Optimization Approaches Applied to Mathematical Finance

Transcription

1 Optimization Approaches Applied to Mathematical Finance Tai-Ho Wang Baruch-NSD Summer Camp Lecture 5 August 7, 2017

2 Outline Quick review of optimization problems and duality theory. Optimal option pricing problem (single asset). Distribution free static-arbitrage upper bound for basket option prices. Distribution free static-arbitrage lower bound for basket option prices (2 asset case).

3 Mathematical optimization A mathematical optimization problem has the form minimize f 0 (x) subject to f i (x) b i, i = 1,, m. x = (x 1,, x n ): optimization variable f 0 : R n R: objective function f i : R n R: constraint functions, i = 1,, m b 1,, b m : limits (or bounds) for the constraints

4 Optimal variable A vector x is called optimal, or a solution, to the optimization problem if it has the smallest objective value among all vectors that satisfy the constraints: z with f 1 (z) b 1,, f m (z) b m f 0 (z) f 0 (x ). The collection of such z s is called the feasible set. The optimization problem is called feasible if its feasible set is nonempty.

5 Special types Optimization problems are classified according to the special forms of the objective function and the constraint functions. For example, Convex optimization: objective and constraint functions are convex. (Recall that a function g is called convex if it satisfies g(αx + βy) αg(x) + βg(y) for all x, y and α + β = 1, α, β 0). Linear programming: objective and constraint functions are linear functions. Nonlinear optimization (or nonlinear programming) is the term used to describe an optimization problem when the objective or constraint functions are not linear, but not known to be convex.

6 Lagrange duality theory Recall the optimization problem minimize f 0 (x) subject to f i (x) b i, i = 1,, m. Let X := {x R n f i (x) b i, i = 1,, m} (feasible set) Consider nonnegative linear combinations of the constraint inequalities, i.e., for y i 0, m y i f i (x) i=1 m y i b i for all x X i=1

7 Lagrange duality theory Assume we happen to have that, for some y i 0, f 0 (x) m y i f i (x) for all x R n i=1 Then m i=1 y ib i is a lower bound of f 0 on X. Therefore the optimal value in the problem { max y i y i b i : y 0, i y i f i (x) f 0 (x) x R n } is a lower bound on the values of f 0 on X. This optimization problem is called the dual problem and the original problem is called the primal problem.

8 Primal-dual Primal problem minimize f 0 (x) subject to f 1 (x) b 1,. f m (x) b m. Dual problem maximize y 1 b y m b m subject to y 1,, y m 0, i y if i (x) f 0 (x) x R n.

9 Duality theorem Weak duality: By construction, we always have primal value dual value This relation is called weak duality. Strong duality: If it happens to be the case that the inequality in weak duality is indeed equality, i.e., primal value = dual value we address this case as strong duality.

10 Q and A on duality Q: Why bother with dual problems? A: Simplifying problem, e.g., infinity dimensional problem can be reduced to finite dimensional problem. Q: When strong duality holds? Depends. Usually we require the optimization problem to have special structures, for instance, linear programming.

11 Linear programming A linear programming (LP) is an optimization program of the form { } min c T x Ax b where x R n is the design vector c R n is a given vector of coefficient of the objective function c T x A is a given m n constant matrix and b R m is a given constraint limits. In previous notation, f 0 (x) = c T x and f i (x) = a T i x, where A T = [a 1 a m ] and a j R n.

12 Duality theorem in LP Consider the LP and its dual min{c T x Ax b} max{b T y y 0, A T y = c} The problem dual to dual is equivalent to the primal. weak duality The following are equivalent Primal is feasible and bounded below Dual is feasible and bounded above Both primal and dual are feasible If any of the above cases holds we have strong duality.

13 Farkas lemma - Theorem on alternative The system of linear inequalities (A) Ax 0, c T x < 0 has no solution if and only if the system of linear inequalities (B) A T y + c = 0, y 0 has solution.

14 Sketch of Farkas lemma Apply duality theorem to the LP minimize c T x subject to Ax 0 and its dual maximize 0 subject to A T y + c = 0 y 0

15 Applications to finance For single period financial market model, direct applications of Farkas lemma are There does not exist dominant trading strategy in the market if and only if there exists (at least one) linear pricing measure for the model. There does not exist arbitrage opportunity in the market if and only if there exists (at least one) risk neutral probability measure for the model. Thus, under the assumption of no arbitrage opportunity, the present price of a contingent claim is equal to the expectation of its future payoff under risk neutral probability.

16 Option pricing problem The problem in option (in this case, call option) pricing is to find max (x K) + ν(dx) ν R + subject to the constraints R + xν(dx) = m, R + (x K) + ν(dx) = c, R + ν(dx) = 1.

17 Dual Its dual is subject to the constraints min cu + ν + mv u, ν,v u(x K) + + ν + vx (x K) +, x R +. Note that in primal problem the design variable is of infinite dimensional but finite constraints; whereas in dual the design is of finite dimensional but infinitely many constraints. However, because of the special form of the constraint function (it is actually piecewise linear) in dual problem, we have chance to reduce the dual to finite dimensional LP.

18 Transforming variables Introduce the transformation x = x K, λ = u + v, µ = Kv. Then the problem is transformed, in the new variables into min cλ + ν + F µ, λ,µ, ν where F = m c K, subject to the (implicit) constraints on λ, ν and µ (Kλ µ)( x 1) + + ν + µ x (K x K) + 0, x R +.

19 Characterizing feasible set Define the function f : R + R by f ( x) = (Kλ µ)( x 1) + + ν + µ x (K x K) +. f is a piecewise affine function, it attains its local extrema at the points {0, 1, K K }. In order to find conditions on (λ, ν, µ) so f ( x) 0 for all x R +, require that f be nonnegative at these points and impose the extra conditions λ 1 to ensure the functional is bounded below. Thus the dual problem in this subcase is to find minimize cλ + ν + F µ subject to λ 1 f (0) = ν 0 f (1) ( = ) ν + µ 0 f K K = (K K)λ + ν + µ 0

20 Subcase K K (with f ( x) = (Kλ µ)( x 1) + + ν + µ x (K x K) + ) λ 1 f (0) = ν 0 f (1) = ν + µ 0 ( ) K f = (K K)λ + ν + µ 0 K Notice that condition (K K)λ + ν + µ 0 is redundant because of λ 1, ν + µ 0 and the assumptions K K. Also note that by λ 1, we have cλ + ν + F µ c + ν + F µ

21 Optimal value Ignoring the constant term c temporarily, the problem becomes minimize ν + F µ subject to ν 0 ν + µ 0 The minimum is zero. So, bringing back c, minimum for original is c. Remark This is reasonable for trivial reasons since option price is decreasing in strike. Therefore, in the case K K, the option price of strike K is at most c.

22 Subcase K K (Recall we know C(K), are optimizing C( K)). (with f ( x) = (Kλ µ)( x 1) + + ν + µ x (K x K) + ), the dual problem in this subcase is equivalent to minimize cλ + ν + F µ subject to λ 1 f (0) = ν 0, f (1) = ν + µ K K f ( K K ) = ν + K K µ 0

23 Optimal value Notice that ν + K K µ 0 is redundant since µ + K ν µ + ν K K 0. K Ignoring λ by using λ 1, the problem reduces to minimize ν + F µ subject to ν 0 ν + µ K K

24 Optimal value minimize ν + F µ subject to ν 0 ν + µ K K Note that the region defined by the constraints ν 0, ν + µ 0 (an unbounded polygon) has only one vertex which is ( ν, µ) = (0, K K). Therefore the optimal value is c + (K K)F. Hence the value in this case is, bringing back the constant term c, is c + (K K)F.

25 Upper Bound: Conclusion Conclusion: Upper Bound is c + (K K) + F Similar analysis yields Lower Bound: (c ( K K)F ) +, if K K, c + (KF K) +, if K K

26 Bertsimas-Popescu Given prices C i (K i ) of call options with strikes 0 K 1.. K n on a stock X, the range of all possible prices for a call option with strike K where K (K j, K j+1 ) for some j = 0,, n is [C (K), C + (K)] where C (K) ( K K j 1 K j K = max C j + C j 1, K j K j 1 K k K j 1 ) K j+2 K K K j+1 C j+1 + C j+2 lower bounds K j+2 K j+1 K j+2 K j+1 C + (K) = K j+1 K K K j + C j+1 upper bounds K j+1 K j K j+1 K j

27 Graphic of Bertsimas-Popescu

28 Basket Options The Payoff of a basket option: ( ψ(s 1,, S n ) = w i S i K Price weighted: w i = 1 I (t 0 ). Capitalization Weighted: w i = Cst nb. S i shares outstanding. Total capitalization w i s are readjusted periodically. S&P500, S&P100, Dow Jones i ) +

29 Black-Scholes The most popular model: Multidimensional Black-Scholes ds i t = S i t(r d i )dt + j σ i j S i tdz j t i = 1,, n E[dZ i t dz j t ] = ρ ij dt European Basket Options: t u + s i s j ρ ij σ i σ j si s j u + (r d i )s i si u ru = 0 ( n ) + u(s 1,, s n, T ) = w i s i K i=1

30 Closed Form Solution Black-Scholes Solution at time 0, for maturity T in closed form e rt 1 (2π) n/2 (detv ) 1/2 R n + w i e (r d i σ2 + i 2 )T T X K e 2 1 X t V 1 X dx where V is the covariance matrix {σ i σ i ρ ij } n i,j=1.

31 First Model Independent Results Robert Merton 1973 (Theory of rational option pricing, Bell Journal) established the following result. Given n assets S 1,, S n and n options C i with strike K i. One option per asset. Let C B denote the price of a basket option with positive constant weights w i, i = 1,, n. Suppose that in addition the following condition holds: Then w 1 K 1 + w 2 K w n K n = K C B w 1 C 1 + w 2 C w n C n

32 Merton Result has a simple proof. (w 1 S 1 + w 2 S w n S n K) + = (w 1 (S 1 K 1 ) + w 2 (S 2 K 2 ) + + w n (S n K n )) + w 1 (S 1 K 1 ) + + w 2 (S 2 K 2 ) + + w n (S n K n ) + Taking expectations in the above formula yields C B = e rt E [w 1 S 1 + w 2 S w n S n K] + = e rt w 1 E[(S 1 K 1 ) + ] + + e rt w 1 E[(S n K n ) + ] = w 1 C 1 (K 1 ) + w n C n (K n )

33 Merton Merton did not however characterize the conditions for equality in this relation. This was done later in Laurence-W Risk Magazine 2004, Applied Math Finance Note: Letting D = K w i K i > 0, it is easy to see that the above proof goes through, but, When D < 0 the above bound is no longer optimal.

34 Optimization - Primal Constrained optimization problem for distribution free upper bound. Determine: sup µ ( + w i S i K) µ(ds) i subject to the constraints (S i k (i) j ) + µ(ds) = C (i) (k (i) j ), for i = 1,..., n, j = 1,..., J (i) µ(ds) = 1 Spot price corresponds to zero strike.

35 Optimization - Dual Dual problem subject to inf ν,ψ n J (i) C (i) (k (i) j )ν j i + ψ i=1 j=1 ( ) + w i S i K ( i i,j ν j i R, for i = 1,..., n, ψ R Huge LP problem!! S i k (i) j ) + ν j i + ψ j = 1,..., J (i)

36 Linear Interpolation For 1 i n and 0 j J (i) define (i) j ) ( C (i) ( C (i) k (i) (i) j 1 j = k (i) j k (i) j 1 by (i) 0 = 1 and ) k (i) j Fills-in the missing values of the call price functions by linear interpolation completes the partial information about the marginal to full information. Key observation: the largest convex function passing through given points is the linearly interpolated function. There exists an optimizing portfolio which consists of options on no more than two strikes per asset, and involves at most n + 1 separate options.

37 Linear Interpolation The interpolated call price function. (i) j gives the modulus of the gradient of C (i) over (k (i) j 1, k(i) ). j

38 Hobson-Laurence-W Preliminaries For simplicity of exposition assume all slopes C(i) (u) u (i) u=k j Let I n = {1, 2,, n} where n is the number of assets. There is a privileged index î I n such that: are different as i and j vary. For any model which is consistent with the observed call prices C (i) (k (i) ), the price B(K) for the basket j option is bounded above by B F (K), where Case I: i w i k (i) > K: J (i) B F (K) = i In\î ( ) { ( ) ( )} w i C (i) k (i) + wî (1 θ )C (î) k (î) + θ C (î) k (î) j(i) î j(î) 1 î j (î) θ î is defined as θ î λ λ (φ ) (Kλ /wî ) k (i) î î î j(î) 1 = λ + (φ î ) λ = (φ ) k (i). î j(î) k(i) j(î) 1

39 Hobson-Laurence-W Case II: i w ik (i) J (i) K: B F (K) = i w i C (i) ( k (i) J (i) ) Based on experiments with real data, the second case essentially never arises in practice. Moreover, the upper bound is optimal in the sense that we can find co-monotonic models which are consistent with the observed call prices and for which the arbitrage-free price for the basket option is arbitrarily close to B F (K). So where s the beef in Case I? All the beef in fleshing out the estimate in the first case is in determining the special index î and the indices j(i), i = 1, n.

40 How to Find Which Options to Choose? Possible to show that there is No cash component ψ in the optimal portfolio. So can consider super-replicating portfolios consisting entirely of options with various strikes (some of which may have strike zero). The upper bound is available in quasi-closed form, meaning there is a simple algorithm to determine the solution, modulo a slope ordering algorithm: Order all slopes of all call price functions and cycle through. To get the intuition as to how to proceed, note that if λ i = 1 then i + w i S i K i ( w i S i λ ) + i K w i So that C B (K) i w i C (i) (λ i K/w i ). The λ i are arbitrary and so C B (K) inf λ i 0, w i C (i) (λ i K/w i ). λ i =1 i

41 Intuition We wish to find the infimum of i w i C (i) (λ i K/w i ) over choices λ i satisfying λ i 0, λ i = 1. Define the Lagrangian L(λ, φ) = i w i C (i) (λ i K/w i ) + φ i λ i 1. Objective function is convex but only C 0,1, because each piecewise linear call price functions C (i), is C 0,1, ie. C i j has a jump at each strike K K i, j = 1,, n i. Note that objective functional is separable function of 1-dimensional functions. Therefore for each fixed Lagrange Multiplier φ, the gradient can point in a cone of different directions. In the terminology of convex analysis we have φ/βk C (i) (λ i K/w i ), where is the subdifferential of the function C (i).

42 Algorithm For each φ there is either a unique λ(φ) or an interval [λ (φ), λ + (φ)]. Essentially: [λ(φ), λ(φ) + ] [K j i, K j+1 ] for some i and j. i So Algorithm: Order all the slopes of all call price functions, i.e., if 30 assets and 8 non zero strikes, order 240 slopes Now starting with φ = ɛ << 1 increase φ while monitoring the quantity Λ(φ) = λ + (φ) which starts very large for small φ ( large K j ) and decreases as φ. i The first time Λ(φ) crosses 1. STOP! Optimal value of φ = φ has been reached.

43 Real business: DJX contract DJX DJX UB BS Price BS Price BS Price BS Price BS Price Strikes Prices ρ = 0 ρ =.5 ρ =.75 ρ =.9 ρ =

44 Real business: DJX contract DJX DJX UB BS Price BS Price BS Price BS Price BS Price Strikes Prices ρ = 0 ρ =.5 ρ =.75 ρ =.9 ρ =

45 Mathematical Formulation - LB Constrained optimization problem inf (x + y K) + µ(dxdy) µ R 2 + subject to the constraints on the marginal distributions R+ (x k 1 ) + µ X (dx) = C X (k 1 ), R+ (y k 2 ) + dµ Y (dy) = C Y (k 2 ), R 2 + µ(dx, dy) = 1. µ X and µ Y are the marginal distributions.

46 Mathematical Formulation - Dual sup ν 1,ν 2,λ C X (k 1 )ν 1 (k 1 ) + R + C Y (k 2 )ν 2 (k 2 ) + λ R + subject to the constraints on the marginal distributions [ ] (x + y K) + (x k 1 ) + ν 1 (dk 1 ) (y k 2 ) + ν 1 (dk 1 ) λ µ(dxdy) 0 for all µ M + where ν i s range over all finite signed measure and λ R

47 Weak Duality For any feasible primal variable µ and any dual variables ν 1, ν 2 and λ the corresponding primal value is always greater than or equal to the corresponding dual value. Namely, (x + y K) + µ(dxdy) R 2 + C X (k 1 )ν 1 (k 1 ) + R + C Y (k 2 )ν 2 (k 2 ) + λ R + Hence, dual variables ν 1, ν 2 and λ represent a sub-replicating portfolio for basket option consists of (buying or selling) individual options and cash.

48 Optimality Find feasible primal variable µ and dual variables ν 1, ν 2 and λ with the same value, i.e., (x + y K) + µ(dxdy) = R 2 + C X (k 1 ) ν 1 (k 1 ) + C Y (k 2 ) ν 2 (k 2 ) + λ R + R +

49 Use of Copulas A copula is joint distribution C(x 1, x 2,, x n ) = P (X 1 x 1,, X n x n ), with uniform marginals and obeying certain natural structure conditions.

50 Sklar s theorem Any joint distribution with continuous marginal distribution functions F i, i = 1,, n, can be expressed as C(F 1 1, F 1 2,, F 1 n ) where C(x 1,, x n ) is a copula and where F 1 is the generalized inverse of F, ie F 1 (t) = inf{x R F (x) t}

51 Copula2 Minimization problem with fixed marginals can be reduced to problem of finding optimal copula. This problem was solved in the case n = 2 by Rapuch and Roncalli (Crédit Lyonnais web site) based on earlier results of Muller and Scarsini and Chen. The Frechet Copulas C (u 1, u 2 ) and C + (u 1, u 2 ) given by C = max(u 1 + u 2 1, 0) C + = min(u 1, u 2 ) Let C (M 1, M 2 ) and C + (M 1, M 2 ) be the corresponding call option prices. Then for a generic basket option on two assets with the same marginals we have C (M 1, M 2 ) C(M 1, M 2 ) C + (M 1, M 2 )

52 Muller-Scarsini-Chen Their result is deduced from the following more general result of Muller and Scarsini. Theorem Let F 1 and F 2 be the probability distribution functions of X 1 and X 2. Let E C [f (X 1, X 2 )] denote the expectation of the function f (X 1, X 2 ) when the copula of the random vector (X 1, X 2 ) is C. If C 1 C 2 (concordance order, same as ptwise in 2-D) then E C1 [f (X 1, X 2 )] E C2 [f (X 1, X 2 )] for all supermodular functions f such that the expectations exist. Supermodular is a natural generalization of non-negative mixed second derivative supermodular function is convex. Supermodular 2 f 0. Ie. a C 2 x 1 x 2 (2) f = f (x 1 + ɛ 1, x 2 + ɛ 2 ) f (x 1 + ɛ 1, x 2 ) f (x 1, x 2 + ɛ 2 ) + f (x 1, x 2 ) 0

53 Rapuch-Roncalli Basket Option Payoff is supermodular. Intuitive Proof: Use the fact that 2 (w 1 S 1 + w 2 S 2 K) + S 1 S 2 = w 1 w 2 δ({w 1 S 1 + w 2 S 2 = K}) is a positive distribution. So can apply theorem. Therefore optimal upper bound for basket option with prescribed marginals is associated with the Fréchet upper bound. Also optimal lower bound for basket option with prescribed marginals is associated with the Fréchet lower bound.

54 Optimal Primal Variable Let F X and F Y be the marginal cdfs of X and Y respectively. The optimal primal variable µ is characterized as (X, Y ) (F 1 X (U), F 1 X (1 U)) where U Uniform(0, 1) Equivalently, the joint cdf F of X and Y is F (x, y) = (F X (x) + F Y (y) 1) + That is, X and Y are antimonotonic. Feasibility of µ is immediate, since marginals are right ones by construction.

55 Auxiliary Function From the marginals distribution functions, construct an auxiliary function φ. C + : the right derivative of call price function C the left derivative. φ(x) := C X + (x) + C Y (K x) + 1 = F + X (x) + F y + (K x) + 1 φ is only defined on [0, K] and right-continuous. φ is of finite variation and hence has at most countably many crossings over zero. A := {x : φ(x) > 0} is a countable union of disjoint intervals, A = j A j.

56 Illustrative Case X and Y are continuous random variables with strictly positive densities on [0, K]. Y = G(X ) := F 1 Y (1 F X (X )) (X and Y are antimonotonic.) Note that y = G(x) < K x F X (x) + F Y (K x) 1 > 0 φ(x) > 0 Hence the points where G crosses the line x + y = K are exactly the boundary points of the set A For general case, take X = F 1 1 X (U) and Y = FY (1 U) where U Uniform[0, 1]

57 Countermonotonicity Distribution

58 Sub Replication from Copula Suppose that the joint cdf of X and Y are given by F (x, y) = max{f X (x)+f Y (y) 1, 0} = Let E = {u (0, 1) : F 1 X ( ) C X + (x) + C + + Y (y) + 1 (u) + F 1 (1 u) K > 0} Y = = 1 (F 1 E[(X + Y K) + 1 ] = X (u) + FY (1 u) K)+ du 0 (F 1 1 X (u) + FY (1 u) K)du E (F 1 X (u) K)du + F 1 Y (1 u)du E E

59 Sub Replication from Copula where = C X (0) + C Y (0) K + = σ(x) = + y (. A ) x (. A ) ( 1) σ(x) C X (x) ( 1) σ(y)+1 C Y (K y) + A C X (k 1 )ν 1 (dk 1 ) + C Y (k 2 )ν 2 (dk 2 ) + λ R + R + { 1 if x is a left endpoint in (A ); 0 if x is a right endpoint in (A ), and recall that A = {0 < x < K : φ(x) > 0}

60 Optimal Dual: Sheep-Track Portfolio ν 1 (dk 1 ) = δ 0 (k 1 )dk 1 + ν 2 (dk 2 ) = δ 0 (k 2 )dk 2 + λ = n i=1 2n i=1 2n i=1 (K 2i 1 K 2i 1 1 ) K = ( 1) i δ K i 1 (k 1 )dk 1, ( 1) i δ K i 2 (k 2 )dk 2, n i=1 (K 2i 2 K 2i 1 2 ) K. We call such portfolio STP, short for sheep-track portfolios, since the graph of such a portfolio is reminiscent of such tracks on British hillsides.

61 Sheep Tract portfolio

62 Summary The optimal lower bound and optimal sub-replicating strategies is unknown in the discrete strike case. Optimal lower bound is open even in the full marginals prescribed (cts strike case) when n 3. We have treated only a 1 period model. Multi-period models are open. Corresponding problems in the American Basket option case are open Upper bound could be made closer if add additional constraints. Which ones? Correlation prescribed? Entropy constraints?

63 THANK YOU FOR YOUR PATIENCE