Arbitrage bounds for weighted variance swap prices Mark Davis - PowerPoint PPT Presentation

Analysis, Stochastics and Applications Walter Schachermayer, 60th Birthday Vienna, 12-16 July 2010 � Arbitrage bounds for weighted variance swap prices � Mark Davis Department of Mathematics Imperial College London www.ma.ic.ac.uk/ ∼ mdavis l´ Joint work with Vimal Raval and Jan Ob� oj . 1

AGENDA • Introduction - Model-free bounds for puts and calls - Variance swap + one put option • General formulation - Weighted realized variance and convex payoffs - Lower and upper arbitrage bounds for convex payoffs - Application to weighted variance swaps - Example - Empirical results • Concluding remarks • Left-wing information 2

Model-free bounds on European option prices Given a set of traded option prices, is there an arbitrage opportunity? Note: no model is given a priori . There are three possibilities • There is a model-independent arbitrage . We can realize a profit by trading at time zero. Example: butterfly spread with negative price. • The prices are consistent with absence of arbitrage. There is a model such that for all options ( p i , H i ) we have p i = D E [ H i ], where H i is the possibly path-dependent exercise value. A model for an asset price S t is simply a filtered probability space (Ω , F t , Q ) carrying an adapted process S t such that S t = F t M t where F t is the forward price and M t is a Q -martingale with M 0 = 1. Example: a put option has model price p K = D E [( K − S T ) + ] . In normalized units r = p K /DF, k = K/F this is r = E [( k − M T ) + ] . 3

• There is a model-dependent arbitrage. Prices are inconsistent with any model but more information is needed to determine the arbitrage strategy. Example: two call options with different strikes but the same price. We say there is weak arbitrage if the 1st or 3rd cases hold. We then have a dichotomy between ‘weak arbitrage’ and ‘consistency with absence of arbitrage’. Standing assumptions • Liquid market in underlying asset S t , t ∈ [0 , T ]. • No interest rate volatility; time-0 discount factors are p (0 , t ) = D t (usually D ≡ D T ) • Uniquely determined forward price F t (e.g. deterministic dividend yield q ) Usually F = F T . • Options are traded at time 0 at quoted prices. In this talk all options are European with the same exercise time T . 4

Put and call bounds DF p1 p2 p3 K K1 F K2 K3 Call options. Normalized prices ( k i , c i ) (with ( k 0 , c 0 ) = (0 , 1))are consistent with absence of arbitrage if the linear interpolant is strictly decreasing and convex and lies above the line c = 1 − k . Model-dependent arbitrage if not strictly decreasing. Else model-free arbitrage. 5

Put options By put-call parity, the conditions are as shown in the figure. In particular D ( K − F ) + ≤ P < DK or in normalized units ( k − 1) + ≤ p < k. kn' kn k1 k2 k3 1 The slope is < 1 unless there is some n ′ such that p n ′ = k n ′ − 1. (In this case P [ S T /F T > k n ′ ] = 0 in any model.) 6

Adding a Variance swap From now on we make—at least—the standing assumption (A) The price S t is a positive continuous function of t ∈ [0 , T ]. Important point: In considering puts and calls, we essentially determine a prob- ability law µ for M T = S T /F T . To create a ‘model’ satisfying (A), we need to specify a continuous martingale whose law at time T is µ . Let B t be Brownian motion with B 0 = S 0 . By Skorohod embedding, for any law µ there is a stop- ping time τ such that B τ ∼ µ and B t ∧ τ is a u.i. martingale. We can now take t M t = B ( T − t ∧ τ ). The key point is that imposing (A) does not change the arbitrage conditions for plain-vanilla puts and calls. 7

Variance swap By standard convention, a variance swap is deemed to be a forward contract in which a cash payment of p vs is exchanged at time T for the realized quadratic variation of returns < log S > T . By Ito � T dS t < log S > t = − 2 log( S T /S 0 ) + 2 , S t 0 and hence in any model p vs = − 2 E [log( S T )] + 2 log( S 0 ) . However, without further assumptions we have no model-free definition of the variance swap contract (other than the actual market definition!) For this we need calcul d’Ito sans probabilit´ es . Strengthen (A) to ( A ′ ) The price S t is a positive continuous function on [0 , T ] having the quadratic variation property. 8

Definition: A partition π of [0 , T ] is a finite sequence 0 ≤ t 0 < t 1 < ∙ ∙ ∙ < t k ≤ T . The mesh size is max 1 ≤ j ≤ k ( t j − t j − 1 ). A function S : [0 , T ] → R has the quadratic variation property (QVP) if there is some sequence π n of partitions such that the mesh size converges to zero and the sequence of measures � ( S ( t j +1 ) − S ( t j )) 2 δ t j µ n = t j ∈ π n converges weakly to a measure on [0 , T ] whose distribution function is denoted < S > t . If S has the QVP and X t = F ( S t ) for F ∈ C 1 then X has the quadratic variation property and � t ( F ′ ( S u )) 2 d < S > u . < X > t = 0 9

In particular if S satisfies ( A ′ ) and F = log we have � t 1 < log S > t = d < S > u (1) S 2 0 u Applying the F¨ ollmer-Ito formula to log S t and using (1) we obtain � t 1 < log S > t = − 2 log( S t /S 0 ) + 2 dS u . S u 0 Remark . If S t is a continuous semimartingale on some probability space then almost all paths have the QVP. Hence if the sample paths do not have the QVP then there is an arbitrage opportunity with or without options—no equivalent martingale measure. So strengthening ( A ) to ( A ′ ) is ‘harmless’. 10

Next question: What’s the relation between the prices of the log option and other traded options? Start with one put option, strike K . We obtain bounds by considering super- replicating strategies. b x0 x1 K log x We can superhedge the log option by a static portfolio containing cash, the underlying asset and a short position in the put option with strike K . For minimum cost the payoff profile of the superhedging portfolio consists of two lines tangent to the log curve as shown in the figure. 11

The two lines have slopes 1 /x 0 and 1 /x 1 , so that b + 1 x 1 = 1 + b = log x 1 x 1 and hence x 1 = e 1+ b . For x 0 we have log x 0 + 1 ( K − x 0 ) = b + 1 K = b + Ke − (1+ b ) , x 0 x 1 which implies that log x 0 + K = 1 + b + K = log x 1 + K = 1 + b + Ke − (1+ b ) . (2) x 0 x 1 x 1 Thus x 0 and x 1 are the two solutions of f ( x ) = z where f ( x ) = log x + K/x and z = z ( b ) is the expression on the right of (2). We find that z ( b ) ≥ 1 + log K for all b . 12

log x + K/x z 1+log K x x1 x0 K The value of the superhedging portfolio at time T is � 1 � v T = b + 1 − 1 ( K − S T ) + S T − x 1 x 0 x 1 so its value at time 0 is � 1 v 0 = Db + 1 − 1 � DF − p K x 1 x 0 x 1 where p K is the time-0 put option price. Since the portfolio superhedges, there is an arbitrage opportunity unless v log ≤ v 0 . We obtain the tightest bound by minimizing v 0 over the one remaining free parameter b , or equivalently z . 13

Writing y 0 = 1 /x 0 , y 1 = 1 /x 1 we have, since y 1 = e − (1+ b ) , v 0 + D = D ( Fy 1 − log y 1 ) − ( y 0 − y 1 ) p K = D (( F − K ) y 1 + z ) − ( y 0 − y 1 ) p K . Now dy j /dz = 1 / ( K − 1 /y j ) = 1 / ( K − x j ) , j = 0 , 1 so � � d 1 ( F − x 1 ) − x 0 − x 1 dz ( v 0 + D ) = p K . K − x 1 K − x 0 At the minimum point the derivative is zero, i.e. p K = D x 1 − F ( K − x 0 ) . (3) x 1 − x 0 Letting Q be the distribution of S T given by the two-point probability measure Q = qδ x 0 + (1 − q ) δ x 1 with q = ( x 1 − F ) / ( x 1 − x 0 ) we have • E Q [ S T ] = F • p K = D E Q [ K − S T ] + , from (3) • v 0 = D E Q [log S T ], since v T = log S T a.s. 14

S 105 F 107.12 D 0.95123 K 100 T 1 Table 1: Parameter values Example Parameter values shown correspond to an interest rate of 5% and a dividend yield of 3%. We find that if the put option price is p K = 88 . 02 then v 0 is minimized at b = 14 . 407 and the minimum value is 1 . 912. The values of x 0 and x 1 are 7 . 465 and 4 . 91 × 10 6 . We conclude that if there is a quoted log-option price of 1.912 in the market (this corresponds roughly to the Black-Scholes value with σ = 25%) then the put price cannot be more than 88 . 02, since the minimum v 0 decreases with p K . This contrasts with the maximum put value DK = 95 . 13 in the absence of the log option. 15

Interpretation The above problem is a semi-infinite linear program: Find v P = inf { c ′ z | z ∈ Z } where Z = { z ∈ R 3 : a ( s ) z ≥ b ( s ) ∀ s ∈ R + } . Here a ( s ) is the vector of exercise values a ( s ) = (1 , s, ( K − s ) + ), b ( s ) = log s and c is the vector of asset prices c = ( D, DF, p K ) Formally the LP dual is � Find v D = sup R + b ( s ) µ ( ds ), where the supremum is taken over positive � measures µ satisfying the equality constraints c = a ( s ) µ ( ds ), i.e. �� � � � ( K − s ) + dµ ( D, DF, p K ) = 1 dµ, s dµ, � � � ≡ ( a 1 ( s ) µ ( ds ) , a 2 ( s ) µ ( ds ) , a 3 ( s ) µ ( ds )) . Our calculation shows there is no duality gap. 16

The Karlin-Isii Theorem Define the moment cone � M = { c = ( c 1 , . . . , c m ) : c i = a i ( s ) µ ( ds ) , µ ∈ M} where M is the set of positive measures. Suppose 1. ( a 1 , . . . , a m ) are linearly independent. 2. c is an interior point of M . 3. v D is finite. Then v P = v D and the primal problem has a solution. 17