Smiling for the Delayed Volatility Swaps

Anatoliy Swishchuk1 and Nelson Vadori2

1 Corresponding author. University of Calgary, Department of Mathematics, 2500University Drive NW, Calgary, AB, Canada T2N 1N4;e-mail: aswish@ucalgary.ca;Tel.: 1(403)220-3274; Fax: 1(403) 282-5150.

2 University of Calgary, Department of Mathematics, e-mail: nvadori@ucalgary.ca.

Abstract. We present a variance drift adjusted version of the Heston model whichleads to a significant improvement of the market volatility surface fitting (compared toHeston). The numerical example we performed with recent market data shows a sig-nificant reduction of the average absolute calibration error 1 (calibration on 12 datesranging from Sep. 19th to Oct. 17th 2011 for the FOREX underlying EURUSD).Our model has two additional parameters compared to the Heston model, can beimplemented very easily and was initially introduced for volatility derivatives pricingpurpose. The main idea behind our model is to take into account some past historyof the variance process in its (risk-neutral) diffusion. Using a change of time methodfor continuous local martingales, we derive a closed formula for the Brockhaus&Longapproximation of the volatility swap price in this model. We also consider dynamichedging of volatility swaps using a portfolio of variance swaps.

Keywords: variance swap; volatility swap; stochastic volatility with delay; Hestonmodel; change of time; dynamic hedging.

1 Introduction

The volatility process is an important concept in financial modeling as it quanti-fies at each time t how likely the modeled asset log-return is to vary significantlyover some short immediate time period [t, t+ ε]. This process can be stochas-tic or deterministic, e.g. local volatility models in which the (deterministic)volatility depends on time and spot price level. In quantitative finance, weoften consider the volatility process

√Vt (where Vt is the variance process) to

be stochastic as it allows to fit the observed vanilla option market prices withan acceptable bias as well as to model the risk linked with the future evolutionof the volatility smile (which deterministic model cannot), namely the forwardsmile. Many derivatives are known to be very sensitive to the forward smile,one of the most popular example being the cliquet options (options on futureasset performance, see Kruse and Nogel [18] for example).

Heston model (Heston [11]; Heston and Nandi [12]) is one of the most popularstochastic volatility models in the industry as semi-closed formulas for vanillaoption prices are available, few (five) parameters need to be calibrated, and it

1 The average absolute calibration error is defined to be the average of the absolutevalues of the differences between market and model implied Black & Scholes volatili-ties.

accounts for the mean-reverting feature of the volatility.

One might be willing, in the variance diffusion, to take into account not onlyits current state but also its past history over some interval [t − τ, t], whereτ > 0 is a constant and is called the delay. Starting from the discrete-timeGARCH(1,1) model (Bollerslev [4]), a first attempt was made in this direc-tion in Kazmerchuk et al. [16], where a non-Markov delayed continuous-timeGARCH model was proposed (St being the asset price at time t, and γ, θ, αsome positive constants):

dVtdt

= γθ2 +α

τln2

(StSt−τ

)− (α+ γ)Vt, (1)

this model being inherited from its discrete-time analogue (where L is a positiveinteger):

σ2n = γθ2 +

α

Lln2

(Sn−1Sn−1−L

)+ (1− α− γ)σ2

n−1. (2)

The parameter θ2 (resp. γ) can be interpreted as the value of the long-rangevariance (resp. variance mean-reversion speed) when the delay is equal to 0 (wewill see that introducing delay modifies the value of these two model features),and α a continuous-time equivalent of the variance ARCH(1,1) autoregressivecoefficient. In fact, we can interpret the right-hand side of previous diffusionequation as the sum of two terms:

• the delay-free term γ(θ2−Vt) which accounts for the mean-reverting featureof the variance process

• α(

1τ ln2

(StSt−τ

)− Vt

)which is a pure (noisy) delay term, i.e. that vanishes

when τ → 0 and takes into account the past history of the variance (via

the term ln(

StSt−τ

)). The autoregressive coefficient α can be seen as the

amplitude of this pure delay term.

In Swishchuk [23] and Swishchuk and Li [21], the authors point out the im-

portance to incorporate the real world P−drift dP(t, τ) :=∫ tt−τ (µ− 1

2Vu)du of

ln(

StSt−τ

)in the model (where µ stands for the real world P−drift of the stock

price St), transforming the variance dynamics into:

dVtdt

= γθ2 +α

τ

[ln

(StSt−τ

)− dP(t, τ)

]2− (α+ γ)Vt. (3)

The latter diffusion (3) was introduced in Swishchuk [23] and Kazmerchuk etal. [15], and the proposed model was proved to be complete and to accountfor the mean-reverting feature of the volatility process. This model is also nonMarkov as the past history (Vu)u∈[t−τ,t] of the variance appears in its diffusion

equation via the term ln(

StSt−τ

), as it is shown in Swishchuk [23]. In the conti-

nuity of this approach, a series of papers were published by one of the authorsfocusing on the pricing of variance swaps in this delayed framework: one-factor

stochastic volatility with delay has been presented in Swishchuk [23]; multi-factor stochastic volatility with delay in Swishchuk [24]; one-factor stochasticvolatility with delay and jumps in Swishchuk and Li [21]; and finally localLevy-based stochastic volatility with delay in Swishchuk and Malenfant [26].

Other papers related to the concept of delay are also of interest. For exam-ple, Kind et al. [17] obtained a diffusion approximation result for processessatisfying some equations with past dependent coefficients, with application tooption pricing. Arriojas et al. [1] derived a Black&Scholes formula for call op-tions assuming the stock price follows a Stochastic Delay Differential Equation(SDDE). Mohammed and Bell have also published a series of papers in whichthey investigate various properties of SDDE (see e.g. [2], [3]).

Unfortunately, the model (3) doesn’t lead to (semi-)closed formulas for thevanilla options, making it difficult to use for practitioners willing to calibrateon vanilla market prices. Nevertheless, one can notice that the Heston modeland the delayed continuous-time GARCH model (3) are very similar in the sensethat the expected values of the variances are the same - when we make the delaytend to 0 in (3). As mentioned before, the Heston framework is very conve-nient, and therefore it is naturally tempting to adjust the Heston dynamics inorder to incorporate the delay introduced in (3). In this way, we considered ina first approach adjusting the Heston drift by a deterministic function of timeso that the expected value of the variance under the delayed Heston model isequal to the one under the delayed GARCH model (3). In addition to makingour delayed Heston framework coherent with (3), this construction makes thevariance process diffusion dependent not on its past history (Vu)u∈[t−τ,t], but

on the past history of its risk-neutral expectation (EQ0 (Vu))u∈[t−τ,t], preserving

the Markov feature of the Heston model (where we denote EQt (·) := EQ(·|Ft)

for some filtration (Ft)t≥0). The purpose of sections 2 and 3 is to present theDelayed Heston model as well as some calibration results on call option prices,with a comparison to the Heston model. In sections 4 and 5, we will considerthe pricing and hedging of volatility and variance swaps in this model.

Volatility and variance swaps are contracts whose payoff depend (respectivelyconvexly and linearly) on the realized variance of the underlying asset over somespecified time interval. They provide pure exposure to volatility, and thereforemake it a tradable market instrument. Variance Swaps are even consideredby some practitioners to be vanilla derivatives. The most commonly tradedvariance swaps are discretely sampled and have a payoff PVn (T ) at maturity Tof the form:

PVn (T ) = N

[252

n

n∑i=0

ln2

(Si+1

Si

)−Kvar

],

where Si is the asset spot price on fixing time ti ∈ [0, T ] (usually there is onefixing time each day, but there could be more, or less), N the notional amountof the contract (in currency per unit of variance) and Kvar the strike specified

in the contract. The corresponding volatility swap payoff P vn (T ) is given by:

P vn (T ) = N

√√√√252

n

n∑i=0

ln2

(Si+1

Si

)−Kvol

.One can also consider continuously sampled volatility and variance swaps (onwhich we will focus in this article), which payoffs are respectively defined asthe limit when n → +∞ of their discretely sampled versions. Formally, if wedenote (Vt)t≥0 the stochastic volatility process of our asset, adapted to somebrownian filtration (Ft)t≥0, then the continuously-sampled realized varianceVR from initiation date of the contract t = 0 to maturity date t = T is given

by VR = 1T

∫ T0Vsds. The fair variance strike Kvar is calculated such that the

initial value of the contract is 0, and therefore is given by:

EQ0

[e−rT (VR −Kvar)

]= 0⇒ Kvar = EQ

0 (VR).

In the same way, the fair volatility strike Kvol is given by:

EQ0

[e−rT (

√VR −Kvol)

]= 0⇒ Kvol = EQ

0 (√VR).

The volatility swap fair strike might be difficult to compute explicitly as wehave to compute the expectation of a square-root. In Brockhaus and Long [7],the following approximation - based on a Taylor expansion - was proposed tocompute the expected value of the square-root of an almost surely non negativerandom variable Z:

E(√Z) ≈

√E(Z)− V ar(Z)

8E(Z)32

. (4)

We will refer to this approximation in our paper as the Brockhaus&Long ap-proximation.

There exists a vast literature on volatility and variance swaps. We provide inthe following lines a selection of papers covering important topics. Carr and Lee[8] provides an overview of the current market of volatility derivatives. Theysurvey the early literature on the subject. They also provide relatively sim-ple proofs of some fundamental results related to variance swaps and volatilityswaps. Pricing of variance swaps for one-factor stochastic volatility is presentedin Swishchuk [22]. Variance and volatility swaps in energy markets are con-sidered in Swishchuk [25]. Broadie and Jain [6] covers pricing and dynamichedging of volatility derivatives in the Heston model. Moreover, various pa-pers deal with the VIX Index - the Chicago Board Options Exchange MarketVolatility Index - which is a popular measure of the one month implied volatil-ity on the S&P 500 index (see e.g. Zhang and Zhu [27], Hao and Zhang [10] orFilipovic [9]).

The paper is organized as follows: in section 2, we present the Delayed Hestonmodel; in section 3, we present calibration results (for underlying EURUSDon 12 dates ranging from Sep. 19th to Oct. 17th 2011 2011) as well as a

comparison with the Heston model. In section 4, we compute the price processXt(T ) := EQ

t (VR) of the floating leg of the variance swap of maturity T , as wellas the Brockhaus&Long approximation of the price process Yt(T ) := EQ

t (√VR)

of the floating leg of the volatility swap of maturity T . This leads in particularto closed formulas for the fair volatility and variance strikes. In section 5, weconsider - in this model - dynamic hedging of volatility swaps using varianceswaps.

2 Presentation of the Delayed Heston model

Throughout this paper, we will assume constant risk-free rate r, dividend yieldq and finite time-horizon T . We fix (Ω,F ,P) a probability space and we con-sider a stock whose price process is denoted by (St)t≥0. We let Q be a risk-

neutral measure and we let (ZQt )t≥0 and (WQ

t )t≥0 be two correlated standardbrownian motions on (Ω,F ,Q). We let the natural filtration associated tothese brownian motions Ft := σ(ZQ

t ,WQt ) and we denote EQ

t (·) := EQ(·|Ft)and V arQt (·) := V arQ(·|Ft).

We assume the following risk-neutral Q− stock price dynamics :

dSt = (r − q)Stdt+ St√VtdZ

Qt . (5)

The well-known Heston model has the following Q−dynamics for the varianceVt:

dVt = γ(θ2 − Vt)dt+ δ√VtdW

Qt , (6)

where θ2 is the long-range variance, γ the variance mean-reversion speed, δ thevolatility of the variance and ρ the brownian correlation coefficient (

⟨WQ, ZQ⟩

t=

ρt). We also assume S0 = s0 a.e. and V0 = v0 a.e., for some positive constantsv0, s0.

As explained in the introduction, the following delayed continuous-time GARCHdynamics have been introduced for the variance in Swishchuk [23]:

dVtdt

= γθ2 +α

τ

[∫ t

t−τ

√VsdZ

Qs − (µ− r)τ

]2− (α+ γ)Vt, (7)

where µ stands for the real world P−drift of the stock price St. We notice thatθ2 (resp. γ) has been defined in introduction for the delayed continuous-timeGARCH model as the value of the long-range variance (resp. variance mean-reversion speed) when τ = 0, therefore it has the same meaning as the Hestonlong-range variance (resp. variance mean-reversion speed). That is why we usethe same notations in both models.

We can see that the two models are very similar. Indeed, they both give thesame expected value for Vt as the delay goes to 0 in (7), namely θ2 + (V0 −

θ2)e−γt. The idea here is to adjust the Heston dynamics (6) in order to ac-count for the delay introduced in (7). Our approach is to adjust the drift bya deterministic function of time so that the expected value of Vt under theadjusted Heston model is the same as under (7). This approach can be seenas a correction by a pure delay term of amplitude α of the Heston drift by adeterministic function in order to account for the delay.

Namely, we assume the adjusted Heston dynamics:

dVt =[γ(θ2 − Vt) + ετ (t)

]dt+ δ

√VtdW

Qt , (8)

ετ (t) := ατ(µ− r)2 +α

τ

∫ t

t−τvsds− αvt, (9)

with vt := EQ0 (Vt). It was shown in Swishchuk [23] that vt solves the following

equation:

dvtdt

= γθ2 + ατ(µ− r)2 +α

τ

∫ t

t−τvsds− (α+ γ)vt, (10)

and that we have the following expression for vt:

vt = θ2τ + (V0 − θ2τ )e−γτ t, (11)

with:

θ2τ := θ2 +ατ(µ− r)2

γ. (12)

By (11) and (15) (see below), we have limt→∞ vt = θ2τ and therefore the pa-rameter θ2τ can be interpreted as the adjusted value of the limit towards vttends to as t→∞, that has been (positively) shifted from its original value θ2

because of the introduction of delay. We have θ2τ → θ2 when τ → 0, which iscoherent. We will see below that we can interpret the parameter γτ > 0 as theadjusted mean-reversion speed. This parameter is given in Swishchuk [23] bya (nonzero) solution to the following equation:

γτ = α+ γ +α

γττ(1− eγττ ). (13)

By (9), (11) and (13) we get an explicit expression for the drift adjustment:

ετ (t) = ατ(µ− r)2 + (V0 − θ2τ )(γ − γτ )e−γτ t. (14)

The following simple property gives us some information about the correctionterm ετ (t) and the parameter γτ , that will be useful for interpretation purposeand in the derivation of the semi-closed formulas for call options in AppendixA. Indeed, given (15) and (11), the parameter γτ can be interpreted as the ad-justed variance mean-reversion speed because it quantifies the speed at whichvt tends to θ2τ as t→∞, and we have by using a Taylor expansion in (13) thatγτ → γ when τ → 0, which is coherent.

Property 1: γτ is the unique solution to (13) and:

0 < γτ < γ, limτ→0

supt∈R+

|ετ (t)| = 0. (15)

Proof : Let’s show γτ ≥ 0. If γτ < 0 then by (13) we have α+γ+ αγτ τ

(1− eγτ τ ) < 0,

i.e. 1−eγτ τ +γττ > − γαγττ . But τ > 0 so ∃x0 > 0 s.t. 1−e−x0−x0 > γ

αx0. A simple

study shows that is impossible whenever γα≥ 0, which is what we have by assumption.

Therefore γτ ≥ 0, and in fact γτ > 0 since it is a nonzero solution of (13). If γ ≤ γτthen by (13) γττ + 1− eγτ τ ≥ 0. But γττ > 0 therefore ∃x0 > 0 s.t. x0 + 1− ex0 ≥ 0.

A simple study shows that is impossible. The uniqueness comes from a similar simple

study. Now, because γτ > 0, we have supt∈R+

|ετ (t)| ≤ ατ(µ− r)2 + |(V0 − θ2τ )(γ − γτ )|

and (V0 − θ2τ )(γ − γτ ) = (1) by (13). So limτ→0

ατ(µ− r)2 + |(V0 − θ2τ )(γ − γτ )| = 0.

Using (14) and (12), we can rewrite (8) as a time-dependent Heston model withtime-dependent long-range variance θ2t :

dVt = γ(θ2t − Vt)dt+ δ√VtdW

Qt , (16)

θ2t := θ2τ + (V0 − θ2τ )(γ − γτ )

γe−γτ t. (17)

The parameter θ2τ is - as we mentioned above - the adjusted value of the limittowards which vt tends to as t → ∞. For this reason, it is coherent that it isalso the limiting value of the time-dependent long-range variance θ2t as t→∞(by (17) and (15)).

3 Calibration on call option prices and comparison tothe Heston model

Following Kahl and Jackel [13] and Mikhailov and Noegel [20], it is possible toget semi-closed formulas for call options in our delayed Heston model. Indeed,our model is a time-dependent Heston model with time-dependent long-rangevariance θ2t . We refer to Appendix A for the procedure to derive such semi-closed formulas.

We perform our calibration on September 30th 2011 for underlying EURUSDon the whole volatility surface (maturities from 1M to 10Y, strikes ATM, 25DCall/Put, 10D Call/Put). The implied volatility surface, the Zero Couponcurves EUR Vs. Euribor 6M and USD Vs. Libor 3M and the spot price aretaken from Bloomberg (mid prices). The drift µ = 0.0188 is estimated from7.5Y of daily close prices (source: www.forexrate.co.uk).

The calibration procedure is a least-squares minimization procedure that weperform via MATLAB (function lsqnonlin that uses a trust-region-reflective

algorithm). The Heston integral (83) is computed via the MATLAB func-tion quadl that uses a recursive adaptive Lobatto quadrature. The integral∫ t0e−γτsD(s, u)ds in (81) is computed via a composite Simpson’s rule with 100

points.

The calibrated parameters for delayed Heston are:

(V0, γ, θ2, δ, ρ, α, τ) = (0.0343, 3.9037, 10−8, 0.808,−0.5057, 71.35, 0.7821),

and for Heston, they are:

(V0, γ, θ2, δ, ρ) = (0.0328, 0.5829, 0.0256, 0.3672,−0.4824).

We notice that we cannot compare straightforwardly the parameters θ2 of bothmodels. Indeed, as mentioned above, the Delayed Heston model has a time-dependent long range variance θ2t which has been shifted away from its originalvalue θ2 because of the introduction of the delay τ . When τ = 0, θ2t = θ2 butwhen τ > 0, θ2t and θ2 differ. Therefore, to be coherent, one should comparethe Heston long range-variance θ2 = 0.0256 with the Delayed Heston time-dependent long-range variance θ2t . Below we give the value of θ2t for differentmaturities t:

Maturity θ2t1M 0.03252M 0.03223M 0.03196M 0.0311Y 0.02942Y 0.03645Y 0.018410Y 0.0102

Table 1: Parameter θ2t for different maturities t.

We remark that the short and medium term values (less than 2Y) of θ2t aresimilar to the value of θ2 in the Heston model, but that for long maturities, thevalue of θ2t decreases significantly. Allowing this time-dependence of the thelong-range variance could be an explanation why the Delayed Heston modeloutperforms the Heston model especially for long maturities (see the discus-sion below). Similarly, the Heston mean-reversion speed γ = 0.58 has to becompared with the Delayed Heston adjusted mean-reversion speed γτ , which isgiven by (13) and is approximately equal to 0.12 on our calibration date. Fo-cusing on the delay parameters α and τ , they were expected to be significantlynon zero because as we will see below, the Delayed Heston model significantlyoutperforms the Heston model in terms of calibration error (and standard de-viation of the calibration errors): if α and τ were close to 0, the calibrationerrors would have been approximately the same for both models, because again,the Delayed Heston model reduces to the Heston model when the delay termvanishes, i.e. when τ = 0 or α = 0.

The calibration errors for all call options (expressed as the absolute value of thedifference between market and model implied Black & Scholes volatilities, in

bp) for the Heston model and our Delayed Heston model are given below. Theresults show a 44% reduction of the average absolute calibration error (46bpfor delayed Heston, 81bp for Heston). It is to be noted that we didn’t useany weight matrix in our calibration procedure, i.e. the calibration aims atminimizing the sum of the (squares of the) errors of each call option, equallyweighted. In practice, one might be willing to give more importance to ATMoptions for instance, or options of a certain range of maturities. The optimiza-tion algorithm aims at minimizing the sum of the squares of the errors: inother words, it aims at minimizing the average absolute calibration error. Forthis reason, it might be the case that for some specific option (e.g. ATM 6M,see table below), the Heston model has a lower model error than the DelayedHeston model. But the total calibration error for the Delayed Heston model isalways expected to be lower than for the Heston model.

On our calibration date, the Delayed Heston model seems to outperform theHeston model specifically for long maturities (≥ 3Y): if we consider only theseoptions, the average absolute error is of 79bp for the Heston model and 33bpfor the Delayed Heston model, which represents a 58% reduction of the cali-bration error. We can also note that for ATM options only, the improvementis significant too (43bp Vs. 92bp, i.e. an error reduction of 54%). For mediummaturity options (6M to 2Y), the Delayed Heston model still outperforms theHeston model but less significantly (53bp Vs. 75bp, i.e. an error reduction of30%), and we have the same observation for very out of the money options (10Delta Call and Put, 51bp Vs. 79bp, i.e. an error reduction of 35%).

Another very interesting observation we can make is that the standard devi-ation of the calibration errors is much lower for the Delayed Heston modelcompared to the Heston model (34bp Vs. 52bp, which represents a 35% reduc-tion of the standard deviation): in addition to improving the average absolutecalibration error, it also improves the distance of the individual errors to theaverage error, which is highly appreciable in practice because it means that youwon’t face the case where some options are priced really poorly by the modelwhereas some others are priced almost perfectly.

ATM 25D Call 25D Put 10D Call 10D Put1M 152 192 41 193 672M 114 139 15 136 813M 89 109 3 110 924M 48 61 17 67 1016M 5 15 34 29 859M 59 42 63 2 851Y 107 83 102 31 961.5Y 141 116 111 42 732Y 166 137 127 54 683Y 145 124 77 52 04Y 96 95 18 37 665Y 29 47 52 7 1387Y 39 10 112 28 18610Y 100 67 168 58 225

Table 2: Heston Absolute Calibration Error (in bp of the Black & Scholes volatility).

ATM 25D Call 25D Put 10D Call 10D Put1M 116 91 109 128 1152M 44 24 59 54 883M 14 3 32 36 604M 18 28 1 5 296M 31 37 23 19 39M 45 45 56 37 571Y 51 47 82 50 1041.5Y 29 30 79 49 1292Y 24 23 83 47 1393Y 11 9 29 30 904Y 41 28 14 17 385Y 76 55 59 5 167Y 71 49 58 1 1410Y 26 8 18 47 24

Table 3: Delayed Heston Absolute Calibration Error (in bp of the Black & Scholes volatility).

Delayed Heston HestonStandard Deviation ofthe calibration errors(bp)

33.67 (35%) 52.12

Table 4: Standard Deviation of the calibration errors in bp.The reduction of this error is indicated in brackets.

In order to i) check that our calibration on September 30th 2011 was not anexception and ii) investigate the stability of the calibrated parameters, we per-formed calibrations on 11 additional dates evenly spaced around September30th 2011, ranging from September 19th 2011 to October 17th 2011. We chosea one month window because from the past experience of the authors in thefinancial industry, it can happen that the parameters are recalibrated by fi-nancial institutions every month only, and not every day (because it would betoo time-consuming) and therefore the choice of a one month window seemsreasonable to investigate the stability of the parameters. We summarize thefindings in the tables below. We find that the Delayed Heston model alwaysoutperforms significantly the Heston model (average calibration error reductionvarying from 29% to 56%), and that the Delayed Heston model is performantespecially for long maturities (≥ 3Y, calibration error reduction varying from40% to 66%) and ATM options (calibration error reduction varying from 42%to 67%). Finally, the standard deviation of the calibrations errors is always re-duced significantly by the Delayed Heston model (reduction varying from 23%to 49%).

Date (2011) Sep. 19 Sep. 21 Sep. 23 Sep. 27 Sep. 29 Oct. 3 Oct. 5 Oct. 7 Oct. 11 Oct. 13 Oct. 17Total Error Re-duction (%)

44 45 56 47 38 51 42 37 38 29 39

Long MaturityError Reduction(%)

58 63 65 55 53 66 55 51 48 40 59

ATM Error Re-duction (%)

57 55 67 62 55 65 56 51 50 42 53

Table 5: Summary of the calibration error reductions.

Date (2011) Sep. 19 Sep. 21 Sep. 23 Sep. 27 Sep. 29 Oct. 3 Oct. 5 Oct. 7 Oct. 11 Oct. 13 Oct. 17Calibration er-rors St. Dev.Reduction (%)

43 45 49 46 31 49 40 29 29 23 29

Table 6: Summary of the calibration errors St. Dev reductions.

In order to investigate the stability of the model parameters, we present belowthe calibrated parameters for the Heston model and the Delayed Heston modelfrom September 19th 2011 to October 17th 2011.

Date (2011) Sep. 19 Sep. 21 Sep. 23 Sep. 27 Sep. 29 Oct. 3 Oct. 5 Oct. 7 Oct. 11 Oct. 13 Oct. 17V0 0.0313 0.0337 0.0384 0.0354 0.0335 0.0368 0.0344 0.0295 0.0279 0.0271 0.0283γ 3.99 3.72 3.82 3.72 4.52 3.47 3.86 3.71 3.13 3.08 3.39

θ2 5 e-4 7 e-6 2e-4 1e-8 1e-8 1e-5 3e-4 2e-3 1e-3 5e-3 4e-3δ 0.79 0.75 0.82 0.81 0.89 0.78 0.81 0.76 0.68 0.67 0.80ρ -0.51 -0.51 -0.52 -0.50 -0.49 -0.51 -0.51 -0.51 -0.51 -0.51 -0.49α 82.2 77.5 64.5 166.7 124 66.6 76.5 90.2 125.1 83.7 77.3τ 0.86 0.77 0.71 0.32 0.59 0.72 0.78 0.90 0.67 1.00 0.81

Table 7: Calibrated Parameters for the Delayed Heston model.

Date (2011) Sep. 19 Sep. 21 Sep. 23 Sep. 27 Sep. 29 Oct. 3 Oct. 5 Oct. 7 Oct. 11 Oct. 13 Oct. 17V0 0.0298 0.0322 0.0369 0.0338 0.0311 0.0351 0.0326 0.0283 0.0271 0.0262 0.0269γ 0.54 0.46 0.45 0.43 0.35 0.44 0.43 0.89 1.22 1.13 0.92

θ2 0.0258 0.026 0.0258 0.0264 0.0276 0.0265 0.0275 0.0265 0.0249 0.0254 0.024δ 0.34 0.33 0.35 0.35 0.32 0.34 0.34 0.41 0.45 0.43 0.39ρ -0.49 -0.49 -0.49 -0.48 -0.48 -0.50 -0.49 -0.50 -0.50 -0.49 -0.51

Table 8: Calibrated Parameters for the Heston model.

We can see that in average, the parameters stay relatively stable throughoutthis 1 month time window. In this case, it would be reasonable to use thesame parameters throughout the 1 month time window as some financial insti-tutions do (from the past experience of the authors in the financial industry).Of course, there are some periods of high volatility in which not recalibratingthe model parameters often enough might lead to a significant mispricing ofthe call options by the model.

4 Pricing Variance and Volatility Swaps

In this section, we derive a closed formula for the Brockhaus&Long approxima-tion of the volatility swap price using the change of time method introduced inSwishchuk [22], as well as the price of the variance swap. Precisely, in Brock-haus and Long [7], the following approximation was presented to compute theexpected value of the square-root of an almost surely non negative random vari-

able Z: E(√Z) ≈

√E(Z) − V ar(Z)

8E(Z)32

. We denote VR := 1T

∫ T0Vsds the realized

variance on [0, T ].

We let Xt(T ) := EQt (VR) (resp. Yt(T ) := EQ

t (√VR)) the price process of the

floating leg of the variance swap (resp. volatility swap) of maturity T .

Theorem 1: The price process Xt(T ) of the floating leg of the variance swapof maturity T in the delayed Heston model (5)-(8) is given by:

Xt(T ) =1

T

∫ t

0

Vsds+T − tT

θ2τ + (Vt − θ2τ )

(1− e−γ(T−t)

γT

)+(V0 − θ2τ )e−γτ t

(1− e−γτ (T−t)

γτT− 1− e−γ(T−t)

γT

).

(18)

Proof: By definition, Xt(T ) = EQt ( 1

T

∫ T0Vsds) = 1

T

∫ t0Vsds+ 1

T

∫ Tt

EQt (Vs)ds. In the

previous integral, the interchange between the expectation and the integral is justifiedby the use of Tonelli’s theorem, as the variance process (t, ω) → Vt(ω) is a.e. non-negative and measurable. Let s ≥ t. Then we have by (8) that EQ

t (Vs−Vt) = EQt (Vs)−

Vt =∫ stγ(θ2 − EQ

t (Vu)) + ετ (u)du+ EQt (∫ st

√VudW

Qu ). Again, the interchange of the

expectation and the integral EQt (∫ stγ(θ2−Vu)+ετ (u)du) =

∫ stγ(θ2−EQ

t (Vu))+ετ (u)duis obtained the following way:

EQt (

∫ s

t

γ(θ2 − Vu) + ετ (u)du) =

∫ s

t

γθ2 + ετ (u)du− γEQt (

∫ s

t

Vudu). (19)

Then again, by Tonelli’s theorem we get EQt (∫ stVudu) =

∫ stEQt (Vu)du, which justifies

the interchange.

Now, (√Vt)t≥0 is an adapted process (to our filtration (Ft)t≥0) s.t. EQ(

∫ T0Vudu) =∫ T

0EQ(Vu)du < +∞ (by Tonelli’s theorem), therefore

∫ t0

√VudW

Qu is a martingale

and we have EQt (∫ st

√VudW

Qu ) = 0. Therefore ∀s ≥ t ≥ 0, the function s → EQ

t (Vs)

is a solution of y′s = γ(θ2 − ys) + ετ (s) with initial condition yt = Vt. This gives us

EQt (Vs) = θ2τ + (Vt − θ2τ )e−γ(s−t) + (V0 − θ2τ )e−γτ t(e−γτ (s−t) − e−γ(s−t)). Integrating

the latter in the variable s via∫ Tt

EQt (Vs)ds completes the proof.

Corollary 1: The price Kvar of the variance swap of maturity T at initiationof the contract t = 0 in the delayed Heston model (5)-(8) is given by:

Kvar = θ2τ + (V0 − θ2τ )1− e−γτT

γτT. (20)

Proof: By definition, Kvar = X0(T ).

Now, let:

xt := −(V0 − θ2τ )e(γ−γτ )t + eγt(Vt − θ2τ ). (21)

Then by Ito’s Lemma we get:

dxt = δeγt√

(xt + (V0 − θ2τ )e(γ−γτ )t)e−γt + θ2τdWQt . (22)

Which is of the form dxt = f(t, xt)dWQt with:

f(t, x) := δeγt√

(x+ (V0 − θ2τ )e(γ−γτ )t)e−γt + θ2τ . (23)

Indeed, since xt = g(t, Vt) with g(t, x) := −(V0 − θ2τ )e(γ−γτ )t + eγt(x− θ2τ ), themultidimensional version of Ito’s lemma reads:

dxt = dg(t, Vt) = gt(t, Vt)dt+ gx(t, Vt)dVt +1

2gxx(t, Vt)d 〈V, V 〉t , (24)

where 〈V, V 〉t is the quadratic variation of the process (Vt)t≥0 (see e.g. [14],theorem 3.6. of section 3.3). Since gxx(t, x) = 0, gt(t, Vt) = −(γ − γτ )(V0 −θ2τ )e(γ−γτ )t+γeγt(Vt− θ2τ ) and gx(t, Vt) = eγt, we get, using (8), (12) and (14):

dxt =gt(t, Vt)dt+ gx(t, Vt)dVt (25)

=− (γ − γτ )(V0 − θ2τ )e(γ−γτ )tdt+ γeγt(Vt − θ2τ )dt+ eγtdVt (26)

=− eγt(ετ (t)− γ(θ2τ − θ2))dt+ γeγt(Vt − θ2τ )dt (27)

+ eγt[γ(θ2 − Vt) + ετ (t)

]dt+ eγtδ

√VtdW

Qt (28)

=eγtδ√VtdW

Qt . (29)

The fact that Vt = (xt + (V0 − θ2τ )e(γ−γτ )t)e−γt + θ2τ by definition of xt (21)completes the proof.

Because dxt = f(t, xt)dWQt , the process (xt)t≥0 is a continuous local martin-

gale, and even a true martingale since EQ(∫ T0f2(s, xs)ds) =

∫ T0EQ(f2(s, xs))ds <

∞ (again, the interchange between expectation and integral follows from Tonelli’stheorem). We can use the change of time method introduced in Swishchuk [22]and we get xt = Wφt , where Wt is a Fφ−1

t− adapted Q−Brownian motion,

which is based on the fact that every continuous local martingale can be repre-sented as a time-changed brownian motion. The process (φt)t≥0 is a.e. increas-ing, non negative, Ft− adapted and is called the change of time process. Thisprocess is also equal to the quadratic variation 〈x〉t of the (square-integrable)continuous martingale xt (see [14], section 3.2, Proposition 2.10.).

Expressions of φt, φ−1t and Wt are given by:

φt = 〈x〉t =

∫ t

0

f2 (s, xs) ds, (30)

Wt =

∫ φ−1t

0

f(s, xs)dWQs , (31)

φ−1t =

∫ t

0

1

f2(φ−1s , xφ−1

s

)ds. (32)

To see that φ−1t has the following form, observe that:

φ−1φt =

∫ φt

0

1

f2(φ−1s , xφ−1

s

)ds. (33)

Now make the change of variable s = φu, so that ds = dφu = f2 (u, xu) du. Weget:

φ−1φt =

∫ t

0

f2 (u, xu)

f2(φ−1φu , xφ−1

φu

)du =

∫ t

0

f2 (u, xu)

f2 (u, xu)du = t. (34)

This immediately yields:

Vt = θ2τ + (V0 − θ2τ )e−γτ t + e−γtWφt . (35)

Lemma 1: For s, t ≥ 0 we have:

EQt (Wφs) = Wφt∧s , (36)

and for s, u ≥ t:

EQt (WφsWφu) = x2t + δ2

[θ2τ

(e2γ(s∧u) − e2γt

2γ

)+(V0 − θ2τ )

(e(2γ−γτ )(s∧u) − e(2γ−γτ )t

2γ − γτ

)+ xt

(eγ(s∧u) − eγt

γ

)].

(37)

Proof: (36) comes from the fact that xt = Wφt is a martingale. Let s ≥ u ≥ t. Thenby iterated conditioning: EQ

t (WφsWφu) = EQt (EQ

u(WφsWφu)) = EQt (WφuEQ

u(Wφs)) =EQt (W 2

φu), because xt = Wφt is a martingale. Now, by definition of the quadratic

variation, x2u−〈x〉u is a martingale and therefore EQt (W 2

φu) = x2t −〈x〉t +EQt (〈x〉u) =

x2t − φt + EQt (φu) = x2t − φt + φt + EQ

t (∫ utf2 (s, xs) ds). We can again interchange

expectation and integral by Tonelli’s theorem. By definition of f2 (s, xs) (the lat-ter is a linear function of xs) and since xt martingale, then we have (for s ≥ t)EQt (f2 (s, xs)) = f2 (s, xt), and therefore EQ

t (WφsWφu) = x2t +∫ utf2 (s, xt) ds. We

use the fact that, by definition of f in (23):

f2 (s, xt) = δ2e2γs[(xt + (V0 − θ2τ )e(γ−γτ )s)e−γs + θ2τ ], (38)

to integrate the latter expression with respect to s to complete the proof.

The following theorem gives the expression of the Brockhaus&Long approxi-mation of the volatility swap floating leg price process Yt(T ).

Theorem 2: The Brockhaus&Long approximation of the price process Yt(T )of the floating leg of the volatility swap of maturity T in the delayed Hestonmodel (5)-(8) is given by:

Yt(T ) ≈√Xt(T )− V arQt (VR)

8Xt(T )32

, (39)

where Xt(T ) is given by equation (18) of Theorem 1 and:

V arQt (VR) =xtδ

2

γ3T 2

[e−γt

(1− e−2γ(T−t)

)− 2(T − t)γe−γT

]+

δ2

2γ3T 2

[2θ2τγ(T − t)+ 2(V0 − θ2τ )

γ

γτe−γτ t + 4θ2τe

−γ(T−t) − θ2τe−2γ(T−t) − 3θ2τ

]− δ2(V0 − θ2τ )

γ2T 2(γ2τ + 2γ2 − 3γγτ )

[ 2(γτ − 2γ)e−γ(T−t)−γτ t

+(γ − γτ )e−2γ(T−t)−γτ t + 2γ2

γτe−γτT

].

(40)

Proof: The (conditioned) Brockhaus&Long approximation gives us:

Yt(T ) = EQt (√VR) ≈

√EQt (VR)− V arQt (VR)

8EQt (VR)

32

=√Xt(T )− V arQt (VR)

8Xt(T )32

.

Furthermore:

V arQt (VR) = EQt ((VR − EQ

t (VR))2)

=1

T 2EQt

((∫ T

0

(Vs − EQt (Vs))ds

)2).

(41)

From (35) we have Vt = θ2τ +(V0−θ2τ )e−γτ t+e−γtWφt , and since Wφt is a martingale,Vs − EQ

t (Vs) = 0 if s ≤ t, and Vs − EQt (Vs) = e−γs(Wφs − xt) if s > t.

Therefore:

V arQt (VR) =1

T 2EQt

((∫ T

t

e−γs(Wφs − xt)ds)2)

=1

T 2x2t

(∫ T

t

e−γsds

)2

+1

T 2EQt

((∫ T

t

e−γsWφsds

)2)

− 2

T 2xt

(∫ T

t

e−γsEQt (Wφs)ds

)(∫ T

t

e−γsds

).

(42)

The interchange of expectation and integral in the last equation is justified the fol-lowing way: by definition of Wφs = xs in (21), we get:

EQt (

∫ T

t

e−γsWφsds) = EQt (

∫ T

t

−(V0 − θ2τ )e−γτ s + Vs − θ2τds) (43)

=

∫ T

t

−(V0 − θ2τ )e−γτ s − θ2τds+ EQt (

∫ T

t

Vsds). (44)

We can interchange expectation and integral in the latter expression by Tonelli’stheorem, which gives:

EQt (

∫ T

t

e−γsWφsds) =

∫ T

t

−(V0 − θ2τ )e−γτ s − θ2τds+

∫ T

t

EQt (Vs)ds (45)

=

∫ T

t

e−γsEQt (Wφs)ds. (46)

Now we continue our computation to get:

V arQt (VR) = − 1

T 2x2t

(∫ T

t

e−γsds

)2

+1

T 2EQt

((∫ T

t

e−γsWφsds

)2)

=1

T 2

∫ T

t

∫ T

t

e−γ(s+u)EQt (WφsWφu)dsdu− 1

T 2x2t e−2γt

(1− e−γ(T−t)

γ

)2

.

(47)

The interchange expectation-integral:

EQt (

∫ T

t

∫ T

t

e−γ(s+u)WφsWφudsdu) =

∫ T

t

∫ T

t

e−γ(s+u)EQt (WφsWφu)dsdu (48)

is justified the same way as above, using the definition of Wφt = xt in (21) togetherwith Tonelli’s theorem. Finally, we use equation (37) of Lemma 1 and integrate theexpression with respect to s and u to complete the proof.

Corollary 2: The Brockhaus&Long approximation of the volatility swap priceKvol of maturity T at initiation of the contract t = 0 in the delayed Hestonmodel (5)-(8) is given by:

Kvol ≈√Kvar −

V arQ0 (VR)

8K32var

, (49)

where Kvar is given by formula (20) of Corollary 1 and:

V arQ0 (VR) =δ2e−2γT

2T 2γ3

[θ2τ

(2γTe2γT + 4eγT − 3e2γT − 1

)+

γ

2γ − γτ(V0 − θ2τ )(

2e2γT(

2γ

γτ− 1

)− 4γeγT

(e(γ−γτ )T − 1

γ − γτ

)+ 4eγT

(1− γ

γτe(γ−γτ )T

)− 2

)].

(50)

We notice that letting τ → 0 (and therefore γτ → γ) we get the formula ofSwishchuk [22].

Proof: We have by definition Kvol = Y0(T ), therefore the result is obtained fromequation (40) of Theorem 2.

5 Volatility Swap Hedging

In this section, we consider dynamic hedging of volatility swaps using varianceswaps, as the latter are a fairly liquid, easy to trade derivatives. In the spiritof Broadie and Jain [6], we consider a portfolio containing at time t one unit ofvolatility swap and βt units of variance swaps, both of maturity T . Thereforethe value Πt of the portfolio at time t is:

Πt = e−r(T−t) [Yt(T )−Kvol + βt(Xt(T )−Kvar)] . (51)

The portfolio is self-financing, therefore:

dΠt = rΠtdt+ e−r(T−t) [dYt(T ) + βtdXt(T )] . (52)

The price processes Xt(T ) and Yt(T ) can be expressed, denoting It :=∫ t0Vsds

the accumulated variance at time t (known at this time):

Xt(T ) = EQt

[1

TIt +

1

T

∫ T

t

Vsds

]= g(t, It, Vt), (53)

Yt(T ) = EQt

√ 1

TIt +

1

T

∫ T

t

Vsds

= h(t, It, Vt). (54)

Remembering that θ2t = θ2τ +(V0−θ2τ ) (γ−γτ )γ e−γτ t and noticing that dIt = Vtdt,

by Ito’s lemma we get:

dXt(T ) =

[∂g

∂t+∂g

∂ItVt +

∂g

∂Vtγ(θ2t − Vt) +

1

2

∂2g

∂V 2t

δ2Vt

]dt+

∂g

∂Vtδ√VtdW

Qt ,

(55)

dYt(T ) =

[∂h

∂t+∂h

∂ItVt +

∂h

∂Vtγ(θ2t − Vt) +

1

2

∂2h

∂V 2t

δ2Vt

]dt+

∂h

∂Vtδ√VtdW

Qt .

(56)

As conditional expectations of cashflows at maturity of the contract, the priceprocesses Xt(T ) and Yt(T ) are by construction martingales, and therefore weshould have:

∂g

∂t+∂g

∂ItVt +

∂g

∂Vtγ(θ2t − Vt) +

1

2

∂2g

∂V 2t

δ2Vt = 0, (57)

∂h

∂t+∂h

∂ItVt +

∂h

∂Vtγ(θ2t − Vt) +

1

2

∂2h

∂V 2t

δ2Vt = 0. (58)

The second equation, combined with some appropriate boundary conditions,was used in Broadie and Jain [6] to compute the value of the price processYt(T ), whereas we focus on its Brockhaus&Long approximation.

Therefore we get:

dXt(T ) =∂g

∂Vtδ√VtdW

Qt , (59)

dYt(T ) =∂h

∂Vtδ√VtdW

Qt . (60)

and so:

dΠt = rΠtdt+ e−r(T−t)[∂h

∂Vtδ√VtdW

Qt + βt

∂g

∂Vtδ√VtdW

Qt

]. (61)

In order to dynamically hedge a volatility swap of maturity T , one shouldtherefore hold βt units of variance swap of maturity T , with:

βt = −∂h∂Vt∂g∂Vt

= −∂Yt(T )∂Vt

∂Xt(T )∂Vt

. (62)

Remembering that V arQ0 (VR), Kvar are given respectively in Corollary 2 and1, the initial hedge ratio β0 is given by:

β0 = −∂Y0(T )∂V0

∂X0(T )∂V0

, (63)

∂X0(T )

∂V0=

1− e−γτT

γτT, (64)

∂Y0(T )

∂V0≈

∂X0(T )∂V0

2√Kvar

−Kvar

∂V arQ0 (VR)

∂V0− 3

2∂X0(T )∂V0

V arQ0 (VR)

8K52var

, (65)

∂V arQ0 (VR)

∂V0=δ2e−2γT

T 2γ3

γ

2γ − γτ

[e2γT

(2γ

γτ− 1

)−2γeγT

(e(γ−γτ )T − 1

γ − γτ

)+ 2eγT

(1− γ

γτe(γ−γτ )T

)− 1

].

(66)

Remembering that V arQt (VR), Xt(T ) are given respectively in Theorems 2 and1, the hedge ratio βt for t > 0 is given by:

βt = −∂Yt(T )∂Vt

∂Xt(T )∂Vt

, (67)

∂Xt(T )

∂Vt=

1− e−γ(T−t)

γT, (68)

∂Yt(T )

∂Vt≈

∂Xt(T )∂Vt

2√Xt(T )

−Xt(T )

∂V arQt (VR)

∂Vt− 3

2∂Xt(T )∂Vt

V arQt (VR)

8Xt(T )52

, (69)

∂V arQt (VR)

∂Vt=

δ2

γ3T 2

[1− e−2γ(T−t) − 2(T − t)γe−γ(T−t)

]. (70)

We take the parameters that have been calibrated in section 3 on September30th 2011 and we plot the naive Volatility Swap strike

√Kvar together with

the adjusted Volatility Swap strike√Kvar − V arQ(VR)

8K32var

along the maturity di-

mension, as well as the convexity adjustment V arQ(VR)

8K32var

:

Figure 1: Naive Volatility Swap Strike Vs. Adjusted Volatility Swap Strike

Figure 2: Convexity Adjustment

The naive Volatility Swap Strike represents the initial fair value of the volatilityswap contract obtained without taking into account the convexity adjustmentV arQ(VR)

8K32var

linked to the Brockhaus&Long approximation, whereas the adjusted

Volatility Swap Strike represents this initial fair value when we do take intoaccount the convexity adjustment. The difference between the former and thelatter is quantified by the convexity adjustment and is represented on the sec-ond graphic. We see that neglecting the convexity adjustment leads to anoverpricing of the volatility swap. On this example, the overpricing is espe-cially significant for maturities less than 2Y, with a peak difference of morethan 2% between the naive and adjusted strikes for maturities around 6M. Theposition of this local extremum (here, around 6M) is linked to the values of the

calibrated parameters and therefore varies depending on the date we performthe calibration at.

We also plot the initial hedge ratio β0 along the maturity dimension:

Figure 3: Initial Hedge Ratio

This initial hedge ratio β0 represents the quantity of variance swap contractswe need to buy (if β0 > 0) or sell (if β0 < 0) to hedge our position on onevolatility swap contract of the same maturity. Of course, in order to cancel therisk, β0 has to be negative if we buy a volatility swap contract, and positive ifwe sell one. Here we have assumed that we hold a long position on a volatilityswap contract, i.e. that we have bought one such contract. The plot tells usthat for one volatility swap contract bought, we need to sell approximately 3variance swap contracts of the same maturity (depending of the maturity ofthe contract) to hedge our position on the volatility swap, i.e. to cancel the riskinherent to our position. We say that we hold a short position on the varianceswap contracts. The trend is that the higher the maturity of the volatilityswap contract, the more variance swap contracts we need to sell in order tohedge our position. This was to be expected because for such pure volatilitycontracts, the longer the maturity, the higher the probability that the volatilityvaries significantly, i.e. the higher the risk.

6 Conclusions

In this paper, we introduced a variance drift adjusted version of the Hestonmodel based on the concept of delay, the Delayed Heston model (section 2).As explained in the introduction, this model makes a bridge between the pop-ular Heston model and the delayed stochastic volatility model considered bySwishchuk in [23]. Our model has two additional parameters compared to theHeston model and since it can be seen as a time-dependent Heston model withtime-dependent long-range variance θ2t , it can be implemented very easily, for

both Monte Carlo simulation and pricing of call options via the semi-closedformulas which can be derived (see Appendix A). We calibrated our model on12 dates ranging from Sep. 19th to Oct. 17th 2011 for the FOREX underlyingEURUSD (section 3). Our findings were twofold: the Delayed Heston modelalways outperformed significantly the Heston model in terms of average (abso-lute) calibration error (especially for long maturities and ATM options), butalso in terms of the standard deviation of the calibration errors. The latter ishighly desirable in practice as we do not want to face the case of very poorlypriced options on one hand, and almost perfectly priced options on the other:it is better that each individual calibration error corresponding to each calloption is close to the average calibration error, and therefore that the standarddeviation of the calibration errors is low. In sections 4 and 5, we consideredrespectively the pricing of variance and volatility swaps, and the dynamic hedg-ing of a position on a volatility swap by a position on variance swaps, the latterbeing very liquid financial derivatives. We obtained a closed formula for boththe price process of the variance swap and the Brockhaus&Long approximation(which is a 2nd order approximation) of the price process of the volatility swap.Finally, to illustrate these last sections, we displayed 3 graphics showing theimportance of taking into account the convexity adjustment (corresponding tothe Brockhaus&Long approximation) when pricing a volatility swap, and thattaking naively the volatility swap strike Kvol to be

√Kvar may lead to a sig-

nificant mispricing of the volatility swap. Regarding future research directions,it would be interesting to understand how the Delayed Heston model could beimproved by considering a Stochastic Local Volatility (SLV) approach, or howwe could improve calibration on call option prices using the Generalized Meth-ods of Moments (GMM), for example. Another interesting direction would beto use our model for forecasting purposes (instead of pricing), and estimateour model parameters via the computation of the Delayed Heston volatilitylikelihood function, which would have to be derived.

A Semi-closed formulas for call options in the DelayedHeston Model

From Kahl&Jackel [13], we get equations (71) to (74) for the price of a calloption with maturity T and strike K in the time-dependent long-range varianceHeston model:

C0 = e−rT[

1

2(F −K) +

1

π

∫ ∞0

(Fh1(u)−Kh2(u))du

], (71)

h1(u) = <(e−iu ln(K)ϕ(u− i)

iuF

), (72)

h2(u) = <(e−iu ln(K)ϕ(u)

iu

), (73)

with F = S0e(r−q)T and:

ϕ(u) = eC(T,u)+V0D(T,u)+iu ln(F ). (74)

By Michailov&Noegel [20], we have that C(t, u) and D(t, u) solve the followingdifferential equations:

dC(t, u)

dt= γθ2tD(t, u), (75)

dD(t, u)

dt− δ2

2D2(t, u) + (γ − iuρδ)D(t, u) +

1

2(u2 + iu) = 0, (76)

C(0, u) = D(0, u) = 0. (77)

The Riccati equation for D(t, u) doesn’t depend on θ2t , therefore its solution isjust the solution of the classical Heston model given in Kahl&Jackel [13]:

D(t, u) =γ − iρδu+ d

δ2

[1− edt

1− gedt

], (78)

g =γ − iρδu+ d

γ − iρδu− d, (79)

d =√

(γ − iρδu)2 + δ2(iu+ u2). (80)

Given D(t, u) and the definition of θ2t , we can compute C(t, u) from (75) and(77):

C(t, u) = γθ2τf(t, u) + (V0 − θ2τ )(γ − γτ )

∫ t

0

e−γτsD(s, u)ds. (81)

Where f(t, u) =∫ t0D(s, u)ds is given in Kahl&Jackel [13]:

f(t, u) =1

δ2

((γ − iρδu+ d)t− 2 ln

(1− gedt

1− g

)). (82)

Unfortunately, the integral∫ t0e−γτsD(s, u)ds in (81) cannot be computed di-

rectly as∫ t0D(s, u)ds. The logarithm in f(t, u) can be handled as suggested in

Kahl&Jackel [13], as well as the integration of the Heston integral, namely:

C0 = e−rT∫ 1

0

y(x)dx, (83)

y(x) =1

2(F −K) +

Fh1(− ln(x)C∞

)−Kh2(− ln(x)C∞

)

xπC∞, (84)

where C∞ > 0 is an integration constant.

The following limit conditions are given in Kahl&Jackel [13]:

limx→0

y(x) =1

2(F −K), (85)

limx→1

y(x) =1

2(F −K) +

FH1 −KH2

πC∞, (86)

Hj = limu→0

hj(u) = ln

(F

K

)+ cj(T ) + V0dj(T ), (87)

where:

d1(t) = =(∂D

∂u(t,−i)

), (88)

c1(t) = =(∂C

∂u(t,−i)

), (89)

d2(t) = =(∂D

∂u(t, 0)

), (90)

c2(t) = =(∂C

∂u(t, 0)

). (91)

Expressions for d1(t) and d2(t) are the same as in Kahl&Jackel [13] as θ2t doesn’tplay any role in them. Given (75) and (77), we compute c1(T ) and c2(T ) inour time-dependent long-range variance Heston model by:

cj(T ) = γ

∫ T

0

θ2t dj(t)dt. (92)

After computing the integrals we get:

If γ − ρδ 6= 0 and γ − ρδ + γτ 6= 0:

d1(T ) =1− e−(γ−ρδ)T

2(γ − ρδ), (93)

c1(T ) = γθ2τe−(γ−ρδ)T − 1 + (γ − ρδ)T

2(γ − ρδ)2(94)

+(V0 − θ2τ )(γ − γτ )

2(γ − ρδ)

(−e−γτT − 1

γτ+e−(γ−ρδ+γτ )T − 1

γ − ρδ + γτ

). (95)

If γ − ρδ 6= 0 and γ − ρδ + γτ = 0:

d1(T ) =1− e−(γ−ρδ)T

2(γ − ρδ), (96)

c1(T ) = γθ2τe−(γ−ρδ)T − 1 + (γ − ρδ)T

2(γ − ρδ)2(97)

+(V0 − θ2τ )(γ − γτ )

2(γ − ρδ)

(−e−γτT − 1

γτ− T

). (98)

If γ − ρδ = 0:

d1(T ) =T

2, (99)

c1(T ) = γθ2τT 2

4+

(V0 − θ2τ )(γ − γτ )

2

(−Te

−γτT

γτ+

1− e−γτT

γ2τ

), (100)

and:

d2(T ) =e−γT − 1

2γ, (101)

c2(T ) = γθ2τ1− e−γT − γT

2γ2(102)

+(V0 − θ2τ )(γ − γτ )

2γ

(−1− e−γτT

γτ− e(−γτ−γ)T − 1

γτ + γ

). (103)

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