Explicit approximations for option prices via Malliavin calculus for the Stochastic Verhulst volatility model
aa r X i v : . [ q -f i n . M F ] J un EXPLICIT APPROXIMATIONS OF OPTION PRICES VIA MALLIAVINCALCULUS FOR THE STOCHASTIC VERHULST VOLATILITY MODEL
KAUSTAV DAS † AND NICOLAS LANGREN´E ‡ Abstract.
We consider explicit approximations for European put option prices within the sto-chastic Verhulst model with time-dependent parameters, where the volatility process follows thedynamics d V t = κ t ( θ t − V t ) V t d t + λ t V t d B t . Our methodology involves writing the put optionprice as an expectation of a Black-Scholes formula, reparameterising the volatility process andthen performing a number of expansions. The difficulties faced are computing a number of ex-pectations induced by the expansion procedure explicitly. We do this by appealing to techniquesfrom Malliavin calculus. Moreover, we deduce that our methodology extends to models withmore generic drift and diffusions for the volatility process. We obtain the explicit representa-tion of the form of the error generated by the expansion procedure, and we provide sufficientingredients in order to obtain a meaningful bound. Under the assumption of piecewise-constantparameters, our approximation formulas become closed-form, and moreover we are able to es-tablish a fast calibration scheme. Furthermore, we perform a numerical sensitivity analysis toinvestigate the quality of our approximation formula in the stochastic Verhulst model, and showthat the errors are well within the acceptable range for application purposes.Keywords: Stochastic volatility model, Closed-form expansion, Closed-form approximation,Malliavin calculus, Stochastic Verhulst, Stochastic Logistic, XGBM Introduction
In this article, we consider the European put option pricing problem in the stochastic Ver-hulst model with time-dependent parameters, namely, where the volatility process satisfies theSDE d V t = κ t ( θ t − V t ) V t d t + λ t V t d B t . The main contributions of this article are an explicitsecond-order approximation for the price of a European put option, an explicit form for theerror induced in the approximation, as well as a fast calibration scheme. Furthermore, the ap-proximation formulas are written in terms of certain iterated integral operators, which, underthe assumption of piecewise-constant parameters, are closed-form, yielding a closed-form ap-proximation to the European put option price. Additionally, we notice that our methodologyeasily extends to models where the volatility’s drift and diffusion coefficients have a general formthat satisfy some regularity properties. We provide the approximation formula in this generalcase too, as well as a closed-form expression when parameters are piecewise-constant. We obtainsufficient conditions needed in order to obtain a meaningful bound on the error term induced bythe approximation procedure, these ingredients essentially become moments involved with the † School of Mathematics, Monash University, Victoria, 3800 Australia. ‡ CSIRO Data61, RiskLab, Victoria, 3008 Australia.
E-mail addresses : [email protected], [email protected] .Research supported by an Australian Government Research Training Program (RTP) Scholarship. volatility process. Lastly, we perform a numerical sensitivity analysis in the stochastic Verhulstmodel in order to assess our approximation procedure in practice. Our approximation method-ology involves appealing to the mixing solution methodology (see Hull and White [15], Romanoand Touzi [26], Willard [33]), which reduces the dimensionality of the pricing problem, small vol-of-vol expansion techniques, as well as Malliavin calculus machinery, in the lines of Benhamouet al. [5], Langren´e et al. [18].It is well known that implied volatility is heavily dependent on the strike and maturity of Eu-ropean option contracts. This phenomenon is called the volatility smile, a feature the seminalBlack-Scholes model fails to address, due to the assumption of a constant volatility [6]. In re-sponse, a number of frameworks have been proposed with the intention of accurately modellingthe smile effect. In particular, stochastic volatility models are one of a number of classes ofmodels which were developed in order to achieve this. In a stochastic volatility model, thevolatility (or variance) process is modelled as a stochastic process itself, possibly correlated withthe spot. Empirical evidence has demonstrated that stochastic volatility models are significantlymore realistic than models with deterministic volatility. However, with this added complexitycomes a cost, as it is usually not possible to compute the prices of even the simplest contractsin a closed-form manner.Affine stochastic volatility models are a subclass of stochastic volatility models which possess acertain amount of tractability. Specifically, affine models are those in which the characteristicfunction of the log-spot can be computed explicitly , see for example the Heston and Sch¨obel-Zhu models [14, 27]. As a consequence, in such models, it can be shown that it is possibleto express European option prices in a quasi closed-form fashion. However, non-affine modelshave been shown to be substantially more realistic than their affine counterparts. This empir-ical evidence has been presented in a number of studies, see for example Christoffersen et al.[8], Gander and Stephens [11], Kaeck and Alexander [16]. For this reason, recently there hasbeen a significant push in the industry towards favouring non-affine models. When one studies anon-affine model, this usually means that prices cannot be represented in any sort of convenientway. Consequently, numerical procedures such as PDE and Monte Carlo methods have beensubstantially developed in the literature, see [2, 28].Closed-form approximations are an alternative methodology for option pricing, where the op-tion price is approximated by a closed-form expression. The purpose of obtaning a closed-formapproximation is to achieve a ‘best of both worlds’ scenario; one can utilise a realistic and sophis-ticated model, yet still have a means to obtain prices of option prices rapidly. Moreover, sincetransform methods are usually not utilised, time-dependent parameters can often be handledwell. One motivation for quick option pricing formulas is calibration, where the option pricemust be computed several times within an optimisation procedure.Over the past 20 years, closed-form approximation results have been extensively studied in the More precisely, an affine model is one where the log of the characteristic function of the log-spot is an affinefunction. literature. For example, Lorig et al. [21] derive a general closed-form expression for the price ofan option via a PDE approach, as well as its corresponding implied volatility. Hagan et al. [13]use singular perturbation techniques to obtain an explicit approximation for the option priceand implied volatility in their SABR model. Al`os [1] show that from the mixing solution, onecan approximate the put option price by decomposing it into a sum of two terms, one beingcompletely correlation independent and the other dependent on correlation. However, neitherterms are explicit. Furthermore, Antonelli et al. [3], Antonelli and Scarlatti [4] show that underthe assumption of small correlation, an expansion can be performed with respect to the mixingsolution, where the resulting expectations can be computed using Malliavin calculus techniques.Similarly, in the case of the time-dependent Heston model, Benhamou et al. [5] consider themixing solution and expand around vol-of-vol, performing a combination of Taylor expansionsand computing the resulting terms via Malliavin calculus techniques. Langren´e et al. [18] adaptsthe methodology of Benhamou et al. [5] to the Inverse-Gamma model.Stochastic volatility models usually either model the volatility directly, or indirectly via thevariance process. A critical assumption is that volatility or variance has some sort of meanreversion behaviour, and this is supported by empirical evidence, see for example Gatheral [12].Specifically, for modelling the variance, a large class of one-factor stochastic volatility models isgiven by d S t = S t (( r dt − r ft )d t + p V t d W t ) , S , d V t = κ t ( θ t V ˆ µt − V ˜ µt )d t + λ t V µt d B t , V = v , d h W, B i t = ρ t d t, whereas for modelling the volatility, this class is of the formd S t = S t (( r dt − r ft )d t + V t d W t ) , S , d V t = κ t ( θ t V ˆ µt − V ˜ µt )d t + λ t V µt d B t , V = v , d h W, B i t = ρ t d t, for some ˜ µ, ˆ µ and µ ∈ R . . Some popular models in the literature include:Model Variance/Volatility Dynamics of V ˆ µ ˜ µ µ Heston [14] Variance d V t = κ t ( θ t − V t )d t + λ t √ V t d B t V t = κ t ( θ t − V t )d t + λ t d B t V t = κ t ( θ t − V t )d t + λ t V t d B t V t = κ t ( θ t − V t )d t + λ t V t d B t V t = κ t ( θ t V t − V t )d t + λ t V / t d B t V t = κ t ( θ t V t − V t )d t + λ t V t d B t There exist other classes of stochastic volatility models. For example, the exponential Ornstein-Uhlenbeck modelWiggins [32] is not included in either of these classes.
In this article, we study the stochastic Verhulst model with time-dependent parameters, here onin referred to as the Verhulst model. Specifically, the dynamics of the spot S with volatility V are given by d S t = ( r dt − r ft ) S t d t + V t S t d W t , S , d V t = κ t ( θ t − V t ) V t d t + λ t V t d B t , V = v , d h W, B i t = ρ t d t, (1.1)where W and B are Brownian motions with deterministic, time-dependent instantaneous cor-relation ( ρ t ) ≤ t ≤ T , defined on the filtered probability space (Ω , F , ( F t ) ≤ t ≤ T , Q ). Here T is afinite time horizon where ( r dt ) ≤ t ≤ T and ( r ft ) ≤ t ≤ T are the deterministic, time-dependent do-mestic and foreign interest rates respectively. In addition, ρ t ∈ [ − ,
1] for any t ∈ [0 , T ]. Thetime-dependent parameters ( κ t ) ≤ t ≤ T , ( θ t ) ≤ t ≤ T and ( λ t ) ≤ t ≤ T are all assumed to be positive forall t ∈ [0 , T ] and bounded. Furthermore, ( F t ) ≤ t ≤ T is the filtration generated by ( W, B ) whichsatisfies the usual assumptions. In the following, E ( · ) denotes the expectation under Q , where Q is a domestic risk-neutral measure which we assume to be chosen. Remark 1.1 (Stochastic Verhulst model heuristics) . The process V occuring in the SDE forthe volatility we call the stochastic Verhulst process, here on in referred to as the Verhulstprocess. This process is reminiscent of the deterministic Verhulst/Logistic model which mostfamously arises in population growth models. The deterministic Verhulst/Logistic model wasfirst introduced by Verhulst in 1838 [29, 30, 31], then rediscovered and revived by Pearl and Reedin 1920 [24, 25]. The process behaves intuitively in the following way. Focusing on the driftterm of the volatility V in eq. (1.1), specifically κ ( θ − V ) V , we notice that there is a quadraticterm. The interpretation here is that V mean reverts to level θ at a speed of κV . That is, themean reversion speed of V depends on V itself, and is thus stochastic. Contrasting this with theregular linear type mean reversion drift coefficients, namely κ ( θ − V ), we have that the meanreversion level is still θ , however the mean reversion speed is κ , and is not directly influencedby V . For an in depth discussion of the Verhulst model for option pricing, we refer the readerto Lewis [20].In what follows, we will successfully obtain an explicit second-order expression for the price ofa European put option in the Verhulst model eq. (1.1). Here, the price of a put option withlog-strike k is given by Put Verhulst = e − R T r dt d t E ( e k − S T ) + . The methodology utilised in this article has been previously implemented for the subsequentmodels: Commonly the stochastic Verhulst model also goes by the stochastic Logistic model, see Carr and Willems [7].It is also referred to as the XGBM model, short for ‘extended Geometric Brownian motion’, see Lewis [20]. Our formulation is for FX market purposes, but can be adapted to equity and fixed income markets easily. Meaning that ( F t ) ≤ t ≤ T is right continuous and augmented by Q null-sets. • For the Heston model d S t = S t (( r dt − r ft )d t + p V t d W t ) , d V t = κ t ( θ t − V t )d t + λ t p V t d B t , d h W, B i t = ρ t d t, this has been studied by Benhamou et al. [5]. • For the Inverse-Gamma (IGa) modeld S t = S t (( r dt − r ft )d t + V t d W t ) , d V t = κ t ( θ t − V t )d t + λ t V t d B t , d h W, B i t = ρ t d t, this has been tackled by Langren´e et al. [18].The purpose of this paper is to extend the methodology utilised in these aforementioned papersto that of pricing under the Verhulst model, as well as developing an associated fast calibrationscheme. In fact, we will deduce that our expansion methodology can be easily adapted to volatil-ity processes with arbitrary drifts and diffusions that satisfy some regularity conditions (givenin Assumption A and Assumption B). Additionally, we will present the explicit approximationformula for the price of a put option in this general framework. The sections are organised asfollows: • Section 2 details some preliminary calculations. First, we reparameterise the volatility processin terms of a small perturbation parameter, obtaining the process (cid:16) V ( ε ) t (cid:17) . Then, we rewritethe expression for the price of a put option by utilising the mixing solution. • In Section 3 we implement our expansion procedure. Namely, we combine a Taylor expansionof a Black-Scholes formula with a small vol-of-vol expansion of the function ε V ( ε ) t and itsvariants. This gives a second-order approximation to the price of a put option. • Section 4 is dedicated to the explicit calculation of terms induced by our expansion procedurefrom Section 3. In particular, we utilise Malliavin calculus techniques in order to reduce thecorresponding terms down into expressions which are in terms of certain iterated integraloperators. • In Section 5 we present the explicit form for the error in our expansion methodology. Inparticular, we provide sufficient ingredients in order to obtain a meaningful bound on theerror term. • Section 6 is dedicated to the study of the iterated integral operators in terms of which ourapproximation formulas are expressed, albeit under the assumption of piecewise-constant pa-rameters. We deduce that under this assumption, our approximation formulas are closed-form.Moreover, this allows us to establish a fast calibration scheme. • Section 7 is dedicated to a numerical sensitivity analysis of our put option approximationformula in the Verhulst model.
Remark 1.2.
In this article, the phrase ‘explicit expression’ will be used to refer to expressionsthat can be represented mathematically with an equation that does not involve the subject. Theword ‘closed-form’ will be used to describe an explicit expression which only involves elementaryfunctions, and does not involve complicated infinite sums or complicated integrals. For example,let g be a function with no closed-form anti-derivative and let f ( x ) = Z x g ( u )d u, (1.2) f ( x ) = e sin( x ) , (1.3) f ( x ) = Z x g ( f ( u ))d u. (1.4)Then eq. (1.2) is explicit, eq. (1.3) is closed-form and eq. (1.4) is neither.2. Preliminaries
Although we are concerned with the pricing of a put option under the Verhulst model eq. (1.1),we start by considering the following general model,d S t = ( r dt − r ft ) S t d t + V t S t d W t , S , d V t = α ( t, V t )d t + β ( t, V t )d B t , V = v , d h W, B i t = ρ t d t. (2.1)The details and assumptions for the general model eq. (2.1) are otherwise exactly the same asthe those for the Verhulst model eq. (1.1). Notice that in the general model eq. (2.1), whenwe set α ( t, x ) = κ t ( θ t − x ) x and β ( t, x ) = λ t x , we recover the Verhulst model eq. (1.1). Thereasoning for this change in framework is twofold. Firstly, this will make the notation cleanerand more aesthetically pleasing, which will be important during the expansion procedure, asthe calculations become quite involved. Furthermore, we will deduce that our methodology ofobtaining an expression for the put option price will extend to more general drifts and diffusions.Thus, we will only need to restrict the drift and diffusion if the expansion procedure demandsit. In anticipation of this, for the rest of this article, we will enforce the following assumptionson the regularity of the drift and diffusion coefficients of V in eq. (2.1). Assumption A.
For t ∈ [0 , T ]:(A1) α is Lipschitz continuous in x , uniformly in t .(A2) β is H¨older continuous of order ≥ / x , uniformly in t .(A3) There exists a solution of V (weak or strong).(A4) α and β satisfy the growth bounds xα ( t, x ) ≤ K (1 + | x | ) and | β ( t, x ) | ≤ K (1 + | x | )uniformly in t , where K is a positive constant. Assumption B.
The following properties hold:(B1) The second derivative α xx exists and is continuous a.e. in x and t ∈ [0 , T ].(B2) The first derivative β x exists and is continuous a.e. in x and t ∈ [0 , T ].The purpose of Assumption A is to guarantee the existence of a pathwise unique strong solutionto V in eq. (1.1), see the Yamada-Watanabe theorem [34], as well as to guarantee that solutionsdo not explode in finite time. We will comment on the purpose of Assumption B in full detailin Remark 3.2. Clearly item (B2) implies item (A1); nonetheless we include item (A1) for thepurpose of clarity. Notice that the Verhulst process from eq. (1.1) does not satisfy item (A1),as its drift coefficient is only locally Lipschitz continuous. However, this is not a problem as thediffusion coefficient is Lipschitz continuous. Hence, we can appeal to the usual Itˆo style resultson existence and uniqueness for SDEs. Proposition 2.1 (Explicit pathwise unique strong solution to the Verhulst process) . Suppose Y solves the SDE d Y t = a t ( b t − Y t ) Y t d t + c t Y t d B t , Y = y > , (2.2)where ( a t ) ≤ t ≤ T , ( b t ) ≤ t ≤ T and ( c t ) ≤ t ≤ T are strictly positive and bounded on [0 , T ]. Then theexplicit pathwise unique strong solution is given by Y t = F t (cid:18) y − + Z t a u F u d u (cid:19) − ,F t = exp (cid:18)Z t (cid:18) a u b u − c u (cid:19) d u + Z t c u d B u (cid:19) . (2.3) Proof.
Both the drift and diffusion coefficients in the SDE eq. (2.2) are locally Lipschitz, uni-formly in t ∈ [0 , T ]. Clearly the diffusion coefficient obeys the linear growth condition, ( c t x ) ≤ K (1 + | x | ) uniformly in t , for some constant K >
0. In addition, we have that x [ a t ( b t − x ) x ] ≤ K (1 + | x | ) uniformly in t ∈ [0 , T ], and thus any potential of explosion in finite time is miti-gated (this somewhat non-standard restriction on the growth of the drift is given in Kloedenand Platen [17] Section 4.5, page 135). It remains to be seen that the solution is indeed givenby eq. (2.3). Utilising Itˆo’s formula with f ( x ) = x − yields a linear SDE, which results inthe explicit solution, namely eq. (2.3). Clearly this solution remains strictly positive in finitetime. (cid:3) Mixing solution.
Denote the price of a put option on S in the general model eq. (2.1) byPut G . Namely, Put G = e − R T r dt d t E ( e k − S T ) + . Let the process X be the log-spot. Specifically, X t := ln S t . Now perturb X in the followingway: for ε ∈ [0 , X ( ε ) t = (cid:16) r dt − r ft −
12 ( V ( ε ) t ) (cid:17) d t + V ( ε ) t d W t , X ( ε )0 = ln S =: x , d V ( ε ) t = α ( t, V ( ε ) t )d t + εβ ( t, V ( ε ) t )d B t , V ( ε )0 = v , d h W, B i t = ρ t d t. (2.4) We can recover the original diffusion from eq. (2.4) by noticing (
S, V ) = (exp( X (1) ) , V (1) ).Denote the filtration generated by B as ( F Bt ) ≤ t ≤ T and let ˜ X ( ε ) t := X ( ε ) t − R t ( r du − r fu )d u . Bywriting W t = R t ρ u d B u + R t p − ρ u d Z u , where Z is a Brownian motion independent of B , itcan be seen that ˜ X ( ε ) T |F BT d = N (ˆ µ ε ( T ) , ˆ σ ε ( T )) , with ˆ µ ε ( T ) := x − Z T
12 ( V ( ε ) t ) d t + Z T ρ t V ( ε ) t d B t , ˆ σ ε ( T ) := Z T (1 − ρ t )( V ( ε ) t ) d t. Let g ( ε ) := e − R T r du d u E ( e k − e X ( ε ) T ) + . Then g (1) is the price of a put option in the general model eq. (2.1). That is, g (1) = Put G . Proposition 2.2.
The function g can be expressed as g ( ε ) = E n e − R T r du d u E [( e k − e X ( ε ) T ) + |F BT ] o = E h P BS (cid:0) ˆ µ ε ( T ) + 12 ˆ σ ε ( T ) , ˆ σ ε ( T ) (cid:1)i , where explicitly ˆ µ ε ( T ) + 12 ˆ σ ε ( T ) = x − Z T ρ t ( V ( ε ) t ) d t + Z T ρ t V ( ε ) t d B t , ˆ σ ε ( T ) = Z T (1 − ρ t )( V ( ε ) t ) d t, and P BS ( x, y ) := e k e − R T r dt d t N ( − d ln − ) − e x e − R T r ft d t N ( − d ln+ ) ,d ln ± := d ln ± ( x, y ) := x − k + R T ( r dt − r ft )d t √ y ± √ y, (2.5)where N ( · ) denotes the standard normal distribution function. Proof.
This result is a consequence of the mixing solution methodology. A derivation can befound in Appendix A. (cid:3) Expansion procedure
In this section, we detail our expansion procedure. The notation is similar to that in Benhamouet al. [5], however there are some differences. The expansion procedure can be briefly summarisedby two main steps.(1) First, we expand the function P BS up to second-order. This step is given in Section 3.2. (2) Then, we expand the functions ε V ( ε ) t and ε (cid:16) V ( ε ) t (cid:17) up to second-order. This step isgiven in Section 3.1 and Section 3.3.We then combine both these expansions in order to obtain a second-order approximation for theput option price, which is given in Theorem 3.1. However, this approximation is still in terms ofexpectations. In Theorem 4.1, we will obtain the explicit second-order approximation in termsof iterated integral operators (defined in Definition 4.1), which will be convenient for our fastcalibration scheme in Section 6. Remark 3.1.
Let ( t, ε ) ξ ( ε ) t be a C ([0 , T ] × [0 , R ) function smooth in ε . Denote by ξ ( ε ) i,t := ∂ i ξ∂ε i its i -th derivative in ε , and let ξ i,t := ξ ( ε ) i,t | ε =0 . Then by a second-order Taylorexpansion around ε = 0, we have the representation ξ ( ε ) t = ξ ,t + ξ ,t + 12 ξ ,t + Θ ( ε )2 ,t ( ξ )where Θ is the second-order error term given by Taylor’s theorem. Specifically, for i ≥ ( ε ) i,t ( ξ ) := Z ε i ! ( ε − u ) i ξ ( u ) i +1 ,t d u. Expanding processes ε V ( ε ) t and ε (cid:16) V ( ε ) t (cid:17) . Using the notation from Remark 3.1,we can now represent the functions ε V ( ε ) t and ε (cid:16) V ( ε ) t (cid:17) via a Taylor expansion around ε = 0 to second-order. V ( ε ) t = v ,t + εV ,t + 12 ε V ,t + Θ ( ε )2 ,t ( V ) , ( V ( ε ) t ) = v ,t + 2 εv ,t V ,t + ε (cid:0) V ,t + v ,t V ,t (cid:1) + Θ ( ε )2 ,t ( V ) , (3.1)where v ,t := V ,t . Lemma 3.1.
The processes ( V ,t ) and ( V ,t ) satisfy the SDEsd V ,t = α x ( t, v ,t ) V ,t d t + β ( t, v ,t )d B t , V , = 0 , (3.2)d V ,t = (cid:16) α xx ( t, v ,t )( V ,t ) + α x ( t, v ,t ) V ,t (cid:17) d t + 2 β x ( t, v ,t ) V ,t d B t , V , = 0 , (3.3)with explicit solutions V ,t = e R t α x ( z,v ,z )d z Z t β ( s, v ,s ) e − R s α x ( z,v ,z )d z d B s , (3.4) V ,t = e R t α x ( z,v ,z )d z (cid:26)Z t α xx ( s, v ,s )( V ,s ) e − R s α x ( z,v ,z )d z d s + Z t β x ( s, v ,s ) V ,s e − R s α x ( z,v ,z )d z d B s (cid:27) . (3.5) Proof.
We give a sketch of the proof for ( V ,t ). First, we writed V ( ε )1 ,t = d (cid:16) ∂ ε V ( ε ) t (cid:17) = ∂ ε (d V ( ε ) t ) . The SDE for V ( ε ) t is given in eq. (2.4). By differentiating, we obtaind V ( ε )1 ,t = α x ( t, V ( ε ) t ) V ( ε )1 ,t d t + h εβ x ( t, V ( ε ) t ) V ( ε )1 ,t + β ( t, V ( ε ) t ) i d B t , V ( ε )1 , = 0 . Letting ε = 0 yields the SDE eq. (3.2). Since the SDE is linear, it can be solved explicitly. Thisgives the result eq. (3.4). The calculations for ( V ,t ) are similar. (cid:3) Remark 3.2.
We now comment on the purpose of Assumption B. • As explained before, Assumption A guarantees a pathwise unique strong solution for V whichwill not explode in finite time. • For Lemma 3.1 to be valid, it is clear that we will require the existence of α xx and β x as wellas their continuity a.e. in x and t ∈ [0 , T ]. This is assumed via item (B1) and item (B2)respectively in Assumption B.3.2. Expanding P BS . Let˜ P ( ε ) T := x − Z T ρ t (cid:16) V ( ε ) t (cid:17) d t + Z T ρ t V ( ε ) t d B t , ˜ Q ( ε ) T := Z T (1 − ρ t ) (cid:16) V ( ε ) t (cid:17) d t. Immediately we have ˜ P ( ε ) T = ˆ µ ε ( T ) + ˆ σ ε ( T ) and ˜ Q ( ε ) T = ˆ σ ε ( T ). Hence from Proposition 2.2 g ( ε ) = E (cid:16) P BS (cid:16) ˜ P ( ε ) T , ˜ Q ( ε ) T (cid:17)(cid:17) . (3.6)As g (1) corresponds to the price of a put option, we are interested in approximating the function P BS at (cid:16) ˜ P (1) T , ˜ Q (1) T (cid:17) . To do this we will expand P BS around the point (cid:16) ˜ P (0) T , ˜ Q (0) T (cid:17) = (cid:16) x − Z T ρ t v ,t d t + Z T ρ t v ,t d B t , Z T (1 − ρ t ) v ,t d t (cid:17) and evaluate at (cid:16) ˜ P (1) T , ˜ Q (1) T (cid:17) . Additionally, introduce the functions P ( ε ) T := ˜ P ( ε ) T − ˜ P (0) T = Z T ρ t ( V ( ε ) t − v ,t )d B t − Z T ρ t (cid:18)(cid:16) V ( ε ) t (cid:17) − v ,t (cid:19) d t,Q ( ε ) T := ˜ Q ( ε ) T − ˜ Q (0) T = Z T (1 − ρ t ) (cid:18)(cid:16) V ( ε ) t (cid:17) − v ,t (cid:19) d t. and the short-hand ˜ P BS := P BS (cid:16) ˜ P (0) T , ˜ Q (0) T (cid:17) ,∂ i + j ˜ P BS ∂x i ∂y j := ∂ i + j P BS (cid:16) ˜ P (0) T , ˜ Q (0) T (cid:17) ∂x i ∂y j . Proposition 3.1.
By a second-order Taylor expansion, the expression P BS (cid:16) ˜ P (1) T , ˜ Q (1) T (cid:17) can beapproximated to second-order as P BS (cid:16) ˜ P (1) T , ˜ Q (1) T (cid:17) ≈ ˜ P BS + (cid:16) ∂ x ˜ P BS (cid:17) P (1) T + (cid:16) ∂ y ˜ P BS (cid:17) Q (1) T + 12 (cid:16) ∂ xx ˜ P BS (cid:17) (cid:16) P (1) T (cid:17) + 12 (cid:16) ∂ yy ˜ P BS (cid:17) (cid:16) Q (1) T (cid:17) + (cid:16) ∂ xy ˜ P BS (cid:17) P (1) T Q (1) T . Expanding functions ε P ( ε ) T , ε Q ( ε ) T and its variants. The next step in ourexpansion procedure is to approximate the functions ε P ( ε ) T , ε (cid:16) P ( ε ) T (cid:17) , ε Q ( ε ) T , ε (cid:16) Q ( ε ) T (cid:17) and ε P ( ε ) T Q ( ε ) T . By Remark 3.1 we can write P ( ε ) T = P ,T + εP ,T + 12 ε P ,T + Θ ( ε )2 ,T ( P ) , ( P ( ε ) T ) = P ,T + 2 εP ,T P ,T + ε (cid:0) P ,T + P ,T P ,T (cid:1) + Θ ( ε )2 ,T ( P ) , (3.7) Q ( ε ) T = Q ,T + εQ ,T + 12 ε Q ,T + Θ ( ε )2 ,T ( Q ) , ( Q ( ε ) T ) = Q ,T + 2 εQ ,T Q ,T + ε (cid:0) Q ,T + Q ,T Q ,T (cid:1) + Θ ( ε )2 ,T ( Q ) , (3.8)and P ( ε ) T Q ( ε ) T = P ,T Q ,T + ε ( Q ,T P ,T + P ,T Q ,T )+ 12 ε ( Q ,T P ,T + P ,T Q ,T + 3 ( Q ,T P ,T + P ,T Q ,T )) + Θ ( ε )2 ,T ( P Q ) . (3.9)This results in the following lemma. Lemma 3.2.
Equations (3.7) to (3.9) can be rewritten as P ( ε ) T = εP ,T + 12 ε P ,T + Θ ( ε )2 ,T ( P ) , ( P ( ε ) T ) = ε P ,T + Θ ( ε )2 ,T ( P ) , (3.10) Q ( ε ) T = εQ ,T + 12 ε Q ,T + Θ ( ε )2 ,T ( Q ) , ( Q ( ε ) T ) = ε Q ,T Θ ( ε )2 ,T ( Q ) , (3.11)and P ( ε ) T Q ( ε ) T = ε P ,T Q ,T + Θ ( ε )2 ,T ( P Q ) , (3.12) respectively, where P ,T = Z T ρ t V ,t d B t − Z T ρ t v ,t V ,t d t,P ,T = Z T ρ t V ,t d B t − Z T ρ t (cid:0) V ,t + v ,t V ,t (cid:1) d t,Q ,T = 2 Z T (1 − ρ t ) v ,t V ,t d t,Q ,T = 2 Z T (1 − ρ t ) (cid:0) V ,t + v ,t V ,t (cid:1) d t. Proof.
First, notice that by their definitions, P ,T = P (0) T = ˜ P (0) T − ˜ P (0) T = 0, and similarly Q ,T = 0. We will show how to obtain the form of P ,T , the rest being similar. By definition P ( ε )1 ,T = ∂ ε (cid:16) P ( ε ) T (cid:17) = ∂ ε (cid:18) x − Z T ρ t (cid:16) V ( ε ) t (cid:17) d t + Z T ρ t V ( ε ) t d B t (cid:19) = Z T ρ t V ( ε )1 ,t d B t − Z T ρ t V ( ε ) t V ( ε )1 ,t d t. By putting ε = 0 we obtain P ,T , namely P ,T = Z T ρ t V ,t d B t − Z T ρ t v ,t V ,t d t. (cid:3) Theorem 3.1 (Second-order put option price approximation) . Denote by Put (2)G the second-order approximation to the price of a put option in the general model eq. (2.1). ThenPut (2)G = E ˜ P BS ( C x :=) + E ∂ x ˜ P BS Z T ρ t (cid:18) V ,t + 12 V ,t (cid:19) d B t − Z T ρ t (cid:0) v ,t V ,t + (cid:0) V ,t + v ,t V ,t (cid:1) (cid:1) d t ! ( C y :=) + E ∂ y ˜ P BS (cid:18)Z T (1 − ρ t ) (cid:0) v ,t V ,t + (cid:0) V ,t + v ,t V ,t (cid:1) (cid:1) d t (cid:19) ( C xx :=) + 12 E ∂ xx ˜ P BS (cid:18)Z T ρ t V ,t d B t − Z T ρ t v ,t V ,t d t (cid:19) ( C yy :=) + 12 E ∂ yy ˜ P BS (cid:18)Z T (1 − ρ t )(2 v ,t V ,t )d t (cid:19) ( C xy :=) + E ∂ xy ˜ P BS (cid:18)Z T ρ t V ,t d B t − Z T ρ t v ,t V ,t d t (cid:19)(cid:18)Z T (1 − ρ t )(2 v ,t V ,t )d t (cid:19) . Additionally, Put G = Put (2)G + E ( E ), where E denotes the error in the expansion. Proof.
From Proposition 3.1, consider the two-dimensional Taylor expansion of P BS around (cid:16) ˜ P (0) T , ˜ Q (0) T (cid:17) evaluated at (cid:16) ˜ P (1) T , ˜ Q (1) T (cid:17) . Then, substitute in the second-order expressions of P (1) T , (cid:16) P (1) T (cid:17) , Q (1) T , (cid:16) Q (1) T (cid:17) and P (1) T Q (1) T from Lemma 3.2. As this is a second-order expression, the remainder terms Θ are neglected.Taking expectation yields Put (2)G . (cid:3) Remark 3.3.
The explicit expression for E and the analysis of it is left for Section 5.4. Explicit price
The goal now is to express the terms C x , C y , C xx , C yy , C xy in terms of the following integraloperators. Definition 4.1 (Integral operator) . To this end, define the following integral operator ω ( k,l ) t,T := Z Tt l u e R u k z d z d u. In addition, we define the n -fold iterated integral operator through the following recurrence. ω ( k ( n ) ,l ( n ) ) , ( k ( n − ,l ( n − ) ,..., ( k (1) ,l (1) ) t,T := ω (cid:0) k ( n ) ,l ( n ) w ( k ( n − ,l ( n − , ··· , ( k (1) ,l (1)) · ,T (cid:1) t,T , n ∈ N . Assumption C. β ( t, x ) = λ t x µ for µ ∈ [1 / , λ is bounded over [0 , T ].We will comment on the reasoning behind Assumption C in Remark D.2. Theorem 4.1 (Explicit second-order put option price) . Under Assumption C, the explicitsecond-order price of a put option in the general model eq. (2.1) is given byPut (2)G = P BS (cid:18) x , Z T v ,t d t (cid:19) + 2 ω ( − α x ,ρλv µ +10 , · ) , ( α x ,v , · )0 ,T ∂ xy P BS (cid:18) x , Z T v ,t d t (cid:19) + ω ( − α x ,λ v µ , · ) , (2 α x , ,T ∂ y P BS (cid:18) x , Z T v ,t d t (cid:19) + 2 ω ( − α x ,ρλv µ +10 , · ) , ( − α x ,ρλv µ +10 , · ) , (2 α x , ,T ∂ xxy P BS (cid:18) x , Z T v ,t d t (cid:19) + ω ( − α x ,λ v µ , · ) , ( α x ,α xx ) , ( α x ,v , · )0 ,T ∂ y P BS (cid:18) x , Z T v ,t d t (cid:19) + ( ω ( − α x ,ρλv µ +10 , · ) , ( − α x ,ρλv µ +10 , · ) , ( α x ,α xx ) , ( α x ,v , · )0 ,T + 2 µω ( − α x ,ρλv µ +10 , · ) , (0 ,ρλv µ − , · ) , ( α x ,v , · )0 ,T ) ∂ xxy P BS (cid:18) x , Z T v ,t d t (cid:19) + 2 ω ( − α x ,ρλv µ +10 , · ) , (0 ,ρλv µ , · ) , ( α x ,v , · )0 ,T ∂ xxy P BS (cid:18) x , Z T v ,t d t (cid:19) + 4 ω ( − α x ,λ v µ , · ) , ( α x ,v , · ) , ( α x ,v , · )0 ,T ∂ yy P BS (cid:18) x , Z T v ,t d t (cid:19) + 2 (cid:18) ω ( − α x ,ρ λ v µ +10 , · ) , ( α x ,v , · )0 ,T (cid:19) ∂ xxyy P BS (cid:18) x , Z T v ,t d t (cid:19) where the partial derivatives of P BS are given in Appendix C. Proof.
The proof is given in Appendix D. (cid:3) For example ω ( k (3) ,l (3) ) , ( k (2) ,l (2) ) , ( k (1) ,l (1) ) t,T = Z Tt l (3) u e R u k (3) z d z (cid:18)Z Tu l (2) u e R u k (2) z d z (cid:18)Z Tu l (1) u e R u k (1) z d z d u (cid:19) d u (cid:19) d u . Remark 4.1.
In Theorem 4.1, we enforce Assumption C. This means we can obtain the second-order pricing formula for different models by choosing a specific α ( t, x ) that adheres to Assump-tion A and Assumption B, as well as a µ ∈ [1 / , α ( t, x ) = κ t ( θ t − x ), then this drift satisfies Assumption A and Assump-tion B. By choosing some µ ∈ [1 / , V t = κ t ( θ t − V t )d t + λ t V µt d B t , V = v . In particular, to obtain the explicit second-order put option price in the Inverse-Gamma model,choose α ( t, x ) = κ t ( θ t − x ) and µ = 1, so that α x ( t, x ) = − κ t and α xx ( t, x ) = 0. Indeed, thisgives the desired result for the second-order put option price in the Inverse-Gamma model asseen in Langren´e et al. [18].4.1. Verhulst model explicit price.
We now return to the Verhulst model eq. (1.1). Recallthat the volatility process obeys the dynamicsd V t = κ t ( θ t − V t ) V t d t + λ t V t d B t , V = v . Lemma 4.1 (Verhulst model explicit second-order put option price) . Under the Verhulst modeleq. (1.1), the explicit second-order price of a put option is given byPut (2)Verhulst = P BS (cid:18) x , Z T v ,t d t (cid:19) + 2 ω ( − ( κθ − κv , · ) ,ρλv , · ) , ( κθ − κv , · ,v , · )0 ,T ∂ xy P BS (cid:18) x , Z T v ,t d t (cid:19) + ω ( − κθ − κv , · ) ,λ v , · ) , (2( κθ − κv , · ) , ,T ∂ y P BS (cid:18) x , Z T v ,t d t (cid:19) + 2 ω ( − ( κθ − κv , · ) ,ρλv , · ) , ( − ( κθ − κv , · ) ,ρλv , · ) , (2( κθ − κv , · ) , ,T ∂ xxy P BS (cid:18) x , Z T v ,t d t (cid:19) + ω ( − κθ − κv , · ) ,λ v , · ) , ( κθ − κv , · , − κ ) , ( κθ − κv , · ,v , · )0 ,T ∂ y P BS (cid:18) x , Z T v ,t d t (cid:19) + ( ω ( − ( κθ − κv , · ) ,ρλv , · ) , ( − ( κθ − κv , · ) ,ρλv , · ) , ( κθ − κv , · , − κ ) , ( κθ − κv , · ,v , · )0 ,T + 2 ω ( − ( κθ − κv , · ) ,ρλv , · ) , (0 ,ρλv , · ) , ( κθ − κv , · ,v , · )0 ,T ) ∂ xxy P BS (cid:18) x , Z T v ,t d t (cid:19) + 2 ω ( − ( κθ − κv , · ) ,ρλv , · ) , (0 ,ρλv , · ) , ( κθ − κv , · ,v , · )0 ,T ∂ xxy P BS (cid:18) x , Z T v ,t d t (cid:19) + 4 ω ( − κθ − κv , · ) ,λ v , · ) , ( κθ − κv , · ,v , · ) , ( κθ − κv , · ,v , · )0 ,T ∂ yy P BS (cid:18) x , Z T v ,t d t (cid:19) + 2 (cid:18) ω ( − ( κθ − κv , · ) ,ρλv , · ) , ( κθ − κv , · ,v , · )0 ,T (cid:19) ∂ xxyy P BS (cid:18) x , Z T v ,t d t (cid:19) . Proof.
Under Assumption C, the approximation formula in the general model eq. (2.1) is givenin Theorem 4.1. Notice under the Verhulst model we have α ( t, x ) = κ t ( θ t − x ) x, which yields α x ( t, x ) = κ t θ t − κ t x,α xx ( t, x ) = − κ t x. Furthermore, the diffusion is β ( t, x ) = λ t x , meaning we take µ = 1 in Theorem 4.1. Substitutingthese expressions in the formula from Theorem 4.1 gives the result. (cid:3) Remark 4.2.
Currently the second-order approximations Put (2)G and Put (2)Verhulst are expressedin terms of iterated integral operators and partial derivatives of P BS . When parameters areassumed to be piecewise-constant, then the iterated integral operators can be expressed in aclosed-form manner, which we prove in Section 6.5. Error analysis
This section is dedicated to the explicit representation and analysis of the error induced by ourexpansion procedure in Section 3. The section is divided into two parts.(1) Section 5.1 is devoted to the explicit representation of the error term induced by the expan-sion procedure.(2) Section 5.2 details how one would approach bounding the error term induced by the ex-pansion procedure in terms of the remainder terms generated by the approximation of theunderlying volatility/variance process.In this section we will make extensive use of the following notation: • L p := L p (Ω , F , Q ) denotes the vector space of equivalence classes of random variables withfinite L p norm, given by k · k p = [ E | · | p ] /p . • For an n -tuple α := ( α , . . . , α n ) ∈ N n , then | α | = P ni =1 α i denotes its 1-norm.5.1. Explicit expression for error.
Recall from Theorem 3.1 that the price of a put option inthe general model eq. (2.1) was Put G = Put (2)G + E ( E ), where Put (2)G is the second-order closed-form price. As our expansion methodology was contingent on the use of Taylor polynomials, theterm E evidently appears due to the truncation of Taylor series. To represent E we will needexplicit expressions for the error terms. These are given by Taylor’s theorem, which we willpresent here to fix notation. We only consider the results up to second-order. Theorem 5.1 (Taylor’s theorem for g : R → R ) . Let A ⊆ R , B ⊆ R and let g : A → B bea C ( R ; R ) function in a closed ball about the point ( a, b ) ∈ A . Then Taylor’s theorem statesthat the Taylor series of g around the point ( a, b ) is given by g ( x, y ) = g ( a, b ) + g x ( a, b )( x − a ) + g y ( a, b )( y − b )+ 12 g xx ( a, b )( x − a ) + 12 g yy ( a, b )( y − b ) + g xy ( a, b )( x − a )( y − b ) + R ( x, y ) , where R ( x, y ) = X | α | =3 | α | α ! α ! E α ( x, y )( x − a ) α ( y − b ) α E α ( x, y ) = Z (1 − u ) ∂ ∂x α ∂y α g ( a + u ( x − a ) , b + u ( y − b ))d u. Now recall from Section 3 the functions˜ P ( ε ) T = x − Z T ρ t (cid:16) V ( ε ) t (cid:17) d t + Z T ρ t V ( ε ) t d B t , ˜ Q ( ε ) T = Z T (1 − ρ t ) (cid:16) V ( ε ) t (cid:17) d t and P ( ε ) T := ˜ P ( ε ) T − ˜ P (0) T = Z T ρ t ( V ( ε ) t − v ,t )d B t − Z T ρ t (cid:18)(cid:16) V ( ε ) t (cid:17) − v ,t (cid:19) d t,Q ( ε ) T := ˜ Q ( ε ) T − ˜ Q (0) T = Z T (1 − ρ t ) (cid:18)(cid:16) V ( ε ) t (cid:17) − v ,t (cid:19) d t. Furthermore, recall the short hand ˜ P BS = P BS (cid:16) ˜ P (0) T , ˜ Q (0) T (cid:17) ,∂ i + j ˜ P BS ∂x i ∂y j = ∂ i + j P BS (cid:16) ˜ P (0) T , ˜ Q (0) T (cid:17) ∂x i ∂y j . Theorem 5.2 (Explicit error term) . The error term E in Theorem 3.1 induced from the expan-sion procedure can be decomposed as E = E P + E V where E P = X | α | =3 | α | α ! α ! E α (cid:16) ˜ P (1) T , ˜ Q (1) T (cid:17) (cid:16) P (1) T (cid:17) α (cid:16) Q (1) T (cid:17) α ,E α (cid:16) ˜ P (1) T , ˜ Q (1) T (cid:17) = Z (1 − u ) ∂ α P BS ∂x α ∂y α (cid:16) (1 − u ) ˜ P T (0) + u ˜ P T (1) , (1 − u ) ˜ Q T (0) + u ˜ Q T (1) (cid:17) d u, and E V = X | α | =1 ∂ ˜ P BS ∂x α ∂y α Θ (1)2 ,T ( P α Q α ) + 12 X | α | =2 | α | α ! α ! ∂ ˜ P BS ∂x α ∂y α Θ (1)2 ,T ( P α Q α ) , where Θ (1)2 ,T ( P Q x ) := Θ (1)2 ,T ( Q x ) and Θ (1)2 ,T ( P y Q ) := Θ (1)2 ,T ( P y ), for x, y ∈ { , } . Here, E P corresponds to the error in the approximation of the function P BS , and E V corresponds to theerror in the approximation of the functions ε V ( ε ) t and ε (cid:16) V ( ε ) t (cid:17) . Proof.
The decomposition E = E P + E V is a clear consequence of Taylor’s theorem. The nexttwo subsections are dedicated to representing E P and E V explicitly.5.1.1. Explicit E P . First we will derive E P explicitly, the error term corresponding to the second-order approximation of P BS . In our expansion procedure, we expand P BS up to second-orderaround the point (cid:16) ˜ P (0) T , ˜ Q (0) T (cid:17) = (cid:16) x − Z T ρ t v ,t d t + Z T ρ t v ,t d B t , Z T (1 − ρ t ) v ,t d t (cid:17) and evaluate at (cid:16) ˜ P (1) T , ˜ Q (1) T (cid:17) . Thus in the Taylor expansion of P BS , the terms will be of the form ∂ | α | ˜ P BS ∂x α ∂y α (cid:16) P (1) T (cid:17) α (cid:16) Q (1) T (cid:17) α for | α | = 0 , ,
2. By Theorem 5.1 (Taylor’s theorem) we can write the second-order Taylorpolynomial of P BS (cid:16) ˜ P (1) T , ˜ Q (1) T (cid:17) with error term as P BS (cid:16) ˜ P (1) T , ˜ Q (1) T (cid:17) = ˜ P BS + (cid:16) ∂ x ˜ P BS (cid:17) P (1) T + (cid:16) ∂ y ˜ P BS (cid:17) Q (1) T + 12 (cid:16) ∂ xx ˜ P BS (cid:17) (cid:16) P (1) T (cid:17) + 12 (cid:16) ∂ yy ˜ P BS (cid:17) (cid:16) Q (1) T (cid:17) + (cid:16) ∂ xy ˜ P BS (cid:17) P (1) T Q (1) T + X | α | =3 | α | α ! α ! E α (cid:16) ˜ P (1) T , ˜ Q (1) T (cid:17) (cid:16) P (1) T (cid:17) α (cid:16) Q (1) T (cid:17) α | {z } Error term (5.1)with E α (cid:16) ˜ P (1) T , ˜ Q (1) T (cid:17) = Z (1 − u ) ∂ P BS ∂x α ∂y α (cid:16) ˜ P T (0) + uP T (1) , ˜ Q T (0) + uQ T (1) (cid:17) d u = Z (1 − u ) ∂ P BS ∂x α ∂y α (cid:16) (1 − u ) ˜ P T (0) + u ˜ P T (1) , (1 − u ˜ Q T (0) + u ˜ Q T (1) (cid:17) d u. Taking expectation gives Put G . Thus the explicit form for the error term E P is E P = X | α | =3 | α | α ! α ! E α (cid:16) ˜ P (1) T , ˜ Q (1) T (cid:17) (cid:16) P (1) T (cid:17) α (cid:16) Q (1) T (cid:17) α . Explicit E V . Now we derive E V explicitly, the error corresponding to the second-orderapproximation of the functions ε V ( ε ) t and ε (cid:16) V ( ε ) t (cid:17) . Recall from Lemma 3.2, since P ,T = ˜ P (0) T − ˜ P (0) T = 0 and similarly Q ,T = 0, we could write P ( ε ) T = P ,T + 12 ε P ,T + Θ ( ε )2 ,T ( P ) , ( P ( ε ) T ) = ε P ,T + Θ ( ε )2 ,T ( P ) ,Q ( ε ) T = εQ ,T + 12 ε Q ,T + Θ ( ε )2 ,T ( Q ) , ( Q ( ε ) T ) = ε Q ,T + Θ ( ε )2 ,T ( Q ) , and P ( ε ) T Q ( ε ) T = P ,T Q ,T + Θ ( ε )2 ,T ( P Q ) , where P ,T = Z T ρ t V ,t d B t − Z T ρ t v ,t V ,t d t,P ,T = Z T ρ t V ,t d B t − Z T ρ t (cid:0) V ,t + v ,t V ,t (cid:1) d t,Q ,T = 2 Z T (1 − ρ t ) v ,t V ,t d t,Q ,T = 2 Z T (1 − ρ t ) (cid:0) V ,t + v ,t V ,t (cid:1) d t. The idea then is to approximate the functions ε P ( ε ) T , ε Q ( ε ) T and their variants by theirsecond-order expansions. For example, in the expansion of P BS in eq. (5.1) if we focus on theterm corresponding to the first derivative of P BS in its second argument, we have( ∂ y ˜ P BS ) Q (1) T = ( ∂ y ˜ P BS )( Q ,T + 12 Q ,T + Θ (1)2 ,T ( Q ))= ( ∂ y ˜ P BS )( Q ,T + 12 Q ,T ) + ( ∂ y ˜ P BS )(Θ (1)2 ,T ( Q ) | {z } Error term ) . For the term corresponding to the second derivative of P BS in its second argument, we wouldhave 12 ( ∂ yy ˜ P BS ) (cid:16) Q (1) T (cid:17) = 12 ( ∂ yy ˜ P BS ) (cid:16) Q ,T + Θ (1)2 ,T ( Q ) (cid:17) = 12 ( ∂ yy ˜ P BS ) (cid:0) Q ,T (cid:1) + 12 ( ∂ yy ˜ P BS ) (cid:16) Θ (1)2 ,T ( Q ) (cid:17)| {z } Error term . Following this pattern, we can see that the error term E V can be written explicitly as E V = X | α | =1 ∂ ˜ P BS ∂x α y α Θ (1)2 ,T ( P α Q α ) + 12 X | α | =2 | α | α ! α ! ∂ ˜ P BS ∂x α y α Θ (1)2 ,T ( P α Q α ) . (cid:3) As the second-order price of a put option is the expectation of our expansion, the goal is tobound E in L for a specific volatility process V .5.2. Bounding error term.
Our objective is to appeal to the explicit representation of theerror term E as seen in Theorem 5.2 and bound it in L under the general model eq. (2.1). Inorder to obtain an L bound on the error term E , it is sufficient to obtain ingredients given inthe following proposition. Proposition 5.1.
In order to obtain an L bound on the error term E , it is sufficient to obtain:1 Bounds on k Θ (1)2 ,T ( P α Q α ) k , where | α | = 1 , k P (1) T k p and k Q (1) T k p for p ≥ Lemma 5.1.
DefinePut BS ( x, y ) := Ke − R T r dt d t N ( − d − ) − xe − R T r ft d t N ( − d + ) ,d ± ( x, y ) := d ± := ln( x/K ) + R T ( r dt − r ft )d t √ y ± √ y. Consider the third-order partial derivatives of Put BS , ∂ Put BS ∂x α ∂y α , where α + α = 3 as well as thelinear functions h , h : [0 , → R + such that h ( u ) = u ( d − c ) + c and h ( u ) = u ( d − c ) + c .Assume there exists no point a ∈ (0 ,
1) such thatlim u → a ln( h ( u ) /K ) + R T ( r dt − r ft )d t p h ( u ) = 0 and lim u → a h ( u ) = 0 . Then there exists functions M α bounded on R such thatsup u ∈ (0 , (cid:12)(cid:12)(cid:12)(cid:12) ∂ Put BS ∂x α ∂y α ( h ( u ) , h ( u )) (cid:12)(cid:12)(cid:12)(cid:12) = M α ( T, K ) . Furthermore, the behaviour of M α for fixed K and T is characterised by the functions ζ and η respectively, where ζ ( T ) = ˆ Ae − R T r ft d t e − E ˜ r ( T ) e − E ˜ r ( T ) n X i =0 c i ˜ r i ( T ) , with ˜ r ( T ) := R T ( r dt − r ft )d t and E > E ∈ R , ˆ A ∈ R , n ∈ N and c , . . . , c n are constants, and η ( K ) = ˜ AK − D ln( K )+ D N X i =0 C i ( − i ln i ( K ) , with D > , D ∈ R , ˜ A ∈ R , N ∈ N and C , . . . , C N are constants. Proof.
See Lemma 6.1 in Das and Langren´e [10]. (cid:3) Lemma 5.2.
Consider the third-order partial derivatives of P BS , ∂ P BS ∂x α ∂y α , where α + α = 3as well as the linear functions h , h : [0 , → R + such that h ( u ) = u ( d − c ) + c and h ( u ) = u ( d − c ) + c . Assume there exists no point a ∈ (0 ,
1) such thatlim u → a h ( u ) − k + R T ( r dt − r ft )d t p h ( u ) = 0 and lim u → a h ( u ) = 0 . (5.2)Then there exists functions B α bounded on R + × R such thatsup u ∈ (0 , (cid:12)(cid:12)(cid:12)(cid:12) ∂ P BS ∂x α ∂y α ( h ( u ) , h ( u )) (cid:12)(cid:12)(cid:12)(cid:12) = B α ( T, k ) . Furthermore, the behaviour of B α for fixed k and T is characterised by the functions ζ and ν respectively, where ζ ( T ) = ˆ Ae − R T r ft d t e − E ˜ r ( T ) e − E ˜ r ( T ) n X i =0 c i ˜ r i ( T ) , with ˜ r ( T ) := R T ( r dt − r ft )d t and E > E ∈ R , ˆ A ∈ R , n ∈ N and c , . . . , c n are constants, and ν ( k ) = ˜ Ae − D k + D k N X i =0 C i ( − i k i , with D > , D ∈ R , ˜ A ∈ R , N ∈ N and C , . . . , C N are constants. Proof.
Lemma 5.2 is very similar to Lemma 5.1, where the latter is the equivalent lemma forthe function Put BS . In fact, we will show that Lemma 5.1 implies Lemma 5.2. In the following,we will repeatedly denote by F or G to be an arbitrary polynomial of some degree, as well as A to be an arbitrary constant. That is, they may be different on each use.First, as a function of x and y , notice from Appendix C that the third-order partial derivatives ∂ P BS ∂x α ∂y α , where α + α = 3 can be written as A e x φ ( d ln+ ) y m/ G ( d ln+ , d ln − , √ y ) , m ∈ N (5.3)except for when α = (3 , A e x φ ( d ln+ ) y m/ G ( d ln+ , d ln − , √ y ) + Ae x φ ( d ln+ )( N ( d ln+ ) − , m ∈ N . (5.4)Similarly, it can seen that as a function of x and y , the third-order partial derivatives of Put BS can be written as A φ ( d + ) x n y m/ F ( d + , d − , √ y ) , n ∈ Z , m ∈ N . (5.5) Recall d ± = d ± ( x, y ) = ln( x/K ) + R T (cid:16) r dt − r ft (cid:17) d t √ y ± √ y,d ln ± = d ln ± ( x, y ) = x − k + R T (cid:16) r dt − r ft (cid:17) d t √ y ± √ y,φ ( x ) = 1 √ π e − x / . Let us consider the cases for which α = (3 , k = ln( K ). Noticethat d ln ± ( x, y ) = d ± ( e x , y ). Take n = − f and g are ‘of the same form’ if they are equal up to constant values. Furthermore,we will denote this relation by f C ∼ g . Then comparing eq. (5.3) and eq. (5.5), the form of thepartial derivatives of P BS are the same as the partial derivatives of Put BS composed with thefunction e x in its first argument. Specifically, we can write ∂ P BS ∂x α ∂y α ( x, y ) C ∼ ∂ Put BS ∂x α ∂y α ( e x , y ) . Now, consider arbitrary functions f, b : R → R such thatsup x ∈ R | f ( x ) | = L < ∞ . Then it is true that sup x ∈ R | f ( b ( x )) | = ˜ L ≤ L < ∞ . Thus sup u ∈ (0 , (cid:12)(cid:12)(cid:12)(cid:12) ∂ P BS ∂x α ∂y α ( h ( u ) , h ( u )) (cid:12)(cid:12)(cid:12)(cid:12) C ∼ sup u ∈ (0 , (cid:12)(cid:12)(cid:12)(cid:12) ∂ Put BS ∂x α ∂y α ( e h ( u ) , h ( u )) (cid:12)(cid:12)(cid:12)(cid:12) . Under the assumption in eq. (5.2), and then using Lemma 5.1, this supremum will not blow up.Clearly, sup u ∈ (0 , ∂ Put BS ∂x α ∂y α ( e h ( u ) , h ( u )) is a function of T and K . By substituting k = ln( K )in the result of Lemma 5.1, we obtain the form of ζ and ν .Now for the case of α = (3 , ∂ P BS ∂x ( x, y ) C ∼ ∂ Put BS ∂x ( e x , y ) + A e x φ ( d ln+ )( N ( d ln+ ) − | {z } =: H ( x,y ) . Now | H ( x, y ) | = | e x φ ( d ln+ )( N ( d ln+ ) − | ≤ e x φ ( d ln+ ) . Thus sup x ∈ R | H ( x, y ) | = sup x ∈ R | e x φ ( d ln+ )( N ( d ln+ ) − | ≤ sup x ∈ R e x φ ( d ln+ ) < ∞ and also sup y ∈ R + | H ( x, y ) | = sup y ∈ R + | φ ( d ln+ )( N ( d ln+ ) − | ≤ sup y ∈ R + φ ( d ln+ ) < ∞ . Hence sup u ∈ (0 , H ( h ( u ) , h ( u )) ≤ sup u ∈ (0 , e h ( u ) φ ( d ln+ ( h ( u ) , h ( u ))) = ˆ m ( T, k ) , where ˆ m is a bounded function on R + × R . By direct computation, it is clear that for fixed T the form of ˆ m is given by Ae − ˆ D k e ˆ D k , where ˆ D > D ∈ R . For fixed k , it is given by Ae − ˆ E ˜ r ( T ) e ˆ E ˜ r ( T ) , where ˆ E > E ∈ R . Thussup u ∈ (0 , (cid:12)(cid:12)(cid:12)(cid:12) ∂ P BS ∂x ( h ( u ) , h ( u )) (cid:12)(cid:12)(cid:12)(cid:12) C ∼ sup u ∈ (0 , (cid:12)(cid:12)(cid:12)(cid:12) ∂ Put BS ∂x ( e h ( u ) , h ( u )) + AH ( h ( u ) , h ( u )) (cid:12)(cid:12)(cid:12)(cid:12) ≤ sup u ∈ (0 , (cid:12)(cid:12)(cid:12)(cid:12) ∂ Put BS ∂x ( e h ( u ) , h ( u )) (cid:12)(cid:12)(cid:12)(cid:12) + A sup u ∈ (0 , | H ( h ( u ) , h ( u )) | C ∼ B (3 , ( T, k ) + A ˆ m ( T, k ) . But the form of ˆ m is exactly that of B α without the polynomial expression. Thus, the sum ofthem is again of the form of B α . (cid:3) Bounding E V . We first consider bounding the term E V from Theorem 5.2 in L . Theterms of interest to bound are ∂ | α | ˜ P BS ∂x α ∂y α Θ (1)2 ,T ( P α Q α ) , | α | = 1 , . Now the second argument of ˜ P BS is ˜ Q (0) T , which is strictly positive. By considering the linearfunction u (1 − u ) Q (0) T + uQ (0) T , then by Lemma 5.2 this implies ∂ | α | ˜ P BS ∂x α ∂y α ≤ B α ( T, k ). Thus (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∂ | α | ˜ P BS ∂x α ∂y α Θ (1)2 ,T ( P α Q α ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ B α ( T, k ) (cid:13)(cid:13)(cid:13) Θ (1)2 ,T ( P α Q α ) (cid:13)(cid:13)(cid:13) . (5.6)Equation (5.6) suggests that obtaining an L bound on the remainder term Θ (1)2 ,T ( P α Q α ) for | α | = 1 , Bounding E P . The terms of interest are E α (cid:16) ˜ P (1) T , ˜ Q (1) T (cid:17) (cid:16) P (1) T (cid:17) α (cid:16) Q (1) T (cid:17) α , | α | = 3 . We now define J ( u ) := (1 − u ) ˜ P T (0) + u ˜ P T (1) ,K ( u ) := (1 − u ) ˜ Q T (0) + u ˜ Q T (1) , so that ∂ P BS ∂x α ∂y α ( J ( u ) , K ( u )) = ∂ P BS ∂x α ∂y α (cid:16) (1 − u ) ˜ P T (0) + u ˜ P T (1) , (1 − u ) ˜ Q T (0) + u ˜ Q T (1) (cid:17) . Proposition 5.2.
There exists functions B α with α + α = 3 as in Lemma 5.2 such thatsup u ∈ (0 , (cid:12)(cid:12)(cid:12)(cid:12) ∂ P BS ∂x α ∂y α ( J ( u ) , K ( u )) (cid:12)(cid:12)(cid:12)(cid:12) ≤ B α ( T, k ) Q a.s. . Proof.
Since J and K are linear functions, then from Lemma 5.2, this claim is immediately trueif we can show that K is strictly positive Q a.s.. Recall K ( u ) = (1 − u ) (cid:18)Z T (1 − ρ t ) v ,t d t (cid:19) + u Z T (1 − ρ t ) V t d t.K corresponds to the linear interpolation of R T (1 − ρ t ) v ,t d t and R T (1 − ρ t ) V t d t . It is clearsup t ∈ [0 ,T ] (1 − ρ t ) >
0. As V corresponds to the variance process, this is always chosen to bea non-negative process such that the set { t ∈ [0 , T ] : V t > } has non-zero Lebesgue measure.Thus these integrals are strictly positive and hence K is strictly positive Q a.s.. (cid:3) By Proposition 5.2 (cid:12)(cid:12)(cid:12) E α (cid:16) ˜ P (1) T , ˜ Q (1) T (cid:17)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z (1 − u ) ∂ | α | P BS ∂x α ∂y α ( J ( u ) , K ( u )) d u (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ B α ( T, k ) . Thus (cid:13)(cid:13)(cid:13) E α (cid:16) ˜ P (1) T , ˜ Q (1) T (cid:17) (cid:16) P (1) T (cid:17) α (cid:16) Q (1) T (cid:17) α (cid:13)(cid:13)(cid:13) ≤ B α ( T, k ) (cid:13)(cid:13)(cid:13)(cid:16) P (1) T (cid:17) α (cid:13)(cid:13)(cid:13) (cid:13)(cid:13)(cid:13)(cid:16) Q (1) T (cid:17) α (cid:13)(cid:13)(cid:13) . (5.7)Looking at the second and third term on the RHS of eq. (5.7), it is clear one of our objectivesis to bound P (1) T and Q (1) T in L p for p ≥
2. This validates item (2) in Proposition 5.1.
Lemma 5.3.
The terms from Proposition 5.1 can be bounded if the following quantities can bebounded:(1) k Θ (1)0 ,t ( V ) k p and k Θ (1)0 ,t ( V ) k p for p ≥ k Θ (1)1 ,t ( V ) k p and k Θ (1)1 ,t ( V ) k p for p ≥ k Θ (1)2 ,t ( V ) k p and k Θ (1)2 ,t ( V ) k p for p ≥ Proof.
We will make extensive use of the following integral inequality: (cid:18)Z T | f ( u ) | d u (cid:19) p ≤ T p − Z T | f ( u ) | p d u, p ≥ . (5.8)For the rest of this proof, assume that p ≥
2. We will denote by C p and D p generic constantsthat solely depend on p . They may be different on each use. Notice P (1) T = Z T ρ t Θ (1)0 ,t ( V )d B t − Z T ρ t Θ (1)0 ,t ( V )d t,Q (1) T = Z T (1 − ρ t )Θ (1)0 ,t ( V )d t. Applying the Minkowski and Burkholder-Davis-Gundy inequalities, as well as the integral in-equality eq. (5.8), we obtain k P (1) T k p ≤ C p T − p (cid:18)Z T ρ pt k Θ (1)0 ,t ( V ) k pp d t (cid:19) /p + 12 D p T − p (cid:18)Z T ρ pt k Θ (1)0 ,t ( V ) k pp d t (cid:19) /p and k Q (1) T k p ≤ C p T − p (cid:18)Z T (1 − ρ t ) p k Θ (1)0 ,t ( V ) k pp d t (cid:19) /p . Now also (cid:16) V (1) t (cid:17) = (cid:16) v ,t + Θ (1)0 ,t ( V ) (cid:17) = v ,t + 2 v ,t Θ (1)0 ,t ( V ) + (cid:16) Θ (1)0 ,t ( V ) (cid:17) , so thatΘ (1)0 ,t ( V ) = 2 v ,t Θ (1)0 ,t ( V ) + (cid:16) Θ (1)0 ,t ( V ) (cid:17) . This suggests that finding an L p bound on the remainder term Θ (1)0 ,t ( V ) is sufficient in order tobound P (1) T and Q (1) T in L p . This validates item (1). We can write the following remainder terms of P and Q asΘ (1)2 ,T ( P ) = Z T ρ t Θ (1)2 ,t ( V )d B t − Z T ρ t Θ (1)2 ,t ( V )d t, Θ (1)2 ,T ( Q ) = Z T (1 − ρ t )Θ (1)2 ,t ( V )d t, Θ (1)2 ,T ( P ) = (cid:16) P (1) T (cid:17) − P ,T = ( P (1) T − P ,T )( P (1) T + P ,T )= (cid:18)Z T ρ t Θ (1)1 ,t ( V )d B t − Z T ρ t Θ (1)1 ,t ( V )d t (cid:19) · (cid:18)Z T ρ t (2Θ (1)0 ,t ( V ) − Θ (1)1 ,t ( V ))d B t − Z T ρ t (2Θ (1)0 ,t ( V ) − Θ (1)1 ,t ( V ))d t (cid:19) , Θ (1)2 ,T ( Q ) = (cid:16) Q (1) T (cid:17) − Q ,T = ( Q (1) T − Q ,T )( Q (1) T + Q ,T )= (cid:18)Z T (1 − ρ t )Θ (1)1 ,t ( V )d t (cid:19) (cid:18)Z T (1 − ρ t ) h (1)0 ,t ( V ) − Θ (1)1 ,t ( V ) i d t (cid:19) . (5.9)Furthermore, noticeΘ (1)1 ,t ( V ) = Θ (1)0 ,t ( V ) + 2 v ,t (cid:16) Θ (1)1 ,t ( V ) − Θ (1)0 ,t ( V ) (cid:17) , Θ (1)2 ,t ( V ) = Θ (1)1 ,t ( V ) − v ,t (cid:16) Θ (1)1 ,t ( V ) − Θ (1)2 ,t ( V ) (cid:17) − (cid:16) Θ (1)0 ,t ( V ) − Θ (1)1 ,t ( V ) (cid:17) . Then, by application of the Minkowski, Burkholder-Davis-Gundy and Cauchy-Schwarz inequal-ities, it is sufficient to obtain L p bounds on Θ (1)1 ,t ( V ) and Θ (1)2 ,t ( V ) in order to obtain L p boundson the remainders of P and Q from eq. (5.9). For the cross remainder term, we have k Θ (1)2 ,T ( P Q ) k p ≤ k P (1) T k p k Q (1) T k p + k P (1)1 ,T k p k Q (1)1 ,T k p . We just need to check how to obtain L p bounds on P (1)1 ,T and Q (1)1 ,T . Notice k P ,T k p ≤ C p T − p (cid:18)Z T ρ pt k Θ (1)0 ,t ( V ) − Θ (1)1 ,t ( V ) k pp d t (cid:19) /p + 12 D p T − p (cid:18)Z T ρ pt k Θ (1)0 ,t ( V ) − Θ (1)1 ,t ( V ) k pp d t (cid:19) /p and k Q ,T k p ≤ D p T − p (cid:18)Z T (1 − ρ t ) p k Θ (1)0 ,t ( V ) − Θ (1)1 ,t ( V ) k pp d t (cid:19) /p . Again, all we need to obtain L p bounds on the cross remainder term are L p bounds on Θ (1)1 ,t ( V )and Θ (1)2 ,t ( V ). This validates item (2) and item (3). (cid:3) Fast calibration procedure
In this section, we present a fast calibration scheme in the Verhulst model. To do this, we appealto the results from Section 4 on the approximation of prices of put options, namely Lemma 4.1.We recognise that the approximation of the put option price is expressed in terms of iteratedintegral operators, defined in Definition 4.1. Our goal is to show that when parameters areassumed to piecewise-constant, these iterated integral operators • are closed-form, and • obey a convenient recursive property.Recall the integral operator from Definition 4.1, ω ( k,l ) t,T = Z Tt l u e R u k z d z d u, (6.1)and its n -fold iterated extension ω ( k ( n ) ,l ( n ) ) , ( k ( n − ,l ( n − ) ,..., ( k (1) ,l (1) ) t,T = ω (cid:0) k ( n ) ,l ( n ) w ( k ( n − ,l ( n − , ··· , ( k (1) ,l (1)) · ,T (cid:1) t,T , n ∈ N . (6.2)Let T = { T , T , . . . , T N − , T N = T } , where T i < T i +1 be a collection of maturity dates on[0 , T ], with ∆ T i := T i +1 − T i and ∆ T ≡
1. When the dummy functions are piecewise-constant,that is, l ( n ) t = l ( n ) i on t ∈ [ T i , T i +1 ) and similarly for k ( n ) , we can recursively calculate the integraloperators eq. (6.1) and eq. (6.2). Furthermore, let ˜ T = { , ˜ T , . . . , ˜ T ˜ N − , T } , where ˜ T i < ˜ T i +1 such that ˜ T ⊇ T . Let ∆ ˜ T i := ˜ T i +1 − ˜ T i with ∆ ˜ T ≡
1. Consider the ODE for ( v ,t ) in theVerhulst model eq. (1.1), d v ,t = κ t ( θ t − v ,t ) v ,t d t, v , = v . (6.3)We have the following Euler approximation to the ODE: ⇒ v ,t ≈ v , ˜ T i + κ i ( θ i − v , ˜ T i ) v , ˜ T i ∆ ˜ T i ˜ γ i ( t ) , t ∈ [ ˜ T i , ˜ T i +1 ) , where ˜ γ i ( t ) := ( t − ˜ T i ) / ∆ ˜ T i . It is true that an explicit solution exists for this ODE eq. (6.3).However, the solution is non-linear, which is problematic as we will need to utilise the linearityof the Euler approximation for our calibration scheme. Define e ( k ( n ) ,...,k (1) ) t := e R t P nj =1 k ( j ) z d z ,e ( h ( n ) ,...,h (1) ) v,t := e R t v ,z P nj =1 h ( j ) z d z ,ϕ ( k,h,p ) t,T i +1 := Z T i +1 t γ pi ( u ) e R uTi k z + h z v ,z d z d u, where γ i ( u ) := ( u − T i ) / ∆ T i and p ∈ N ∪ { } . In addition, define the n-fold extension of ϕ ( · , · , · ) · , · , ϕ ( k ( n ) ,h ( n ) ,p n ) ,..., ( k (1) ,h (1) ,p ) t,T i +1 := Z T i +1 t γ p n i ( u ) e R uTi k ( n ) z + h ( n ) z v ,z d z · ϕ ( k ( n − ,h ( n − ,p n − ) ,..., ( k (2) ,h (2) ,p ) , ( k (1) ,h (1) ,p ) u,T i +1 d u, where p n ∈ N ∪ { } We now assume that the dummy functions are piecewise-constant on T .However, to make this recursion simpler, we will assume that we are working on the finer grid ˜ T rather than T , since if the dummy functions are piecewise-constant on T , then there exists anequivalent parameterisation on ˜ T . For example, let k i be the constant value of k on [ T i , T i +1 ).Then there exists ˜ T ˜ i , ˜ T ˜ i +1 , . . . , ˜ T ˜ j such that ˜ T ˜ i = T i and ˜ T ˜ j = T i +1 . Then let ˜ k m := k i for m = ˜ i, . . . , ˜ j . Thus, without loss of generality, we can assume that we are working on ˜ T andwe will suppress the tilde from now on. With the assumption that the dummy functions arepiecewise-constant on T , we can obtain the integral operator at time T i +1 expressed by termsat T i . ω ( k (1) + h (1) v , · ,l (1) v q , · )0 ,T i +1 = ω ( k (1) + h (1) v , · ,l (1) v q , · )0 ,T i + l (1) i v q ,T i e ( k (1) ) T i e ( h (1) ) v,T i ϕ ( k (1) ,h (1) , T i ,T i +1 ,ω ( k (2) + h (2) v , · ,l (2) v q , · ) , ( k (1) + h (1) v , · ,l (1) v q , · )0 ,T i +1 = ω ( k (2) + h (2) v , · ,l (2) v q , · ) , ( k (1) + h (1) v , · ,l (1) v q , · )0 ,T i + l (1) i v q ,T i e ( k (1) ) T i e ( h (1) ) v,T i ϕ ( k (1) ,h (1) , T i ,T i +1 ω ( k (2) + h (2) v , · ,l (2) v q , · )0 ,T i + l (2) i l (1) i v q + q ,T i e ( k (2) ,k (1) ) T i e ( h (2) ,h (1) ) v,T i ϕ ( k (2) ,h (2) , , ( k (1) ,h (1) , T i ,T i +1 ,ω ( k (3) + h (3) v , · ,l (3) v q , · ) ,..., ( k (1) + h (1) v , · ,l (1) v q , · )0 ,T i +1 = ω ( k (3) + h (3) v , · ,l (3) v q , · ) ,..., ( k (1) + h (1) v , · ,l (1) v q , · )0 ,T i + l (1) i v q ,T i e ( k (1) ) T i e ( h (1) ) v,T i ϕ ( k (1) ,h (1) , T i ,T i +1 ω ( k (3) + h (3) v , · ,l (3) v q , · ) , ( k (2) + h (2) v , · ,l (2) v q , · )0 ,T i + l (2) i l (1) i v q + q ,T i e ( k (2) ,k (1) ) T i e ( h (2) ,h (1) ) v,T i ϕ ( k (2) ,h (2) , , ( k (1) ,h (1) , T i ,T i +1 ω ( k (3) + h (3) v , · ,l (3) v q , · )0 ,T i + l (3) i l (2) i l (1) i v q + q + q ,T i e ( k (3) ,k (2) ,k (1) ) T i e ( h (3) ,h (2) ,h (1) ) v,T i ϕ ( k (3) ,h (3) , , ( k (2) ,h (2) , , ( k (1) ,h (1) , T i ,T i +1 ,ω ( k (4) + h (4) v , · ,l (4) v q , · ) ,..., ( k (1) + h (1) v , · ,l (1) v q , · )0 ,T i +1 = ω ( k (4) + h (4) v , · ,l (4) v q , · ) ,..., ( k (1) + h (1) v , · ,l (1) v q , · )0 ,T i + l (1) i v q ,T i e ( k (1) ) T i e ( h (1) ) v,T i ϕ ( k (1) ,h (1) , T i ,T i +1 ω ( k (4) + h (4) v , · ,l (4) v q , · ) , ( k (3) + h (3) v , · ,l (3) v q , · ) , ( k (2) + h (2) v , · ,l (2) v q , · )0 ,T i + l (2) i l (1) i v q + q ,T i e ( k (2) ,k (1) ) T i e ( h (2) ,h (1) ) v,T i ϕ ( k (2) ,h (2) , , ( k (1) ,h (1) , T i ,T i +1 ω ( k (4) + h (4) v , · ,l (4) v q , · ) , ( k (3) + h (3) v , · ,l (3) v q , · )0 ,T i + l (3) i l (2) i l (1) i v q + q + q ,T i e ( k (3) ,k (2) ,k (1) ) T i e ( h (3) ,h (2) ,h (1) ) v,T i ϕ ( k (3) ,h (3) , , ( k (2) ,h (2) , , ( k (1) ,h (1) , T i ,T i +1 ω ( k (4) + h (4) v , · ,l (4) v q , · )0 ,T i + l (4) i l (3) i l (2) i l (1) i v q + q + q + q ,T i e ( k (4) ,k (3) ,k (2) ,k (1) ) T i e ( h (4) ,h (3) ,h (2) ,h (1) ) v,T i ϕ ( k (4) ,h (4) , , ( k (3) ,h (3) , , ( k (2) ,h (2) , , ( k (1) ,h (1) , T i ,T i +1 . For example ϕ ( k (3) ,p ,q ) , ( k (2) ,p ,q ) , ( k (1) ,p ,q ) t,T i +1 = Z T i +1 t γ p i ( u ) v q ,u e R u Ti k (3) z d z (cid:18)Z T i +1 u γ p i ( u ) v q ,u e R u Ti k (2) z d z (cid:18)Z T i +1 u γ p i ( u ) v q ,u e R u Ti k (1) z d z d u (cid:19) d u (cid:19) d u . The only terms here that are not closed-form are the functions e ( · ,..., · ) · , e ( · ,..., · ) v, · and ϕ ( · , · , · ) ,..., ( · , · , · ) t,T i +1 .For t ∈ ( T i , T i +1 ], we can derive the following: e ( k ( n ) ,...,k (1) ) t = e ( k ( n ) ,...,k (1) ) T i e ∆ T i γ i ( t ) P nj =1 k ( j ) i = e P i − m =0 ∆ T m P nj =1 k ( j ) m e ∆ T i γ i ( t ) P nj =1 k ( j ) i ,e ( h ( n ) ,...,h (1) ) v,t = e ( h ( n ) ,...,h (1) ) v,T i e ∆ T i γ i ( t ) v ,Ti P nj =1 h ( j ) i = e P i − m =0 ∆ T m v ,Tm P nj =1 h ( j ) m e ∆ T i γ i ( t ) v ,Ti P nj =1 h ( j ) i , where e ( k ( n ) ,...,k (1) )0 = 1 and e ( h ( n ) ,...,h (1) ) v, = 1. Let ˜ k i := k i + h i v ,T i and ˜ k ( n ) i := k ( n ) i + h ( n ) i v ,T i .Then ϕ ( k,h,p ) t,T i +1 = k i (cid:16) e ˜ k i ∆ T i − γ pi ( t ) e ˜ k i ∆ T i γ i ( t ) − p ∆ T i ϕ ( k,h,p − t,T i +1 (cid:17) , ˜ k i = 0 , p ≥ , k i (cid:16) e ˜ k i ∆ T i − e ˜ k i ∆ T i γ i ( t ) (cid:17) , ˜ k i = 0 , p = 0 , p +1 ∆ T i (cid:16) − γ p +1 i ( t ) (cid:17) ˜ k i = 0 , p ≥ . In addition, for n ≥ ϕ ( k ( n ) ,h ( n ) ,p n ) ,..., ( k (1) ,h (1) ,p ) t,T i +1 = k ( n ) i (cid:16) ϕ ( k ( n ) + k ( n − ,h ( n ) + h ( n − ,p n + p n − ) , ( k ( n − ,h ( n − ,p n − ) ,..., ( k (1) ,h (1) ,p ) t,T i +1 − p n ∆ T i ϕ ( k ( n ) ,h ( n ) ,p n − , ( k ( n − ,h ( n − ,p n − ) ,..., ( k (1) ,h (1) ,p ) t,T i +1 − γ p n i ( t ) e ˜ k ( n ) i ∆ T i γ i ( t ) ϕ ( k ( n − ,h ( n − ,p n − ) ,..., ( k (1) ,h (1) ,p ) t,T i +1 (cid:17) , ˜ k ( n ) i = 0 , p n ≥ , k ( n ) i (cid:16) ϕ ( k ( n ) + k ( n − ,h ( n ) + h ( n − ,p n − ) , ( k ( n − ,h ( n − ,p n − ) ,..., ( k (1) ,h (1) ,p ) t,T i +1 − e ˜ k ( n ) i ∆ T i γ i ( t ) ϕ ( k ( n − ,h ( n − ,p n − ) ,..., ( k (1) ,h (1) ,p ) t,T i +1 (cid:17) , ˜ k ( n ) i , = 0 , p n = 0 , ∆ T i p n +1 (cid:16) ϕ ( k ( n − ,h ( n − ,p n + p n − +1) , ( k ( n − ,h ( n − ,p n − ) ,..., ( k (1) ,h (1) ,p ) t,T i +1 − γ p n +1 i ( t ) ϕ ( k ( n − ,h ( n − ,p n − ) ,..., ( k (1) ,h (1) ,p ) t,T i +1 (cid:17) , ˜ k ( n ) i = 0 , p n ≥ . Remark 6.1 (Fast calibration scheme) . To implement our fast calibration scheme, one executesthe following algorithm. Let µ t ≡ µ = ( µ (1) , µ (2) , . . . , µ ( n ) ) be an arbitrary set of parametersand denote by ω t an arbitrary integral operator. • Calibrate µ over [0 , T ) to obtain µ . This involves computing ω T . In general ω ( k ( n ) + h ( n ) v , · ,l ( n ) v qn , · ) ,..., ( k (1) + h (1) v , · ,l (1) v q , · )0 ,T i +1 = n +1 X m =1 ω ( k ( n ) + h ( n ) v , · ,l ( n ) v qn , · ) ,..., ( k ( m ) + h ( m ) v , · ,l ( m ) v qm , · )0 ,T i · m − Y j =0 l ( j +1) i ! v P m − j =1 q j ,T i e ( k ( n − m +2) ,...,k (1) ) T i e ( h ( n − m +2) ,...,h (1) ) v,T i ϕ ( k ( n − m +2) ,h ( n − m +2) , ,..., ( k (1) ,h (1) , T i ,T i +1 , where whenever the index goes outside of { , . . . , n } , then that term is equal to 1. • Calibrate µ over [ T , T ) to obtain µ . This involves computing ω T which is in terms of ω T ,the latter already being computed in the previous step. • Repeat until time T N . Remark 6.2.
In the general model eq. (2.1), the approximation of the put option price is givenby Theorem 4.1, which is expressed in terms of these integral operators eq. (6.1) and eq. (6.2). Itis clear that as long as a unique solution for ( v ,t ) exists, then the same procedure outlined abovewill result in a fast calibration scheme. This is due to the fact that the Euler approximationto the ODE for ( v ,t ) yields a linear equation, which is all that is necessary to obtain the fastcalibration scheme. Thus, this fast calibration scheme can be adapted to the general modeleq. (2.1) immediately. This extends the fast calibration scheme presented in Langren´e et al.[18], where they require the ODE for ( v ,t ) to be linear.7. Numerical tests and sensitivity analysis
We test our approximation method by considering the sensitivity of our approximation withrespect to one parameter at a time. Specifically, for an arbitrary parameter set ( µ , µ , . . . , µ n ),we vary only one of the µ i at a time and keep the rest fixed. Then, we compute implied volatilitiesvia our approximation method as well as the Monte Carlo for strikes corresponding to Put 10,25 and ATM deltas. Specifically,Error( µ ) = σ IM − Approx ( µ, K ) − σ IM − Monte ( µ, K )for K corresponding to Put 10, Put 25 and ATM.For all our simulations, we use 2,000,000 Monte Carlo paths, and 24 time steps per day. This isto reduce the Monte Carlo and discretisation errors sufficiently well.The safe parameter set is ( S , v , r d , r f ) = (100 . , . , . ,
0) with( κ, θ, λ, ρ ) = (8 . , . , . , − . , T = 1M , (8 . , . , . , − . , T = 3M , (8 . , . , . , − . , T = 6M , (8 . , . , . , − . , T = 1Y . In our analysis, we vary one of the parameters ( κ, θ, λ, ρ ) at a time and keep the rest fixed.7.1.
Varying κ . We vary κ over { , , , , , , , } . Table 7.1. κ : Error for ATM implied volatilities in basis points κ Table 7.2. κ : Error for Put 25 implied volatilities in basis points κ Table 7.3. κ : Error for Put 10 implied volatilities in basis points κ Varying θ . We vary θ over the set { . , . , . , . , . , . , . , . } . Table 7.4. θ : Error for ATM implied volatilities in basis points θ Table 7.5. θ : Error for Put 25 implied volatilities in basis points θ Table 7.6. θ : Error for Put 10 implied volatilities in basis points θ Varying λ . We vary λ over the set { . , . , . , . , . , . , . , . } . Table 7.7. λ : Error for ATM implied volatilities in basis points λ Table 7.8. λ : Error for Put 25 implied volatilities in basis points λ Table 7.9. λ : Error for Put 10 implied volatilities in basis points λ Varying ρ . We vary ρ over the set {− . , − . , − . , − . , − . , − . , − . , } . Table 7.10. ρ : Error for ATM implied volatilities in basis points ρ -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 01M 7.93 5.67 3.75 2.180 0.970 0.100 -0.410 -0.563M 19.34 13.81 9.17 5.41 2.54 0.54 -0.59 -0.836M 30.76 22.13 14.87 8.97 4.40 1.17 -0.73 -1.301Y 29.67 19.23 10.51 3.49 -1.84 -5.49 -7.47 -7.77 Table 7.11. ρ : Error for Put 25 implied volatilities in basis points ρ -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 01M 6.79 5.27 3.92 2.73 1.71 0.870 0.210 -0.263M 17.74 14.28 11.09 8.18 5.59 3.34 1.48 0.046M 31.23 25.86 20.76 15.97 11.55 7.54 4.02 1.071Y 35.19 28.44 21.91 15.68 9.81 4.39 -0.49 -4.72 Table 7.12. ρ : Error for Put 10 implied volatilities in basis points ρ -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 01M -0.36 0.38 0.90 1.23 1.34 1.24 0.92 0.383M 5.57 7.09 8.05 8.43 8.22 7.38 5.88 3.656M 21.95 23.90 24.78 24.58 23.27 20.77 17.01 11.851Y 37.53 40.42 41.55 40.86 38.30 33.72 26.97 17.75The sensitivity analysis displays errors that are on average approximately 10-50bps out, withsmall errors being exhibited for reasonable parameter values, and large errors for more unrea-sonable parameter values. The errors also behave as we expect. For example, large values ofvol-of-vol should intuitively result in an error which is large, since the expansion procedure wascontingent on vol-of-vol being small. This behaviour is exhibited in the above numerical results.A high mean reversion speed should intuitively result in an error which is lower, which is alsoseen in the κ numerical sensitivity analysis.8. Conclusion
We have provided a second-order approximation for the price of a put option in the Verhulstmodel with time-dependent parameters, as well as an associated fast calibration scheme. Inaddition, we deduce that our expansion methodology can easily extend to models with general drift and power type diffusions that satisfy some regularity conditions. We provide the formulafor the second-order put option price in this general setting too. Moreover, when parameters areassumed to be piecewise-constant, our approximation formula is closed-form. In addition, thisassumption allows us to devise a fast calibration scheme by exploiting recursive properties ofthe iterated integral operators in terms of which our approximation formulas are expressed. Weestablish the explicit form of the error term induced by the expansion. We determine sufficientingredients for obtaining a meaningful bound on this error term, these ingredients essentiallybeing higher order moments pertaining to the volatility process. Lastly, we perform a numericalsensitivity analysis for the approximation formula in the Verhulst model, and show that theerror is small, behaves as we expect with respect to parameter changes, and is within therange for application purposes. In particular, it is of our opinion that the general second-orderapproximation formula will be very useful for practitioners, as obtaining the pricing formula fordifferent models requires just some basic differentiation, the formula is essentially instantaneousto compute, the fast calibration scheme can be used to calibrate models rapidly, and the erroris sufficiently low for application purposes. References [1] Al`os, E. (2006). A generalization of the Hull and White formula with applications to optionpricing approximation.
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Appendix A. Mixing solution
In this appendix, we present a derivation of the result referred to as the mixing solution by Hulland White [15]. This result is crucial for the expansion methodology implemented in Section 3.Hull and White first established the expression for the case of independent Brownian motions W and B . Later on, this was extended to the correlated Brownian motions case, see Romanoand Touzi [26], Willard [33]. Theorem A.1 (Mixing solution) . Under a chosen domestic risk-neutral measure Q , supposethat the spot S with volatility V are given as the solution to the general model eq. (2.1). Define X as the log-spot and k the log-strike. Namely, X t = ln S t and k = ln K . ThenPut = e − R T r dt d t E ( e k − e X T ) + = E n e − R T r dt d t E (cid:2) ( e k − e X T ) + |F BT (cid:3)o = E (cid:18) P BS (cid:18) x − Z T ρ t V t d t + Z T ρ t V t d B t , Z T V t (1 − ρ t )d t (cid:19)(cid:19) , where P BS is given in eq. (2.5). Proof.
By writing the driving Brownian motion of the spot as W t = R t ρ u d B u + R t p − ρ u d Z u ,where Z is a Brownian motion under Q which is independent of B , this yields the explicit strongsolution of X as X T = x + Z T (cid:18) r dt − r ft − V t (cid:19) d t + Z T ρ t V t d B t + Z T V t q − ρ t d Z t . First, notice that V is adapted to the filtration ( F Bt ) ≤ t ≤ T . Thus, it is evident that X T |F BT willhave a normal distribution. Namely, X T |F BT ∼ N (cid:0) ˆ µ ( T ) , ˜ σ ( T ) (cid:1) , ¯ µ ( T ) := x + Z T (cid:16) r dt − r ft (cid:17) d t − Z T V t d t + Z T ρ t V t d B t , ˜ σ ( T ) := Z T V t (1 − ρ t )d t. Also, let ˜ µ ( T ) := ¯ µ ( T ) − R T ( r dt − r ft )d t . Hence the calculation of e − R T r dt d t E (( e k − e X T ) + |F BT )will result in a Black-Scholes like formula. e − R T r dt d t E (( e k − e X T ) + |F BT )= e k e − R T r dt d t N (cid:18) k − ˆ µ ( T )˜ σ ( T ) (cid:19) − e − R T r dt d t e ˆ µ ( T )+ ˜ σ ( T ) N (cid:18) k − ˆ µ ( T ) − ˜ σ ( T )˜ σ ( T ) (cid:19) = e k e − R T r dt d t N k − ˆ µ ( T ) − ˜ σ ( T )˜ σ ( T ) + 12 ˜ σ ( T ) ! − e ˜ µ ( T )+ ˜ σ ( T ) e − R T r ft d t N k − ˆ µ ( T ) − ˜ σ ( T )˜ σ ( T ) −
12 ˜ σ ( T ) ! = e k e − R T r dt d t N k − (˜ µ ( T ) + ˜ σ ( T )) − R T ( r dt − r ft )d t ˜ σ ( T ) + 12 ˜ σ ( T ) ! − e ˜ µ ( T )+ ˜ σ ( T ) e − R T r ft d t N k − (˜ µ ( T ) + ˜ σ ( T )) − R T ( r dt − r ft )d t ˜ σ ( T ) −
12 ˜ σ ( T ) ! . It is now immediate that e − R T r dt d t E (( e k − e X T ) + |F BT ) = P BS (cid:0) ˜ µ ( T ) + ˜ σ ( T ) , ˜ σ ( T ) (cid:1) . (cid:3) Appendix B. Malliavin calculus machinery
In the following appendix we give a short excerpt on Malliavin calculus. This is predominantlyto fix notation. We point the reader towards the lecture notes by Nualart [23] for a completeand accessible source on Malliavin calculus.The underlying framework of Malliavin calculus involves a zero-mean Gaussian process ˜ W in-duced by an underlying separable Hilbert space H . Specifically, we have that ˜ W = { ˜ W ( h ) : h ∈ H } is a zero-mean Gaussian process such that E ( ˜ W ( h ) ˜ W ( g )) = h h, g i H .We need only make use of Malliavin calculus when the underlying Hilbert space is H = L ([0 , T ]) := L ([0 , T ] , B ([0 , T ]) , λ ∗ ) , where λ ∗ is the one-dimensional Lebesgue measure. Thus the inner product on H is h h, g i H = Z T h t g t λ ∗ (d t ) = Z T h t g t d t. Our Gaussian process ˜ W will be explicitly given as ˜ W ( h ) := R T h t d ˜ B t , where ˜ B is a Brownianmotion with natural filtration ( F ˜ Bt ) ≤ t ≤ T and h ∈ L ([0 , T ]). By use of the zero-mean and Itˆoisometry properties of the Itˆo integral, it can be seen that such a Hilbert space H and Gaussianprocess ˜ W satisfy the framework for Malliavin calculus. Definition B.1 (Malliavin derivative) . Let S n := (cid:26) F = f (cid:18)Z T h ,t d ˜ B t , . . . , Z T h n,t d ˜ B t (cid:19) : f ∈ C ∞ p ( R n ; R ) , h i, · ∈ H (cid:27) and S := S n ≥ S n . Here C ∞ p ( R n ; R ) is the space of smooth Borel measurable functions f : R n / B ( R n ) → R / B ( R ) which have at most polynomial growth. Thus, the elements of S n arerandom variables. For F ∈ S n , the Malliavin derivative D is an operator from S → L ([0 , T ] × Ω)and is given by ( DF ) t := n X i =1 ∂ i f (cid:18)Z T h ,u d ˜ B u , . . . , Z T h n,u d ˜ B u (cid:19) h i,t . Proposition B.1 (Extending domain of D ) . Define the space D ,p as the completion of S withrespect to the norm k F k ,p := E | F | p + E (cid:20)Z T ( D t F ) p d t (cid:21) p ! /p where F ∈ S and p ≥
1. Then the operator D is closable to D ,p , and D : D ,p → L p ([0 , T ] × Ω).
Proof.
See Nualart [23]. (cid:3)
The Malliavin derivative satisfies a duality relationship.
Proposition B.2 (Malliavin duality relationship) . Let G ∈ D , and α ∈ L ([0 , T ] × Ω) suchthat α is adapted to the filtration ( F ˜ Bt ) ≤ t ≤ T . Then E (cid:18)Z t α s ( DG ) s d s (cid:19) = E (cid:18) G Z t α s d ˜ B s (cid:19) for any t < T . Proof.
See Nualart [23]. (cid:3)
Lemma B.1 (Malliavin integration by parts) . Let ˜ T ≤ T and ˆ T ≤ T . Also, let α ∈ L ([0 , T ] × Ω)such that α is adapted to ( F ˜ Bt ) ≤ t ≤ T . Then, E " l Z ˜ T h u d ˜ B u ! Z ˆ T α u d ˜ B u ! = E " l ′ Z ˜ T h u d ˜ B u ! Z ˜ T ∧ ˆ T h u α u d u ! . In particular, for ˜ T = T and ˆ T = t < T , E (cid:20) l (cid:18)Z T h u d ˜ B u (cid:19) (cid:18)Z t α u d ˜ B u (cid:19)(cid:21) = E (cid:20) l ′ (cid:18)Z T h u d ˜ B u (cid:19) (cid:18)Z t h u α u d u (cid:19)(cid:21) . Proof.
Let G = l (cid:16)R ˜ T h u d ˜ B u (cid:17) . Then G ∈ S ⊆ D , and ( DG ) t = l ′ (cid:16)R ˜ T h u d ˜ B u (cid:17) h t { t ≤ ˜ T } .Then the result follows by a consequence of Proposition B.2. (cid:3) Appendix C. P BS partial derivatives This appendix contains some partial derivatives for the Black-Scholes put option formula P BS One can think of these partial derivatives as being analogous to the Black-Scholes Greeks.However, these are slightly different as our Black-Scholes formulas are parametrised with respectto log-spot and integrated variance rather than spot and volatility respectively. C.1.
First-order P BS . ∂ x P BS = e x e − R T r fu d u (cid:16) N ( d ln+ ) − (cid:17) ,∂ y P BS = e x e − R T r fu d u φ ( d ln+ )2 √ y . C.2.
Second-order P BS . ∂ xx P BS = e x e − R T r fu d u φ ( d ln+ ) √ y + ∂ x P BS = e x e − R T r fu d u φ ( d ln+ ) √ y + e x e − R T r fu d u (cid:16) N ( d ln+ ) − (cid:17) ,∂ xy P BS = ( − e x e − R T r fu d u φ ( d ln+ ) d ln − y ,∂ yy P BS = e x e − R T r fu d u φ ( d ln+ )4 y / ( d ln − d ln+ − . C.3.
Third-order P BS . ∂ xxx P BS = e x e − R T r fu d u φ ( d ln+ ) y ( √ y − d ln+ ) + ∂ xx P BS = e x e − R T r fu d u φ ( d ln+ ) y (2 √ y − d ln+ ) + e x e − R T r fu d u ( N ( d ln+ ) − ,∂ xxy P BS = ( − e x e − R T r fu d u φ ( d ln+ )2 y / (cid:16) d ln − √ y + (1 − d ln − d ln+ ) (cid:17) ,∂ xyy P BS = e x e − R T r fu d u φ ( d ln+ )4 y (cid:16) (2 d ln+ − √ y ) + (1 − d ln − d ln+ )( d ln+ − √ y ) (cid:17) ,∂ yyy P BS = e x e − R T r fu d u φ ( d ln+ )8 y / (cid:16) ( d ln − d ln+ − − ( d ln − + d ln+ ) + 2 (cid:17) . C.4.
Fourth-order P BS . ∂ xxxx P BS = ( − e x e − R T r fu d u φ ( d ln+ ) y / (1 − ( d ln+ − √ y ) ) + ∂ xxx P BS = e x e − R T r fu d u φ ( d ln+ ) y / h ( d ln+ − √ y ) + 2 y − d ln+ √ y − i + e x e − R T r fu d u ( N ( d ln+ ) − ,∂ xxxy P BS = e x e − R T r fu d u φ ( d ln+ )2 y " ( √ y − d ln+ )( d ln − d ln+ −
2) + ( √ y + d ln − ) − d ln − y − √ y (1 − d ln − d ln+ ) ,∂ xxyy P BS = ( − e x e − R T r fu d u φ ( d ln+ )2 y / " d ln − d ln+ + 12 ( d ln − ) d ln+ √ y −
12 ( d ln − ) ( d ln+ ) + 12 y − √ y (cid:16) d ln − + d ln+ (cid:17) − ,∂ xyyy P BS = e x e − R T r fu d u φ ( d ln+ )8 y / " y / ( d ln − d ln+ − d ln+ − √ y ) − √ y ( d ln − + d ln+ )+ √ y ( √ y − d ln+ ) (cid:16) ( d ln − d ln+ − − ( d ln − + d ln+ ) + 2 (cid:17) ,∂ yyyy P BS = e x e − R T r fu d u φ ( d ln+ )8 y /
12 ( d ln − d ln+ − ( d ln − d ln+ − − ( d ln − d ln+ − d ln − + d ln+ ) −
12 ( d ln − + d ln+ ) ( d ln − d ln+ −
7) + ( d ln − d ln+ − ! . Appendix D. Proof of Theorem 4.1
In this appendix, we provide the proof of Theorem 4.1. In order to do so, we will utilise resultsfrom Malliavin calculus extensively. A short treatment of Malliavin calculus is presented in Ap-pendix B. In addition to Malliavin calculus machinery, we will require the following ingredients.
Proposition D.1 ( P BS partial derivative relationship) . ∂ y P BS ( x, y ) = 12 ( ∂ xx P BS ( x, y ) − ∂ x P BS ( x, y )) . Proof.
A simple application of differentiation yields the result. (cid:3)
In addition, we will make extensive use of the stochastic integration by parts formula, which wewill list here for convenience.
Remark D.1 (Stochastic integration by parts) . Let X and Y be semimartingales with respectto a filtration ( ˜ F t ). Then we have X T Y T = Z T X t d Y t + Z T Y t d X t + Z T d h X, Y i t , given that the above Itˆo integrals exist. In particular, if X t = R t x u d ˜ X u and Y t = R t y u d ˜ Y u , where˜ X and ˜ Y are semimartingales and x and y are stochastic processes adapted to the underlyingfiltration ( ˜ F t ) such that X and Y exist, then the stochastic integration by parts formula readsas Z T x t d ˜ X t Z T y t d ˜ Y t = Z T (cid:18)Z t x u d ˜ X u (cid:19) y t d ˜ Y t + Z T (cid:18)Z t y u d ˜ Y u (cid:19) x t d ˜ X t + Z T x t y t d h ˜ X, ˜ Y i t . Lemma D.1.
Let Z be a semimartingale such that Z = 0 and let f be a Lebesgue integrabledeterministic function. Then Z T f t Z t d t = Z T ω (0 ,f ) t,T d Z t . Proof.
A simple application of Remark D.1 (stochastic integration by parts) gives the desiredresult. (cid:3)
Using Lemma D.1 we can obtain the following lemma.
Lemma D.2.
The following equalities hold: E (cid:18) l (cid:18)Z T ρ t v ,t d B t (cid:19) Z T ξ t V ,t d t (cid:19) = ω ( − α x ,ρλv µ +10 , · ) , ( α x ,ξ )0 ,T E (cid:18) l (1) (cid:18)Z T ρ t v ,t d B t (cid:19)(cid:19) , (D.1) E (cid:18) l (cid:18)Z T ρ t v ,t d B t (cid:19) Z T ξ t V ,t d t (cid:19) = 2 ω ( − α x ,ρλv µ +10 , · ) , ( − α x ,ρλv µ +10 , · ) , (2 α x ,ξ )0 ,T E (cid:18) l (2) (cid:18)Z T ρ t v ,t d B t (cid:19)(cid:19) + ω ( − α x ,λ v µ , · ) , (2 α x ,ξ )0 ,T E (cid:18) l (cid:18)Z T ρ t v ,t d B t (cid:19)(cid:19) , (D.2) E (cid:18) l (cid:18)Z T ρ t v ,t d B t (cid:19) Z T ξ t V ,t d t (cid:19) = ω ( − α x ,λ v µ , · ) , ( α x ,α xx ) , ( α x ,ξ )0 ,T E (cid:18) l (cid:18)Z T ρ t v ,t d t (cid:19)(cid:19) + ( ω ( − α x ,ρλv µ +10 , · ) , ( − α x ,ρλv µ +10 , · ) , ( α x ,α xx ) , ( α x ,ξ )0 ,T + 2 µω ( − α x ,ρλv µ +10 , · ) , (0 ,ρλv µ − , · ) , ( α x ,ξ )0 ,T ) E (cid:18) l (2) (cid:18)Z T ρ t v ,t d B t (cid:19)(cid:19) , (D.3) E l (cid:18)Z T ρ t v ,t d B t (cid:19) (cid:26)Z T ξ t V ,t d t (cid:27) ! = 2 ω ( − α x ,λ v µ , · ) , ( α x ,ξ ) , ( α x ,ξ )0 ,T E (cid:18) l (cid:18)Z T ρ t v ,t d B t (cid:19)(cid:19) + (cid:18) ω ( − α x ,ρλv µ +10 , · ) , ( α x ,ξ )0 ,T (cid:19) E (cid:18) l (2) (cid:18)Z T ρ t v ,t d B t (cid:19)(cid:19) . (D.4)Here we write α x := α x ( · , v , · ) and α xx := α xx ( · , v , · ) for readability purposes. Proof.
We will only show how to obtain eq. (D.1). Equations (D.2) to (D.4) can be obtained ina similar way. First, we replace V ,t with its explicit form from eq. (D.5). Thus we can writethe left hand side of eq. (D.1) as E (cid:18) l (cid:18)Z T ρ t v ,t d B t (cid:19) Z T ξ t e R t α x ( z,v ,z )d z (cid:18)Z t λ s v µ ,s e − R s α x ( z,v ,z )d z d B s (cid:19) d t (cid:19) . Using Lemma D.1 with f t = ξ t e R t α x ( z,v ,z )d z and Z t = R t λ s v µ ,s e − R s α x ( z,v ,z )d z d B s , we get E (cid:18) l (cid:18)Z T ρ t v ,t d B t (cid:19) Z T ξ t e R t α x ( z,v ,z )d z (cid:18)Z t λ s v µ ,s e − R s α x ( z,v ,z )d z d B s (cid:19) d t (cid:19) = E (cid:18) l (cid:18)Z T ρ t v ,t d B t (cid:19) Z T ω ( α x ,ξ ) t,T λ t v µ ,t e − R t α x ( z,v ,z )d z d B t (cid:19) . Lastly, appealing to the Malliavin integration by parts Lemma B.1 we obtain E (cid:18) l (1) (cid:18)Z T ρ t v ,t d B t (cid:19) Z T ω ( α x ,ξ ) t,T ρ t λ t v µ +10 ,t e − R t α x ( z,v ,z )d z d t (cid:19) = (cid:18)Z T ω ( α x ,ξ ) t,T ρ t λ t v µ +10 ,t e − R t α x ( z,v ,z )d z d t (cid:19) E (cid:18) l (1) (cid:18)Z T ρ t v ,t d B t (cid:19)(cid:19) = ω ( − α x ,ρλv µ +10 , · ) , ( α x ,ξ )0 ,T E (cid:18) l (1) (cid:18)Z T ρ t v ,t d B t (cid:19)(cid:19) . In addition, to obtain eq. (D.4), notice the following integral property holds: (cid:16) ω ( k (2) ,l (2) ) , ( k (1) ,l (1) )0 ,T (cid:17) = 2 ω ( k (2) ,l (2) ) , ( k (1) ,l (1) ) , ( k (2) ,l (2) ) , ( k (1) ,l (1) )0 ,T + 4 ω ( k (2) ,l (2) ) , ( k (2) ,l (2) ) , ( k (1) ,l (1) ) , ( k (1) ,l (1) )0 ,T . (cid:3) Now we can proceed with the proof of Theorem 4.1, which will be broken up over a number ofsubsections. For the time being we will not yet enforce Assumption C.D.1. E ˜ P BS . Notice that E ˜ P BS = g (0) = E ( e k − e X (0) T ) + . Since the perturbed volatility process V ( ε ) t is deterministic when ε = 0, then g (0) will just be a Black-Scholes formula. Thus we have E ˜ P BS = P BS (cid:18) x , Z T v ,t d t (cid:19) . D.2. C x . Using Lemma B.1 (Malliavin integration by parts), E ∂ x ˜ P BS Z T ρ t (cid:18) V ,t + 12 V ,t (cid:19) d B t = E ∂ xx ˜ P BS Z T ρ t v ,t (cid:18) V ,t + 12 V ,t (cid:19) d t. Furthermore, using Proposition D.1 ( P BS partial derivative relationship), E ∂ xx ˜ P BS Z T ρ t v ,t (cid:18) V ,t + 12 V ,t (cid:19) d t = E (2 ∂ y + ∂ x ) ˜ P BS Z T ρ t v ,t (cid:18) V ,t + 12 V ,t (cid:19) d t. Thus C x = 2 E ∂ y ˜ P BS Z T ρ t v ,t (cid:18) V ,t + 12 V ,t (cid:19) d t − E ∂ x ˜ P BS Z T ρ t V ,t d t. D.3. C xx . For C xx we first use Remark D.1 (stochastic integration by parts) to reduce thisexpression. C xx = 12 E ∂ xx ˜ P BS (cid:18)Z T ρ t V ,t d B t − Z T ρ t v ,t V ,t d t (cid:19) = 12 E ∂ xx ˜ P BS (cid:18)Z T ρ t V ,t d B t (cid:19) − E ∂ xx ˜ P BS (cid:18)Z T ρ t v ,t V ,t d t (cid:19) (cid:18)Z T ρ t V ,t d B t (cid:19) + 12 E ∂ xx ˜ P BS (cid:18)Z T ρ t v ,t V ,t d t (cid:19) = E ∂ xx ˜ P BS (cid:18)Z T (cid:26)Z t ρ s v ,s V ,s d s (cid:27) ρ t v ,t V ,t d t (cid:19) − E ∂ xx ˜ P BS (cid:18)Z T (cid:26)Z t ρ s v ,s V ,s d s (cid:27) ρ t V ,t d B t + Z T (cid:26)Z t ρ s V ,s d B s (cid:27) ρ t v ,t V ,t d t (cid:19) + E ∂ xx ˜ P BS (cid:18)Z T (cid:26)Z t ρ s V ,s d B s (cid:27) ρ t V ,t d B t (cid:19) + 12 E ∂ xx ˜ P BS (cid:18)Z T ρ t V ,t d t (cid:19) = E ∂ xx ˜ P BS (cid:18)Z T (cid:26)(cid:16) Z t ρ s V ,s d B s − Z t ρ s v ,s V ,s d s (cid:17)(cid:16) ρ t V ,t d B t − ρ t v ,t V ,t d t (cid:17)(cid:27)(cid:19) + 12 E ∂ xx ˜ P BS (cid:18)Z T ρ t V ,t d t (cid:19) . Using Lemma B.1 (Malliavin integration by parts), C xx = − E ∂ xx ˜ P BS (cid:18)Z T (cid:26)(cid:16) Z t ρ s V ,s d B s − Z t ρ s v ,s V ,s d s (cid:17) ρ t v ,t V ,t d t (cid:27)(cid:19) + E ∂ xxx ˜ P BS (cid:18)Z T (cid:16) Z t ρ s V ,s d B s − Z t ρ s v ,s V ,s d s (cid:17) ρ t v ,t V ,t d t (cid:19) + 12 E ∂ xx ˜ P BS (cid:18)Z T ρ t V ,t d t (cid:19) = E ( ∂ xxx ˜ P BS − ∂ xx ˜ P BS ) (cid:18)Z T (cid:16) Z t ρ s V ,s d B s − Z t ρ s v ,s V ,s d s (cid:17) ρ t v ,t V ,t d t (cid:19) + 12 E ∂ xx ˜ P BS (cid:18)Z T ρ t V ,t d t (cid:19) . Then using Proposition D.1 ( P BS partial derivative relationship), C xx = 2 E ∂ xy ˜ P BS (cid:18)Z T (cid:16) Z t ρ s V ,s d B s − Z t ρ s v ,s V ,s d s (cid:17) ρ t v ,t V ,t d t (cid:19) + 12 E ∂ xx ˜ P BS (cid:18)Z T ρ t V ,t d t (cid:19) . Adding the terms C x , C xx and C y yields C x + C xx + C y = E ∂ y ˜ P BS (cid:18)Z T v ,t V ,t + V ,t + v ,t V ,t d t (cid:19) + 2 E ∂ xy ˜ P BS (cid:18)Z T (cid:18)Z t ρ s V ,s d B s − Z t ρ s v ,s V ,s d s (cid:19) ρ t v ,t V ,t d t (cid:19) . D.4. C xy . For C xy we use Remark D.1 (stochastic integration by parts) to obtain E ∂ xy ˜ P BS (cid:18)Z T ρ t V ,t d B t − Z T ρ t v ,t V ,t d t (cid:19) (cid:18)Z T (1 − ρ t )(2 v ,t V ,t )d t (cid:19) = 2 E ∂ xy ˜ P BS (cid:18)Z T (cid:18)Z t (1 − ρ s ) v ,s V ,s d s (cid:19) (cid:0) ρ t V ,t d B t − ρ t v ,t V ,t d t (cid:1)(cid:19) + 2 E ∂ xy ˜ P BS Z T (cid:18)Z t ρ s V ,s d B s − Z t ρ s v ,s V ,s d s (cid:19) (1 − ρ t ) v ,t V ,t d t = 2 E ∂ xy ˜ P BS (cid:18)Z T (cid:18)Z t (1 − ρ s ) v ,s V ,s d s (cid:19) (cid:0) ρ t V ,t d B t − ρ t v ,t V ,t d t (cid:1)(cid:19) − E ∂ xy ˜ P BS Z T (cid:18)Z t ρ s v ,s V ,s d s (cid:19) v ,t V ,t d t + 2 E ∂ xy ˜ P BS (cid:18)Z T (cid:18)Z t ρ s V ,s d B s (cid:19) v ,t V ,t d t (cid:19) − E ∂ xy ˜ P BS (cid:18)Z T (cid:18)Z t ρ s V ,s d B s − Z t ρ s v ,s V ,s d s (cid:19) ρ t v ,t V ,t d t (cid:19) . Furthermore, using Proposition B.2 (Malliavin duality relationship),ˆ C xy := 2 E ∂ xy ˜ P BS (cid:18)Z T v ,t V ,t (cid:18)Z t ρ s V ,s d B s (cid:19) d t (cid:19) = 2 Z T E ∂ xy ˜ P BS v ,t V ,t (cid:18)Z t ρ s V ,s d B s (cid:19) d t = 2 Z T E (cid:18)Z t ρ s V ,s D Bs ( ∂ xy ˜ P BS v , · V , · )d s (cid:19) d t. Using the definition of the Malliavin derivative, we obtain D Bs ( ∂ xy ˜ P BS v , · V , · ) = ∂ xxy ˜ P BS v ,t V ,t ρ s v ,s { s ≤ T } + ∂ xy ˜ P BS D Bs ( v , · V , · )= ∂ xxy ˜ P BS v ,t V ,t ρ s v ,s { s ≤ T } + ∂ xy ˜ P BS v ,t (cid:16) e R t α x ( u,v ,u )d u β ( s, v ,s ) e − R s α x ( z,v ,z )d z { s ≤ t } (cid:17) , where we have used the explicit form for V ,t from eq. (3.4). Thus using Proposition B.2 (Malli-avin duality relationship),2 Z T E (cid:18)Z t ρ s V ,s D Bs ( ∂ xy ˜ P BS v , · V , · )d s (cid:19) d t = 2 Z T E ∂ xxy ˜ P BS (cid:18)Z t ρ s v ,s V ,s d s (cid:19) v ,t V ,t d t + 2 Z T E ∂ xy ˜ P BS (cid:18)Z t ρ s V ,s β ( s, v ,s ) e − R s α x ( z,v ,z )d z d s (cid:19) v ,t e R t α x ( z,v ,z )d z d t = 2 E ∂ xxy ˜ P BS Z T (cid:18)Z t ρ s v ,s V ,s d s (cid:19) v ,t V ,t d t + 2 E ∂ y ˜ P BS Z T e R t α x ( z,v ,z )d z v ,t (cid:18)Z t v − ,s β ( s, v ,s ) V ,s e − R s α x ( z,v ,z )d z d B s (cid:19) d t. Remark D.2.
We now comment on the purpose of Assumption C.(1) β ( t, x ) = λ t x µ for µ ≥ / ≥ / x uniformly in t ∈ [0 , T ],and β x ( t, x ) = λ t µx µ − is continuous a.e. in x and t ∈ [0 , T ]. Thus, Assumption A andAssumption B are satisfied.(2) Such a diffusion coefficient is common in application, see for example SABR model Haganet al. [13] and CEV model Cox [9].Truthfully, we could leave β as an arbitrary diffusion coeffcient that solely obeys the items inAssumption A and Assumption B. However, in terms of application purposes and also for ourfast calibration scheme in Section 6, it will be more insightful to have this form for β . For theinterested reader, all the following calculations still remain valid solely under Assumption A andAssumption B.We now enforce Assumption C. Hence, we can rewrite V ,t and V ,t from Lemma 3.1 as V ,t = e R t α x ( z,v ,z )d z Z t λ s v µ ,s e − R s α x ( z,v ,z )d z d B s , (D.5) V ,t = e R t α x ( z,v ,z )d z (cid:26)Z t α xx ( s, v ,s )( V ,s ) e − R s α x ( z,v ,z )d z d s + Z t µλ s v µ − ,s V ,s e − R s α x ( z,v ,z )d z d B s (cid:27) . (D.6)Then we obtainˆ C xy = 2 E ∂ xxy ˜ P BS Z T (cid:18)Z t ρ s v ,s V ,s d s (cid:19) v ,t V ,t d t + 2 E ∂ y ˜ P BS Z T v ,t e R t α x ( z,v ,z )d z (cid:18)Z t λ s v µ − ,s V ,s e − R s α x ( z,v ,z )d z d B s (cid:19) d t. Hence C xy = 2 E ∂ xy ˜ P BS (cid:18)Z T (cid:18)Z t (1 − ρ s ) v ,s V ,s d s (cid:19) (cid:0) ρ t V ,t d B t − ρ t v ,t V ,t d t (cid:1)(cid:19) − E ∂ xy ˜ P BS Z T (cid:18)Z t ρ s v ,s V ,s d s (cid:19) v ,t V ,t d t + 2 E ∂ xxy ˜ P BS Z T (cid:18)Z t ρ s v ,s V ,s d s (cid:19) v ,t V ,t d t + 2 E ∂ y ˜ P BS Z T v ,t e R t α x ( z,v ,z )d z (cid:18)Z t λ s v µ − ,s V ,s e − R s α x ( z,v ,z )d z d B s (cid:19) d t − E ∂ xy ˜ P BS (cid:18)Z T (cid:18)Z t ρ s V ,s d B s − Z t ρ s v ,s V ,s d s (cid:19) ρ t v ,t V ,t d t (cid:19) . D.5. C yy . C yy is given by Remark D.1 (stochastic integration by parts) as4 E ∂ yy ˜ P BS (cid:18)Z T (cid:26)Z t (1 − ρ s ) v ,s V ,s d s (cid:27) (1 − ρ t ) v ,t V ,t d t (cid:19) . D.6.
Adding C x , C y , C xx , C xy and C yy . Now we add up all the terms after manipulation fromAppendices D.1 to D.5( C x + C y + C xx ) + C xy + C yy = E ∂ y ˜ P BS (cid:18)Z T v ,t V ,t + V ,t + v ,t V ,t d t (cid:19) + 2 E ∂ y ˜ P BS Z T v ,t e R t α x ( z,v ,z )d z (cid:18)Z t λ s v µ − ,s V ,s e − R s α x ( z,v ,z )d z d B s (cid:19) d t + 2 E ∂ xy ˜ P BS (cid:18)Z T (cid:18)Z t (1 − ρ s ) v ,s V ,s d s (cid:19) (cid:0) ρ t V ,t d B t − ρ t v ,t V ,t d t (cid:1)(cid:19) − E ∂ xy ˜ P BS Z T (cid:18)Z t ρ s v ,s V ,s d s (cid:19) v ,t V ,t d t + 2 E ∂ xxy ˜ P BS Z T (cid:18)Z t ρ s v ,s V ,s d s (cid:19) v ,t V ,t d t + 4 E ∂ yy ˜ P BS (cid:18)Z T (cid:26)Z t (1 − ρ s ) v ,s V ,s d s (cid:27) (1 − ρ t ) v ,t V ,t d t (cid:19) . Then C x + C y + C xx + C xy + C yy = E ∂ y ˜ P BS (cid:18)Z T v ,t V ,t + V ,t + v ,t V ,t d t (cid:19) + 2 E ∂ y ˜ P BS Z T v ,t e R t α x ( z,v ,z )d z (cid:18)Z t λ s v µ − ,s V ,s e − R s α x ( z,v ,z )d z d B s (cid:19) d t + 2 E ∂ xy ˜ P BS (cid:18)Z T (cid:18)Z t (1 − ρ s ) v ,s V ,s d s (cid:19) (cid:0) ρ t V ,t d B t − ρ t v ,t V ,t d t (cid:1)(cid:19) + 4 E ∂ yy ˜ P BS Z T (cid:18)Z t ρ s v ,s V ,s d s (cid:19) v ,t V ,t d t + 4 E ∂ yy ˜ P BS (cid:18)Z T (cid:26)Z t (1 − ρ s ) v ,s V ,s d s (cid:27) (1 − ρ t ) v ,t V ,t d t (cid:19) = E ∂ y ˜ P BS (cid:18)Z T v ,t V ,t + V ,t + v ,t V ,t d t (cid:19) + 2 E ∂ y ˜ P BS Z T v ,t e R t α x ( z,v ,z )d z (cid:18)Z t λ s v µ − ,s V ,s e − R s α x ( z,v ,z )d z d B s (cid:19) d t + 4 E ∂ yy ˜ P BS (cid:18)Z T (cid:18)Z t (1 − ρ s ) v ,s V ,s d s (cid:19) ρ t v ,t V ,t d t (cid:19) + 4 E ∂ yy ˜ P BS Z T (cid:18)Z t ρ s v ,s V ,s d s (cid:19) v ,t V ,t d t + 4 E ∂ yy ˜ P BS (cid:18)Z T (cid:26)Z t (1 − ρ s ) v ,s V ,s d s (cid:27) (1 − ρ t ) v ,t V ,t d t (cid:19) = E ∂ y ˜ P BS (cid:18)Z T v ,t V ,t + V ,t + v ,t V ,t d t (cid:19) + 2 E ∂ y ˜ P BS Z T v ,t e R t α x ( z,v ,z )d z (cid:18)Z t λ s v µ − ,s V ,s e − R s α x ( z,v ,z )d z d B s (cid:19) d t + 4 E ∂ yy ˜ P BS (cid:18)Z T (cid:18)Z t v ,s V ,s d s (cid:19) v ,t V ,t d t (cid:19) , where we have used the partial derivative relationship Proposition D.1 ( P BS partial derivativerelationship), Proposition B.2 (Malliavin duality relationship) and partial derivative relationshipProposition D.1 ( P BS partial derivative relationship), then simplification for the first, second andthird equalities respectively. Lastly, notice by Remark D.1 (stochastic integration by parts)2 (cid:18)Z T (cid:18)Z t v ,s V ,s d s (cid:19) v ,t V ,t d t (cid:19) = (cid:18)Z T v ,t V ,t d t (cid:19) . Proposition D.2.
In view of the calculations in Appendices D.1 to D.6, and under Assump-tion C, we obtain the simpler form of the second-order approximation Put (2)G from Theorem 3.1 as Put (2)G = P BS (cid:18) x , Z T v ,t d t (cid:19) + E ∂ y ˜ P BS (cid:18)Z T v ,t V ,t + V ,t + v ,t V ,t d t (cid:19) + 2 E ∂ y ˜ P BS (cid:18)Z T v ,t e R t α x ( z,v ,z )d z (cid:18)Z t λ s v µ − ,s V ,s e − R s α x ( z,v ,z )d z d B s (cid:19) d t (cid:19) + 2 E ∂ yy ˜ P BS (cid:18)Z T v ,t V ,t d t (cid:19) . (D.7)D.7. Eliminating the processes ( V ,t ) and ( V ,t ) . The last step is to reduce these remainingexpectations in Proposition D.2 down by eliminating the stochastic processes ( V ,t ) and ( V ,t ).To do so, we utilise Lemma B.1 (Mallavin integration by parts) and eq. (D.1), which yields2 E ∂ y ˜ P BS (cid:18)Z T v ,t e R t α x ( z,v ,z )d z (cid:18)Z t λ s v µ − ,s V ,s e − R s α x ( z,v ,z )d z d B s (cid:19) d t (cid:19) = 2 ω ( − α x ,ρλv µ +10 , · ) , (0 ,ρλv µ , · ) , ( α x ,v , · )0 ,T E ∂ xxy ˜ P BS ..