Computational Optimization Problems in Practical Finance

Transcription

1 Computational Optimization Problems in Practical Finance Computer Science and Applied Mathematics Director, Cornell Theory Center (CTC) Cornell University

2 Co-workers in Computation Optimization & Finance Students Research Associates/Consultants Siddharth Alexander Shirish Chinchalker Katharyn Boyle Yohan Kim Jay Henniger Yuying Li Changhong He Peter Mansfield Dimitriy Levchenkov Cristina Patron

3 Is computational finance computational optimization?

4 No, but {computational optimization} I {computational finance} = a significant set

5 Objectives of this talk 1. Entertainment 2. Intro to some of the practical problems of computational finance 3. Illustration of the important role that optimization can play (but be careful! Look out for solution sensitivity to problem parameters, robustness, conditioning of problem).

6 5 Computational Finance Problems (with optimization solutions) Problem 1: The Implied Volatility Surface Problem Problem 2: The Incomplete Market Hedging Problem (A: local) [Problem 3: The Incomplete Market Hedging Problem (B: global)] Problem 4: The Portfolio of Derivatives Hedging Problem Problem 5: The Optimal VaR/CVaR Problem

7 Problem 1: The Implied Volatility Surface Problem

8 PDE Calculation Cubic Splines Local Volatility Calculation Automatic Differentiation Optimization Method

9 Elementary (solved) questions 1. How to fairly price (vanilla) options? 2. How to determine the volatility parameter (needed for 1)? σ Useful for pricing other (exotic options), hedging, Background: Vanilla put option The buyer has the option (not the obligation) to sell the underlying at strike price K at time (maturity) T. Vanilla call option The buyer has the option (not the obligation) to buy the underlying at strike price K at time (maturity) T.

10 The answer to 1: Assuming geometric Brownian motion ds S t t = µ dt + σdw t + complete market, no arbitrage, constant (future) volatility.. the unique fair price is given by unobservable W : Brownian motion µ : constant, the drift σ : constant, the volatility

11 Black-Scholes Solution 2 v v v + ( r q ) s + σ s = r v 2 t s 2 s r q : risk -free in terest rate : dividend rate natural boundary conditions N o µ!

12 Solving The B-S B S Equation Given volatility, B-S is easy to solve: e.g., evaluating a discretized PDE backwards in time evaluating a binomial/trinomial tree explicit soln (using l cumulative normal dist n lookup) The problem: how to get σ

13 Implied Volatility The answer: assume todays vanilla options are well-priced by the market and solve the inverse problem! F( σ ) value = 0 But this leads to a non-constant (i.e., different data points yield diff. answers) In fact, it appears σ Known, trusted σ = σ ( S, t) t

14 Option pricing model: 1-Factor Continuous Diffusion Approach ds S t t = µ ( S, t) dt+ σ ( S, t) dw t t t W: standard Brownian motion µ, σ: deterministic functions σ ( s, t): local volatility function

15 Fair price for vanilla option : Generalized Black -Scholes: v + r q s v t s + 1 ( ) σ ( st, ) s 2 2 v s = rv No µ!

16 Evaluation Given the vol surface σ = σ ( S, t) t Numerical approaches can be used to solve the generalized B-S equation. But, σ = σ ( S, t) How to get? t

17 Optimization answer 1 (bad) Take todays (trusted) prices and invert GBS model to extract vol surface (generalize the 1-D case): GBS min F( σ ) values unknown surface 2 trusted prices Why bad? 1. too curvaceous (can be smoothed but ) 2. too many optimization variables (number of grid points when σ = σ ( S, t) is discretized (too few values)) t

18 Optimization answer 2 (better) Add as smoothing (regularization) term: min F( σ ) values + smoothing term 2 But: 1. Still thousands of variables (nonlinear obj fcn) 2. How to balance the 2 objectives

19 Optimization answer 3 (best) Model the implied vol surface by a bi-cubic spline form, with p unknown knot values: σ σ σ = (,..., ) 1 p

20 Optimization answer 3 (best) Model the implied vol surface by a bi-cubic spline form, with p unknown knot values: σ = σ1 σ p (,..., ) Once σ = ( σ 1,..., σ p ) is known, and boundary values, the entire σ ( st, ) surface is determined.

21 Optimization answer 3 (best) Model the implied vol surface by a bi-cubic spline form, with p unknown knot values: σ = σ1 σ p (,..., ) Once σ = ( σ1,..., σ p ) is known, and boundary values, the entire surface. is determined σ (,) st σ = σ1 σ p (,..., ) To determine solve min f( σ) F( v( σ)) 2

22 The optimization solution The problem min f( σ) F( v( σ)) 2 Is nonlinear least-squares (as before). But the number of unknowns is the number of knot points p which can be chosen p #option values available Smoothness: built- in! A bit of art needed here

23 Example

24 Problem 1 moral: Design an optimization approach, if possible, so that the number of optimization variables is small but appropriate to the available information. Emphasize smoothness, not just matching the data.

25 Problem 2: The Incomplete Market Hedging Problem (A: local)

26 2 philosophical points 1. Many hedging strategies assume a complete market (which implies continuous hedging). Then, after all the theorems: in practise we will (of course) only hedge at discrete times (which implies an incomplete market). So, perhaps better to assume reality to begin with (but of course fewer theorems, fewer papers, ) 2. Least-squares minimization has many advantages, especially theoretical (more theorems!). But absolute-value minimization pays less attention to outliers and can yield better average case results.

27 The Setting. T>0: Expiry of a European option Discrete hedging dates: Hence, incomplete market prob. space with filtration ( Ω, FP, ) trivial: F 0 = {, Ω} ( ) : Fk X k k= 0, M measurable discounted asset price process Bond price B=1 F K M = t < t < < t = T H: an measurable random payoff for an option T ( ) F k k = 0, M

28 The Setting. Hedging portfolio value at : t V k k = ε kx k + η k ε k : Units of underlying held at t k η k : Units of bonds held at t k Where, ( ε ) and ( η ) k k= 0, M k k= 0, M denote a hedging strategy

29 2 Definitions: Accumulated Gain (change in value of the hedging portfolio due to change in stock price before any changes in the portfolio): k 1 G = ε ( X X ), 1 k M, G = 0 k j j+ 1 j 0 j = 0 Cumulative Cost : C = V G, 0 k M k k k (Self-financing if C0 = C1 =... = CM ( ε ε ) X + η η = 0 i.e., ) k+ 1 k k+ 1 k+ 1 k

30 Local risk minimization The cost at M is H ; let V = H ( η = H) M M General idea: Choose a hedging strategy so that C k 1 C k 0, 0 + k M 1 E.g., local quadratic risk minimization: for k = M 1, M 2,...,0, 2 min Ε(( Ck+ 1 Ck) Fk)

31 An alternative L1 Incremental Cost Quadratic measure may be less than ideal: Larger incremental costs heavily weighted Not in monetary units L1-measure: V = H, for k = M 1, M 2,...,0 M min E( C C F ) k+ 1 k k

32 2 methods: local L1 minimization, local L2 minimization Method 1: min E( C C F ) k+ 1 k k Method 2: 2 min E(( C k+ 1 Ck) Fk)

33 Implementation Suppose stock price modeled using a binomial tree with N periods Hedging can take place at M << N times at dates 0 = i <... < i < N : = i 0 M 1 M Hence, at time i there are n = i + 1 k k k possible states for the stock price. i k Given state j at time the stock price can only move to α k = ik+ 1 ik + 1 possible states

34 .Implementation Flow of optimization problems State j t=0 i 0 i 1 i 2 i k t=t

35 Implementation j S k j k 1 j S = α k+ 1 u Sk S u d S j + i 1 j α k 1 j j k+ 1 = k X j k = S B j k j k j + α k α k j Sk+ = d Sk

36 Implementation So at each hedging time for each state j The optimization problem to solve is t k min E( C C X = + X ) j k 1 k k k For k=m-1,,0 For each state j αk 1 j+ 1 j+ 1 j j+ 1 j j j p, l Xk 1 ε k 1 ε ε k + ηk 1 ηk k η k l= 0 min ( ) ( )

37 Implementation With a bit of manipulation For k=m-1,,0 For each state j min j j j 2 A z b z R 1 A Where matrix is j α k -by-2

38 Computational results dst Assume = µ dt + σdz where Z is a standard Brownian motion S t t A binomial tree is calibrated to this process, e.g., CRR t Assume T = 1, S = µ =.2, σ =.2 r =.1, #periods=600 Consider European put options with different strike prices

39 Performance measures (Discounted) incremental cost (risk): 1 M M 1 k = 0 C C k+ 1 k

40 Multiple Rebalancing Times Less frequent Rebalance every K periods Strike Md n s 0 = 100 Average Incremental Cost, Risk (500 Simulations)

41 Histogram of incremental Costs(risks): strike =95, monthly rebal. 300 Method Method simulations mean: mean:

42 Strike = 95 and Monthly Rebalancing Total Cost: Method 1: 70% less than mean. 55% less than ½ mean Method2: 51% less than mean. 12% less than ½ mean Incremental Cost: Method 1: 69% less than mean. 58% less than ½ mean Method 2: 63% less than mean. 30% less than ½ mean.

43 Problem 2 Moral 1. Many hedging strategies assume a complete market (which implies continuous hedging). Then, after all the theorems: in practise we will (of course) only hedge at discrete times (which implies an incomplete market). So, perhaps better to assume reality to begin with (but of course fewer theorems, fewer papers, ) 2. Least-squares minimization has many advantages, especially theoretical (more theorems!). But absolute-value minimization pays less attention to outliers and can yield better average case results.

44 [Problem 3: The Incomplete Market Hedging Problem (B: global)]

45 Problem 4: The Portfolio of Derivatives Hedging Problem

46 Philosophical motivating points 1. Derivative portfolio hedging problems are often ill-posed 2. Hedge risk minimization can be preferable to hedging by sensitivities 3. Watch out for stochastic vol

47 The setting and the problem The problem: Effectively hedge a large portfolio of derivative instruments Formalize: Risk factors: d S R Hedging instruments: { L } V,, V, V( S, t) value at time t 1 n i Value of hedging portfolio: π (,, ) = where = [, L, ] xst Vx V V1 V n Value of target portfolio at time t: π 0 ( St, )

48 Sensitivities of hedging instruments V V V,, t L t t V V V,, S L S S 1 n 1 n = R 1 n d n = R 2 2 d n Vi V i Γ= [ Γ1, L, Γn] R, where Γ i =, L, 2 2 S1 Sd = n T d = n d * for simplicity of presentation we assume here that each hedging instrument V depends on i exactly one risk factor

49 The Hedge Risk Minimization Approach We measure risk as the expected quadratic replicating error at time t: n min n risk( x) Ε [ xv i i( S, t) π ( S, t)] x R 11 = 0 2

50 The problem is ill-posed To see that this problem is also ill-posed, suppose first that the hedging change is specified by delta-gamma approximation: V( S, t) V( S,0) i i 0 T 2 i i 1 T i δ 2 V V V = t+ S + ( S) ( )( S) t S 2 S Vi Vi 1 T = δt+ S + Γi ( S) t S 2 T 2

51 Infinite number of hedge risk minimizers In this delta-gamma setting, if each hedge instrument depends on a single risk factor then n > 2d + 1 infinite number of risk minimizers In this delta-gamma setting, allowing each hedge instrument to depend on several risk factors: n> d + sparsity indicator* infinite number of risk minimizers *defined in paper

52 More generally, the problem is very illconditioned If we move away from the delta-gamma setting, the resulting problem can be very ill-conditioned. For example, assume a single risk factor, a stock price defined by ds S t t = µ dt + σdx t Experiments defined by S0 = 100, σ =.2, µ =.1, r =.04 And 21 hedging instruments: underlying + vanilla calls with maturities 1,2,3,6 months and strikes [90,95,100,105,110].

53 Continue example Hedge risk minimization becomes: 1 min n risk( x) = V x b, where x R 2 m V( 1 S,) t V( 2 S,) t L V( n S,) t π ( S,) t V = M M L M, b = M m m m 0 m V( 1 S,) t V( 2 S,) t V( n S,) t π ( S,) t L

54 Continue example ill-conditioned matrix and (impractically) large positions Choosing m=20,000 and hedge horizon = 1 month, cond(v) For different target portfolios of 100 vanilla options P, binary options P, barrier options, P, and mix (plus some Asian options) V bi ba P m Pv Pbi Pba Pm risk 7.27e e e e-2 x* e e e e+7

55 The point of this example Minimizing hedge risk alone yields massively ill-conditioned problems, and ridiculously large holdings. However, incorporating realistic costs and bounds can yield better problems, more practical solutions

56 Adding management costs and bounds l x u Bounds, x x, can limit extreme positions and can help control initial formulation costs. Management costs are related to both the number of different instruments in the portfolio, and the size of the positions. Our approach to address both problems simultaneously: min { risk( x) + α c x : l x u } x R n n i= 1 i i x x Balance between risk and cost Per unit cost

57 Why 1-norm penalty? There exists a finite threshold value of αc i For which the optimal solution has a zero holding of instrument i α So, as increases, the number of zero holdings increases. control An alternative formulation: min c x n n x R i i i = 1 l x u x x risk(x) µ r control

58 Example results: Binary options Model 0 Model 1 Model 2 # active instruments risk* 3.05e e e-1 x* e e+2 risk 0 (0) = E( π ) = cost0 = Method 0: no constraints, Method 1: bounds only Method 2: 1-norm + bounds e

59 Incorporating volatility uncertainty Note that we have assumed that the future implied vol, σ t, is the same as current implied vol. Suppose we assume this in our computation but in reality, σ N( σ, σ ) t 0 vol Implied vol at t=0 Standard deviation

60 Sensitivity to errors in future vol Suppose it is assumed, in our computation, that future vol is the same as current vol and x is chosen by solving n min n risk( x) Ε [ xv i i( S, t) π ( S, t)] x R 11 = Next assume that in reality σ N( σ, σ ) t 0 vol 0 2 And compute ( x, σ, S) ( x, σ, S) * 0

61 model 0 is a disaster under vol error Risk when σ N(.2,.005) t Model 0 Model 1 Model 2 Pv 6.28e e e+1 Pbi 1.46e e e-1 Pba 1.73e e e+1 Pm 2.77e e e+1 Why so large!?

62 More model 0 under vol error Extreme sensitivity due to large positions model 0 incurs combined with ill-conditioning of the problem, combined with minimization using just a single value of σ Minimization does do a good job reducing risk if future σ equals current value.

63 0.02 If vol is constant vanilla σ = $σ=$ Difference between target and hedging portfolios S (at T)

64 Model 0 sensitivity to vol error vanilla x 10 4 Difference between target and hedging portfolios σ (at T) S (at T) 130

65 More model 0 under vol error To help ameliorate this effect, assume volatility is stochastic: Still, but 1 risk( x) = Vx b m V( 1 S, σ, t) V 2( S, σ, t) L V n( S, σ, t) π ( S, σ, t) V = M M L M, b = M 0 V( m 1 S, σ m, t) V 2( S m, σ m, t) V n( S m, σ m, t) π ( S m, σ m, t) L

66 Now compute difference between target and hedging portfolios. ( x, σ, S) ( x, σ, S) * 0

67 Not quite as good around initial vol 4 vanilla $σ=$ 0.2 Difference between target and hedging portfolios S (at T)

68 ..yields a much flatter variation surface vanilla σ vol =.5% Difference between target and hedging portfolios σ (at T) S (at T) 130

69 Example results when stochastic vol is included (binary option portfolio) Model 0 Model 1 Model 2 # active instruments risk* 9.03e e-1 1.4e-1 x* e e e+1

70 Problem 4 Moral 1. Hedging a portfolio of derivatives is often ill-posed 2. Adding bounds and management costs (in the 1-norm formulation) can stabilize and yield practical solutions (fewer instruments, smaller positions) 3. Further stabilizing can be achieved with incorporation of stochastic volatility

71 Problem 5: The Optimal VaR/CVaR Problem

72 The Problem V1, V2,..., Vn Given a set of derivative instruments (with values ), dependent on a set d of risk factors, how to choose an investment S n x R R where is the amount invested in i instrument I, to minimize the (conditional) value-at-risk. (The x worst (5%) losses.)

73 Some definitions = T 0 Portfolio loss function: f( x, S) x ( V V) Probability density of S: p( S) Cumulative distribution function: ψ( x, α) = p( S) ds f ( x, α) α

74 .more definitions Value-at-risk (VaR) of a portfolio β, α ( x) = inf{ α R: ψ( x, α) β} β x for a confidence level Conditional Value-at-risk (CVaR): mean of a the tail loss distribution φ α β α 1 + β( x) = inf α( + (1 ) Ε( f ) ))

75

76 Porfolio CVaR Optimization Rockafellar & Uryasev: 1999,2002: min φ ( x) = min F ( x, α), where x β ( x, α) X R β F x f 1 + β (, α) = α + (1 β) Ε[( α) ] If f(, S) and X are convex then min F ( x, ) ( x, α) X R β α is a convex nonlinear programming problem.

77 But, Similar to the previous example, CVaR/VaR minimization for portfolios of derivatives is ill-posed. To see the effect of this ill-posedness, consider a typical CVaR solution:

78 uppe Holdings Hedging ratio lower Instrument index

79 Properties: Properties and problems of the optimal portfolio 1. The optimal portfolio contains all 192 instruments 2. 77% of the instruments are at their bounds Practical Problems: 1. Large management and transaction costs 2. Magnification of the model error A Solution: Add cost consideration to the CVaR objective: n + min ( φ ( x) c x ), where c are pos. weights. x B i i i 11 =

80 Holdings upper Hedging Ratio lower Includes transaction costs Instrument index

81 Minimizing CVaR (for portfolios of derivatives) {( δ V ) } m = 1 Can be formed as a large LP (m simulations for ) : min V ( xyz,,, α ) x 0 T ( ) 1 T ( Ε ( δv)) x= r 1 m(1 β ) i= 1 T y ( δv) x α, y 0, i= 1,..., m i i i l x u α + = m y i i i

82 LP Efficiency Mosek (cpu sec) CPLEX (cpu sec) m n=8 n=48 n=200 n=8 n=48 n=

83 Removing the dependence on m Note that m 1 T + m + Fβ ( x, α) = α + [( δv) i x α] m(1 β ) i= 1 F x α = α + β Ε f x S α 1 + β (, ) (1 ) (( (, ) ) ) And assuming continuity of Ψ( S, α), F ( x, α) iscontinuously differentiab le. β

84 A smooth approximation Let p ( z) max(0, z). Given ε >0, p ( z) is the continuously ε differentiable function: z if z ε 2 z 1 1 pε ( z) = + z+ ε if ε z ε 2 4 4ε 0 otherwise ε m 1 T F% β( x, α) = α + pε( ( δv) i x α) m(1 β ) i= 1 F% ( x, α) is a continuously differentiable approximation to F β β ( x, α)

85 Smooth approximations Smooth approx Decrease tolerance Piecewise linear

86 A piecewise quadratic convex program: This leads to the following piecewise quadratic convex program, with O(n) independent variables and constraints: n ( x, ) F% α β x α + cj xj j= 1 min (, ) 0 T ( V ) x= 1 T subject to ( E[ δv]) x= r l x u

87 Efficiency: Lp vs smoothing technique m n Mosek w=0 smth Mosek w=.01 smth

88 Moral of Problem 5 1. Look out for ill-posedness in the formulation of optimization problems. Correct it. 2. Optimal CVaR problems naturally lead to VERY large LPs. However, the LPs actually approximate a smooth function (as # scenarios increase.). Therefore, it can be cost effective to approximate this smooth function directly, reducing the number of constraints and the number of variables. W/o this reduction the problems quickly become intractable.

89 Concluding Remarks

90 Concluding Remarks 1. Optimization ideas and methods can/do play a central role in the solution of problems in computational finance and financial engineering. Some to the problems are internal.

91 Concluding Remarks 1. Optimization ideas and methods can/do play a central role in the solution of problems in computational finance and financial engineering. Some to the problems are internal. 2. Computational finance/fe yield many interesting optimization problems to be solved (many of the discete constraints are soft and can be handled through the use of continuous methodologies).

92 Concluding Remarks 1. Optimization ideas and methods can/do play a central role in the solution of problems in computational finance and financial engineering. Some to the problems are internal. 2. Computational finance/fe yield many interesting optimization problems to be solved (many of the discrete constraints are soft and can be handled through the use of continuous methodologies) 3. To effectively apply optimization methodology to finance, the financial setting must be well understood!

93 Concluding Remarks 1. Optimization ideas and methods can/do play a central role in the solution of problems in computational finance and financial engineering. Some to the problems are internal. 2. Computational finance/fe yield many interesting optimization problems to be solved (many of the discrete constraints are soft and can be handled through the use of continuous methodologies) 3. To effectively apply optimization methodology to finance, the financial setting must be well understood! 4. To effectively apply optimization methodology to finance, the methods/tools, strengths/weaknesses of optimization must be will understood!

94 Thank you for listening! Feel free to me with follow-up questions, etc: