Path-dependent volatility models
PPATH-DEPENDENT VOLATILITY MODELS
ANTOINE JACQUIER AND CHLO´E LACOMBE
Abstract.
We provide a thorough analysis of the path-dependent volatility model introduced byGuyon [13], proving existence and uniqueness of a strong solution, characterising its behaviour at bound-ary points, and deriving large deviations estimates. We further develop a numerical algorithm in orderto jointly calibrate S&P 500 and VIX market data. Introduction
Stochastic volatility models have been used extensively over the past three decades in order to re-produce particular features of market data, on Equities, FX and Fixed Income markets, both under thehistorical measure and for pricing purposes. Most of the literature and the models used in practice arebased on a Markovian assumption for the underlying process, essentially for mathematical convenience,as PDE techniques and Monte Carlo schemes are more readily available then. However, recent modelshave departed from this Markovian confinement and have shown to provide extremely accurate fit tomarket data. One approach considers instantaneous volatility driven by fractional Brownian motion,giving rise to the rough volatility generation and its numerous descendents. A less strodden, yet very in-tuitive, path, originally introduced by Engle [7] and Bollerslev [2] in the early 1980s suggested to considermodels where volatility depends on the past history of the stock price process. Their approach, though,was under the historical measure, and Duan [5] investigated these discrete-time models in the contextof option pricing. With this in mind, Hobson and Rogers [15] extended this approach to continuoustime, suggesting that instantaneous volatility depends on exponentially weighted moments of the stockprice. Contrary to stochastic volatility models (rough or not), the market here is complete. Hobsonand Rogers [15] showed that such models generate implied volatility smiles and skews consistent withmarket data. Further results investigated some theoretical properties of these models, in particular [20]proving existence and uniqueness of strong solutions. This path has recently been given new highlightsby Guyon [12], who concentrated on the following setup for the stock price process S :d S t S t = σ ( t, S t , Y t ) l ( t, S t )d W t , S := s > , where W is a standard Brownian motion, Y an adapted process and l ( · ) the leverage function ensuring(similar to local volatility models) that European options are fully recovered. Inspired by Hobson andRogers [15], Guyon [13] suggested to choose Y as an exponentially weighted moving average of the stockprice. Not only does this model calibrate perfectly to the observed smile, but the diffusion map σ ( · ) canbe chosen in such a way that joint calibration with VIX data becomes feasible, a notoriously hard taskso far. We take up on this challenge set by Guyon, and analyse theoretical and empirical properties ofthis class of models. In particular, we provide a full characterisation of the behaviour of the process atits boundaries and derive precise small-time large deviations asymptotics. Only scarce related results forsystems with memory are available in the literature, for example by Azencott, Geiger and Ott [1] forsystems with finite discrete-time delay or Ma, Ren, Touzi and Zhang [18] for non-Markovian stochasticdifferential equations with random coefficients. Date : January 16, 2020.2010
Mathematics Subject Classification.
Key words and phrases.
Path-dependent volatility, large deviations, implied volatility asymptotics.The authors would like to thank Alexander Kalinin for his insightful remarks about the behaviour of solutions at theboundary and Julien Guyon for introducing us to this exciting problem. a r X i v : . [ q -f i n . P R ] J a n ANTOINE JACQUIER AND CHLO´E LACOMBE
The structure of the paper is as follows. In Section 2, we set the notations and present the model.Section 3 gathers the main theoretical results, proving existence and uniqueness of a strong solution(Section 3.1), deriving the stationary distribution (Section 3.2) and presenting pathwise large deviationsestimates in Section 3.3, from which implied volatility asymptotics follow readily. Finally, in Section 4we construct a numerical scheme to estimate the coefficients of the system for a joint calibration to S&P500 and VIX market data. 2.
Set up and notations
The goal of this project is to develop and analyse a model describing the relationship between the VIXindex and the VVIX, a volatility of volatility index. Figure 1 below shows a scatter plot of one versusthe other over a five-year period. The approximate linear relationship highlighted by the least-squareregression fit was first noted by Guyon [13], and we follow his recommendations here. The underlying
Figure 1.
Historical VVIX vs historical VIX (13/4/12 - 8/5/17).
Source: CBOE data. process S , describing the evolution of the S&P index follows the general dynamicsd S t S t = σ ( Y t )d W t , S := s > W generating a filtration F = ( F t ) t ≥ , where σ : R ∗ + → R is nonanticipative. Following Guyon [13] and Hobson and Rogers [15], we assume that the process Y is adaptedto F and is a function of the past history of the stock S , making the latter non-Markovian, in the sense(1) Y t := S t S ht , for t ∈ T , where S ht := 1 h (cid:90) t −∞ exp (cid:26) − t − uh (cid:27) S u d u is the exponential weighted moving average (EWMA) of the stock price process. Here, T = [0 , T ] for somefixed time horizon T . The constant h >
0, denoting the length of the time window, is left unspecified fornow. Using Itˆo’s formula, we can summarise the dynamics for the couple (
S, Y ) as(2) d S t = S t σ ( Y t )d W t , S = s > , d Y t = b ( Y t )d t + (cid:101) σ ( Y t )d W t , Y = y > , with b ( y ) := h y (1 − y ) and (cid:101) σ ( y ) := yσ ( y ) for y >
0. Guyon [13] showed that, for the linear relationshipbetween the VIX and the VVIX to hold, one needs to consider a diffusion coefficient of the form(3) σ ( x ) := − αβ + γx − β , with α, β, γ >
0. In that case, (cid:101) σ is null at y σ := (cid:16) βγα (cid:17) /β , and (cid:101) σ (0) = not defined , if β > , , if β < ,γ, if β = 1 . ATH-DEPENDENT VOLATILITY MODELS 3
Before diving into the asymptotic behaviour of the process, we first concentrate on the existence anduniqueness of a strong solution for (2), depending on the values of α , β and γ , and classify the singularpoints 0, y σ and infinity.3. Theoretical analysis of the path-dependent volatility model
Boundary conditions.
We start the analysis of the model by a detailed study of the behaviourof the system (2) at the boundary points 0, y σ and ∞ . Recall [3, Definition 2.3] that a point d ∈ R iscalled singular if | b | (cid:101) σ is not locally integrable in a neighbourhood of d . A point that is not singular iscalled regular. A singular point is further called isolated if it admits a deleted neighbourhood consistingof regular points. Introduce the hitting times τ x := inf { t ≥ Y t = x } and τ x,y := min ( τ x , τ y ). Theprecise classification of boundary points is rather delicate and we only give below an informal help, andrefer the interested reader to [3, Sections 2.3 and 2.4] for full and precise details: Definition 3.1.
Let a > • A point x is said to have right-type 0, and we write x ∈ B a , if for any y ∈ [ x, x + a ], there existsa unique solution defined up to τ x,a . This solution reaches x with strictly positive probability, E ( τ x,x + a ) is finite, and P (cid:0) Y τ x,x + a = 0 (cid:1) > • A point x is said to have right-type 1, and we write x ∈ B a , if for any y ∈ [ x, x + a ], there existsa unique solution defined up to τ x,a . This solution reaches x with strictly positive probability.Any solution may only leave x in the left direction. • A point x is said to have right-type 3 if for any y ∈ ( x, x + a ], there exists a unique solutiondefined up to τ x + a . This solution never reaches x and E ( τ x + a ) is finite. We write x ∈ B a . • Infinity is called a recurrent boundary point if a solution cannot explode there. Moreover, if thereare no singular points between y and a point z < y , then the solution reaches z almost surely.The left-types B a − and B a − are defined similarly.The following proposition, proved in Appendix A.1, characterises the behaviour of the solution to (2)at the origin and at infinity. Proposition 3.2.
Infinity is a recurrent boundary, and the origin is regular if and only if β > , and • for β > , zero is an exit, non-entrance boundary point and belongs to B y σ ; • for ≤ β ≤ , zero is an exit, non-entrance boundary point and belongs to B y σ . The following theorem, proved in Section A.2, is more involved and fully characterises the solution tothe SDE (2).
Theorem 3.3. y σ is an isolated singular point, and for a > , the following holds:Left boundary of y σ > y Right boundary of y σ < y y σ < B a − B a y σ ≥ B a − B a Stationary distribution and pricing PDE.
We now prove that the process Y introduced in (2)admits a stationary distribution and we derive the pricing PDE associated to (2).3.2.1. Stationary distribution.
We recall [9, Section 3.2] that a process ( Y t ) t> is ergodic if it admits aunique, stationary distribution Π, and for any measurable bounded function g , the almost sure limitlim t ↑∞ t (cid:90) t g ( Y s )d s = (cid:90) g ( y )Π(d y )holds. If it exists, an ergodic solution must satisfy L ∗ Π = 0, where L ∗ is the adjoint of the infinitesimalgenerator L , defined [9, Section 1.5.3] via the equality(4) (cid:90) Ψ( ξ ) L φ ( ξ )d ξ = (cid:90) φ ( ξ ) L ∗ Ψ( ξ )d ξ, for any rapidly decaying smooth test functions φ and Ψ. ANTOINE JACQUIER AND CHLO´E LACOMBE
Proposition 3.4.
The infinitesimal generator L Y of Y in (2) and its dual L ∗ Y read ( L Y f )( y ) = 1 h y (1 − y ) ∂∂y + 12 y σ ( y ) ∂ ∂y and ( L ∗ Y f )( y ) = − ∂∂y (cid:18) h y (1 − y ) f ( y ) (cid:19) + 12 ∂ ∂y (cid:0) y σ ( y ) f ( y ) (cid:1) . Proof.
Using (4) and integration by part, we obtain the expression for L ∗ . Given y ∈ R and f, g : R → R twice continuously differentiable functions with bounded derivatives, we have (cid:104) f, L ∗ Y g (cid:105) = (cid:90) R f ( y ) (cid:20) − ∂∂y (cid:18) h y (1 − y ) g ( y ) (cid:19) + 12 ∂ ∂y (cid:0) y σ ( y ) g ( y ) (cid:1)(cid:21) d y, = − (cid:90) R ∂f∂y ( y ) (cid:26) − h y (1 − y ) g ( y ) + 12 ∂∂y (cid:0) y σ ( y ) g ( y ) (cid:1)(cid:27) d y, = 1 h (cid:90) R ∂f∂y ( y ) y (1 − y ) g ( y )d y − (cid:90) R ∂f∂y ( y ) ∂∂y (cid:0) y σ ( y ) g ( y ) (cid:1) d y, = 1 h (cid:90) R ∂f∂y ( y ) y (1 − y ) g ( y )d y + 12 (cid:90) R ∂ f∂y ( y ) y σ ( y ) g ( y )d y, = (cid:90) R g ( y ) (cid:20) h y (1 − y ) ∂f∂y ( y ) + 12 y σ ( y ) ∂ f∂y ( y ) (cid:21) d y, = (cid:104)L Y f, g (cid:105) . The rapidly decaying smooth test functions f and g ensure that the boundary terms in the integrationby parts are equal to zero. (cid:3) In our setting, the process Y in (2) admits at least one stationary distribution, which can be provedeasily following the arguments in [21]. Proposition 3.5.
The SDE (2) admits a stationary distribution Φ . However, since the map σ is notbounded away from zero, Φ might not be the unique solution to the Poisson equation L ∗ Φ = 0 . Remark 3.6.
For f : R → R , finding the explicit solution of the Poisson equation is tedious. Indeed,( L ∗ Y f )( y ) = − ∂∂y (cid:18) h y (1 − y ) f ( y ) (cid:19) + 12 ∂ ∂y (cid:0) y σ ( y ) f ( y ) (cid:1) = 0 , is equivalent to12 y f (cid:48)(cid:48) ( y ) (cid:34)(cid:18) αβ (cid:19) − αγβ y − β + γ y − β (cid:35) + yf (cid:48) ( y ) (cid:34) (cid:40)(cid:18) αβ (cid:19) − αγβ (2 − β ) y − β + (1 − β ) γ y − β (cid:41) − h (1 − y ) (cid:35) + f ( y ) (cid:34)(cid:18) αβ (cid:19) + αγβ (1 − β )(2 α + β ) y − β + γ ( β − (cid:0) α ( β −
1) + β (cid:1) y − β − h (1 − y ) (cid:35) = 0 , with the constraint (cid:82) R f ( y )d y = 1. This is a highly non-linear problem, which does not admit any explicitsolution in general.3.2.2. Pricing PDE.
Consider an option with payoff H ( S T , Y T ) at expiry T , and denote its price P ( t, S t , Y t )at time t ≤ T . Since the market is complete, we can construct a self-financing and riskless replicatingportfolio consisting of the derivative itself and − ( SP S + Y P Y ) /S units of the underlying asset S to findthe corresponding pricing PDE. Proposition 3.7.
Under the risk-neutral measure Q , the pricing PDE associated to (2) is (5) (cid:18) ∂ t + rs∂ S + (cid:18) − yh + r (cid:19) y∂ y + s σ ( y ) ∂ ss + sxσ ( y ) ∂ sy + y σ ( y ) ∂ yy − r (cid:19) P ( t, s, y ) = 0 , for all ( t, s, y ) ∈ [0 , T ) × (0 , ∞ ) , with terminal condition P ( T, s, y ) = H ( s, y ) . ATH-DEPENDENT VOLATILITY MODELS 5
Equation (5) can be rewritten as ( L Y + L + L ) P = 0, with L Y defined in Proposition 3.4 and L := syσ ( y ) ∂ ∂s∂y + ry ∂∂y , L := ∂∂t + s σ ( y ) ∂ ∂s + rs ∂∂s − r. The operator L is the Black-Scholes infinitesimal generator with volatility σ ( y ). Unfortunately, this pric-ing PDE does not admit an obvious explicit solution. Approximate solutions can be found by expandingthe solution using perturbation methods, as developed in [9], but we leave this to future endeavours.3.3. Small-time asymptotics.
We now investigate the small-time behaviour of the solution to (2)using large deviations techniques. Consider the log-stock price process X := log S . For ε > t ∈ T ,introduce the small-time rescaling ( X εt , Y εt ) := ( X εt , Y εt ). The model becomes(6) (cid:40) d X εt = − ε σ ( Y εt )d t + √ εσ ( Y εt )d W t , X ε := x = log( S ) , d Y εt = εb ( Y εt )d t + √ ε (cid:101) σ ( Y εt )d W t , Y ε = y > . Let H denote the space of absolutely continuous functions starting at the origin, with square integrablederivatives, such that H := (cid:26) f : T → R such that f = (cid:90) g ( s )d s on T for some g ∈ L ( T ) , and inf t ∈T f t ≥ α log (cid:18) − y β αβγ (cid:19)(cid:27) . Remark 3.8.
When y ≥ y σ , the condition(7) inf t ∈T f t ≥ α log (cid:18) − y β αβγ (cid:19) , is automatically satisfied and the space H boils down to the usual Cameron-Martin space. When y < y σ ,Condition (7) is needed to ensure that the solution of the controlled ODE introduced below is positive.The main result here is the following theorem, proved in Appendix A.5, which states a pathwiselarge deviations principle for the log-stock price process. With x := ( x , y ), introduce the map I X,Y on C ( T , R × R ∗ + ) by I X,Y (g) := inf (cid:8) Λ( f ) , f ∈ H , S x ( f ) = g (cid:9) , where Λ is the usual rate function of the standard Brownian motion:Λ( f ) := (cid:90) T (cid:13)(cid:13)(cid:13) ˙ f t (cid:13)(cid:13)(cid:13) d t, if f ∈ H , + ∞ , otherwise , and S x ( f ) on T is the solution to the controlled ODE ˙g t = ( σ (g t ) , (cid:101) σ (g t )) (cid:62) · ( ˙ f t , ˙ f t ), starting from g = x . Theorem 3.9.
The rescaled log -stock price process X ε satisfies a pathwise large deviations principle on C ( T , R ) as ε tends to zero with speed ε and rate function (8) I X ( g ) := inf (cid:110) I X,Y (h) , h := ( g, l ) , l ∈ C ( T , R ∗ + ) , l = y (cid:111) , The proof of the theorem relies on first obtaining a large deviations principle for the rescaled process Y ε ,which we state below (and defer its proof to Appendix A.4). Similarly to above, denote S y ( f )( t ) thesolution to the controlled ODE ˙ g t = (cid:101) σ ( g t ) ˙ f t , with g = y . Proposition 3.10.
The rescaled process Y ε satisfies a pathwise large deviations principle on C ( T , R ∗ + ) as ε tends to zero with speed ε and rate function (9) I Y ( g ) := inf (cid:8) Λ( f ) , f ∈ H , S y ( f ) = g (cid:9) . Large deviations have been used extensively in Mathematical Finance in order to derive asymp-totic behaviour of the implied volatility [10]. The latter, Σ t ( k ), is the unique non-negative solutionto C BS ( t, e k , Σ t ( k )) = C obs ( t, e k ), with C obs ( t, e k ) the observed Call option prices with maturity t andstrike e k , and C BS is the Call option price in the Black-Scholes model. ANTOINE JACQUIER AND CHLO´E LACOMBE
Corollary 3.11.
The small-time asymptotic behaviour of the implied volatility is given by lim t ↓ Σ t ( k ) = k (cid:18) inf y ≥ k I X ( g ) | g (1)= y (cid:19) − , if k > ,k (cid:18) inf y ≤ k I X ( g ) | g (1)= y (cid:19) − , if k < . Proof.
We only consider k >
0, the other case being symmetric. It follows from [11, Corollary 7.1] thattaking ε = t , from Theorem 3.9, X satisfies a large deviations principle with speed t and rate function I X :lim t ↓ t log P ( X ≥ k ) = − inf y ≥ k I X ( g ) | g (1)= y . In the Black Scholes model, we have the following small-time implied volatility behaviour:lim t ↓ t Σ t ( k ) log P ( X ≥ k ) = − k , and the result follows from [11, Corollary 7.1]. (cid:3) Numerical estimation of the coefficients ( α, β γ )We now devise an algorithm, using VIX and VVIX data, to find the optimal parameters ( α, β, γ )appearing in the definition of the diffusion function σ in (3). We consider the continuously monitoredformula for the VIX:VIX t := E (cid:34) (cid:90) t +∆ t d (cid:104) X s , X s (cid:105) d s | F t (cid:35) = 1∆ E (cid:34)(cid:90) t +∆ t σ ( Y u )d u | F t (cid:35) , where ∆ is equal to 30 days. In order to simulate the VIX on [0 , T ], we must first approximate theintegral and the conditional expectation, and we denote (cid:100) VIX t its approximation: along a discretisationgrid U t := ( u i ) i =0 ,...,N with u i := t + i ∆ N , assume that σ ( Y u ) is constant on each interval, and writeVIX t = 1∆ N − (cid:88) i =0 (cid:90) u i +1 u i E t (cid:2) σ ( Y u ) (cid:3) d u (cid:39) N − (cid:88) i =0 E t (cid:2) σ ( Y u i ) (cid:3) ( u i +1 − u i ) = 1 N N − (cid:88) i =0 E t (cid:2) σ ( Y u i ) (cid:3) The second step is to approximate the conditional expectation. Since the process Y is Markovian, then E (cid:2) σ ( Y u i ) | F t (cid:3) = E (cid:2) σ ( Y u i ) | Y t (cid:3) , for any u i ∈ U t . We thus approximate the conditional expectationusing the empirical mean, by simulating M paths ( Y (1) , · · · , Y ( M ) ) of Y , so that(10) 1 N N − (cid:88) i =0 E t (cid:2) σ ( Y u i ) (cid:3) (cid:39) N M M (cid:88) j =1 N − (cid:88) i =0 σ ( Y ( j ) u i ) =: (cid:100) VIX t . Note that, in order to simulate (cid:100)
VIX t , one first needs to obtain the starting point Y t , from which all thepoints ( Y u i ) i =1 ,...,N can then be determined. Since Y t := S t S ht from (1), with S ht := 1 h (cid:90) t −∞ e − t − vh S v d v (cid:39) h I − (cid:88) i =0 (cid:90) v i +1 v i e − t − vh S v d v (cid:39) e − t/h h ∆ t I − (cid:88) i =0 e v i /h S v i , with V t := ( v i ) i =0 ,...,I a time-discretisation of ( −∞ , t ], such that v is the first time for which we canobserve the S&P 500, and v I = t ; we further set the grid size to be v i +1 − v i =: ∆ t for all i = 0 , ..., I .In our simulations, we will consider v to be 25 / / t to be as small as possible. However here, dealing with daily datafor the S&P500 imposes ∆ t = 1 day. We leave an extended analysis of this with high-frequency data tofuture work. Hence(11) Y t (cid:39) S t S ht (cid:39) S t e − t/h h ∆ t (cid:80) I − i =0 e v i /h S v i (cid:39) h e t/h S t ∆ t (cid:80) I − i =0 e v i /h S v i . Note that for any t ∈ [0 , T ], the two time grids V t and U t do not overlap, the former taking into accounttime before t , the latter time after t . The algorithm to approximate (cid:0) VIX t (cid:1) t ∈ [0 ,T ] is as follows: ATH-DEPENDENT VOLATILITY MODELS 7
Algorithm: (i) Compute ( Y t ) t ∈ [0 ,T ] using (11), with S obtained from S&P 500 data;(ii) For t ∈ [0 , T ], using Y t from (i), simulate the auxiliary process ( (cid:98) Y s ) s>t on the grid U t as (cid:98) Y u i +1 := ψ ( (cid:98) Y u i , W u i )11 { ψ ( (cid:98) Y ui ,W ui ) >D } , where ψ ( x, w ) := x (cid:18) t h (1 − x ) + σ ( x )∆ w (cid:19) , with ∆ t := and ∆ W u i := W u i +1 − W u i (cid:39) √ N (0 , N (cid:80) N − i =0 σ ( (cid:98) Y u i );(iii) Repeat step (ii) M times to obtain (cid:100) VIX t from (10);(iv) Repeat steps (ii)-(iii) on the grid to obtain (cid:100) VIX on V T ∩ [0 , T ]. Remark 4.1.
In the Euler scheme in step (ii) of the algorithm, we introduced the threshold D to preventthe simulations from ‘exploding’ when Y approaches zero. We could alternatively apply the thresholdon σ ( Y t ), but we leave the precise analysis of the (weak or strong) convergence of our discretisation forfuture work, hints thereabout can be picked from [17].4.1. Simulations.
We use daily data to simulate VIX , between January 2nd, 2013 and May 8th, 2017(corresponding to T = 1200 days). The simulations are run for h ∈ { , , } days with differentparameters ( α, β, γ ), first taken arbitrarily, but optimised over later. Table 1 summarises them: h = 180 h = 30 h = 5 α . . β . . γ . . . N
500 500 500 M
500 500 500 D .
01 1
Table 1.
Parameters used in the simulationsNote first from (2) that the process is mean reverting to 1. The smaller the h , the larger the meanreversion 1 /h , which is clearly visible in Figure 2. (a) h = 180 days (b) h = 30 days (c) h = 5 days Figure 2.
Process Y simulated using the Euler scheme between 2/1/2013 and 8/5/2017.Using the algorithm above, the simulations for the VIX , respectively for h = 180, 30 and 5 days, aredepicted in Figure 3.In all cases, the simulated VIX seems to have jumps similar to those visible in the historical VIX .However, when h = 5 days, the variations of the simulated VIX are very small compared to the historicalone. In order to obtain similar magnitudes, for h = 5 days, we had to consider a threshold D = 1, meaning ANTOINE JACQUIER AND CHLO´E LACOMBE (a) h = 180 days (b) h = 30 days (c) h = 5 days Figure 3.
Simulated (blue) and historical (red) VIX, between 2/1/2013 and 8/5/2017.that many simulated paths were discarded, thus creating a clear bias. We also chose ( α, β, γ ) = (5 , , . σ (1) = 0 .
66 and confirming that the representation (3)works best for h > ∆ = 30. In order to analyse the results, we plot in Figure 4 the simulated VIX andthe historical VIX against the historical VVIX as well as against the recent trend of S&P 500, that is,the process Y simulated using its definition Y t := S t S ht for t ∈ [0 , T ] and h ∈ { , , } . VV I X SimulationMarket (a) h = 180 days VV I X SimulationMarket (b) h = 30 days VV I X SimulationMarket (c) h = 5 days Figure 4.
Simulated (blue crosses) and historical (yellow circles) VIX against historicalVVIX, between 2/1/13 and 8/5/17. The red line is the linear regression fit.In the first two figures (for h = 180 and 30 days), it seems to fit the regression of the simulated dataas well. However, for h = 5 days, the simulated values clearly follow a different trend. This might beexplained by the presence of a large threshold D in the simulations, and by the fact that the approximationfor σ using the VIX data is only valid for h > ∆. Finally, Figure 5 shows the simulated VIX and historicalVIX against the recent trend of S&P 500. The latter are computed using the definition of Y changingthe window h to adapt it to each case. We observe that the model does not fit market data for h = 5,which is consistent with the assumption that the representation of the function σ is only valid for h > ∆. V I X SimulationMarket (a) h = 180 days V I X SimulationMarket (b) h = 30 days V I X SimulationMarket (c) h = 5 days Figure 5.
Simulated (blue crosses) and historical (yellow circles) VIX against the recenttrend of S&P500, between 2/1/13 and 8/5/17.
Source: CBOE VIX Index.
ATH-DEPENDENT VOLATILITY MODELS 9
Optimisation of coefficients.
The calibration of the model is performed by minimising the leastsquared error (LSE) LSE( α, β, γ ) := (cid:88) t ∈V T ∩ [0 ,T ] (cid:12)(cid:12)(cid:12) VIX t − (cid:100) VIX t ( α, β, γ ) (cid:12)(cid:12)(cid:12) over the coefficients ( α, β, γ ). Table 2 summarises the results of the optimisation, indicating the optimisedparameters as well as the initial guesses and the number of runs. The optimal coefficients are close tothe initial parameters ( α , β , γ ). This is due to starting with good guesses, in order to reduce theactual computation time, quite involved in the numerics (which can be seen from the small number ofsimulations ( N, M ) = (100 , h = 180 h = 30 h = 5 α . . . β . . . γ . . . α ∗ .
88 1 .
96 3 . β ∗ .
52 1 .
21 8 . γ ∗ .
83 1 .
76 0 . T
500 200 100 N
100 200 100 M
200 100 200 D .
01 0
Table 2.
Initial and calibrated parameters
Appendix A. Additional Proofs
A.1.
Proof of Proposition 3.2.
We start with the behaviour at the origin. Regarding the first state-ment, for ε > (cid:90) ε − ε | b ( x ) | (cid:101) σ ( x ) d x ≤ (cid:20) x ∈ [ − ε,ε ] | b ( x ) | (cid:21) (cid:90) ε d x (cid:101) σ ( x ) , ≤ (cid:2) x ∈ [ − ε,ε ] | b ( x ) | (cid:3) γ + δ (cid:90) ε d xx − β ) , as there exists δ ∈ ( − γ , ∞ ) and x > x ≤ x , γ − αγβ x β + α β x β ≥ γ + δ >
0. Thisintegral is finite if and only if β > . Now, for x >
0, we havelim x ↓ + b ( x ) h x = lim x ↓ + (1 − x ) = 1 and lim x ↓ + (cid:101) σ ( x ) γx − β = lim x ↓ + (cid:18) − αβγ x β (cid:19) = 1 . We can then apply [3, Theorem 5.3] for γ := 2 β − (cid:54) = − β >
0, and conclude.Regarding the behaviour at infinity, for x >
0, we have, as β > x ↑∞ b ( x ) − h x = 1 and lim x ↑∞ (cid:101) σ ( x ) − αβ x = lim x ↑∞ (cid:26) − βγα x − β (cid:27) = 1 . We can then apply [3, Theorem 5.7] for γ = 0 > − λ := − β hα < ∞ has type A.A.2. Proof of Theorem 3.3(i).
We first show that y σ is an isolated point. For ε > (cid:90) y σ + εy σ | b ( x ) | (cid:101) σ ( x ) d x ≥ (cid:90) y σ + εy σ d x (cid:101) σ ( x ) = (cid:90) y σ + εy σ d xx (cid:16) − αβ + γx − β (cid:17) ≥ K ε (cid:90) y σ + εy σ d x (cid:16) − αβ x β + γ (cid:17) = K ε γ (cid:90) ε d x (cid:18) − αβγ x βσ (cid:16) xy σ (cid:17) β (cid:19) (12) with K ε := ( y σ + ε ) − − β ) for β ∈ (0 , K ε := y β − σ for β > K ε := 1 for β = 1.Introduce the following quantities:(13) A β := ( β − β − ,B β := A β (cid:20) A β − β − (cid:21) ,C β := A β (cid:20) − ( β − β − β − β − β − − A β (cid:21) . A straightforward Taylor expansion around the origin then yields (cid:34) − αβγ x βσ (cid:18) xy σ (cid:19) β (cid:35) − = y σ β x (cid:26)(cid:20) − A β xy σ + (2 B β + A β ) x y σ (cid:21) + (cid:20) C β − A β B β ) x y σ + o ( x ) (cid:21)(cid:27) , which is not integrable around zero. We thus conclude about the right behaviour of y σ by noting thatthe last term in (12) diverges. Using similar arguments, (cid:90) y σ y σ − ε | b ( x ) | (cid:101) σ ( x ) d x ≥ (cid:90) y σ y σ − ε d x (cid:101) σ ( x ) = (cid:90) y σ y σ − ε d xx (cid:16) − αβ + γx − β (cid:17) ≥ K ε (cid:90) y σ y σ − ε d x (cid:16) − αβ x β + γ (cid:17) = ∞ , with K ε := y − − β ) σ for β ∈ (0 , K ε := ( y σ − ε ) β − for β > K ε := 1 for β = 1.Introduce, for a > x ∈ (0 , a ], the following functions:(14) ρ ( x ) := exp (cid:18)(cid:90) ax b ( y ) σ ( y ) d y (cid:19) , s ( x ) := (cid:90) x ρ ( y )d y, if y σ ≥ , − (cid:90) ax ρ ( y )d y, if y σ < , and ϕ ( x ) := 1 + | b ( x ) | σ ( x ) . We rely here on the analysis in [3, Section 2.3]. The key ingredient is the following lemma:
Lemma A.1.
The following hold: lim x ↓ ρ ( x ) = 0 , (cid:90) a ρ ( y )d y < ∞ , (cid:90) a ϕ ( x ) ρ ( x ) d x = ∞ , (cid:90) a ϕ ( x ) s ( x )d x < ∞ , if y σ ≥ , lim x ↓ ρ ( x ) = ∞ , (cid:90) a ρ ( y )d y = ∞ , (cid:90) a ϕ ( x ) ρ ( x ) | s ( x ) | d x < ∞ , if y σ < . Introduce now the process Z := ( Y − y σ ), satisfying the SDEd Z t = b ( Z t )d t + σ ( Z t )d W t , Z := y − y σ > . with b ( x ) := b ( x + y σ ) and σ ( x ) := (cid:101) σ ( x + y σ ) for x >
0. Armed with Lemma A.1, the right behaviourof Z at the origin, which corresponds to the right behaviour for the original process Y at y σ . The case y σ ≥ y σ < Proof of Lemma A.1.
We start with the limiting behaviour of the function ρ and its integral. A straight-forward Taylor expansion around the origin yields(15) (cid:34) − αβγ x βσ (cid:20) yy σ (cid:21) β (cid:35) − = y σ β y (cid:20)(cid:20) − A β yy σ + (2 B β + A β ) y y σ (cid:21) + (cid:20) C β − A β B β ) y y σ + o ( x ) (cid:21)(cid:21) , with A β , B β , C β introduced in (13). Introduce K := x β +1 σ (1 − y σ ) hβ γ ≤ y σ ≥ K aβ := − a − A β y σ log( a ) + B β + A β y σ a .When y σ ≥
1, using (15), we obtain the asymptotic behaviour, as x approaches zero,(16) 2 b ( x ) σ ( x ) = Kx (cid:32) − A β xy σ + (2 B β + A β ) (cid:18) xy σ (cid:19) + 2( C β − A β B β ) (cid:18) xy σ (cid:19) + o ( x ) (cid:33) . ATH-DEPENDENT VOLATILITY MODELS 11
As the expansion is uniform on [ x, a ], one obtains(17) ρ ( x ) = exp (cid:18) (cid:90) ax b ( y ) σ ( y ) d y (cid:19) = exp (cid:18) Kx (cid:18) A β y σ x log x (cid:19)(cid:19) exp (cid:32) KK aβ − B β + A β y σ Kx + o ( x ) (cid:33) = exp (cid:18) Kx + KK aβ (cid:19) x AβKyσ (1 + o ( x )) . Since K ≤ ρ ( x ) tends to zero as x tends to zero from above, and (cid:82) a ρ ( x )d x is finite.In the case y σ <
1, the expansion (17) is still valid, albeit with K ≥
0. Therefore ρ explodes at theorigin and (cid:82) a ρ ( y )d y is infinite.Now, it is straightforward to see that (cid:90) a | b ( x ) | ρ ( x ) σ ( x ) d x ≥ (cid:90) a d xρ ( x ) (cid:101) σ ( x + y σ ) ≥ (cid:90) a d x (cid:101) σ ( x + y σ ) , which is clearly infinite., because ρ is bounded above by 1 on (0 , a ] and from the asymptotic behaviourof the integrand around the origin, analysed in (12).We now prove the last statement of the lemma, and start with the case y σ ≥
1. With this, (cid:82) a ρ ( y )d y is finite and s ( x ) = (cid:82) x ρ ( y )d y for x ∈ (0 , a ]. Using (17), we can write the asymptotic behaviour of s ( · )around the origin by integrating the asymptotic behaviour of ρ ( · ) around zero. Classical asymptoticexpansions for integrals [19, Chapter 3.3] (note that the leading contribution arises at the right boundaryof the integration domain) yields, after the change of variable y (cid:55)→ zx , s ( x ) = (cid:90) x ρ ( y )d y = e KK aβ x AβKyσ +1 (cid:90) exp (cid:18) Kzx (cid:19) z AβKyσ d z = e KK aβ x AβKyσ +1 exp (cid:26) Kx (cid:27) (cid:18) − K x + o ( x ) (cid:19) , as x tends to zero. Combining this with (15) and (17), we obtain s ( x ) ρ ( x ) σ ( x ) = x βσ β γ x (cid:18) − K x + o ( x ) (cid:19) x (1 + o (1)) = − x βσ Kβ γ (1 + o (1)) , which is integrable on (0 , a ] and concludes the proof, using the fact that (cid:90) a | b ( x ) | ρ ( x ) σ ( x ) s ( x )d x ≤ (cid:18) x ∈ (0 ,a ] | b ( x ) | (cid:19) (cid:90) a s ( x ) ρ ( x ) σ ( x ) d x < ∞ . Consider now the case where y σ < (cid:82) a | b ( x ) | ρ ( x ) σ ( x ) | s ( x ) | d x ≤ (cid:2) (0 ,a ] | b | (cid:3) (cid:82) a | s ( x ) | ρ ( x ) σ ( x ) d x . Sinceboth | s ( x ) | and ρ ( x ) σ ( x ) diverge to infinity at the origin, we need to study the behaviour of the integrandaround zero to be able to conclude. We already proved that (cid:82) a ρ ( x )d x is infinite for y σ <
1. Hence,for x ∈ (0 , a ], we obtain s ( x ) := − (cid:82) ax ρ ( y )d y . For δ > x < δ < a and x > (cid:82) ax ρ ( y )d y = (cid:82) δx ρ ( y )d y + (cid:82) aδ ρ ( y )d y . The second integral exists as the integral of a continuous function over a closedinterval in R + . Regarding the first one, classical asymptotic expansions for integrals and (17), yield, afterthe change of variable y (cid:55)→ zx , (cid:90) δx ρ ( y )d y = e KK aβ x AβKyσ (cid:90) δ/x exp (cid:18) Kxz (cid:19) z AβKyσ d z = e KK aβ x AβKyσ e Kx (cid:16) xK + o ( x ) (cid:17) , and the asymptotic behaviour of the integrand around the origin becomes | s ( x ) | ρ ( x ) σ ( x ) = x βσ β γ (cid:18) x K + o ( x ) (cid:19) x (1 + o (1)) = x βσ β γ K (1 + o (1)) , which is integrable at the origin, and the claim is proved. (cid:3) A.3.
Proof of Theorem 3.3(ii).
The strategy to prove the proposition is similar, albeit with differentcomputations, to the previous case. Introduce, for a > x ∈ (0 , a ], ρ ( x ) := exp (cid:18)(cid:90) ax b ( y ) σ ( y ) d y (cid:19) and s ( x ) := (cid:90) x ρ ( y )d y, if y σ < , − (cid:90) ax ρ ( y )d y, if y σ ≥ . and ϕ ( x ) := 1 + | b ( x ) | σ ( x ) . In order to determine the left behaviour of y σ , we first state the following lemmas. They will enable usto decide which theorem to use from [3, Section 2.3]. Lemma A.2. lim x ↓ + ρ ( x ) = ∞ , (cid:90) a ρ ( y )d y = ∞ , (cid:90) a ϕ ( x ) ρ ( x ) | s ( x ) | d x < ∞ , if y σ ≥ , lim x ↓ + ρ ( x ) = 0 , (cid:90) a ρ ( y )d y < ∞ , (cid:90) a ϕ ( x ) ρ ( x ) d x = ∞ , (cid:90) a ϕ ( x ) s ( x )d x < ∞ , if y σ < . Similarly to before, introduce the process Z := y σ − Y satisfying the SDEd Z t = b ( Z t )d t + σ ( Z t )d W t , Z := y σ − y > , as well as the maps b ( x ) := − b ( y σ − x ) and σ ( x ) := − (cid:101) σ ( y σ − x ) for x >
0. With Lemma A.2, we obtainthe left behaviour of Z at the origin, corresponding to the left behaviour of y σ for Y . The case y σ < y σ ≥ Proof of Lemma A.2.
A straightforward Taylor expansion around the origin yields(18) (cid:34) − αx βσ βγ (cid:20) − yy σ (cid:21) β (cid:35) − = y σ β y (cid:26)(cid:20) A β yy σ + (2 B β + A β ) y y σ (cid:21) + (cid:20) A β B β − C β ) y y σ + o ( x ) (cid:21)(cid:27) , with A β , B β , C β defined in (13). We start with the behaviour of the function ρ and its integrated version.Consider first the case y σ ≥
1. We split the range of possibilities into three possible intervals for a :(i) If a < y σ −
1, then y σ − x ≥ y σ − a > b is negative on [ y σ − a, y σ − x ]. Then, for x ∈ (0 , a ], (cid:90) ax b ( y ) σ ( y ) d y = (cid:90) ax − b ( y σ − y ) (cid:101) σ ( y σ − y ) d y = − (cid:90) ax b ( y σ − y )d y ( y σ − y ) − β ) γ (cid:16) − αβγ x βσ (1 − yy σ ) β (cid:17) = 1 h (cid:90) ax ( y σ − y ) β − ( y σ − y − γ (cid:16) − αβγ x βσ (1 − yy σ ) β (cid:17) d y ≥ ( y σ − a ) β − ( y σ − a − γ h (cid:90) ax d y (cid:16) − αβγ x βσ (1 − yy σ ) β (cid:17) , as min y ∈ [ x,a ] (cid:2) ( y σ − y ) β − ( y σ − y − (cid:3) = ( y σ − a ) β − ( y σ − a − >
0. Indeed the map y (cid:55)→ y β − ( y −
1) is increasing on [ y σ − a, y σ − x ] because y σ − a >
1. Noting that (18) is uniform on[ x, a ], we obtain, as x approaches zeroexp y σ − a ) β − ( y σ − a − γ h (cid:90) ax d y (cid:16) − αβγ x βσ (1 − yy σ ) β (cid:17) = exp (cid:26) Kx + KK aβ (cid:27) x − AβKyσ (1 + o ( x )) , with K := ( y σ − a ) β − ( y σ − a − y σ hβ γ > K aβ := − a + A β y σ log( a ) + B β + A β y σ a , and thereforelim x ↓ ρ ( x ) = ∞ and (cid:82) a ρ ( x )d x = ∞ .(ii) If y σ − ≤ a < y σ , then for x ∈ (0 , a ], (cid:90) ax b ( y ) σ ( y ) d y = (cid:90) y σ − x b ( y ) σ ( y ) d y + (cid:90) ay σ − b ( y ) σ ( y ) d y. Similarly to (i), one can prove that lim x ↓ (cid:82) y σ − x b ( y ) σ ( y ) d y = ∞ . Then, on ( y σ − , a ], σ does not goto zero and is continuous, thus bounded; similarly, b is negative and continuous, hence boundedon ( y σ − , a ]. Therefore (cid:82) ay σ − b ( y ) σ ( y ) d y < ∞ , for x ∈ (0 , a ], andlim x ↓ ρ ( x ) = exp (cid:18)(cid:90) ay σ − b ( y ) σ ( y ) d y (cid:19) exp (cid:18)(cid:90) y σ − x b ( y ) σ ( y ) d y (cid:19) = ∞ . ATH-DEPENDENT VOLATILITY MODELS 13 (iii) If y σ ≤ a , then for x ∈ (0 , a ], (cid:90) ax b ( y ) σ ( y ) d y = (cid:90) y σ − x b ( y ) σ ( y ) d y + (cid:90) y σ y σ − b ( y ) σ ( y ) d y + (cid:90) ay σ b ( y ) σ ( y ) d y. Similarly to (ii), it is easy to show that (cid:82) y σ y σ − b ( y ) σ ( y ) d y = − (cid:82) b ( y ) (cid:101) σ ( y ) d y ≤
0. Then, as b is positiveon ( y σ , a ], (cid:82) ay σ b ( y ) σ ( y ) d y ≥
0, and lim x ↓ ρ ( x ) = ∞ .Consider now the case y σ <
1. Using (18), we have the asymptotic behaviour, as x approaches zero(19) 2 b ( x ) σ ( x ) = (cid:101) Kx (cid:32) A β xy σ + (2 B β + A β ) (cid:18) xy σ (cid:19) + 2( a β B β − C β ) (cid:18) xy σ (cid:19) + o ( x ) (cid:33) , with (cid:101) K := x β +1 σ ( y σ − hβ γ < y σ <
1. Since the expansion is again uniform on [ x, a ], we obtain ρ ( x ) = exp (cid:26) (cid:90) ax b ( y ) σ ( y ) d y (cid:27) = exp (cid:40) (cid:101) Kx (cid:18) − A β y σ x log x (cid:19)(cid:41) exp (cid:40) (cid:101) KK aβ − B β + A β y σ (cid:101) Kx + o ( x ) (cid:41) , = exp (cid:40) (cid:101) Kx + (cid:101) KK aβ (cid:41) x − Aβ (cid:102) Kyσ (1 + o ( x )) . (20)Since (cid:101) K <
0, we easily deduce that lim x ↓ ρ ( x ) = 0, and (cid:82) a ρ ( x )d x is finite.The middle statement in the lemma is straightforward. When x ∈ (0 , a ], (cid:82) ax b ( y ) σ ( y ) d y = − (cid:82) y σ − xy σ − a b ( y ) (cid:101) σ ( y ) d y .Since 0 < y σ − a < y σ − x < b is positive on [ y σ − a, y σ − x ] and the above integral is therefore negative.Hence, ρ is bounded by 1 on (0 , a ], and (cid:90) a | b ( x ) | σ ( x ) d x ≥ (cid:90) a d x (cid:101) σ ( y σ − x ) = ∞ , using (18), which concludes the proof.The final integrals in the lemma are rather delicate to analyse. We start with the case y σ < (cid:82) a ρ ( x )d x < ∞ , s ( x ) = (cid:82) x ρ ( y )d y . Using (20), one can obtain the asymptotic behaviour of s ( · )around zero by integrating the asymptotic behaviour of ρ ( · ) around 0. Classical asymptotic expansionsfor integrals (note that the leading contribution arises at the right boundary of the integration domain)yields, after the change of variable y (cid:55)→ xz ,(21) s ( x ) = (cid:90) x ρ ( y )d y = e (cid:101) KK aβ x − Aβ (cid:102) Kyσ (cid:90) exp (cid:32) (cid:101) Kxz (cid:33) z − Aβ (cid:102) Kyσ d z, = e (cid:101) KK aβ x − Aβ (cid:102) Kyσ exp (cid:32) (cid:101) Kx (cid:33) (cid:18) − x (cid:101) K + o ( x ) (cid:19) , as x tends to zero . Therefore, combining (18), (20) and (21), we obtain s ( x ) ρ ( x ) σ ( x ) = x (cid:18) − (cid:101) K x + o ( x ) (cid:19) x βσ β γ x (1 + o (1)) = − x βσ β γ (cid:101) K (1 + o (1)) , which is integrable on (0 , a ] and concludes the proof.We now move on to the case where y σ ≥
1. In that case, (cid:90) a | b ( x ) | ρ ( x ) σ ( x ) | s ( x ) | d x ≤ (cid:20) (0 ,a ] | b | (cid:21) (cid:90) a | s ( x ) | ρ ( x ) σ ( x ) d x. As lim x ↓ | s ( x ) | = ∞ and lim x ↓ ρ ( x ) σ ( x ) = ∞ , one needs to study the behaviour of the integrandaround zero to conclude. As (cid:82) a ρ ( x )d x = ∞ , for x ∈ (0 , a ], s ( x ) := − (cid:82) ax ρ ( y )d y . For δ > x < δ < a and x > (cid:82) ax ρ ( y )d y = (cid:82) δx ρ ( y )d y + (cid:82) aδ ρ ( y )d y . Note that the second integral is convergent asthe integral of a continuous function over a closed interval of R . Classical asymptotic expansions for integrals [19, Chapter 3.3] and (20), yield, after mapping y (cid:55)→ zx , (cid:90) δx ρ ( y )d y = e (cid:101) K (cid:101) K aβ x − Aβ (cid:102) Kyσ (cid:90) δ/x exp (cid:32) (cid:101) Kxz (cid:33) z − Aβ (cid:102) Kyσ d z = e (cid:101) K (cid:101) K aβ x − Aβ (cid:102) Kyσ e (cid:102) Kx (cid:18) x (cid:101) K + o ( x ) (cid:19) , and the asymptotic behaviour of the integrand around the origin is given by | s ( x ) | ρ ( x ) σ ( x ) = x βσ β γ (cid:18) x (cid:101) K + o ( x ) (cid:19) o (1) x = x βσ β γ (cid:101) K (1 + o (1)) , which is integrable at the origin, and concludes the proof. (cid:3) A.4.
Proof of Proposition 3.10.
As the process Y ε is in R ∗ + instead of R , we adapt the proof of [22,Theorem 2.9] to show that it satisfies a large deviations principle with speed ε and rate function I Y .Note first that for y σ >
0, and in both cases y ≥ y σ and y < y σ , the function (cid:101) σ is locally Lipschitzcontinuous on R ∗ + . Furthermore, for f ∈ H , the Picard-Lindel¨of Theorem implies that the controlledODE ˙ g t = (cid:101) σ ( g t ) ˙ f t , with g = x admits the solution S x ( f )( t ) = (cid:18) βγα (cid:19) β (cid:20) e − α (cid:82) t ˙ f u d u (cid:18) x β αβγ − (cid:19) + 1 (cid:21) /β , for t ∈ T , x > . This formulation requires the term (cid:104) e − α (cid:82) t ˙ f u d u (cid:16) x β αβγ − (cid:17) + 1 (cid:105) to be positive for all x > • If y ≥ y σ , then y β αβγ − ≥ S y ( f ) is positive on T ; • If y < y σ , then y β αβγ − < S y ( f ) is positive on T if and only if Condition (7) holds.The crucial step in [22, Theorem 2.9] is [22, Theorem 2.7], which states that if √ εW is close to f ∈ H ,then Y ε should be close to S y ( f ), the solution of the controlled ODE. The case of bounded and globallyLipschitz coefficients follows directly from [22, Theorem 2.7]. In order to deal with locally Lipschitzcoefficients here, we need to localise. For 0 < r σ < r b , the functions b ( x ) := b ( x ) , (cid:107) x (cid:107) ≤ r b ,b (cid:18) rx (cid:107) x (cid:107) (cid:19) , (cid:107) x (cid:107) > r b , and σ ( x ) := (cid:101) σ ( x ) , (cid:107) x (cid:107) > r σ , (cid:101) σ (cid:18) rx (cid:107) x (cid:107) (cid:19) , (cid:107) x (cid:107) ≤ r σ . are bounded and globally Lipschitz continuous on R ∗ + , and clearly εb ( · ) converges uniformly to zero on R ∗ + as ε goes to zero. Moreover, given η ∈ ( r b , r σ ), there exists 0 < r ≤ min { η − r σ , η − r b } such that the δ -tube around S y ( f ) is contained in B r ( η ). In order for this radius r to exist, one simply needs to makesure that the solution S y ( f ) of the controlled ODE never reaches zero (explosion is impossible as infinityis recurrent), which is obvious when y ≥ y σ , and guaranteed by Condition (7) when y < y σ .Denote Y ε the solution to d Y εt = εb ( Y εt )d t + √ εσ ( Y εt )d W t with Y ε = y >
0. Then the two sequences( Y ε ) ε> and ( Y ε ) ε> are identical on B r ( η ). Thus, for each δ, λ >
0, there exist ξ, ζ > f ∈ H with Λ( f ) ≤ λ and x ∈ B ξ ( y ), P (cid:2) (cid:107) Y ε − S y ( f ) (cid:107) ∞ > δ, (cid:107)√ εW − f (cid:107) ∞ ≤ ζ (cid:3) = P (cid:104) (cid:107) Y ε − S y ( f ) (cid:107) ∞ > δ, (cid:107)√ εW − f (cid:107) ∞ ≤ ζ (cid:105) . The constants ξ, ζ >
R, δ, λ >
0, there exist ζ, ξ, ε > f ∈ H with Λ( f ) ≤ λ , x ∈ B ξ ( y ), ε ≤ ε , P (cid:2) (cid:107) Y ε − S y ( f ) (cid:107) ∞ > δ, (cid:107)√ εW − f (cid:107) ∞ ≤ ζ (cid:3) ≤ exp (cid:26) − Rε (cid:27) holds, and the proof follows from [22, Theorem 2.9].A.5. Proof of Theorem 3.9.
To obtain a large deviations principle for X ε , a large deviations principlefor the rescaled process X ε := ( X ε , Y ε ) needs to be proved.dX εt = ε b(X εt )d t + √ ε a(X εt )d W t , ATH-DEPENDENT VOLATILITY MODELS 15 with initial condition X ε := x = (cid:18) log s y (cid:19) and the maps b , a : R ∗ + → R defined asb(X εt ) = (cid:18) − σ ( Y εt ) b ( Y εt ) (cid:19) and a(X εt ) = (cid:18) σ ( Y εt ) (cid:101) σ ( Y εt ) (cid:19) . These two maps are both locally Lipschitz continuous on R × R ∗ + . Solving the controlled ODE for Y ε is sufficient to solve the controlled ODE for the process X ε . Using the proof of Proposition 3.10, forf := ( f, f ) with f ∈ H , the controlled ODE ˙g t = a(g t )˙f t , with g = x has a solution g = S x ( f ) on T .For y > y σ , the solution S y is strictly positive and S x ( f ) exists on T for all f ∈ H and x ∈ R × R ∗ + .In this case, H boils down to the Cameron-Martin space. For y < y σ , Condition (7) ensures that S y is positive. Applying [22, Theorem 2.9], the sequence X ε then satisfies a large deviations principle on C ( T , R × R ∗ + ) as ε tends to zero, with speed ε and rate functionI Y,X (g) := inf (cid:8) Λ( f ) , f ∈ H , S x ( f ) = g (cid:9) . To obtain a large deviations principle for the log-stock price X ε , we apply the Contraction Principle [4,Theorem 4.2.1] since the projection on the first component is continuous, and the theorem follows. References [1] R. Azencott, B. Geiger and W. Ott. Large deviations for Gaussian diffusions with delay.
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