Inversion of Convex Ordering: Local Volatility Does Not Maximize the Price of VIX Futures
IINVERSION OF CONVEX ORDERING: LOCAL VOLATILITY DOES NOTMAXIMIZE THE PRICE OF VIX FUTURES
BEATRICE ACCIAIO AND JULIEN GUYON
Abstract.
It has often been stated that, within the class of continuous stochastic volatility models cal-ibrated to vanillas, the price of a VIX future is maximized by the Dupire local volatility model. In thisarticle we prove that this statement is incorrect: we build a continuous stochastic volatility model in whicha VIX future is strictly more expensive than in its associated local volatility model. More generally, in thismodel, strictly convex payoffs on a squared VIX are strictly cheaper than in the associated local volatilitymodel. This corresponds to an inversion of convex ordering between local and stochastic variances, whenmoving from instantaneous variances to squared VIX, as convex payoffs on instantaneous variances are al-ways cheaper in the local volatility model. We thus prove that this inversion of convex ordering, which isobserved in the SPX market for short VIX maturities, can be produced by a continuous stochastic volatilitymodel. We also prove that the model can be extended so that, as suggested by market data, the convexordering is preserved for long maturities. Introduction
For simplicity, let us assume zero interest rates, repos, and dividends. Let F t denote the market informationavailable up to time t . We consider continuous stochastic volatility models on the SPX index of the form dS t S t = σ t dW t , S = s , (1.1)where W = ( W t ) t ≥ denotes a standard one-dimensional ( F t ) -Brownian motion, σ = ( σ t ) t ≥ is an ( F t ) -adapted process such that (cid:82) t σ s ds < + ∞ a.s. for all t ≥ , and s > is the initial SPX price. Bycontinuous model we mean that the SPX has no jump, while the volatility process σ may be discontinuous.The local volatility function associated to Model (1.1) is the function σ loc defined by σ ( t, x ) := E [ σ t | S t = x ] . (1.2)The associated local volatility model is defined by: dS loc t S loc t = σ loc ( t, S loc t ) dW t , S loc0 = s . From [10], the marginal distributions of the processes ( S t ) t ≥ and ( S loc t ) t ≥ agree: ∀ t ≥ , S loc t ( d ) = S t . (1.3)Let T ≥ . By definition, the (idealized) VIX at time T is the implied volatility of a 30 day log-contracton the SPX index starting at T . For continuous models (1.1), this translates into VIX T = E (cid:34) τ (cid:90) T + τT σ t dt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) F T (cid:35) = 1 τ (cid:90) T + τT E (cid:2) σ t (cid:12)(cid:12) F T (cid:3) dt, (1.4)where τ = (30 days). In the associated local volatility model, since by the Markov property of ( S loc t ) t ≥ , E [ σ ( t, S loc t ) |F T ] = E [ σ ( t, S loc t ) | S loc T ] , the VIX, denoted by VIX loc ,T , satisfies VIX ,T = 1 τ (cid:90) T + τT E [ σ ( t, S loc t ) | S loc T ] dt = E (cid:34) τ (cid:90) T + τT σ ( t, S loc t ) dt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) S loc T (cid:35) . (1.5) Department of Statistics, London School of Economics and Political ScienceQuantitative Research, Bloomberg L.P.
E-mail address : [email protected], [email protected] . Date : October 15, 2019.
Key words and phrases.
VIX, VIX futures, stochastic volatility, local volatility, convex order, inversion of convex ordering. a r X i v : . [ q -f i n . M F ] O c t NVERSION OF CONVEX ORDERING: LOCAL VOLATILITY DOES NOT MAXIMIZE THE PRICE OF VIX FUTURES 2
Note that
VIX T and VIX ,T have the same mean: E (cid:2) VIX T (cid:3) = E (cid:2) VIX ,T (cid:3) = E (cid:34) τ (cid:90) T + τT σ t dt (cid:35) . (1.6)It has often been stated that, within the class of continuous stochastic volatility models calibrated tovanillas, the price of a VIX future is maximized by Dupire’s local volatility model. For example, in a generaldiscussion in the introduction of [4] about the difficulty of jointly calibrating a stochastic volatility modelto both SPX and VIX smiles, De Marco and Henry-Labordère approximate the VIX by the instantaneousvolatility, i.e., VIX T ≈ σ T and VIX loc ,T ≈ σ loc ( T, S loc T ) , and, using Jensen’s inequality and (1.3), theyconclude that “within [the] class of continuous stochastic volatility models calibrated to vanillas, the VIXfuture is bounded from above by the Dupire local volatility model”: E [VIX T ] ≈ E [ σ T ] = E (cid:20)(cid:113) σ T (cid:21) = E (cid:20) E (cid:20)(cid:113) σ T (cid:12)(cid:12)(cid:12)(cid:12) S T (cid:21)(cid:21) ≤ E (cid:20)(cid:113) E [ σ T | S T ] (cid:21) = E [ σ loc ( T, S T )] = E (cid:2) σ loc ( T, S loc T ) (cid:3) ≈ E [VIX loc ,T ] . Similarly, one would conclude that within the class of continuous stochastic volatility models calibrated tovanillas, the price of convex options on the squared
VIX is minimized by the local volatility model: for anyconvex function f , such as the call or put payoff function,(1.7) E (cid:2) f (cid:0) VIX T (cid:1)(cid:3) ≈ E (cid:2) f (cid:0) σ T (cid:1)(cid:3) = E (cid:2) E (cid:2) f (cid:0) σ T (cid:1)(cid:12)(cid:12) S T (cid:3)(cid:3) ≥ E (cid:2) f (cid:0) E (cid:2) σ T (cid:12)(cid:12) S T (cid:3)(cid:1)(cid:3) = E (cid:2) f (cid:0) σ ( T, S T ) (cid:1)(cid:3) = E (cid:2) f (cid:0) σ ( T, S loc T ) (cid:1)(cid:3) ≈ E (cid:2) f (cid:0) VIX ,T (cid:1)(cid:3) . (The (correct) fact that E (cid:2) f (cid:0) σ T (cid:1)(cid:3) ≥ E (cid:2) f (cid:0) σ ( T, S loc T ) (cid:1)(cid:3) had already been noticed by Dupire in [6].)In this article, we prove that these statements are in fact incorrect. Even if 30 days is a relatively shorthorizon, it cannot be harmlessly ignored. VIX are implied volatilities (of SPX options maturing 30 days later),not instantaneous volatilities. We can actually build continuous stochastic volatility models, i.e., processes ( σ t ) t ≥ , such that E [VIX T ] > E [VIX loc ,T ] (1.8)and, more generally, such that for any strictly convex function f , E (cid:2) f (cid:0) VIX T (cid:1)(cid:3) < E (cid:2) f (cid:0) VIX ,T (cid:1)(cid:3) . (1.9)(Our counterexample actually works for any τ > .) Not only do we find one convex function f such that(1.9) holds, we actually build a model in which (1.9) holds for any strictly convex function f . Actually,we prove an inversion of convex ordering : Despite the fact that σ ( t, S loc t ) is smaller than σ t in convexorder for all t ∈ [ T, T + τ ] (see (1.7)), VIX ,T is strictly larger than VIX T in convex order. Interestingly,Guyon [8] has reported that for short maturities T , the market exhibits this inversion of convex ordering: thedistribution of VIX ,T (computed with the market-implied Dupire local volatility) is strictly larger thanthe distribution of VIX T (implied from the market prices of VIX options) in convex order.Guyon [9] has shown that, when the spot-vol correlation is large in absolute value, stochastic volatilitymodels with fast enough mean reversion, as well as rough volatility models with small enough Hurst exponent,do exhibit this inversion of convex ordering. When mean reversion increases, the distribution of VIX T becomesmore peaked, while the local volatility flattens but not as fast, and as a result it can be numerically checkedthat VIX T is strictly smaller than VIX ,T in convex order for short maturities. Interestingly enough,these models reproduce another characteristic of the SPX/VIX markets: that for larger maturities (say, 3–5months) the two distributions become non-rankable in convex order, and for even larger maturities VIX T seems to become strictly larger than VIX ,T in convex order, i.e., the inversion of convex ordering vanishesas T increases. However, it is very difficult to mathematically prove the inversion of convex ordering in thesemodels. In order to get a proof of this inversion, our idea is to choose a more extreme model, in which thevolatility process σ is such that ( σ t ) t ∈ [ T,T + τ ] is independent of F T , so that VIX T is almost surely constant,but also such that VIX ,T is not a.s. constant. In this case, since these two random variables have the samemean (recall (1.6)), VIX T is strictly smaller than VIX ,T in convex order, and (1.9) holds for any strictlyconvex function f . In particular, applying (1.9) with f ( y ) = −√ y yields (1.8). NVERSION OF CONVEX ORDERING: LOCAL VOLATILITY DOES NOT MAXIMIZE THE PRICE OF VIX FUTURES 3
Clearly, in order for
VIX ,T to be non-constant, the local volatility cannot be constant as a function of S , d t -a.e. in [ T, T + τ ] . There are many ways to achieve this, e.g., through volatility of volatility, and it iseasy to numerically verify that VIX ,T (estimated from (1.5), e.g., using kernel regressions) is non-constant.However, the main mathematical difficulty here is to prove this result. To this end, we will consider modelswhere the non-constant local volatility can be derived in closed form.The remainder of the article is structured as follows. In Section 2 we derive a simple counterexampleinspired by [2] where Beiglböck, Friz, and Sturm use a similar model to prove that local volatility does notminimize the price of options on realized variance. Then we generalize the counterexample in Section 3.Eventually in Section 4 we explain how the model can be extended so that, as suggested by market data, theconvex ordering is preserved for long maturities.2. A simple counterexample
Inspired by [2], we fix
T > and consider the following volatility process: σ t = σ if t < T + τ σ if t ≥ T + τ and U = 1 ,σ if t ≥ T + τ and U = − (2.1)where σ < σ < σ are three positive constants and U denotes the result of a fair coin toss, independent of F T (e.g., known only at a time t ∈ ( T, T + τ ] ). Proposition 1.
The stochastic volatility model in (1.1) , with volatility process described in (2.1) , satisfies (1.9) . In particular, VIX futures are strictly more expensive than in their associated local volatility model.Proof.
Let us denote σ + ( t ) = (cid:40) σ if t < T + τ σ if t ≥ T + τ , σ − ( t ) = (cid:40) σ if t < T + τ σ if t ≥ T + τ , so that σ t is given by σ + ( t ) or σ − ( t ) depending on the coin toss U . Since ( σ t ) t ∈ [ T,T + τ ] is independent of F T , VIX T is a.s. constant: VIX T = E (cid:34) τ (cid:90) T + τT σ t dt (cid:35) = 12 (cid:18) σ + σ + σ (cid:19) . Since this is also the mean of
VIX ,T , to prove (1.9), it is enough to prove that VIX ,T is not a.s. constant.Due to the very simple form of Model (2.1), we know the local volatility in closed form: σ loc ( t, x ) = p + ( t, x ) σ ( t ) + p − ( t, x ) σ − ( t ) p + ( t, x ) + p − ( t, x ) , (2.2)where p ± ( t, · ) is the density of the process ( S ± t ) t ≥ with dynamics dS ± t S ± t = σ ± ( t ) dW t , S = s , i.e., p ± ( t, x ) = 1 x (cid:112) π Σ ± ( t ) exp − (cid:32) ln xs (cid:112) Σ ± ( t ) + 12 (cid:112) Σ ± ( t ) (cid:33) , Σ ± ( t ) = (cid:90) t σ ± ( s ) ds. Figure 2.1 shows the shape of σ loc . Note in particular that σ loc takes values in ( σ, σ ) and that ∀ t ∈ (cid:16) T + τ , T (cid:105) , lim x → + ∞ σ loc ( t, x ) = σ. (2.3)Let us define ψ ( x ) := E (cid:34) τ (cid:90) T + τT σ ( t, S loc t ) dt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) S loc T = x (cid:35) (2.4)so that VIX ,T = ψ (cid:0) S loc T (cid:1) . Note that ∀ x > , ψ ( x ) < (cid:96) := 12 (cid:0) σ + σ (cid:1) . NVERSION OF CONVEX ORDERING: LOCAL VOLATILITY DOES NOT MAXIMIZE THE PRICE OF VIX FUTURES 4
Figure 2.1.
Graph of ( t, x ) (cid:55)→ σ loc ( t, x ) for T = 0 . , σ = 0 . , σ = 0 . , σ = 0 . , s = 1 Since S loc T has support R + , in order to prove that VIX ,T is not a.s. constant, it is enough to prove that ψ tends to (cid:96) when x tends to + ∞ . This follows from the next lemma. (cid:3) Lemma 2.
In Model (2.1), the function ψ defined by (2.4) satisfies lim x → + ∞ ψ ( x ) = (cid:96). Proof.
Note that ψ ( x ) = (cid:0) σ + ϕ ( x ) (cid:1) , where ϕ ( x ) := E (cid:34) τ (cid:90) T + τT + τ σ ( t, S loc t ) dt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) S loc T = x (cid:35) , so it is enough to prove that ϕ ( x ) tends to σ when x tends to + ∞ . Let ε > and ε (cid:48) := (cid:0) (cid:0) σ − σ (cid:1)(cid:1) − ε .Let us denote L t := ln( S loc t ) , whose dynamics is given by dL t = − σ loc (cid:0) t, e L t (cid:1) dt + σ loc (cid:0) t, e L t (cid:1) dW t , L = ln s . Since σ loc is bounded, it is easily checked that c := sup t ∈ [ T,T + τ ] ,x ∈ R E [( L t − L T ) | L T = x ] < + ∞ . Let ∆ := (cid:112) cε (cid:48) . Then ∀ t ∈ [ T, T + τ ] , ∀ x ∈ R , P ( | L t − L T | ≥ ∆ | L T = x ) ≤ E [( L t − L T ) | L T = x ]∆ ≤ c ∆ = ε (cid:48) . (2.5)We have σ − ϕ ( e x ) = 2 τ (cid:90) T + τT + τ E (cid:2)(cid:0) σ − σ ( t, e L t ) (cid:1)(cid:12)(cid:12) L T = x (cid:3) dt = I ( x ) + I ( x ) + I ( x ) , NVERSION OF CONVEX ORDERING: LOCAL VOLATILITY DOES NOT MAXIMIZE THE PRICE OF VIX FUTURES 5 where I ( x ) := 2 τ (cid:90) T + τT + τ E (cid:2)(cid:0) σ − σ ( t, e L t ) L t ≤ L T − ∆ (cid:1)(cid:12)(cid:12) L T = x (cid:3) dt,I ( x ) := 2 τ (cid:90) T + τ (1+ ε (cid:48) ) T + τ E (cid:2)(cid:0) σ − σ ( t, e L t ) L t >L T − ∆ (cid:1)(cid:12)(cid:12) L T = x (cid:3) dt,I ( x ) := 2 τ (cid:90) T + τT + τ (1+ ε (cid:48) ) E (cid:2)(cid:0) σ − σ ( t, e L t ) L t >L T − ∆ (cid:1)(cid:12)(cid:12) L T = x (cid:3) dt. Recall that σ loc takes values in ( σ, σ ) . From (2.5), ≤ I ( x ) ≤ (cid:0) σ − σ (cid:1) ε (cid:48) for all x ∈ R . Obviously, ≤ I ( x ) ≤ (cid:0) σ − σ (cid:1) ε (cid:48) for all x ∈ R . Moreover, it is easy to check that the convergence (2.3) is uniformw.r.t. t ∈ [ T + τ (1 + ε (cid:48) ) , T + τ ] : there exists x ∗ such that ∀ t ∈ (cid:104) T + τ ε (cid:48) ) , T + τ (cid:105) , ∀ x ≥ x ∗ , ≤ σ − σ ( t, e x ) ≤ ε (cid:48) . As a consequence, for all x ≥ x ∗ + ∆ , ≤ I ( x ) ≤ ε (cid:48) . Finally, ∀ x ≥ x ∗ + ∆ , ≤ σ − ϕ ( e x ) ≤ (cid:0) (cid:0) σ − σ (cid:1)(cid:1) ε (cid:48) = ε. We have thus proved that ϕ ( e x ) , hence ϕ ( x ) , tends to σ when x tends to + ∞ . (cid:3) Remark . Note that, if we fix t ∈ (0 , τ ) and define σ t = σ if t < t σ if t ≥ t and U = 1 ,σ if t ≥ t and U = − with U only known at time t , then we have built a model where the inversion of convex ordering holds forevery short maturity T ∈ (0 , t ) . 3. Generalization
In this section, we generalize the example presented in Section 2, to show that the desired inversion ofconvex ordering can be obtained with a more interesting structure for the volatility. We fix < t < τ
For all t ≥ , the following limit holds for the local volatility: (3.2) lim x → + ∞ σ ( t, x ) = 1Λ( A [ t ] ) (cid:90) A [ t ] g ( t ) d Λ( g ) =: σ ( t ) , where (3.3) A [ t ] := { g ∈ D : Σ g ( t ) = Λ - ess sup h ∈D Σ h ( t ) } . Proof.
To study the limit of σ loc ( t, x ) for x → + ∞ , thanks to (3.1) and dominated convergence, we arereduced to consider the limit of q g ( t, x ) . Note that(3.4) q g ( t, x ) − = (cid:90) D F ( g, h, t, x ) d Λ( h ) , where F ( g, h, t, x ) := (cid:113) Σ g ( t )Σ h ( t ) exp (cid:26) − (cid:20)(cid:16) ln xs (cid:17) (cid:16) h ( t ) − g ( t ) (cid:17) + (Σ h ( t ) − Σ g ( t )) (cid:21)(cid:27) . By Fatou’s lemma, lim x → + ∞ (cid:82) D F ( g, h, t, x ) d Λ( h ) = + ∞ as soon as Λ( D g,t ) > , where D g,t := { h ∈ D : Σ h ( t ) > Σ g ( t ) } , which in turn implies lim x → + ∞ q g ( t, x ) = 0 . On the other hand, if Λ( D g,t ) = 0 , then by dominated conver-gence lim x → + ∞ (cid:82) D F ( g, h, t, x ) d Λ( h ) = (cid:82) D lim x → + ∞ F ( g, h, t, x ) d Λ( h ) , being σ bounded and bounded awayfrom zero. Now lim x → + ∞ F ( g, h, t, x ) equals zero when Σ h ( t ) < Σ g ( t ) , and one when Σ h ( t ) = Σ g ( t ) . Thisconcludes the proof, noticing that A [ t ] = { g ∈ D : Λ( D g,t ) = 0 } . (cid:3) Proposition 5.
Consider the stochastic volatility model (1.1) . Let σ be constant equal to σ > in [0 , t ) ,independent of F W in [ t , t ) , admitting only finitely many paths, so that (3.5) Λ = N (cid:88) n =1 u n δ g n , with N ≥ , g n ∈ D , u n > , (cid:80) Nn =1 u n = 1 , with g n ( t ) = σ for all n ∈ { , . . . , N } , t ∈ [0 , t ) . We assume the following non-degeneracy condition of σ ina neighborhood of t : There exist m, n ∈ { , . . . , N } and ε > such that g m ( t ) (cid:54) = g n ( t ) for all t ∈ [ t , t + ε ] .Then, for all maturities T < t , (1.9) holds. In particular, VIX futures are strictly more expensive than inthe associated local volatility model. With an abuse of notation, below we will write q n , Σ n , F ( n, m, t, x ) instead of q g n , Σ g n , F ( g n , g m , t, x ) , toease readability. Proof.
As in the example of Section 2, we consider the function ψ ( x ) := E (cid:34) τ (cid:90) T + τT σ ( t, S loc t ) dt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) S loc T = x (cid:35) and note that our non-degeneracy assumption implies that ∀ x > , ψ ( x ) < τ (cid:90) T + τT σ ( t ) dt =: (cid:96). To prove the inversion of convex ordering for all maturities
T < t , we will show that lim x → + ∞ ψ ( x ) = (cid:96) ,that is,(3.6) (cid:90) T + τt E [ σ ( t, S loc t ) | S loc T = x ] dt −−−−−→ x → + ∞ (cid:90) T + τt σ ( t ) dt. Since Λ is discrete and the functions t (cid:55)→ Σ n ( t ) are continuous and bounded in [ t , t ) , this interval dividesin countably many intervals I k = [ a k , b k ) , k ∈ N , a k < b k , such that, in each open interval ( a k , b k ) , the NVERSION OF CONVEX ORDERING: LOCAL VOLATILITY DOES NOT MAXIMIZE THE PRICE OF VIX FUTURES 7 function σ defined in (3.2) coincides with one or more paths of σ . To be more precise, for every k ∈ N , thesets A [ t ] defined in (3.3) coincide for every t ∈ ( a k , b k ) , say to a set A k , and(3.7) σ ( t ) = g n ( t ) for t ∈ ( a k , b k ) , for all g n ∈ A k . To show the convergence in (3.6), we split the interval [ t , T + τ ] into subintervals (cid:101) I k := [ t , T + τ ] ∩ I k , k ∈ N . Since by dominated convergence lim x → + ∞ (cid:88) k ∈ N (cid:90) (cid:101) I k E [ σ ( t ) − σ ( t, S loc t ) | S loc T = x ] dt = (cid:88) k ∈ N lim x → + ∞ (cid:90) (cid:101) I k E [ σ ( t ) − σ ( t, S loc t ) | S loc T = x ] dt, we are reduced to prove that for all k ∈ N ,(3.8) lim x → + ∞ (cid:90) (cid:101) I k E [ σ ( t ) − σ ( t, S loc t ) | S loc T = x ] dt = 0 . Fix k ∈ N and ε k > , and set ε (cid:48) k := min { ε k ( b k − a k + 3(¯ v − v )) − , ( b k − a k ) / } . We split the interval I k into three subintervals(3.9) J (cid:48) k := [ a k , a k + ε (cid:48) k ] , J k := ( a k + ε (cid:48) k , b k − ε (cid:48) k ) , J (cid:48)(cid:48) k := [ b k − ε (cid:48) k , b k ) , and we are going to show that σ loc ( t, x ) converges uniformly to σ ( t ) w.r.t. t ∈ J k , for x → + ∞ .Let N k := { n ∈ { , ..., N } : g n ∈ A k } and note that the function F ( n, m, t, x ) depends on the paths g n and g m only through Σ n ( t ) and Σ m ( t ) , respectively. Therefore, from (3.4), we have(3.10) q n ( t, x ) = N (cid:88) m =1 F ( n, m, t, x ) u m = F ( n, m k , t, x )Λ( A k ) + (cid:88) m (cid:54)∈ N k F ( n, m, t, x ) u m , t ∈ J k , for any m k ∈ N k , which reduces to Λ( A k )+ (cid:80) m (cid:54)∈ N k F ( n, m, t, x ) u m for n ∈ N k . Now, it is easy to verify that,when x tends to + ∞ , F ( n, m, t, x ) converges to zero uniformly w.r.t. t ∈ J k whenever n ∈ N k and m (cid:54)∈ N k . Inparticular, there is x k such that, for all x ≥ x k , t ∈ J k , and n ∈ N k , (cid:80) m (cid:54)∈ N k F ( n, m, t, x ) u m ≤ ε (cid:48) k Λ( A k ) ¯ v − ,thus(3.11) (cid:12)(cid:12)(cid:12)(cid:12) q n ( t, x ) − A k ) (cid:12)(cid:12)(cid:12)(cid:12) = (cid:80) m (cid:54)∈ N k F ( n, m, t, x ) u m Λ( A k )(Λ( A k ) + (cid:80) m (cid:54)∈ N k F ( n, m, t, x ) u m ) ≤ ε (cid:48) k ¯ v − . Since F ( n, m, t, x ) = F ( m, n, t, x ) − , we also have that, when x tends to + ∞ , F ( n, m, t, x ) converges to + ∞ uniformly w.r.t. t ∈ J k whenever n (cid:54)∈ N k and m ∈ N k . This gives the existence of y k such that, for all x ≥ y k , t ∈ J k , n (cid:54)∈ N k , and m ∈ N k , F ( n, m, t, x ) ≥ ¯ v ( ε (cid:48) k Λ( A k )) − , which by (3.10) implies(3.12) q n ( t, x ) ≤ ε (cid:48) k ¯ v − . Note that in the present setting we have σ loc ( t, x ) = N (cid:88) n =1 u n g n ( t ) q n ( t, x ) and σ ( t ) = 1Λ( A k ) (cid:88) n ∈ N k u n g n ( t ) , t ∈ J k , from (3.7). Now (3.11) and (3.12) imply(3.13) | σ loc ( t, x ) − σ ( t ) | ≤ ε (cid:48) k , for all x ≥ z k := max { x k , y k } and t ∈ J k , which shows the claimed uniform convergence.As in the proof of Lemma 2, we consider the log-price process L t = ln( S loc t ) , and we have c k :=sup t ∈ I k ,x ∈ R E [( L t − L T ) | L T = x ] < + ∞ , k ∈ N , since σ is bounded. Setting ∆ k := (cid:112) c k /ε (cid:48) k , we againobtain(3.14) P ( | L t − L T | ≥ ∆ k | L T = x ) ≤ ε (cid:48) k , for all t ∈ I k and x ∈ R . We are going to show that (cid:90) (cid:101) I k (cid:0) σ ( t ) − E [ σ ( t, S loc t ) | S loc T = e x ] (cid:1) dt = (cid:90) (cid:101) I k E [ σ ( t ) − σ ( t, e L t ) | L T = x ] dt NVERSION OF CONVEX ORDERING: LOCAL VOLATILITY DOES NOT MAXIMIZE THE PRICE OF VIX FUTURES 8 converges to zero for x → + ∞ , by proving that for x big enough this is smaller than the arbitrarily chosen ε k . This in turn implies (3.8), being k ∈ N arbitrary, and concludes the proof of (3.6). In order to do that,we divide (cid:101) I k in three subintervals : (cid:101) J k := [ t , T + τ ] ∩ J k , (cid:101) J (cid:48) k := [ t , T + τ ] ∩ J (cid:48) k , (cid:101) J (cid:48)(cid:48) k := [ t , T + τ ] ∩ J (cid:48)(cid:48) k , k ∈ N , where we used the notation introduced in (3.9). Note that, since σ takes values in [ v, v ] , (cid:90) (cid:101) J (cid:48) k E [ σ ( t ) − σ ( t, e L t ) | L T = x ] dt ≤ ( v − v ) ε (cid:48) k , and the same bound holds when taking the integral over (cid:101) J (cid:48)(cid:48) k . On the other hand, (3.14) implies (cid:90) (cid:101) J k E [( σ ( t ) − σ ( t, e L t )) L t ≤ L T − ∆ k | L T = x ] dt ≤ ( v − v ) ε (cid:48) k , and (3.13) implies (cid:90) (cid:101) J k E [( σ ( t ) − σ ( t, e L t )) L t >L T − ∆ k | L T = x ] dt ≤ ε (cid:48) k | (cid:101) J k | , for all x ≥ ln( z k ) + ∆ k . Altogether, for x ≥ ln( z k ) + ∆ k we have (cid:90) (cid:101) I k E [ σ ( t ) − σ ( t, e L t ) | L T = x ] dt ≤ ( b k − a k + 3( v − v )) ε (cid:48) k ≤ ε k . This concludes the proof. (cid:3) Term-structure of convex ordering
In this section, we extend the model built in Section 3 in order to have the convex ordering preserved forlong maturities, as suggested by market data. To this end, we set dS t S t = σ Y (cid:112) E [ Y | S t ] dW t , t ≥ t , (4.1)where σ ∈ R + and Y is a Bernoulli random variable known in t and independent of anything else. Say Y takes the value y − with probability q − and y + with probability q + = 1 − q − , for some < y − < y + and < q − < . By Jourdain and Zhou [11], as long as the ratio y + /y − is not too large, the stochasticdifferential equation (SDE) (4.1) admits a weak solution (Ω , ( F t ) , P , W, ( S t ) , Y ) , which may not be unique.In the following we use the subscript or superscript P to emphasize that a priori the corresponding quantitiesdepend on the weak solution of (4.1).Note that (4.1) implies that, whatever the weak solution, σ , P ( t, S t ) = σ for t ≥ t . Therefore VIX ,T does not depend on the weak solution and is constant equal to σ for all maturities T ≥ t . We now wantto show that, on the other hand, for any weak solution of (4.1), this is not true for VIX P ,T . This will implythat VIX ,T is strictly smaller than VIX P ,T in convex order for T ≥ t , thus there is no inversion of convexordering for long maturities.For any weak solution of (4.1), we set F P ( s, t, x ) := E P [ Y | S t = s, S t = x ] . Since Y is independent of W , the conditional law of ( S t ) t ≥ t given Y = y ± and S t = s under P agrees withthe (unique) law of a weak solution to the SDE dS s, ± , P t S s, ± , P t = σ y ± (cid:114) F P (cid:16) s, t, S s, ± , P t (cid:17) d (cid:102) W t , t ≥ t , S s, ± , P t = s, living possibly on a different probability space (the weak uniqueness of the solution follows from [13, Theo-rem 3], given that [3, Proposition 5.1] ensures the existence of a measurable version of F P ( s, · , · ) ). Being F P bounded and bounded away from zero, we deduce that, for t > t , the conditional law of S t given Y = y ± and S t = s under P admits a density p P ± ( s, t, x ) , and that p P ± ( s, t, x ) > for all x ∈ R + , which in turn impliesthat F P ( s, t, x ) = q − y − p P − ( s, t, x ) + q + y p P + ( s, t, x ) q − p P − ( s, t, x ) + q + p P + ( s, t, x ) ∈ ( y − , y ) , t > t . (4.2) NVERSION OF CONVEX ORDERING: LOCAL VOLATILITY DOES NOT MAXIMIZE THE PRICE OF VIX FUTURES 9
Then, for T ≥ t , we have VIX P ,T = σ Y τ (cid:90) T + τT E P (cid:20) F P ( S t , t, S t ) (cid:12)(cid:12)(cid:12)(cid:12) F T (cid:21) dt =: σ Y Ψ P . Now, having
VIX P ,T constant (thus necessarily equal to σ ) corresponds to having Y Ψ P ≡ , which is notpossible given that Ψ P takes values in (cid:16) y , y − (cid:17) , by (4.2). This shows that VIX P ,T cannot be constant forany T ≥ t . Remark . To the best of our knowledge, uniqueness of a weak solution of (4.1) is still an open question.More generally, partial results on the existence of a weak solution of a calibrated stochastic local volatility(SLV) model of the form dS t S t = σ Dup ( t, S t ) f ( Y t ) (cid:112) E [ f ( Y t ) | S t ] dW t (4.3)have been obtained in [1, 11], but uniqueness has not been addressed. Note that Lacker et al . [12] haverecently proved the weak existence and uniqueness of a stationary solution of a similar nonlinear SDE withdrift, under some conditions. However, their result does not apply to the calibration of SLV models. Indeed,market-implied risk neutral distributions ( L ( S t )) t ≥ are strictly increasing in convex order and therefore nostationary solution ( S t , Y t ) t ≥ can be a calibrated SLV model.The possible absence of uniqueness of a weak solution of (4.1) or (4.3) is problematic, not only theoreticallybut also practically. It means that the price of a derivative in the calibrated SLV model may not be welldefined. For example, in our case, the VIX may depend on P . More generally, existence and uniqueness of(4.3) for general processes ( Y t ) t ≥ such as Itô processes remain a very challenging, open problem, despite thefact that these models are widely used in the financial industry, in particular thanks to the particle methodof Guyon and Henry-Labordère [7]. Acknowledgements.
We would like to thank Bruno Dupire and Vlad Bally for interesting discussionsand helpful comments.
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