Dynamics of symmetric SSVI smiles and implied volatility bubbles
aa r X i v : . [ q -f i n . P R ] A ug DYNAMICS OF SYMMETRIC SSVI SMILES AND IMPLIED VOLATILITY BUBBLES
MEHDI EL AMRANI, ANTOINE JACQUIER, AND CLAUDE MARTINI
Abstract.
We develop a dynamic version of the SSVI parameterisation for the total implied variance,ensuring that European vanilla option prices are martingales, hence preventing the occurrence of arbitrage,both static and dynamic. Insisting on the constraint that the total implied variance needs to be null at thematurity of the option, we show that no model–in our setting–allows for such behaviour. This naturally givesrise to the concept of implied volatility bubbles, whereby trading in an arbitrage-free way is only possibleduring part of the life of the contract, but not all the way until expiry. Introduction
Implied volatility is at the very core of financial markets, and provides a unifying and homogeneous quotingmechanism for option prices. The literature abounds in stochastic models for stock prices that generateimplied volatility smiles–with various degrees of practical success. Among those, the Heston model [20] inequity and the SABR model [18] in interest rates–together with their ad hoc and in-house improvements–have been of particular importance. Despite this success, these stochastic models do not enjoy the simplicityof closed-form expressions, and dedicated numerical techniques are needed to implement them. One way tobypass this has been to consider approximations of option prices–and the corresponding implied volatilities–in asymptotic regimes; thorough reviews of the latter are available in [10, 11]. A different approach, pioneeredby Gatheral [12], consists in specifying a direct parameterisation of the implied volatility, having the clearadvantage of speeding up computation and calibration times. The original Stochastic Volatility Inspired(SVI) formulation, devised while its inventor was at Merrill Lynch, has proved extremely efficient in fittingvolatility smiles on equity markets. That said, it was only devised as a maturity slice interpolator andextrapolator, and different sets of parameters were needed in order to fit a whole surface (in strike andmaturity). Gatheral and Jacquier [15] extended it to a whole surface, devising tractable sufficient conditionsensuring absence of arbitrage. The design of calibration algorithms is then easy, and this SSVI formulationhas been adopted widely in the financial industry, and has since been extended [6, 19] to a version withmaturity-dependent correlation.SSVI directly tackles option prices (equivalently, implied volatilities), without following the usual route ofspecifying a model for the evolution of the underlying. It is furthermore fully static, as its inputs are marketoption prices at a given point in calendar time, with only strike and expiry allowed to vary. Gatheral andJacquier [14] showed that, as the maturity increases, the SVI parameterisation was in fact the true limitof the Heston smile. A natural question is thus whether there exists a dynamic model such that at eachcalendar time, the option smiles in this model are given by SVI or SSVI.We provide here an answer, albeit through a slightly different lens, as we investigate whether one canimpose stochastic dynamics on the implied volatility, ensuring that arbitrage cannot occur over time. Wework in a simplified and minimal setup, in a perfect market with no interest rates, in continuous time, andconsider European Call/Put options with a fixed expiry, so we will restrict ourselves to the dynamic of afixed smile; we will also assume that the underlying process does not distribute coupons nor dividends anddoes not default. Motivated by the discussion above, we assume that the total implied variance has an SSVIshape at all (calendar) times before maturity, but is allowed to move stochastically, with the condition thatboth the underlying and option prices should be martingales.This is not the first suggested solution to this problem, and several authors have attempted to proposejoint dynamics for the underlying stock price and the implied volatility. Motivated by empirical evidence
Date : August 19, 2020.2010
Mathematics Subject Classification.
Key words and phrases. implied volatility, absence of arbitrage, SSVI, bubbles.We thank Stefano De Marco and Paolo Baldi for useful discussions. that the implied volatility moves over time, the usual approach is to specify a stochastic Itˆo diffusion for thetotal implied variance, as in [4, 17, 25, 26, 28, 31]. However, ‘the problem with market models is the extremelyawkward set of conditions required for absence of arbitrage’ [7]. An important step was made by Schweizerand Wissel [32, 33], who derived general conditions ensuring existence of such market models. Even if theresulting conditions are not easily tractable for modelling purposes, this result is the first positive answer.The only other positive result (for continuous processes) we are aware of can be found in Babbar’s PhDthesis [2], which, building on [26], developed stochastic models for the joint stock price and the total impliedvariance (which she calls the operational time), for a fixed strike, relying on comparison theorems for Besselprocesses. One fundamental catch, though, is that the implied volatility may hit zero strictly before thematurity of the option, making the model degenerate. We shall revisit this degeneracy somehow, giving itsome financial meaning.The Implied Remaining Variance framework [4, 5] shares a common point of view with our approach.There, the shape of the dynamic of the total implied variance is prescribed, whereas we derive it from theshape of the smile (SSVI in our case). Also we identify the terminal condition on the total variance atmaturity as a key property, and prove that there is no process satisfying this condition in our case. Indeedour main result is at first disappointingly negative: starting from an uncorrelated SSVI smile at all times,we show that no Itˆo process, beyond the Black-Scholes model with time-dependent volatility, ensures thatthe option prices are martingales. The by-products of this result, however, are interesting and informative.We obtain explicitly joint dynamics for the underlying and the option prices such that, locally in time (thatis until some time before the true maturity of the option), the underlying price is a martingale, and so areall the vanilla option prices, despite the fact that the option prices are not given by the expectation of thefinal payoffs under an equivalent martingale measure. This implies that until this horizon, it is not possibleto synthesise an arbitrage. Yet, this cannot last until the maturity of the option, and the market will thenchange regime. This naturally gives rise to the new concept of implied volatility bubbles. We believe thisintermediate regime (between the traditional arbitrage-free situation with specified dynamics until maturityand a regime with instantaneous arbitrages) is of interest, and may correspond to real world situations.We introduce precisely the SSVI parameterisation in Section 2, and recall the notions of absence ofarbitrage for a given implied volatility surface. In Section 3, we introduce a new stochastic model describingthe dynamics of the implied volatility surface, and extend the static arbitrage concept to a dynamic version.We show there that unfortunately there cannot be any Itˆo process solution in our setting. However, thisleads us to introduce implied volatility bubbles in Section 4, which we study in detail in the SSVI case.2.
Static arbitrage-free volatility surfaces
We recall in this section the key ingredients of volatility surface parameterisation as well as the differentconcepts of no (static) arbitrage in this setting. This will serve as the basis of our analysis, and allows usto define properly the notion of dynamic arbitrage, and its relation with martingale concepts. In order toset the notations, recall that, in the Black-Scholes model [3] with volatility σ >
0, the price of a Call optionwith strike
K >
T > t ∈ [0 , T ] by(1) C BS ( S t , K, T, σ ) = E [( S T − K ) + |F t ] = S t BS (cid:18) log (cid:18) KS t (cid:19) , σ √ T − t (cid:19) , for any t ∈ [0 , T ], where the function BS : R × R + → R is defined as(2) BS( k, v ) := (cid:26) N ( d + ( k, v )) − e k N ( d − ( k, v )) , if v > , (cid:0) − e k (cid:1) + , if v = 0 , with N the Gaussian cumulative distribution function, and d ± ( u, v ) := − uv ± v . Practitioners generally do notwork with option prices directly, but rather with the Black-Scholes implied volatility map σ t : R + × R + → R + defined through the implicit relationship C obs t ( K, T ) = S t BS (cid:0) k, σ t ( k, T ) √ T − t (cid:1) , where C obs t ( K, T ) denotesthe observed option price at time t for a given strike and maturity, and k := log( K/S t ) is the log-moneyness.There exist different conventions to write the implied volatility; for reasons that will become apparent later,we choose to write it as a function of k . For convenience, we shall in fact work in terms of the total variance ω t ( k, T ) := ( T − t ) σ t ( k, T ), so that the Call price formula (1) can be rewritten, for any t ∈ [0 , T ], as C obs t ( K, T ) = S t BS (cid:16) k, p ω t ( k, T ) (cid:17) . YNAMICS OF SYMMETRIC SSVI SMILES AND IMPLIED VOLATILITY BUBBLES 3
Static arbitrage.
Following [15], we recall the notion of static arbitrage, when for fixed running time t ,we only trade at t for a final maturity T , but not in between. Definition 2.1.
For any fixed t ∈ [0 , T ], • the surface ω t ( · , · ) is free of calendar spread arbitrage if T ω t ( k, T ) is increasing, for any k ∈ R ; • a slice k ω t ( k, T ) is free of butterfly arbitrage if the corresponding density if non-negative.A surface is free of static arbitrage if it is free of both calendar and butterfly arbitrages.As hinted by its very name, this notion of arbitrage is static, in that it only concerns the marginal distributionsof the stock price between t and T , viewed at time t , but does not involve any dynamic behaviour in therunning time t . It is equivalent to the impossibility of locking an arbitrage by trading in the option and thestock t and at the expiry of the option. Before discussing a dynamic version of no-arbitrage, let us recall theSVI parameterisation, a standard on Equity markets, which constitutes the backbone of our analysis.2.2. SSVI parameterisation.
Finding a parametric tractable model for a volatility smile has long been achallenge, and a breakthrough came when Gatheral [12] disclosed the SVI parameterisation(3) ω ( k, T ) = SVI( k ) := a + b (cid:16) ρ ( k − m ) + p ( k − m ) + σ (cid:17) for the total variance. Since we only consider for now a fixed time t , we drop the dependence thereof inthe notation without confusion. Here a, b, ρ, m and σ are parameters. This parameterisation only providesa characterisation of slices, so that the parameters are in principle different for each maturity T . The fitto market data is fairly good, and we refer the reader to [8] for an efficient and robust dimension reductioncalibration method. Necessary and sufficient conditions on the parameters preventing static arbitrage (Def-inition 2.1) have been recently characterized in [27]. To take into account the maturity dimension (hencethe whole volatility surface, still without dynamics), Gatheral and Jacquier [15] extended (3) to the SurfaceSVI (SSVI) parameterisation(4) ω ( k, θ ( T )) := θ ( T )2 (cid:18) ρϕ ( θ ( T )) k + q ( ϕ ( θ ( T )) k + ρ ) + ρ (cid:19) , where T θ ( T ) is a non-decreasing and strictly positive function representing the at-the-money total impliedvariance, ρ ∈ ( − , ρ := p − ρ , and ϕ is a smooth function from R ∗ + to R ∗ + . This formulation in factenables one to find sufficient conditions to ensure absence of static arbitrage: Proposition 2.2. [Theorems 4.1 and 4.2 in [15] ] • There is no calendar spread if ∂ t θ ( t ) ≥ , for all t > , ≤ ∂ θ ( θϕ ( θ )) ≤ ρρ ϕ ( θ ) , for all θ > • There is no butterfly arbitrage if θϕ ( θ ) ≤ min
41 + | ρ | , s θ | ρ | ! for all θ > . Symmetric SSVI.
Setting ρ = 0 leads to a symmetric smile in log-moneyness. Since we consider herea single smile, we only investigate Butterfly arbitrage. In the uncorrelated case ρ = 0, the no-Butterflyarbitrage condition in Proposition 2.2 can be simplified to θϕ ( θ ) ≤ min(4 , √ θ ), for all θ >
0. In this case,a slight improvement, as an explicit necessary and sufficient formulation, was provided in [15]. Define(5) B ( θ ) := A ( θ )11 { θ< } + 16 11 { θ ≥ } from R + to R + , where, for any θ > A ( θ ) = 16 θζ θ ( ζ θ + 1)8 ( ζ θ −
2) + θζ θ ( ζ θ − , with ζ θ := 21 − θ/ s(cid:18) − θ/ (cid:19) + 21 − θ/ . Corollary 2.3. If ρ = 0 , there is no Butterfly arbitrage if and only if ( θϕ ( θ )) ≤ B ( θ ) for all θ > . Since lim θ ↓ p A ( θ ) /θ = c , with c ≈ .
45, we have approximately a gain of a factor two with respectto the simplified sufficient condition. We note in passing that extended versions of SSVI have since beendeveloped [6, 16, 19], where again sufficient conditions are provided to ensure absence of static arbitrage.
MEHDI EL AMRANI, ANTOINE JACQUIER, AND CLAUDE MARTINI Dynamic arbitrage-free volatility surfaces
Static arbitrage is by now well understood and has contributed to providing valid examples to generatemarket options data and to design interpolators and extrapolators of option quotes. In the static settingabove, no dynamics was set for any of the ingredients. We now extend this framework to a dynamic setting,where both the stock price and the implied volatility evolve. We fix a filtered probability space (Ω , F , ( F t ) , Q )on which all processes and Brownian motions are well defined, and consider a stochastic stock price adaptedto ( F t ). The main novelty of our approach, which is common for the moment with the early works by Carrand Sun [4] and Carr and Wu [5], is to impose some dynamics for the total implied variance ( ω t ( k, T )) t ∈ [0 ,T ] .The maturity T > t only makessense for t ∈ [0 , T ). We first introduce the concept of dynamic arbitrage without specifying any dynamics.3.1. Dynamic arbitrage and consistent total variance models.
One key point is that we now write k t := log( K/S t ) instead of k for the log-moneyness, emphasising the importance of the running time t . Definition 3.1.
A consistent total variance model is a couple ( S t , ω t ( k t , T )) t ∈ [0 ,T ] ,K> such that, up to T ,(i) the process S is a strictly positive Q -martingale with continuous sample paths;(ii) for every K >
0, the process ω · ( k · , T ) has continuous paths and is strictly positive on [0 , T );(iii) for every K > ω t ( k t , T ) converges to zero almost surely as t approaches T ;(iv) for every K >
0, the process C defined by C t := S t BS (cid:16) k t , p ω t ( k t , T ) (cid:17) is a Q -martingale.We denote V T the set of all consistent total variance models, and no dynamic arbitrage occurs if V T = ∅ .By Put-Call-Parity we can equivalently replace the last item above by the martingale property of the Putprice process, directly through the Black-Scholes Put pricing function. The following useful remark relaxesCondition (iv) above from, replacing it effectively by a local martingale assumption: Lemma 3.2.
Let
K > and assume that the process ( C t ) t ∈ [0 ,T ] defined by C t := S t BS (cid:16) k t , p ω t ( k t , T ) (cid:17) isa Q -local martingale. Then if S is a martingale, so is C .Proof. Indeed the process P defined as P t := C t − ( S t − K ) is a local martingale which is positive anduniformly bounded by K , hence a martingale, and therefore C = P + ( S − K ) is a martingale as well. (cid:3) At first glance, there is no link between the dynamics of each option contract (indexed by K ), so that theoption could evolve in an inconsistent manner even when starting from a static arbitrage-free configuration;this is actually not the case due to Definition 3.1(iii): Lemma 3.3.
Absence of dynamic arbitrage implies absence of Butterfly arbitrage.Proof.
We claim that the Call price is the conditional expectation of the payoff ( S T − K ) + . Consider a Putoption with price P t := P BS ( S t , K, ω t ( k t , T )), where P BS denotes the Black-Scholes Put option price. Inabsence of dynamic arbitrage, Definition 3.1 implies that P t = E Q t [ P T ] and P T = P BS ( S T , K, ω T ( k T , T )) =P BS ( S T , K,
0) = ( K − S T ) + almost surely. Since the payoff P T is uniformly bounded by K , then domi-nated convergence implies that P t = E t [( K − S T ) + ]. Since C t ( K, T ) = C BS ( S t , K, ω t ( k t , T )) = S t − K − P BS ( S t , K, ω t ( k t , T )) = S t − K − E t [( K − S T ) + ] = E t [( S T − K ) + ], the claim follows. (cid:3) Definition 3.1(iii) is not present in [4] where nothing prevents inconsistent situations to occur. If Defini-tion 3.1-(i)-(iv) hold, each individual price being a martingale, no arbitrage can be exploited from tradingindividually in the options or stocks. Yet Butterfly arbitrage (even in its simple form of the non-monotonicityof the Call option price with respect to the strike) could occur and be exploited. As an example, considertwo Call options with maturity T , strikes K and K , with 0 < K < K , and with dynamics given by C t ( K , T ) = C ( K ) exp (cid:26) σ B (1) t − σ t (cid:27) and C t ( K , T ) = C ( K ) exp (cid:26) σ B (2) t − σ t (cid:27) , for t ∈ [0 , T ], given two independent Brownian motions B (1) and B (2) . Assume further that S follows yetanother Black-Scholes-type dynamics S t = S exp { σ B t − σ t } where B is a Brownian motion independentof B (1) and B (2) , such that ( S − K ) + < C ( K ) < C ( K ) < S . For i = 1 , τ i := inf { t ∈ [0 , T ) : C t ( K i ) / ∈ (( S t − K i ) + , S t ) } , YNAMICS OF SYMMETRIC SSVI SMILES AND IMPLIED VOLATILITY BUBBLES 5 and, for any ε >
0, the crossing time e τ ε := inf { t : C t ( K ) > C t ( K ) + ε } . Then { e τ ε < τ ∧ τ } has positiveprobability, and prices become inconsistent at e τ ε , although each individual option process is a martingale.3.2. The dynamic symmetric SSVI.
In order to be more precise, we now specify some dynamics:(6) d S t = S t √ v t d B St , starting without loss of generality from S = 1, for some Brownian motion B S . Here the process ( v t ) t ≥ isleft unspecified, but regular enough (and non-negative) so that (6) admits a unique weak solution. Since wewant the process S to be a true martingale, we impose the Novikov condition(7) E " exp ( Z T v t d t ) < ∞ . We consider a dynamic version of the uncorrelated SSVI parameterisation (4), namely(8) ω t ( k t , θ t ) = θ t (cid:18) q ϕ t k t (cid:19) . where, similar to (4), θ t accounts now for the at-the-money total implied variance at t for an option maturingat T . We implicitly disentangled here the link between the function ϕ t and the curve T θ t , by introducingthe process notation ( ϕ t ). We keep this terminology from now on, and reverting back to the classical SSVI (4)boils down to a simple change of variables. We further assume that θ and ϕ are diffusion processes given by(9) d θ t = θ ,t d t + θ ,t d B θt , θ > , d ϕ t = ϕ ,t d t + ϕ ,t d B ϕt , ϕ > , d h B θ , B ϕ i t = ̺ d t, where B θ and B ϕ are two Brownian motions. The time-dependent coefficients θ , θ , ϕ , ϕ are left unspec-ified, and may be stochastic, adapted to the filtration ( F t ) t ∈ [0 ,T ] and such that the two stochastic diffusionsadmit unique weak solutions. In our setting, ω t ( k t , θ t ) = θ t (1 + p ϕ t k t ) ≥ θ t >
0. Therefore
Lemma 3.4.
A necessary condition for Definition 3.1(iii) is that θ converges to zero amost surely at time T . We investigate here the existence of consistent total variance models of the form (6)-(8)-(9). Given thesymmetry of the implied volatility (8), we expect θ and ϕ to depend solely on v and on its driving Brownian,assumed independent of B S . The maturity T does not come into play explicitly in our parameterisation,but is present through θ t and ϕ t . The martingale S induces the new measure d Q = S T d P , and Girsanov’stheorem implies that the process f W defined by d f W t := d B St − √ v t d t is a Q -Brownian motion. Hence forany given ( K, T ), a Call option C · ( K, T ) is a P -martingale if and only if the process e C · ( K, T ), defined as e C t ( K, T ) := C t ( K, T ) /S t = BS( k t , p ω t ( k t , T )), is a Q -martingale. The terminal condition on the Call pricesunder P is that S t BS( k t , p ω t ( k t , T )) converges to the intrinsic payoff ( S T − K ) + almost surely as t tendsto T , which is granted as soon as ω t ( k t , T ) converges to zero almost surely. The objective is to find conditionson the parameters θ ,t , θ ,t , ϕ ,t , ϕ ,t and ̺ in (9) ensuring no dynamic arbitrage. Theorem 3.5.
If there is a consistent total variance model, in the sense of Definition 3.1, then necessarily, (10) d θ t = ( θ t ϕ t − θ t ϕ t + 4)16 v t d t − θ t ϕ t √ v t d B t , d ϕ t = (cid:0)
16 + 16 ϕ t θ t − θ t ϕ t (cid:1) ϕ t v t θ t d t + ϕ t √ v t d B t , where B is a Brownian motion independent from B S . We stress that the statement is only necessary. Nothing grants the existence of an actual solution;moreover, even if one exists, the following three conditions should be checked for the solution to be valid: • both processes θ and ϕ should be positive almost surely; • the no-arbitrage Condition 2.3 should hold; • the boundary condition θ T should be null almost surely;Lemma 3.8 below in fact shows that existence is a real issue. An important remark here is that theBrownian motion B may not be related to the dynamics of the variance process ( v t ), as the latter does notcome into play at any stage in the computations. We will discuss this more in detail in Section 4 below. MEHDI EL AMRANI, ANTOINE JACQUIER, AND CLAUDE MARTINI
Proof of Theorem 3.5.
To simplify the computations, introduce the notations Y t := ϕ t k t , η t := h ( θ t ) , γ t := f ( Y t ) , Ω t := γ t η t , f ( y ) := q p y , h ( θ ) := p θ/ . This implies that, in the symmetric SSVI framework (8), the Call price function (1) simplifies to C obs t ( K, T ) = S t BS ( k t , Ω t ) =: S t e C t ( k t , Ω t ) . Itˆo’s formula implies that, for any (fixed) k ∈ R and any t >
0, we can writed e C t = ∂ BS( k t , Ω t )d k t + ∂ BS( k t , Ω t )dΩ t + 12 ∂ BS( k t , Ω t )d h k i t + 12 ∂ BS( k t , Ω t )d h Ω i t + ∂ BS( k t , Ω t )d h k, Ω i t . The derivatives of the BS( · , · ) function are classical and straightforward: ∂ BS( u, v ) = − e u N ( d − ) , ∂ BS( u, v ) = n ( d + ) ,∂ BS( u, v ) = − e u N ( d − ) + n ( d + ) v , ∂ BS( u, v ) = (cid:18) − uv (cid:19) n ( d + ) ,∂ BS( u, v ) = (cid:18) u v − v (cid:19) n ( d + ) . Now, we can write the dynamics for all the processes appearing in this equation asd k t = √ v t d f W t − v t t d h k i t = v t d t, d Y t = ϕ t d k t + Y t ϕ t d ϕ t + d h ϕ, k i t , d h Y i t = ϕ t d h k i t + (cid:18) Y t ϕ t (cid:19) d h ϕ i t + 2 Y t d h ϕ, k i t , d γ t = f ′ ( Y t )d Y t + 12 f ′′ ( Y t )d h Y i t , d h γ i t = f ′ ( Y t ) d h Y i t , dΩ t = η t d γ t + γ t d η t + d h η, γ i t , d h Ω i t = η t d h γ i t + γ t d h η i t + 2 η t γ t d h η, γ i t , d η t = h ′ ( θ t )d θ t + 12 h ′′ ( θ t )d h θ i t , d h η i t = h ′ ( θ t ) d h θ i t , d h k, Ω i t = η t f ′ ( Y t ) (cid:18) ϕ t d h k i t + Y t ϕ t d h k, ϕ i t (cid:19) + γ t d h k, η i t , d h η, γ i t = h ′ ( θ t ) f ′ ( Y t ) (cid:18) ϕ t d h θ, k i t + Y t ϕ t d h θ, ϕ i t (cid:19) d h k, η i t = h ′ ( θ t )d h k, θ i t . In order for the option price to be a martingale, the drift part d e C has to be equal to zero. Assume now, asin the proposition, that h θ, ϕ i t = ψ t = h θ, k i t = 0. Long and tedious computations (that we perform withthe counter checks of Sympy) show that the latter can be written as Z t (1 + Z t ) P ( Z t )d t , where P ( · ) is afifth-order polynomial, where Z t := p Y t . In particular, each coefficient P i of order i = 0 , . . . , P = ( p t − (cid:0) ϕ t θ ,t + θ t ϕ ,t + 2 √ p t χ t (cid:1) , P = − θ t n (8 ϕ t √ p t − p t − ϕ t χ t − ϕ t θ t θ ,t + 16 ϕ t θ t θ ,t − ϕ t θ ,t + 16 ϕ t p t ϕ ,t − p t ϕ ,t − θ t ϕ ,t o , P = ϕ t n − p t ) θ t χ t + ϕ t (cid:2) p t ( p t − v t + p t θ ,t − θ t ϕ ,t − p t θ ,t + 8 ϕ t √ p t θ ,t − θ t ϕ ,t + 16 θ ,t (cid:3) o , P = 2 θ t n (8 ϕ t √ p t − p t − ϕ t χ t + 4 θ t (cid:0) ϕ t θ t v − ϕ t v + 4 ϕ t θ t ϕ ,t − ϕ t θ t ϕ ,t − ϕ ,t (cid:1) o , P = − θ t ( p t + 8 ϕ t √ p t + 16) (cid:0) ϕ t v t − ϕ ,t (cid:1) , P = − p t θ t (cid:0) ϕ t v − ϕ ,t (cid:1) , where we introduced p t := ϕ t θ t , and write χ t for the drift term of the covariation d h θ, ϕ i t .Now, the dependence on K and the running spot S t in the drift of d e C is only through Z t , as both θ t and ϕ t are independent thereof. The derivation above is valid for any fixed K , thus for all K >
0. The onlyway the drift condition can be achieved is therefore that P ( Z t ) is identically null, which holds if and only if P i = 0 for i = 0 , . . . ,
5. This system, with unknown ( θ ,t , θ ,t , ϕ ,t , ϕ ,t , χ t ), is solvable as S + and S − , where S ± = v t (cid:0)(cid:2) ̺ − (cid:3) ϕ t θ t + 8 ϕ t θ t (cid:2) ̺ − (cid:3) − ϕ t θ t [ θ t + 4] ̺ | ̺ | − (cid:1) ± ϕ t θ t √ v t ( | ̺ | ̺/̺ − ̺ ) v t ϕ t θ t (cid:0) ϕ t θ t (cid:2) ̺ (cid:3) − ̺ ϕ t θ t − ̺ + (cid:2) − ϕ t θ t + ϕ t θ t (cid:3) ̺ | ̺ | + 32 (cid:1) ∓ ϕ t √ v t ϕ t θ t v t (cid:0) ̺ | ̺ | − ̺ (cid:1) , YNAMICS OF SYMMETRIC SSVI SMILES AND IMPLIED VOLATILITY BUBBLES 7 and ̺ := p − ̺ . It admits a solution only for ̺ ∈ {− , } , and (10) follows. The two solutions for θ ,t and ϕ ,t are hence of opposite sign, hence equivalent as Brownian increments are symmetric.We now show that the absence of correlation between S and both ϕ and θ is in fact necessary. Denoteby d e C + the expression above for d e C , and let d e C − be the same one, except that we now set Y t := − p Z t − e C + − e C − ) has to be null. This yields the following system: θ t ϕ t d h ϕ, k i t = 0 , (cid:0) θ t ϕ t + 8 θ t ϕ t + 16 (cid:1) θ t d h ϕ, k i t = 0 ,ϕ t (cid:16) d h θ, k i t θ t ϕ t − h θ, k i t θ t ϕ t + 16d h θ, k i t − h ϕ, k i t θ t ϕ t (cid:17) = 0 , (cid:16) d h θ, k i t ϕ t + d h ϕ, k i t θ t (cid:17)(cid:16) θ t ϕ t + 16 (cid:17) = 0 , and it is easy to see that the only valid solution is d h θ, k i t = d h ϕ, k i t = 0. (cid:3) Proposition 3.6. If V T = ∅ and ( θ, ϕ ) solves (10) , then for any t ∈ [0 , T ] , θ t = R Tt v u d u , all the smiles(at t , maturing at T ) are flat, and R Tt v u d u ∈ F t .Proof. The condition θ T = 0 implies ψ T = 0. The SSVI smile is then trivial, equal to θ t for all strikes.Furthermore d θ t = − v t d t , and hence θ t = R Tt v u d u . Since θ t is F t measurable, then so is R Tt v u d u . (cid:3) The Black-Scholes model with deterministic volatility clearly satisfies these conditions, and it is immediateto check from the definition that it is indeed a consistent total variance model.
Remark 3.7.
If the property R Tt v u d u ∈ F t is taken to hold for every T >
0, then v T is F t -measurable,and v deterministic. Note that we only considered here a fixed maturity T , not all of them. One could thenwonder whether the necessary properties above imply that v is deterministic, in which case the only solutionwould be a time-dependent Black-Scholes model. We provide a non-trivial example showing that this is notalways the case: on the filtration of a planar Brownian motion ( W, B ), fix T = 1, 0 < ε , ε <
1, and define v t := v , for t ∈ [0 , / ,v (1 + ε )11 B / > + v (1 − ε )11 B / ≤ , for t ∈ [1 / , / ,v (1 − ε )11 B / > + v (1 + ε )11 B / ≤ , for t ∈ [2 / , . Then clearly R Tt v u d u is F t -measurable for every t ∈ [0 , S T islognormal, and the smiles are flat at every point in time. This in particular yields another example of aconsistent total variance model, different from the time-dependent Black-Scholes.3.3. Non-existence of non-trivial consistent dynamics.
We now give three different proofs that thereis no non-trivial consistent dynamics; they all rely on the core observation that ( θ t ) t ∈ [0 ,T ] should tends tozero at maturity as shown in Lemma 3.4. Notice from (10) that d( θ t ϕ t ) = 0, so that θ t ϕ t = ψ T for any t ∈ [0 , T ] for some constant ψ T . Hence, the SDE for θ can be rewritten more compactly asd θ t = ( ψ T − ψ T + 4)16 v t d t − ψ T √ v t d B t , with boundary condition θ T = 0 . First proof.
With the necessary and sufficient no-Butterfly arbitrage condition ( θϕ ( θ )) ≤ B ( θ ), giventhat θ t ϕ ( θ t ) = ψ T , and that B tends to zero when θ does, there cannot be a consistent dynamic; otherwisethere would be no static arbitrage at any point in time (Lemma 3.3), and since necessarily θ tends to zeroas t approaches T (Lemma 3.4), B ( θ t ) < ( θ t ϕ ( θ t )) and there is a Butterfly arbitrage.3.3.2. Second proof.
We investigate whether the solution in Theorem 3.5 is trivial or not, without using theknowledge of the shape of the exact no-Butterfly arbitrage region.
Lemma 3.8.
Let α := − (cid:16) ψ T − ψ T (cid:17) . If E h exp n α R T v t d t oi is finite (in particular, under the Novikovcondition (7) if α < ), then any solution of (10) on [0 , T ] satisfies θ t = ψ T α log E t (cid:20) exp (cid:18) αθ T ψ T (cid:19)(cid:21) , almost surely for all t ∈ [0 , T ] . In particular, θ T = 0 almost surely entails that θ t = 0 almost surely for all t ∈ [0 , T ] . MEHDI EL AMRANI, ANTOINE JACQUIER, AND CLAUDE MARTINI
Proof.
Setting e θ t := − θ t ψ T yields d e θ t = αv t d t + √ v t d B t , with e θ T = 0 and α := ψ T − ψ T . Itˆo’s formula impliese − α e θ T = e − α e θ t − α Z Tt e − αθ u d e θ u + 2 α Z Tt e − α e θ u v u d u = e − α e θ t − α Z Tt e − α e θ u (cid:2) αv u d u + √ v u d B u (cid:3) + 2 α Z Tt e − α e θ u v u d u = e − α e θ t − α Z Tt e − α e θ u √ v u d B u . Note that − α e θ u > − α e θ u = e − α e θ e − α R u v t d t e − α R u √ v t d B t ; the Novikov condition applied tothe process − α R · √ v t d B t reads E h exp n α R T v t d t oi < ∞ , in which case the stochastic integral is squareintegrable with null expectation, and every term in the expression has finite expectation. Taking expectationsconditional on F t on both sides yields E t h e − α e θ T i = e − α e θ t almost surely for all t ∈ [0 , T ], hence θ t = ψ T α log E t (cid:20) exp (cid:18) αθ T ψ T (cid:19)(cid:21) , almost surely for all t ∈ [0 , T ] . Imposing θ T = 0 almost surely implies that θ t = 0 almost surely for all t ∈ [0 , T ]. (cid:3) Third proof.
Because of the terminal condition θ T = 0, the stochastic differential equation satisfiedby θ is actually a classical example of a BSDE. Consider indeed the process(11) X t = ξ + Z Tt f ( s, Z s )d s − Z Tt Z s d B s , on [0 , T ], with terminal condition X T = ξ , where ξ is a bounded, F BT -measurable random variable, and f : [0 , T ] × R → R has at most quadratic growth. This is the classical (one-dimensional) example of aquadratic BSDE [34, Chapter 10]. In our setting, integrating the SDE for e θ on [ t, T ] reads e θ t = e θ T + Z Tt (cid:18) ψ T − ψ T (cid:19) v s d s − Z Tt √ v s d B s , which is exactly of the form (11) with X = e θ , f ( · , z ) ≡ ( ψ T − ψ T ) z , Z = √ v and ξ = 0 almost surely.From [34, Chapter 10], if ( X, Z ) is a solution to (11) with X bounded, then Z ∈ H , where H := (cid:8) ϕ ∈ H : (cid:13)(cid:13)R · ϕ s d B s (cid:13)(cid:13) BMO < ∞ (cid:9) , with H the space of square integrable martingales, and k M k BMO :=sup {k E [ h M i T − h M i τ |F τ ] k ∞ : τ ∈ T T } , where T T denotes the set of stopping times in [0 , T ]. Since thedriver f is quadratic and smooth, uniqueness of a solution to (11) is guaranteed by [34, Theorem 10.5].Because the terminal condition ξ is bounded here, existence of a unique solution to (11) is then given by [34,Theorem 10.6], with upper bound estimate for the (appropriate) norms of X and Z . This grants that( X, Z ) = (0 ,
0) is the unique solution.
Remark 3.9.
Section 3.3.2 uses computations common in BSDE theory, and Section 3.3.3 is thus a shortcutfor BSDE wizards; so those proofs are essentially one and the same.4.
Smile bubbles
We showed previously that, except for dynamics close to Black-Scholes, there is not dynamic modelfor ( θ t ) t ∈ [0 ,T ] and ( ϕ t ) t ∈ [0 ,T ] of symmetric SSVI up to maturity T . We now investigate what happens if werestrict ourselves to a shorter time horizon; more precisely, we wish to find valid dynamics for θ up to some(possibly stochastic) time horizon τ ∈ (0 , T ). This would constitute a dynamic arbitrage-free model on [0 , τ ],satisfying the martingale condition for the stock and Vanilla option price, with total variance given by SSVI.One caveat is that absence of Butterfly arbitrage is not granted, because the Vanilla prices can a priori nolonger be written as conditional expectations of the terminal payoff, although the individual option pricesare martingales. Now, in the case of symmetric SSVI, we have explicit necessary and sufficient conditions onthe parameters preventing Butterfly arbitrage: so starting in this domain, we will have locally no dynamicnor Butterfly arbitrages up to the exit time τ of this domain. In order to state the definitions below, let T T denote the set of all stopping times with values in (0 , T ]. The following is a localised version of Definition 3.1: YNAMICS OF SYMMETRIC SSVI SMILES AND IMPLIED VOLATILITY BUBBLES 9
Definition 4.1.
A locally consistent total variance model is a couple ( S · , ω · ( k · , T )) such that there exists τ ∈ T T for which, up to τ :(i) the process S is a strictly positive Q -martingale;(ii) for every K >
0, the process ω · ( k · , T ) has continuous paths and is strictly positive;(iii) there is no Butterfly arbitrage.(iv) for every K >
0, the process C defined by C t := S t BS (cid:16) k t , p ω t ( k t , T ) (cid:17) is a Q -martingale.We denote V loc T the set of all locally consistent total variance models, and we say that there is no dynamicarbitrage on [0 , τ ] if V loc T is not empty with τ ∈ V loc T . Remark 4.2.
Here again we assume that the implied volatility of Calls and Puts is the same, so that thePut-Call-Parity holds by assumption, and we can equivalently replace Definition 4.1(iv) by the martingaleproperty of the Put price process, and Definition 4.1(iii) by the corresponding properties of the Put price.Also the martingale property of the option prices is equivalent here again to the local martingale property.Obviously, V T ⊂ V loc T , so that every consistent total variance model can be localised. Conversely, a localconsistent total variance model might or might not be extended to a fully consistent one. Even if it mightbe, this possibility could well be of no use in practice because of the complexity of the overall model. Webelieve therefore that this local approach has a strong interest on its own, and we shall also call such a locallyconsistent model a bubble .If one trades only within a bubble lifespan [0 , τ ], with possibly additionally unwinding trades at theexpiry T , there is no arbitrage to be made. Arbitrage, if some, will follow from purely dynamic strategiesinvolving unwinding positions beyond the bubble lifespan. We believe this set-up, albeit not surprisingfrom a strict mathematical point of view, might be very relevant to account for real life joint underlyingand options dynamics. Indeed, if one goes back to the local dynamic of an SSVI bubble in Theorem 3.5,we stress that the driving Brownian motion of the SSVI parameters should be independent of the drivingBrownian motion of S , with no necessary relation to the Brownian motion (if any) driving the instantaneousvolatility process v . This might correspond to the observations of the high frequency joint dynamics ofthe instantaneous volatility and of option prices on Equity index options in [1], where the complementarynoise driving the non-Delta move of the option prices does not correspond to the idiosyncratic noise of theinstantaneous volatility. In fact, beyond this high-frequency situation, it might be the case that the jointdynamic of the underlying and of the Vanilla option prices is a succession of bubbles, with some bubbles farfrom a fully consistent joint dynamic, and others closer.4.1. Symmetric SSVI smile bubbles.
In the symmetric SSVI dynamics, we just proved that there cannotexist any consistent total variance model. We did so by identifying a local dynamics that ( θ, ϕ ) should satisfyin order for option prices to be local martingales. The only ingredient missing to design a bubble is the no-Butterfly arbitrage condition; in the symmetric SSVI case, we have an explicit description of the no-Butterflydomain in terms of the parameters. So assuming the model parameters start within this domain, and makingthem evolve according to the local dynamics identified in our computations, we indeed obtain a bubble. Thebubble lifespan is then the first time when either θ or ϕ becomes negative, or when we exit the no-Butterflydomain. The explicit description of the no-Butterfly arbitrage region is given by Condition 2.3, so thatcombining it with (10), ψ T ≤ B ( θ t ) holds for all t ∈ [0 , T ]. The function B is smooth and increasingfrom [0 , ∞ ) to [0 , B ← is well defined from [0 ,
16] to [0 , θ ≥ B ← ( ψ T ), implyingthat θ is non-negative. Therefore, if ψ T <
4, then θ > B ← ( ψ T ), and we consider the stopping time(12) τ := inf (cid:8) t ≤ T, θ t < B ← ( ψ T ) (cid:9) ∈ T T . On [0 , τ ] both Vanilla option prices and stock price are martingales, and the no-Butterfly condition holds, sothat we have an explicit description of the symmetric SSVI implied volatility bubbles. These however dependon the parameter ψ T and on the dynamics of the process v , which partially drives θ . Indeed, from (10), θ t = θ + (cid:18) ψ T − (cid:19) Z t v u d u + ψ T Z t √ v u d B θu , where B θ is a Brownian motion independent from B S , which may, or may not, be related to the Brownianmotion driving v . We need to check that θ remains positive, but assuming that θ satisfies the no-Butterflyconstraint in Condition 2.3, the positivity of θ before time τ follows from its definition. Therefore any locally integrable instantaneous process independent from B S defines a symmetric SSVI implied volatility bubbleup to the stopping time τ . Note in passing that the Novikov condition on v grants the martingale propertyof S , but could certainly be weakened in this bubble context.Due to the finite lifespan of the bubble, shorter than the options’ maturity, option prices are not givenby the expectation of the terminal payoff under the risk-neutral dynamic of the underlying. It is natural inthe context of a bubble to then take the conditional expectation at the bubble time boundary. Since optionprices are true martingales, Doob’s optional stopping theorem [29, Chapter II, Section 3], implies that theprice of the option at time t ≤ τ is E t [ S τ BS( k τ , ̟ τ )], where S . = K exp ( − k . ) and ̟ t := 12 (cid:18) θ t + q θ t + ψ T k t (cid:19) . For t ∈ [0 , T ], ̟ t depends on the terminal expiry T through ψ t . Furthermore, the option price is also givenby the Black-Scholes formula composed with SSVI, so that (recall that S = 1, hence k = log K )(13) S BS( k , ̟ ) = E [ S τ BS( k τ , ̟ τ )] . This equality holds irrespective of the dynamics of the variance process, even if the latter directly impacts thejoint law of ( τ, k τ , θ τ ). Since the strike can be chosen arbitrarily, this in turn yields an interesting propertyof this joint law, of which we provide several examples below. We christen (13) the Bubble Master Equation .It will be convenient in some situations to go back to a Brownian setting by time-change: let us restart fromBS( k , ̟ ) = E (cid:2) { τ
0, the process e β defined by e β t = − β θt + ηt is a Q -Brownian motion by Girsanov’s Theorem.Furthermore, τ is now the first hitting time of e β of a := ψ T (cid:0) θ − B ← ( ψ T ) (cid:1) , and hence, by conditioning,(15)BS( k , ̟ ) = E Q (cid:20) exp (cid:26) η e β τ − η τ (cid:27) { τ< V T } BS( k τ , ̟ τ ) + exp (cid:26) η e β V T − η V T (cid:27) { τ ≥V T } BS( k V T , ̟ V T ) (cid:21) . The Black-Scholes-SSVI bubble.
We investigate the Bubble master equation when T goes to infinity,choosing ψ T such that it is constantly equal to a given ψ ∞ <
4. Assume v t = v > t ∈ [0 , T ]; then V T = vT diverges to infinity as T increases. In this limiting case, we deduce from (15): Proposition 4.3.
For any ψ ∞ < , θ > B ← ( ψ ∞ ) , k ∈ R , with β := B ← ( ψ ∞ ) , the equality BS( k , ̟ ) = a e ηa π Z R e − y BS (cid:18) k − y, (cid:18) β + q ψ ∞ ( k − y ) + β (cid:19)(cid:19) s η + 1 y + a K p (4 η + 1) ( y + a )2 ! d y holds, where η := ψ ∞ − ψ ∞ , a := θ − β ψ ∞ > , and K denotes the modified Bessel function of the second kind.Proof. Since a, η >
0, then η e β T < ηa , so that the second term in (15) tends to zero pointwise as T increases,while remaining bounded above by K e ηa . Since τ is finite almost surely, the first term in (15) increases toexp (cid:26) η e β τ − η τ (cid:27) BS( k τ , ̟ τ )11 { τ Under Q , the density of τ is p ( t ) := a √ πt e − a t . Since τ and β S are independent, the right-hand side readse ηa Z ∞ Z R e − η t BS k − x √ t + t , β + s ψ ∞ (cid:18) k − x √ t + t (cid:19) + β p ( t ) e − x √ π d x d t = a e ηa π Z R e − y BS (cid:18) k − y, (cid:18) β + q ψ ∞ ( k − y ) + β (cid:19)(cid:19) (cid:18)Z ∞ e − ( η + ) t e − y a t d tt (cid:19) d y, with y := x √ t − t . The proposition follows from the computation of the inner integral as Z ∞ e − ( η + ) t e − y a t d tt = s η + 1 y + a K p (4 η + 1) ( y + a )2 ! . (cid:3) Remark 4.4. We can in fact apply exactly the same reasoning for any positive level β < θ since we only usethe martingale property of the price under the dynamic of θ : the same identity thus holds for any 0 < β < θ ,and not only for β = B ← ( ψ ∞ ). Observe that the right-hand side does not depend on β , since the left-handside does not. Interestingly enough, this equality does not seem easy to obtain by classical means.4.1.2. The Heston-SSVI bubble. We now investigate the existence of an SSVI bubble in the Heston model,that is when the variance process ( v t ) t ≥ is a Feller diffusion of the formd v t = κ ( v − v t )d t + ξ √ v t d B vt , v , κ, v, ξ > . Yamada-Watanabe conditions [23, Proposition 2.13] guarantee a unique strong solution, and the Fellercondition 2 κv ≥ ξ that the latter never reaches the origin. We consider two cases, depending on whether B v and B θ are fully or anti correlated (and independent of B S ). In the former case, (10) readsd θ t = (cid:18) ψ T − (cid:19) v t d t − ψ T √ v t B θt = (cid:18) ψ T − (cid:19) v t d t − ψ T ξ (d v t − κ ( v − v t )d t ) , so that θ t = θ + (cid:18) ψ T − − κψ T ξ (cid:19) V t − ψ T ξ ( v t − v ) + κvψ T tξ . When h B θ , B v i t = − d t , then θ t = θ + (cid:18) ψ T − κψ T ξ (cid:19) V t + ψ T ξ ( v t − v ) − κvψ T tξ .We can then proceed as in the Black-Scholes case, at least for the innermost conditional expectation; thesubsequent computations are still more intricate, since in general the law of τ is unknown. The symmetricSSVI bubble equation provides in this case some information of the joint law of τ, v τ and V τ .5. 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