Joint Modelling and Calibration of SPX and VIX by Optimal Transport
aa r X i v : . [ q -f i n . M F ] A p r Joint Modelling and Calibration of SPX and VIX byOptimal Transport
Ivan Guo , Grégoire Loeper , Jan Obłój , and Shiyi Wang School of Mathematics, Monash University, Clayton, VIC, Australia Centre for Quantitative Finance and Investment Strategies,Monash University, Clayton, VIC, Australia University of Oxford, Oxford, United Kingdom
Abstract
This paper addresses the joint calibration problem of SPX options and VIX options orfutures. We show that the problem can be formulated as a semimartingale optimal transportproblem under a finite number of discrete constraints, in the spirit of [12]. We introduce a PDEformulation along with its dual counterpart. The optimal processes can then be representedvia the solutions of Hamilton–Jacobi–Bellman equations arising from the dual formulation. Anumerical example shows that the model can be accurately calibrated to the SPX Europeanoptions and the VIX futures simultaneously.
The CBOE Volatility Index (VIX), also known as the stock market’s "fear gauge", reflects theexpectations of investors on the volatility of the S&P500 index (SPX) over the next 30 days.Although the index in itself is not a tradable asset, its derivatives such as futures and optionsare highly liquid. Since the VIX options started trading in 2006, researchers and practitionershave been putting a lot of effort in jointly calibrating the SPX and VIX options prices. This isknown to be a challenging problem. As noted by many authors (e.g. [16, 21]), there might existan inconsistency between the volatility-of-volatility inferred from SPX and VIX.In the literature, the first attempt at jointly calibrating with continuous models was madeby Gatheral [7], who considered a two-factor stochastic volatility model. Other attempts includea Heston model with stochastic volatility-of-volatility by Fouque and Saporito [6] and a regime-switching stochastic volatility model by Goutte et al. [9]. In addition, many authors have triedincorporating jumps into the SPX dynamics, see, e.g., [2, 4, 17, 19, 20]. However, even with jumps,these models have yet to achieve satisfactory accuracy, particularly for short maturities. This leadsto a natural question of whether there exists a continuous model can capture the SPX and VIXsmiles simultaneously. In [1, 14], Acciaio and Guyon provide a necessary condition for the existenceof such continuous models. Their work was followed by the contribution of Gatheral et al. [8] whorecently found an instance of such continuous models called the quadratic rough Heston model.Note that apart from continuous models, Guyon [15] accurately reproduced the SPX and VIXsmiles by modelling the distributions of SPX in discrete time. Continuous models refer to continuous-time models with continuous SPX paths. X whose firstelement X is the logarithm of the SPX price and second element is defined as X t := E ( X T | F t ) .The filtration F t represents the market information available at time t . Then, the payoff of VIXfutures can be expressed in the form of E ( p X t − X t ) . By applying the localisation result of[12], we show that the solutions can be found among a set of Markov processes that the drift anddiffusion are functions of t and X t . We then introduce a PDE formulation along with its dualcounterpart. Furthermore, we show that the solutions can be represented in terms of solutions ofHamilton–Jacobi–Bellman (HJB) equations arising from the dual formulation.There are two motivations for identifying the conditional expectation as a state variable. Firstly,it makes the joint calibration problem fall into the class of the problems studied in [12]. Secondly,taking X as a state variable, we can use conventional Monte Carlo methods or PDE methods tocalculate the prices of VIX derivatives. If we only have X , due to the nonlinearity of the squareroot function, the evaluation of the conditional expectation cannot be computed by direct MonteCarlo methods. It often requires nested Monte Carlo or Least Square Monte Carlo which arecomputationally expensive (see [11] for a Least Square Monte Carlo approach for VIX derivatives).By identifying suitable state variables, our results are applicable to any calibration problem wherethe calibration instruments have payoffs in the form of a function of a conditional expectation.The paper is organised as follows. Section 2 introduces some basic notations and the formulationof the problem. Section 3 presents the main results including the localisation result, the PDEformulation and the dual formulation. Finally, in section 4, we provide a numerical example withsimulated data. Denote by E a Polish space equipped with its Borel σ -algebra. Let C ( E ) be the set of continuousfunctions on E and C b ( E ) be the set of bounded continuous functions. Denote by P ( E ) − w ∗ the set of Borel probability measures endowed with the weak- ∗ topology. Let BV ( E ) be the setof functions of bounded variation and L ( dµ ) be the set of µ -integrable functions. We also write C b ( E, R d ) , BV ( E, R d ) and L ( dµ, R d ) as the vector-valued versions of their corresponding spaces.Let Ω := C ([0 , T ] , R ) be the two-dimensional canonical space with the canonical process X = ( X , X ) , and let F = ( F t ) ≤ t ≤ T be the canonical filtration generated by X . Denote by P the set of Borel probability measures on (Ω , F T ) , T > . Let P ⊂ P denote the subset of2easures such that, for each P ∈ P , X ∈ Ω is a ( F , P ) -semimartingale given by X t = X + A t + M t , h X i t = h M i t = B t , P -a.s., (1)where M is an ( F , P ) -martingale and ( A, B ) is P -a.s. absolutely continuous with respect to t . Inparticular, P is said to be characterised by ( α P , β P ) , which is defined in the following way, α t = dA t dt , β t = dB t dt . Note that ( α P , β P ) is F -adapted and determined up to d P × dt , almost everywhere. In general, ( α P , β P ) takes values in the space R × S , where S is the set of symmetric matrices and S is theset of positive semidefinite matrices of order two. For any A, B ∈ S , we write A : B := tr( A ⊺ B ) .Denote by P ⊂ P a set of probability measures P whose characteristics ( α P , β P ) are P -integrable.In other words, E P Z T | α P t | + | β P t | dt ! < + ∞ , where | · | is the L -norm.Let Λ := [0 , T ] × R . Denote by F : Λ × R × S → R ∪ { + ∞} a cost function, and denote by F ∗ : Λ × R × S → R ∪ { + ∞} the convex conjugate of F with respect to ( α, β ) : F ∗ ( t, x, a, b ) := sup α ∈ R ,β ∈ S { α · a + β : b − F ( t, x, α, β ) } . When there is no ambiguity, we will simply write F ( α, β ) := F ( t, x, α, β ) and F ∗ ( a, b ) := F ∗ ( t, x, a, b ) . Consider a probability measure P ∈ P , then X = ( X , X ) is a ( F , P ) -semimartingale under P .Let X to be the logarithm of the SPX price such that X t := − Z t σ s ds + Z t σ s dW s , where σ is some F -adapted process and W is an one-dimensional P -Brownian motion. Forsimplicity, we assume the interest rates and dividends are zero. Next, define X to be an ( F , P ) -martingale such that X t := E P ( X T | F t ) = X t − E P Z Tt σ s ds ! . (2)The second term on the right-hand side of (2) is indeed a half of the expected total variance of X over [ t, T ] . In particular, we are interested in probability measures P ∈ P characterised by ( α, β ) ∈ R × S such that α = (cid:20) − σ (cid:21) and β = (cid:20) σ β β β (cid:21) , (3)where ( β t ) = d h X t , X t i / dt and ( β t ) = d h X t i / dt .3 emark . Note that we do not specify the dynamics of the volatility σ t . Since we model theSPX price and the expected total variance over [ t, T ] , the semimartingale X can reproduce thevolatility smiles of a wide range of stochastic volatility models.In order to restrict the probability measures to those characterised by ( α, β ) of the form (3),we can define a cost function that penalises characteristics that are not in the following convex set: Γ := { ( α, β ) ∈ R × S : α = − β , α = 0 } . Define the convex cost function F as follows: F ( α, β ) = X i,j =1 ( β ij − ¯ β ij ) if ( α, β ) ∈ Γ , + ∞ otherwise, (4)where ¯ β is a matrix of some reference values for β . Then, F is finite if and only if ( α, β ) is inthe form of (3).The calibration instruments we consider are SPX European options and VIX futures. Themarket prices of these derivatives can be imposed as constraints on X . For SPX European options,let G be a vector of m discounted option payoff functions. For example, if the i -th option is aput option, then the payoff function G i : R → R is defined as G i ( x ) = max( K − x , . Let τ be a vector of maturities and u c ∈ R m be the option prices. We want to restrict P to probabilitymeasures under which X satisfies E P G i ( X τ i ) = u ci , i = 1 , . . . , m . Next, for VIX futures, let ∆ = 30 days and t = T − ∆ . The VIX level from t to T is defined to be the square root of theexpected realised variance of SPX over [ t , T ] , that is V IX t = 100 s E P (cid:18) Z Tt σ t dt (cid:12)(cid:12)(cid:12)(cid:12) F t (cid:19) = 100 √ √ ∆ q X t − X t . (5)Consider a VIX futures contract with the start date t and the end date T . Let u f ∈ R be theprice of the contract. Define function J : R → R such that J ( x ) := 100 √ √ ∆ √ x − x . Then, we want to further restrict P to those under which X also satisfies E P J ( X t ) = u f .To ensure that X T = E ( X T | F T ) = X T , one additional constraint is imposed on the model.Let ξ : R → R be a function such that ξ ( X T ) = 0 if and only if X T = X T . Here, we choose ξ ( x ) := 1 − exp( − ( x − x ) ) and add constraint E P ξ ( X T ) = u ǫ . For numerical reasons, u ǫ is setto be a very small positive number instead of zero.We assume that X is known and the initial marginal of X is a Dirac measure on X . Notethat if X is not observable from the market, we can treat it as a parameter. Now, putting allthe constraints together, we define a set of probability measures P ( X , G, τ, t , u c , u f , u ǫ ) ⊂ P asfollows: P ( X , G, τ, t , u c , u f , u ǫ ) := { P ∈ P : P ◦ X − = δ X , E P G i ( X τ i ) = u ci , i = 1 , . . . , m, E P J ( X t ) = u f , E P ξ ( X T ) = u ǫ } . If P ( X , G, τ, t , u c , u f , u ǫ ) is empty, it means that the model cannot be exactly calibratedto the market data. Ohterwise, for any P ∈ P ( X , G, τ, t , u c , u f , u ǫ ) , the semimartingale X P . Adopting the convention inf ∅ = + ∞ , we formulate thecalibration problem as the following problem, which is a semimartingale optimal transport problemunder a finite number of discrete constraints studied in [12]. Definition 2.2 (Problem 1) . Given X , G, τ, t , u c , u f and u ǫ , minimise V := inf P ∈P ( X ,G,τ,t ,u c ,u f ,u ǫ ) E P Z T F ( α P s , β P s ) ds. (6)The problem is said to be admissible if P ( X , G, τ, t , u c , u f , u ǫ ) is nonempty and the infimum isfinite. Remark . Let Y be an F T -measurable random variable. By identifying X t as E P ( Y | F t ) ,our results apply to any calibration problem where the payoffs of the calibration instruments canbe expressed in the form of g ( X t , X t ) . This section is devoted to present our main results. By following [12], we first present a localisationresult which shows that the optimal transportation cost can be achieved by a set of Markov pro-cesses. Focusing only on these Markov processes, we introduce a PDE formulation. Furthermore,a dual formulation is derived and the optimal characteristics are provided.
In this section, we show that if Problem 1 is admissible then the optimal transportation cost V can be found by minimising (6) over a subset of probability measures under which X t are a set ofMarkov processes that the drift and diffusion are functions of t and X t . Before we proceed, somenotations are introduced for brevity. Denote by E P t,x the conditional expectation E P ( · | X t = x ) .For any square matrix β ∈ S , we write β such that β = β ( β ) ⊺ . Now, let us restate Lemma3.2 of [12]. Lemma 3.1.
Let P ∈ P and ρ P be the law of the semimartingale X t under P , then ρ P is a weaksolution to the Fokker–Planck equation: ∂ t ρ P + ∇ x · ( ρ P E P t,x α P t ) − X i,j ∂ ij ( ρ P ( E P t,x β P t ) ij ) = 0 in [0 , T ] × R ,ρ P = δ X in R . (7) Moreover, there exists a Markov process X ′ such that, under another probability measure P ′ ∈ P , X ′ has law ρ P ′ = ρ P and solves ( dX ′ t = E P t,X ′ t α P t dt + ( E P t,X ′ t β P t ) dW P ′ t ,X ′ = X , (8) where W P ′ is a P ′ -Brownian motion. Let P loc ( X , G, τ, t , u c , u f , u ǫ ) ⊂ P ( X , G, τ, t , u c , u f , u ǫ ) be a subset of probability mea-sures under which X t are Markov processes in the form of (8). In other words, for any P ∈P loc ( X , G, τ, t , u c , u f , u ǫ ) , P is characterised by ( E P ′ t,x α P ′ t , E P ′ t,x β P ′ t ) for some P ′ ∈ P . Moreover,5 has an initial marginal δ X and satisfies E P J ( X t ) = u f , E P ξ ( X T ) = u ǫ and E P G i ( X τ i ) = u ci for all i = 1 , . . . , m . Applying Proposition 3.4 of [12], we have the following proposition for thejoint calibration problem. Proposition 3.2 (Localisation) . Given X , G, τ, t , u c , u f and u ǫ , if Problem 1 is admissible,then V = inf P ∈P ( X ,G,τ,t ,u c ,u f ,u ǫ ) E P Z T F ( α P t , β P t ) dt = inf P ∈P loc ( X ,G,τ,t ,u c ,u f ,u ǫ ) E P Z T F ( α P t , β P t ) dt. For any P ∈ P loc ( X , G, τ, t , u c , u f , u ǫ ) , its characteristics are Markovian to the state variable X t at time t . This Markovian property is very convenient in practice. In order to further utiliseProposition 3.2, we introduce a PDE formulation so that conventional numerical methods can beused. Proposition 3.3. If V is finite, then V = inf ρ,α,β Z T Z R F ( α ( t, x ) , β ( t, x )) dρ ( t, x ) dt, (9) among all ( ρ, α, β ) ∈ C ([0 , T ] , P ( R ) − w ∗ ) × L ( dρ t dt, R ) × L ( dρ t dt, S ) satisfying the followingconstraints in the sense of distributions: ∂ t ρ ( t, x ) + ∇ x · ( ρ ( t, x ) α ( t, x )) − X i,j ∂ ij ( ρ ( t, x ) β ij ( t, x )) = 0 in [0 , T ] × R , (10) Z R G i ( x ) dρ ( τ i , x ) = u ci ∀ i = 1 , . . . , m, (11) Z R J ( x ) dρ ( t , x ) = u f , (12) Z R ξ ( x ) dρ ( T, x ) = u ǫ , (13) and the initial condition ρ (0 , · ) = δ X .Proof. This proposition is a consequence of Lemma 3.1. The interchange of integrals in the objec-tive is justified by Fubini’s theorem. For the weak continuity of measure ρ in time, the reader canrefer to [18].The PDE formulation can be solved by the alternating direction method of multipliers (ADMM)which was originally used in [3] to solve the classical optimal transport. This method was alsoextended in [13] to solve an one-dimensional martingale optimal transport problem. However,the ADMM method requires to solve a PDE with bi-Laplacian operator, which is not a trivialproblem. The standard finite difference method struggles to achieve high accuracy for such fourth-order PDEs in more than one dimensions. Alternatively, we work on a dual formulation that isderived by following the duality argument in [12]. This will be presented in the next subsection.6 .3 Dual formulation Let λ G ∈ R m , λ J ∈ R and λ ǫ ∈ R be the Lagrange multipliers of constraints (11), (12) and (13),respectively. As a direct consequence of Theorem 3.6 and Corollary 3.11 of [12], we introduce thefollowing dual formulation: Theorem 3.4 (Duality) . If V is finite, then the infimum in (9) is attained and it is equal to V = V := sup ( λ G ,λ J ,λ ξ ) ∈ R m +2 λ G · u c + λ J u f + λ ξ u ǫ − φ (0 , X ) , (14) where φ is the viscosity solution to the HJB equation: ∂ t φ ( t, x ) + F ∗ ( ∇ x φ ( t, x ) , ∇ x φ ( t, x )) = − m X i =1 λ Gi G i ( x ) δ ( t − τ i ) − λ J J ( x ) δ ( t − t ) − λ ξ ξ ( x ) δ ( t − T ) in [0 , T ] × R , (15) with the terminal condition φ ( T, · ) = 0 . Moreover, if the supremum is attained by some ( λ G , λ J , λ ξ ) and the solution to (15) is φ ∗ ∈ BV ([0 , T ] , C b ( R )) , then the optimal ( α, β ) is ( α ∗ , β ∗ ) = ∇ F ∗ ( ∇ x φ ∗ , ∇ x φ ∗ ) . (16)Theorem 3.4 is an application of the Fenchel–Rockafellar duality theorem [23, Theorem 1.9].Due to the presence of these Dirac delta functions in the source term, the viscosity solution φ haspossible jump discontinuities at t , T and τ i , i = 1 , . . . , m . The reader can refer to [12, Section3.3] for the definition of viscosity solutions to (15) and the corresponding comparison principle.For the cost function F defined in (4), its F ∗ is given in Lemma A.1.In the dual formulation, the supremum can be solved by a standard optimisation algorithm.As pointed out in Lemma 4.5 of [12], the convergence can be improved by providing the gradientsof the objective. Lemma 3.5.
Define function L : R m +2 → R ∪{ + ∞} such that V = sup ( λ G ,λ J ,λ ξ ) ∈ R m +2 L ( λ G , λ J , λ ξ ) .The gradients of the objective can be formulated as the difference between the market prices andthe model prices: ∂ λ Gi L = u ci − E P ,X G i ( X τ i ) , i = 1 , . . . , m, (17) ∂ λ J L = u f − E P ,X J ( X t ) , (18) ∂ λ ξ L = u ǫ − E P ,X ξ i ( X T ) . (19)In the optimisation process, the gradients are decreasing to zero while the solution is approach-ing the optimal solution. The gradients provide an intuitive interpretation in terms of matchingthe model prices with market prices.Once the optimal ( α ∗ , β ∗ ) has been found, as a result of the Feynman-Kac formula, the modelprices can be found by solving linear pricing PDEs. For example, the price of the VIX futures isequal to E P ,X ( J ( X t )) = φ ′ (0 , X ) where φ ′ is the solution to ∂ t φ ′ + α ∗ · ∇ x φ ′ + 12 β ∗ : ∇ x φ ′ = 0 in [0 , t ] × R , (20)with the terminal condition φ ′ ( t , · ) = J . Similar pricing PDEs can be derived to calculate theexpectations in (17) and (19). 7 Numerical examples
In this section, we present a numerical example to demonstrate our method. The numerical solutionproposed in [12] can be directly applied here. One numerical issue needs to be addressed is how tohandle the term √ X − X for the subset of the computational domain where X − X < . Thisissue can be overcome by performing a change of variables of the form ( X , X ) ( X , X ′ := X − X ) and setting the computational domain of X ′ to be positive. For completeness, wedescribe the algorithm in Appendix B.The model is calibrated to some prices generated by the following Heston model: dS t = p V t S t dW St ,dV t = − . V t − . dt + 0 . p V t dW Vt , h dW St , dW Vt i = − . dt,S = 100 ,V = 0 . , where W St and W Vt are standard Brownian motions. Recall that the interest rates and dividendsare set to zero. Let ∆ t := 1 / denotes 1 trading day. The SPX European options include fivecall options with maturity τ = 1 − t . The start date of the VIX futures is set to t = 1 . Inthis case, T = 1 + 30∆ t . The initial position of the semimartingale X and the reference values of β are shown in Table 1. The results are found in Table 2.As shown in Table 2, the model is perfectly calibrated to the SPX options. Comparing with theerrors for SPX options, the model has a relatively larger error for the VIX futures. When solvingthe pricing PDE (20), we fixed ∇ x φ ( t, · ) to ∇ x J at the boundaries of the computational domainfor all t ∈ [0 , t ] . The results can be improved by imposing a better boundary condition.Parameter Value Interpretation X X X ¯ β β ¯ β β ¯ β β Table 1: The initial position of X and the reference values of β . Acknowledgements
The Centre for Quantitative Finance and Investment Strategies has been supported by BNPParibas. The last author is supported by an Australian Government Research Training Program(RTP) Scholarship. 8trike Target price Model price ErrorSPX call options 80 23.8341 23.8341 2.11e-0690 16.7700 16.7700 -2.30e-06100 11.0333 11.0333 1.11e-06110 6.7706 6.7706 -1.24e-06120 3.9165 3.9165 9.93e-07VIX futures 26.6731 26.6815 -0.0083 E P ξ ( X T ) X . A The convex conjugate F ∗ Lemma A.1.
Define functions A : R × S → R A ( a, b ) := ¯ β + 12 b − a ,B : R × S → R B ( a, b ) := ¯ β + 12 b ,C : R × S → R C ( a, b ) := ¯ β + 12 b ,M : R × S → S M ( a, b ) := (cid:20) A ( a, b ) B ( a, b ) B ( a, b ) C ( a, b ) (cid:21) . The convex conjugate of F is: F ∗ ( a, b ) = ( b − a ) β ∗ + b β ∗ + b β ∗ − X i,j =1 ( β ∗ ij − ¯ β ∗ ij ) . The values of ( β ∗ , β ∗ , β ∗ ) are determined as follows:1. If M ( a, b ) ∈ S , then ( β ∗ , β ∗ , β ∗ ) = ( A, B, C ) .2. Otherwise,(a) if A + C = 0 , then ( β ∗ , β ∗ , β ∗ ) = A + √ A + B , B , √ A + B ! . (b) if A + C = 0 , then ( β ∗ , β ∗ , β ∗ ) = (cid:18) A + C − λ d + A − C λ d , B λ d , A + C − λ d − A − C λ d (cid:19) , where λ d is either λ d + or λ d − : λ d ± := A + 2 B + C ± p (4 B + ( A − C ) )( A + C ) B − AC .
We choose the λ d such that β ∗ ∈ S . Algorithm 1
Let a positive integer N be the grid size of the time interval. We discretise the time intervaland label the grid points in [0 , T ] as t k , k = 0 , . . . , N . We assume that all the SPX options arematuring at τ ∈ [0 , T ] . We also choose a small number ǫ to be the maximum acceptable errorbetween the market prices and the model prices. Algorithm 1:
Numerical method proposed in [12] Set initial ( λ G , λ J , λ ξ ) ; do for k = N − , . . . , do if t k +1 = T then φ t k +1 ← φ t k +1 + λ ξ ξ ; end if t k +1 = t then φ t k +1 ← φ t k +1 + λ J J ; end if t k +1 = τ then φ t k +1 ← φ t k +1 + λ G · G ; end ( α t k , β t k ) ← F ∗ ( ∇ x φ t k +1 , ∇ x φ t k +1 ) ; Solve the HJB equation (15) as a linear PDE at t = t k ; end Calculate the model prices by solving the pricing PDE (20); Calculate the gradients (17), (18) and (19) ; Update ( λ G , λ J , λ ξ ) by an optimisation algorithm; while k ( ∂ λ G L, ∂ λ J L, ∂ λ ξ L ) k ∞ > ǫ ; References [1]
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