Variation Swaps on Time-Changed Lévy Processes

Transcription

1 Variation Swaps on Time-Changed Lévy Processes Bachelier Congress 2010 June 24 Roger Lee University of Chicago Joint with Peter Carr

2 Robust pricing of derivatives Underlying F. Some derivative contract pays Z T, a function of F s path. Ways to find the contract s price Z 0 = EZ T : Specify a model for the underlying F. Compute Parametric : Z 0 = V (model parameters) But we are skeptical of all models. Instead let us find g such that for all models in some universe, we have one of: Nonparametric : Z 0 = Eg(F T ) Semiparametric : Z 0 = V (Eg(F T ), subset of parameters) Nonparametric bounds : Z 0 Eg(F T ) where Eg(F T ) is observable, given prices of options on F T. Note that semiparametric may be more robust than nonparametric.

3 Assumptions Work in (Ω, F, {F u }, P), where P is martingale measure. Underlying F is a positive martingale, for example a forward/futures price, or (under zero rates) a share price. Y t := log(f t /F 0 ), the log-returns process. [Y ] denotes the quadratic variation of Y. (The floating leg of) a continuously-sampled variance swap pays [Y ] T at expiry T. (The floating leg of) a discretely-sampled variance swap pays N 1 n=0 (Y t n+1 Y tn ) 2 where 0 = t 0 < t 1 < < t N = T.

4 Variance swaps Jump risk Variation swaps Pricing variation swaps, with jump risk Share-weighted variation Hedging Discrete Sampling Answers

5 Variance swap valuation standard approach Neuberger (1990), Dupire (92), Carr-Madan (98), Derman et al (99). Let a log contract pay Y T = log(f T /F 0 ). Assume existence. Assume F is continuous Then variance swap value = value of two log contracts E[Y ] T = 2E( Y T ) Widely influential as a reference point for volatility traders This result is the basis for the CBOE s VIX index, and other indicators of options-implied expectations of realized variance (VXN, RVX, VSTOXX, VDAX-NEW, etc). Robust ( model-free ) in that it assumes only the continuity of underlying paths. But empirically jump risk does exist.

6 Extensions Our results (exact semi-parametric pricing formulas) extend the standard theory as follows: Our earlier talk introduced jump risk into F dynamics. Generalize payoffs to G-variation and share-weighted G-variation. Instead of cumulating (dy t ) 2, let us cumulate G(dY t ). Our meta -results provide explanations of: Why does the standard theory work: Why do log contracts price variance swaps? Why two log contracts? Which variance-related contracts admit semi-parametric valuations that have become easy to solve by our methods? Which contracts are still hard to solve semi-parametrically?

7 Variance swaps Jump risk Variation swaps Pricing variation swaps, with jump risk Share-weighted variation Hedging Discrete Sampling Answers

8 Lévy processes An adapted process (X u ) u 0 with X 0 = 0 is a Lévy process if: X v X u is independent of F u for 0 u < v X v X u has same distribution as X v u for 0 u < v X v X u in probability, as v u Lévy -Khintchine: There exist a R, σ 0, and a Lévy measure ν, with ν({0}) = 0 and R (1 x2 )ν(dx) <, such that each X t has CF Ee izxt = e tψ(z) where ψ(z) := iaz 1 2 σ2 z 2 + (e izx 1 izx1 x 1 )ν(dx) R Intuition: ν(a) = E(number of jumps of size A, per unit time).

9 Time-changed exponential Lévy processes A share price could be modeled by an exponential Lévy process F t = F 0 exp(x t ) Indeed, the case that X t = at + σw t gives GBM with drift. But drawbacks: Today s return has same distribution as yesterday s. Today s return is independent of yesterday s.

10 Time-changed exponential Lévy processes Let X be a Lévy process such that Ee X1 <. Let X u := X u u log Ee X1, so that e X is a martingale. Let the time change {τ t } t [0,T ] be an increasing continuous family of stopping times. So τ is a stochastic clock that measures business time : Calendar time t Business time τ t We do not assume that τ and X are independent. Assume Y t = X τ t and F t = F 0 exp(y t ). The time-changed Lévy process Y can exhibit stochastic volatility, stochastic jump intensity, volatility clustering, and leverage effects. By DDS, this family includes all positive continuous martingales.

11 Variance swaps on time-changed Lévy processes Our earlier work introduced jumps: Variance swaps still admit pricing in terms of log contracts. However the correct number of log contracts may not be 2. The correct variance swap multiplier depends only on the dynamics of the Lévy driver X, not on the time change τ. Explicit formula for multiplier: Q X,G = σ 2 + x 2 dν(x) σ 2 /2 + (e x 1 x)dν(x). Whether multiplier is greater or less than 2 depends on skewness of Lévy measure. Effect of discrete sampling

12 Variance swaps Jump risk Variation swaps Pricing variation swaps, with jump risk Share-weighted variation Hedging Discrete Sampling Answers

13 G-variation of a semimartingale Y For a general semimartingale Y, we define the G-variation of Y. Let G(x) = α x + γx 2 + o(x 2 ) be continuous, where α 0 and γ are constants and: Either α = 0 or Y has finite variation. If the latter, then let Y d t := Y t 0<s t Y s, and let TV(Y d ) be total variation of Y d. The o(x 2 ) is for x 0. It can be relaxed if Y c = 0, where Y c is the continuous local martingale part of Y. Then define the (continuously-sampled) G-variation of Y by V Y,G t (where αtv := 0 if α = 0). := αtv(y d ) t + γ[y c ] t + 0<s t G( Y s )

14 G-variation of a semimartingale Y More generally, let G(x) = α x + γx 2 + g(x) where g satisfies any of g(x) = o(x 2 ), g(x) = O( x r ) and r I (1, 2] and Y c = 0 g(x) = O( x r ) and r I (0, 1] and Y d = Y c = 0 where I := {r 0 : ( x r 1)dν Y < for all t > 0} (0,t] R where ν Y denotes the jump compensator of Y

15 Motivation for definition of G-variation For sampling interval n, let us define the discretely-sampled G-variation of Y by V Y,G (n) T := T/ n j=1 G(Y j n Y (j 1) n ) As n, if n 0, we have Skorokhod convergence in probability V Y,G (n) V Y,G. This motivates our definition of V Y,G, and justifies referring to it as continuously-sampled G-variation. Intuition: T V Y,G T = G(dY t ) 0

16 Examples of G-variation swaps Canonical example is a variance swap: G(x) = x 2. Other examples: Total variation swap G(x) = x Simple-returns variance swap Moment swap, for integer p > 1 G(x) = (e x 1) 2 G(x) = x p Absolute moment swap, for real p > 1 G(x) = x p

17 Examples of G-variation swaps Capped-movement versions of above: Replace G with G(min(max(x, a), b)) where a < b. Example: Down semivariance, where G(x) := (x 0) 2, is a statistic of interest to portfolio managers. Capped-G versions of above: Replace G with min(g(x), M) Example: Capped variance G(x) = x 2 M limits the liability of variance sellers.

18 Variation swaps on time-changed Lévy processes We show that: G-variation swaps still admit pricing in terms of log contracts. However the correct number of log contracts may not be 2. The correct variation swap multiplier depends only on G and the dynamics of the Lévy driver X, not on the time change τ.

19 Variance swaps Jump risk Variation swaps Pricing variation swaps, with jump risk Share-weighted variation Hedging Discrete Sampling Answers

20 The multiplier Let X be a nondeterministic Lévy process, with Ee X1 < and E X 1 < and Gdν <. Define the multiplier of (X, G) by Q X,G := EV X,G 1 EV X,G 1 EX 1 = log Ee X1 EX 1 Proposition: Let X have generating triplet (A, σ 2, ν). Then ασ Q X,G 2 /2 + α(e x 1)dν(x) + γσ 2 + G(x)dν(x) = σ 2 /2 +. (e x 1 x)dν(x) Proof: Denominator is sum of EX 1 = A xν(dx) and x 1 log Ee X1 = A + σ 2 /2 + (e x 1 x1 x 1 )ν(dx) Numerator is sum of (ex-jump) drift, Brownian, and jump contributions to EV X,G 1.

21 Pricing variation swaps on time-changed Lévy processes Proposition If Eτ T < then EV Y,G T = Q X,G E( Y T ). Hence the variation swap and Q X,G log contracts have the same value. Proof. V X,G u + Q X,G X u is a Lévy martingale, so by Wald s equation By τ continuity, EV Y,G T E(V X,G τ T + Q X,G X τ T ) = 0. = EV X τ,g T = EV X,G τ T = Q X,G E( Y T ).

22 Proposition: EV Y,G T = Q X,G E( Y T ) Idea of proof: For all fixed times u, by Lévy property of X and V X,G, EV X,G u = Q X,G E( X u ). Replace u with τ T, by a form of the optional stopping theorem: EV X,G τ T = Q X,G E( X τt ). Exchange variation operator and time-change, by continuity of τ: as claimed. X EV τ,g T = Q X,G E( X τt ). In this setting, jumps arise from X jumping, not from clock jumping (although we allow the clock rate to jump).

23 Example: Time-changed geometric Brownian motion Let X be Brownian motion and G(x) = x 2. Then Q X,G = E[X] 1 EX 1 = 1 1/2 = 2 This recovers the 2 multiplier for all positive continuous martingales.

24 Example: Time-changed fixed-size jump diffusion Let X have Brownian variance σ 2 and Lévy measure λ 1 δ c1 + λ 2 δ c2 where δ c denotes a point mass at c, and c 1 > 0 and c 2 < 0. Then Q X,G = α λ 1(e c1 1) + λ 2 (e c2 1) + γσ 2 + λ 1 G(c 1 ) + λ 2 G(c 2 ) σ 2. /2 + λ 1 (e c1 1 c 1 ) + λ 2 (e c2 1 c 2 ) In particular, consider G(x) = x 2. Third-order Taylor expansion in (c 1, c 2 ) about (0, 0), if σ 0: Q X,G 2 2λ 1 3σ 2 c λ 2 3σ 2 c 2 3, increasing in absolute down-jump size, decreasing in up-jump size.

25 Time-changed Kou double-exponential jump-diffusion Let X have Brownian variance σ 2 and Lévy density ν(x) = λ 1 a 1 e a1 x 1 x>0 + λ 2 a 2 e a2 x 1 x<0 where a 1 1 and a 2 > 0. So up-jumps have mean size 1/a 1, down-jumps have mean absolute size 1/a 2. For G(x) = x 2, Q X,G = σ 2 + 2λ 1 /a λ 2 /a 2 2 σ 2 /2 + λ 1 /(a 1 1) λ 2 /(a 2 + 1) λ 1 /a 1 + λ 2 /a 2. Third-order Taylor expansion in (1/a 1, 1/a 2 ) about (0, 0), if σ 0: Q X,G 2 4λ 1/σ 2 a λ 2/σ 2 a 3, 2

26 Example: Time-changed extended CGMY Let X have the extended CGMY Lévy density ν(x) = C n x e G x 1 x<0 + C p 1+Yn x e M x 1 x>0, 1+Yp where C p, C n > 0 and G, M > 0, and Y p, Y n < 2. For G(x) = x 2, Q X,G = C nγ(2 Y n)g Yn 2 C pγ(2 Y p)m Yp 2 C nγ( Y n)[g Yn (G+1)Yn +YnGYn 1 ]+C pγ( Y p)[m Yp (M 1) Yp Y pm Yp 1 ] Expanding the denominator in 1/G and 1/M, Q X,G 2 G Yn 2 (1 2 Yn 3G where ρ := C p Γ(2 Y p )/(C n Γ(2 Y n )). G Yn 2 + ρm Yp ) + ρm Yp 2 (1 + 2 Yp 3M +...).

27 Example: Time-changed VG The Variance Gamma model takes Y = 0. For G(x) = x 2, its multiplier is Q X,G = ν(x) = C x e G x 1 x<0 + C x e M x 1 x>0 1/G 2 + 1/M 2 log(1 + 1/G) + 1/G log(1 1/M) 1/M G 2 + M 2 2 G 2 (1 2 3G +...) + M 2 ( M +...). Note the sign asymmetry between the 2 3G and the + 2 3M.

28 Example: Time-changed normal inverse Gaussian (NIG) Let X have no Brownian component. Let X have Lévy density ν(x) = δα π exp(βx)k 1 (α x ), x where δ > 0, α > 0, β < α, and K 1 = modified Bessel function of the second kind and order 1. Then X has cumulant transform κ(z) = log Ee zx1 = γz + δ( α 2 β 2 α 2 (β + z) 2 ), for some γ that we need not specify. Then for G(x) = x 2, Q X,G = κ (0) κ(1) κ (0) = α 2 /(α 2 β 2 ) α 2 β 2 β (α 2 β 2 )(α 2 (β + 1) 2 ). Small jump-size limit: take α. Expanding in 1/α, Q X,G 2 4β + 1 2α 2. which is decreasing in β, the parameter which controls the tilt.

29 Variance swaps Jump risk Variation swaps Pricing variation swaps, with jump risk Share-weighted variation Hedging Discrete Sampling Answers

30 Definition Define the dual or share-weighted G-variation of Y by Ṽ Y,G t := t 0 e Ys dv Y,G s = t 0 F s dvs Y,G F 0 where the integrals are pathwise Riemann-Stieltjes. Share-weighted variation swaps, which pay Ṽ Y,G t, confer variation exposure proportional to underlying level F. Motivations: Investor may be bullish Investor may have view that the market s downward implied volatility skew is too steep. Investor may be seeking to hedge variation exposure that grows as Y increases, e.g. in dispersion trading Investor may wish to trade single-stock variance without caps

31 Examples of share-weighted variation swaps Share-weighted counterparts exist, for each example of G. Canonical example is the gamma swap: G(x) = x 2 Standard theory: Gamma swap has same value as 2 contracts on (F T /F 0 ) log(f T /F 0 ). Pre-jump share-weighted G-variation swap uses modified weights: t 0 F s dvs Y,G F 0 This is equivalent to using share-weighted variation with respect to the function e x G(x).

32 Pricing share-weighted variation swaps Proposition Again let G(x) = α x + γx 2 + g(x). Under integrability conditions, EṼ Y,G T = Q X,G E((F T /F 0 ) log(f T /F 0 )). where the dual multiplier ασ 2 /2 + α(1 e x )dν(x) + γσ 2 + e x G(x)dν(x) Q X,G := σ 2 /2 +. (1 e x + xe x )dν(x) Proof. Change to share measure P where d P u /dp u = exp X u. Apply unweighted result to X := X and G(x) := G( x).

33 Share-weighted variation swaps, with jump risk We have shown that, for generalized variation, in the presence of jump risk, Share-weighted G-variation swaps still admit pricing in terms of F log F contracts. However the correct number of F log F contracts may not be 2. The correct share-weighted variation swap multiplier depends only on G and the dynamics of the Lévy driver X, not on the time change τ.

34 Skewness impact depends on the contract Q X,G 2 has same sign as EV X,G 1 2E( X 1) = ( ) x3 3 x O(x5 ) dν(x) for G(x) = x 2 ( ) 2x x4 2 + O(x5 ) dν(x) for G(x) = (e x 1) 2 QX,G 2 has same sign as EṼ X,G 1 2E(X 1e X 1 ) = ( ) x x4 6 + O(x5 ) dν(x) for G(x) = x 2 ( ) 2x3 3 x4 4 + O(x5 ) dν(x) for G(x) = e x x 2 ( ) 4x x4 + O(x 5 ) dν(x) for G(x) = (e x 1) 2 6 ( ) x x4 3 + O(x5 ) dν(x) for G(x) = e x (e x 1) 2

35 Multipliers of empirically calibrated processes Variance Simple variance Third moment X Data 1 F t F t 1 F t F t 1 F t F t CGMY Jun CGMY Sep CGMY Dec VG Jun VG Sep VG Dec NIG Jun NIG Sep NIG Dec

36 Variance swaps Jump risk Variation swaps Pricing variation swaps, with jump risk Share-weighted variation Hedging Discrete Sampling Answers

37 Perfect hedging with two jump sizes Let X have jump sizes c 1 > 0 and c 2 < 0 and zero Brownian part, and piecewise constant paths: ν = λ 1 δ c1 + λ 2 δ c2 where λ 1 := (1 e c2 )λ and λ 2 := (e c1 1)λ, for arbitrary λ > 0. Then T Q X,G q X,G log(f 0 /F T ) + df t = V Y,G T. 0 F t where q X,G := (c 2 G(c 1 ) c 1 G(c 2 ))/(c 2 (e c1 1) + c 1 (1 e c2 )). So replicate V Y,G T by holding: Q X,G log contracts, statically q X,G /F t shares, dynamically

38 Variance swaps Jump risk Variation swaps Pricing variation swaps, with jump risk Share-weighted variation Hedging Discrete Sampling Answers

39 Intuition Does decreasing the sampling frequency tend to increase or decrease the expectation of realized variance? Intuition: Consider 1 long sampling period vs. 2 shorter sampling periods. With log-returns of R 1 and R 2 in the two periods, the more-frequently-sampled realized variance is R R 2 2 The less-frequently-sampled realized variance is (R 1 + R 2 ) 2 = R1 2 + R R 1 R 2 So coarser sampling adds 2R 1 R 2 to the realized variance.

40 Intuition The expected impact of less sampling is 2E(R 1 R 2 ) = 2ER 1 ER 2 + 2Cov(R 1, R 2 ) If R would be martingale increments, E(R 1 R 2 ) would vanish. Indeed, realized variance of a martingale M is perfectly replicable, continuously T MT 2 = M M t dm t + [M] T 0 or discretely M 2 t N = M M tn ( M tn ) + ( M tn ) 2 But due to taking logs, E(R 1 R 2 ) > 0 typically.

41 Discrete sampling Let 0 = t 0 < t 1 < < t N = T. Write n Z := Z tn+1 Z tn. If Eτ T < and τ and X are independent then N 1 ( n Y ) 2 = E[Y ] T + (E n Y ) 2 + N 1 E n=0 n=0 N 1 n=0 Var(E( n Y τ)). Proof: M t := Y t τ t EX 1 is a martingale. Sum the following over n: E( [Y ]) = E( [M]) = E( M) 2 = E( Y ( τ)ex 1 ) 2 = E( Y E( Y τ)) 2 = E(Var( Y τ)) = Var( Y ) Var(E( Y τ)) = E( Y ) 2 (E Y ) 2 Var(E( Y τ)). Hence discrete sampling increases variance swap values. Premium depends on squared spreads of log contracts, and Var(E( Y τ)).

42 Variance swaps Jump risk Variation swaps Pricing variation swaps, with jump risk Share-weighted variation Hedging Discrete Sampling Answers

43 Why does standard theory work Why do log contracts price variance swaps? Because, if F is an exponential Lévy process, then log F and [log F ] are both Lévy processes. So the ratio Q of their drifts gives their relative price. This property survives under continuous time change and such time changes generate all continuous positive martingales. Why two log contracts? Because log(gbm) has drift 1/2. So the drift ratio is 2.

44 Extension to jumps The drift-ratio reasoning still holds, but with a different ratio. Variance swap value = a multipler (Q) times log contract value. True for all time-changed Lévy processes. Arbitrary stochastic clock, arbitrary correlation. The Q does not depend on the clock. For continuous underlying paths, Q = 2. In the presence of negatively skewed jump risk, Q > 2 In that case, quotations based on a 2 multiple (including VIX) would underprice the continuously-sampled variance, and typically furthermore underprice the discretely-sampled variance.

45 Extension to other contracts Let G(x) = α x + γx 2 + o(x 2 ). By same techniques, we price a G-variation swap which pays V T = αtv(y d ) T + γ[y c ] T + 0<s T G( Y s ) (subject to conditions on G, Y ), because V t is Lévy if Y is. By same techniques, we price share-weighted G-variation: Ṽ Y,G t := t 0 F s dvs Y,G F 0 in terms of an F T log F T contract, via measure change. Under further conditions, can price volatility derivatives paying h([y ] T ). Different techniques needed, because h([y ] t ) may not be Lévy