illinois Section MAA, April 3, 2004 16 8

Extensions of LP Formulation General model: Find a value function v to min <C,V> s.t. V t (S t ) (K-S t ) +, - LV + ( V/ t) 0, V T (S T ) = (K-S T ) + where C>0 and L denotes the Black-Scholes operator for price changes on a European option. Can consider in linear complementarity framework Solve with various discretizations Finite differences Finite element methods Illinois Section MAA, April 3, 2004 17 The American Option Dual Approach from Haugh/Kogan and Rogers Primal problem: V 0 = sup τ [0,T] E[e -rτ h τ (S τ )] where h(s τ )=(K-S τ ) + Dual problem where E[π τ ]=0 (martingale): sup τ [0,T] E[e -rτ h τ (S τ )] =sup E[e -rτ h τ (S τ )-π τ ] E[sup τ e -rτ h τ (S τ )-π τ ] So, V 0 inf πτ martingale E[sup τ e -rτ h τ (S τ )-π τ ] Illinois Section MAA, April 3, 2004 18 9

Presentation Outline How much to pay? Fundamental Theorem of Asset Pricing Pricing the American option What to buy? Dynamic portfolio optimization How much to lose? Value-at-Risk and the Moment Problem Illinois Section MAA, April 3, 2004 19 Finding Optimal Growth Let W t be wealth at t, W t =R t W t-1 for R t a return process W t = R t R t-1 L R 1 W 0 Take log s log W t = log W 0 + s=1t R s or log(w t /W 0 ) 1/t = (1/t) s=1t log R s By Law of Large Numbers: (1/t) s=1t log R s E[log R 1 ] = m (if i.i.d.) So, log(w t /W 0 ) 1/t m. Implication: Maximize m to maximize growth (W t /W 0 ) (Equivalent to Max E[U(W 1 )] for U = logw ) Illinois Section MAA, April 3, 2004 20 10

Kelly System Suppose double or nothing betting system (bet x and win 2x with prob. p or lose x with prob. 1-p) Let α be fraction to bet of wealth W Max E[U(X 1 )]=p log(α+1)+ (1-p)log(1-α) or α=2p 1. (Luenberger Blackjack.. p=.5075.. time to double = 6440 hands) Illinois Section MAA, April 3, 2004 21 Continuous Portfolio Dynamics Suppose n investments, weights x i on i Continuous time results: dw/w = i x i (ds i /S i ) = i x i µ i dt + x i dz i Result: E[log(W t /W 0 )]=ν t = i x i µ i t (1/2) ij x i σ ij x j t To maximize growth rate: max i x i µ i (1/2) ij x i σ ij x j s. t. i x i = 1. Illinois Section MAA, April 3, 2004 22 11

Optimal Solution Results Log-optimal if riskfree asset with return r f x i s.t. σ ij x j =µ i r f for i =1,, n. Should you do this? µ = 0.1, σ 2 =0.04 for stocks, r f =.04 x stocks = (0.1-0.04)/0.04=1.5 (with borrowing 0.5 at r f ) Illinois Section MAA, April 3, 2004 23 Presentation Outline How much to pay? Fundamental Theorem of Asset Pricing Pricing the American option What to buy? Dynamic portfolio optimization How much to lose? Value-at-Risk and the Moment Problem Illinois Section MAA, April 3, 2004 24 12

Finding Distributions Based on Market Prices How much can you lose? Idea: Assume the market correctly interprets probabilities into prices These will be like the moments in a moment problem Can find what probabilities are implied by market prices Find the probability of a given loss (or the maximum probability of that loss or greater) or Value-at-Risk (VaR, the maximum loss with a given probability) First: implied binomial trees (Rubinstein) Illinois Section MAA, April 3, 2004 25 Implied Binomial Trees Assume that you have a set of option prices: Example: Share price=45, r=5%, T-t=56/365 Observe: Call(45,T)=2, Call(40,T)=5.5 Assume binomial with ending prices: 52, 45, 39 What are the probabilities with these branches? P 2 P 1 P 0 P 0 +P 1 +P 2 =1 P 1 (5)+P 2 (12)=e r(t-t) 5.5 P 2 (7) = e r(t-t) 2 Illinois Section MAA, April 3, 2004 26 13

Implied Trees Example So, P 2 =.28, P 1 =.43, P 0 =.29 Could also go back to find probabilities on branches General idea: create a tree Consistent with market price? (.29(39) +.43(45) +.28(52))e -.05(56/365) =44.9 In general, might have more options than branches Fit the observed prices as closely as possible Illinois Section MAA, April 3, 2004 27 Fitting Implied Trees Suppose additional prices: e.g. Call(35,T) = 10.3, Call(50,T)=0.5 Find P 0, P 1, P 2 to: min i (u i+ + u i- ) s.t. j P j (S j -K i ) + +u i+ -u i- = FV(Call(K i,t)) j P j S j = FV(S t ) j P j = 1, P j 0. Illinois Section MAA, April 3, 2004 28 14

Problems with Implied Binomial Trees Assumes that the binomial is followed Generalization: Use extremal probabilities with generalized linear programming. Results: can find maximum and minimum (riskneutral) probabilities implied by market Example: Suppose we have written a call option at 50 and want to know the maximum probability of paying more than 5 (i.e., price is above 55) Like VaR chance of losing a certain amount (fraction) in a specified time Illinois Section MAA, April 3, 2004 29 Solving the Moment Problem by Linear Programming Semi-infinite L.P. (all prices possible) Initialize: Start with a set of prices S i. g is function of prices to maximize in expectation, other constraints by v Step 1: Generalize to Moment Problem: Master problem: Find p 1 0,,p r 0, l p l =1, to max l=1r g(s l ) p l s.t l v l (S l )p l β i, i=1,,s, l v l (S l )p l = β i, i=s+1,,m; Let {p 1j,..,p rj } attain the max and {σ j,π 1j,,π Mj } be the associated dual multipliers. Illinois Section MAA, April 3, 2004 30 15

Subproblem: Step 2 Subproblem solution: Find new price, S r+1 that maximizes γ(s,σ j,π j ) = g(s) - σ j - i=1m π ij v i (S) If γ(s r+1,σ j,π j )>0, let r=r+1, ν=ν+1 and go to Step 1. Otherwise stop; {p 1j,,p rj } are the optimal probabilities associated with {S 1,,S r }. Illinois Section MAA, April 3, 2004 31 Extremal Probabilities Problem: find p j to Max j Sj 55 p j s.t. j p j = 1 j p j (S j -K i ) + = FV(C(K i,t)) j p j S j = FV(S t ), p j 0 For our example, suppose we have S j = 30, 35,40,45,50,55,60 and C(35)=10.3, C(40)=5.5, C(45)=2, C(50)=0.5, Solve to find: P(60)=0.082, P(50)=.21, P(45)=.40, P(40)=.26, P(30)=.05 with multipliers.165 on K i = 50 constraint. Subproblem: Max { 1 -.165(S-50), S 55; 0 -.165(S-50) +, S < 55} => new S j = 55 Result: P(55)=.10, and multiplier.2 on K=50 constraint Subproblem: Max {1 -.2(S-50), S 55; 0 -.2(S-50) +, S < 55} => optimal Illinois Section MAA, April 3, 2004 32 16

VaR Calculations Generalized programming (moment problem) framework allows calculation of bounds on implied probabilities Note: these are the risk-neutral probabilities How to bound actual probabilities? For values below risk-free rate of growth, with positive risk premium, actual probabilities will be lower so these are also bounds on those probabilities. Illinois Section MAA, April 3, 2004 33 Conclusions Optimization brings value to financial analysis in answering the Big 3 Questions Existing implementations in multiple areas of financial industry Significant potential for research, theory, methodology, and implementation Illinois Section MAA, April 3, 2004 34 17