Pricing Exotic Variance Swaps under

3/2-Stochastic Volatility Models

CHI HUNG YUEN & YUE KUEN KWOK1

Department of Mathematics, Hong Kong University of Science and Technology, Hong Kong

October, 2013

AbstractWe consider pricing of various types of exotic discrete variance swaps, like the gamma swapsand corridor swaps, under the 3/2-stochastic volatility models with jumps. The class ofstochastic volatility models (SVM) that use a constant-elasticity-of-variance (CEV) processfor the instantaneous variance exhibit nice analytical tractability when the CEV parametertakes just a few special values (namely, 0, 1/2, 1 and 3/2). The popular Heston model cor-responds to the choice of the CEV parameter to be 1/2. However, the stochastic volatilitydynamics derived from the Heston model fails to agree with empirical findings from actualmarket data. The choice of 3/2 for the CEV parameter in the SVM shows better agreementwith empirical studies while it maintains a good level of analytical tractability. By usingthe partial integro-differential equation formulation, we manage to derive quasi-closed formpricing formulas for the fair strike values of various types of discrete variance swaps. Pric-ing properties of these exotic discrete variance swaps under different market conditions areexplored.

Keywords: Variance swaps, gamma swaps, corridor swaps, 3/2-volatility model

1 Introduction

Variance and volatility derivatives become more popular in the financial market since theirintroduction in the late nineties. Since volatility is likely to grow when uncertainty and riskincrease, hedge funds and private investors may use these derivatives to manage their expo-sure to the volatility risk associated with their trading positions. Speculators can also placetheir bids on the future movement of the underlying volatility via trading on these instru-ments. Stock options are less than ideal as the instruments to provide exposure to volatilitysince they have exposure to both volatility and direction of the stock price movement. Onemay argue that exposure to the stock price in an option can be hedged. However, deltahedging is at best inaccurate since volatility cannot be estimated accurately. On the otherhand, writers of variance swaps can hedge their positions via replication by using a portfolioof options traded in the markets. The provision of pure exposure of volatility and effectivereplication by traded options provide the impetus for the growth of the markets for swapsand other derivatives on discrete realized variance. Reader may refer to Carr and Madan(1998) for an introduction to the theory of volatility trading.

Variance swaps are essentially forward contracts on discrete realized variance. In recentyears, other variants of variance swaps that target more specific features of variance exposure,like the gamma swaps and corridor swaps (commonly called the third generation varianceswaps), are also structured in the financial markets. The product specifications of these

1Correspondence author; email: maykwok@ust.hk; fax number: 852-2358-1643

1

variance swap products will be presented in Section 3. The potential uses in hedging andbetting the various forms of volatility exposure can be found in numerous articles (Demeterfiet al., 1999; Carr and Lee, 2009; Bouzoubaa and Osseiran, 2010). The pricing of exoticdiscrete variance swaps under time-changed Levy processes and stochastic volatility modelshave been studied by Itkin and Carr (2010), Crosby and Davis (2011) and Zhu and Lian(2011). Zheng and Kwok (2014) study the pricing of highly path dependent corridor andconditional swaps on discrete realized variance under the Heston stochastic model (Heston,1993) with simultaneous jumps in the asset price and variance.

Itkin (2013) considers the class of stochastic volatility models (SVM) whose instantaneousvariance is modeled by a constant-elasticity-of-variance (CEV) process. The number ofanalytically tractable models is rather limited, where the CEV parameter γ takes only 4specific values: 0, 1/2, 1 and 3/2. The Heston model (square root process) corresponds tothe choice of γ = 1/2, and it enjoys the best analytical tractability. Indeed, the Heston modelbelongs to the class of affine models and the methodologies for finding the correspondingjoint moment generating functions are very well developed (Duffie et al., 2000). These niceanalytical tractability properties lead to successful derivation of succinct closed form pricingformulas for discrete variance swaps (Zheng and Kwok, 2014). Unfortunately, the Hestonmodel has been shown to be inconsistent with observations in the variance markets. It leadsto downward sloping volatility of variance smiles, contradicting with empirical findings frommarket data. On the other hand, the 3/2-model (choice of γ = 3/2) exhibits better agreementwith empirical studies while maintains some level of analytical tractability. For example,based on S&P 100 implied volatilities, Jones (2003) and Bakshi et al. (2006) estimate thatγ should be around 1.3, which is close to 3/2 over 1/2 (Heston model). In addition, Jonesconcludes that jumps are needed in the underlying process for short maturity options. In thereal world statistical measure, Javaheri (2004) analyzes the CEV type instantaneous varianceprocess with the exponent of the volatility of variance process either be 1/2, 1 or 3/2 byusing the time series data on S&P 500 daily returns and finds that the 3/2-power performsthe best. Ishida and Engle (2002) estimate the power to be 1.71 for S&P 500 daily returnfor a 30-year period. Chacko and Viceira (2003) employ the technique of the GeneralizedMethod of Moments on a 35-year period of weekly return and a 71-year period of monthlyreturn. They estimate the power to be 1.10 and 1.65, respectively, over the two periods. Byusing S&P 500 index options over a period of 7 years, Poteshman (1998) concludes that thedrift of the instantaneous variance is not of affine structure (like the Heston model).

There have been several recent works that use the 3/2-model for pricing variance andvolatility derivatives. Goard and Mazur (2013) and Drimus (2012) compute the fair strikevalues for the continuously monitored variance and volatility derivatives under the 3/2-modelwithout jumps. Instead of modeling the dynamics of the instantaneous volatility directly,Carr and Sun (2007) adopt a new approach by assuming continuous dynamics for the time-T variance swap rate, taking advantage of the liquidity of the variance swap market. Theymanage to derive the analytic closed form formula for the joint conditional Fourier-Laplacetransform of the terminal log-asset value and its quadratic variation for the 3/2-model. Chanand Platen (2012) derive exact pricing and hedging formulas of continuously monitored longdated variance swaps under the 3/2-model using the benchmark approach, a pricing conceptthat provides minimal fair strike values for variance swaps when an equivalent risk neutralprobability measure does not exist. Itkin and Carr (2010) derive closed form pricing formulasfor discrete variance swaps and options on quadratic variation under a class of time-changedLevy models, which includes the 3/2-power clock change.

In this paper, we consider pricing of various types of discrete variance swaps, includingthe gamma swaps and corridor swaps, under the 3/2-stochastic volatility model with jumps.As a non-affine stochastic volatility model, the level of analytic tractability of the 3/2-model

2

is lower than that of the affine Heston model. Since each term in the accumulated squaredreturns involves asset prices monitored at two successive monitoring instants, the analyticprocedure requires a two-step solution of coupled partial integro-differential equations. Asimilar approach has been used by Rujivan and Zhu (2012) for pricing discrete variance swapsunder the Heston stochastic volatility model. The resulting quasi-closed form formulas forthe fair strikes of the generalized variance swaps are expressed in the form of integrals, thenumerical valuation of them can be performed using standard quadrature to sufficiently highlevel of accuracy.

The later sections of this paper are organized as follows. In the next section, we presentthe model specification of the 3/2-stochastic volatility model with jumps. We then derivethe partial integro-differential equations for pricing contingent claims with payoffs that aredependent on discrete realized variance. We illustrate how to use several skillful integraltransform techniques to find the fundamental solutions of the partial integro-differentialequations. In Section 3, we consider analytic pricing of the third generation swaps: gammaswaps and corridor swaps. The solution procedures involve an elegant combination of tech-niques developed by various earlier papers on pricing similar discrete swap products underthe Heston model (Carr and Sun, 2007; Rujivan and Zhu, 2012; Zheng and Kwok, 2014).In particular, we adopt a two-step procedure that computes the expected discrete variancebased on nested conditional expectation over two successive time intervals. In Section 4,we present the results of our numerical tests that were performed to illustrate the effectivenumerical valuation of the quasi-closed form pricing formulas of the variance swaps. Also,we performed detailed analysis on the pricing properties of the variance swaps with respectto different sets of parameters, like the correlation between asset price and its instanta-neous variance, sampling frequency, volatility of variance, jump distribution. The conclusiveremarks and summary of findings are presented in the last section.

2 3/2-stochastic volatility models and pricing formu-

lation of variance derivatives

In this section, we first present the class of stochastic volatility models (SVM) that usea constant-elasticity-of-variance (CEV) process for the instantaneous variance and discussbriefly the analytical tractability of these SVM under various choices of the CEV parameter.We then provide the justification of the choice of 3/2-model as the underlying asset priceprocess based on empirical studies. We also state a useful formula for the joint conditionalFourier-Laplace transform of the terminal asset price and their quadratic variation for the3/2-model with jumps. Next, we derive the partial integro-differential equation formulationfor pricing a contingent claim with payoff that is dependent on squared asset return. Sincethe squared asset return involves asset prices Sti−1

and Sti at successive monitoring instantsti−1 and ti, and the fair strike value of the variance swap at t0 is to be determined, thebackward induction procedure involves a two-step solution over the successive time intervals[ti−1, ti] and [t0, ti−1]. We show how to apply the appropriate jump condition at the timeinstant ti−1. Once the fundamental solutions to the partial integro-differential equations areobtained, as an illustration of the pricing procedure, we show how to find the fair strike ofa vanilla variance swap under the 3/2-model with jumps.

2.1 3/2-models with jumps

For pricing volatility products and derivatives on discrete realized variance, the first step is toidentify an appropriate stochastic volatility model for the underlying price process. Goard

3

and Mazur (2013) consider the performance of the class of constant-elasticity-of-variance(CEV) process for the instantaneous variance vt and examine their ability to capture thebehavior of VIX, where the dynamics of vt is governed by

dvt = (c1 +c2vt

+ c3vt ln vt + c4vt + c5v2t ) dt+ ǫvγt dZt. (2.1)

Here, the CEV parameter γ, volatility of variance ǫ and other parameters in the drift termare taken to be constants. To achieve analytic tractability, Itkin (2013) identifies the choiceof γ to be limited to four values: 0, 1/2, 1 and 3/2. The choice of γ = 1/2 corresponds tothe renowned Heston model, which has the highest level of analytical tractability due to itsaffine structure. Recently, various empirical studies on the variance exponent γ have shownthat the 3/2-model (with γ = 3/2) is a better choice to fit with market data when comparedwith the Heston model (Jones, 2003; Bakshi et al., 2006). Goard and Mazur (2013) showthat the 3/2-model with either linear drift or quadratic drift are the only acceptable modelsamong the class of stochastic volatility models specified in eq. (2.1) for characterizing theVIX dynamics. Baldeaux and Badran (2012) demonstrate that the 3/2-model is capable toproduce the market observed upward-sloping implied volatility for VIX options. They alsoshow that jumps should be introduced in the asset price process in order to capture thevolatility smile for short-maturity options. Fortunately, the resulting 3/2-model with jumpsremains to be tractable, like its no-jump counterpart. In this paper, based on the aboveobservations, we choose the 3/2-model with jumps as the underlying price process.

We assume that the dynamics of the stock price St and its instantaneous variance vtunder a risk-neutral measure Q is governed by

dSt

St

= (r − d− λm) dt+√vt dW

1t + (eJ − 1) dNt,

dvt = vt[p(t)− qvt] dt+ ǫv3/2t dW 2

t , (2.2)

where r is the riskfree interest rate, d is the dividend yield,W 1t andW 2

t are standard Brownianmotions with dW 1

t dW2t = ρ dt. Also, Nt is a compound Poisson process with constant arrival

rate λ and it is independent of W 1t and W 2

t . For the jump process, J denotes the randomjump size of the log stock price. It has a normal distribution with mean ν and varianceζ2. Also, it is independent to the two Brownian motions and the Poisson process Nt. Thecompensator parameter m is given by EQ[eJ −1] = eν+ζ2/2−1. The parameters q in the driftterm and the correlation coefficient ρ are constant. We may generalize the level parameterp(t) in the dynamics of vt to be a deterministic continuous function of time that allows forflexibility for calibration.

The 3/2-dynamics of the variance process exhibits the mean-reverting feature. The mean-reverting rate depends on the current variance level, thus establishing a more volatile volatil-ity structure than that of the Heston model. We let wt be the reciprocal of the variance vt.By applying Ito’s lemma, wt is seen to follow the time-inhomogeneous Heston model:

dwt = [q + ǫ2 − p(t)wt] dt− ǫ√wt dW

2t . (2.3)

There are certain technical conditions required in order to avoid anomalies in the 3/2-process.For a CIR process (the variance process in the Heston model), it is well known that thecoefficients have to satisfy the Feller’s boundary condition in order to avoid reaching the zerovalue, causing vt in the 3/2-process to explode. In our model, this condition is expressed asq ≥ −ǫ2/2.

When pricing options under the share measure, Lewis (2000) shows that vt has zeroprobability of explosion if and only if ρ < ǫ/2. By imposing the two constraints: q > 0 and

4

ρ < 0 in our model, it becomes sufficient to avoid unboundedness in the variance process.Note that ρ < 0 is not that restrictive since negative correlation between St and vt confirmsquite well with the leverage effect commonly observed in the market: volatility goes up asstock price drops.

The success of analytical tractability of the 3/2-model relies on the availability of closedform representation of the joint conditional Fourier-Laplace transform of the terminal logasset price and the de-annualized realized variance

∫ T

tvs ds. By assuming St to have no

jump component in eq. (2.2), and letting Xt denote lnSte(r−d)(T−t), Carr and Sun (2007)

obtain

E

[eikXT−µ

∫T

tvs ds∣∣∣Xt, vt

]= eikXt

Γ(γ − α)

Γ(γ)

[ 2

ǫ2y(vt, t)

]αM(α, γ,− 2

ǫ2y(vt, t)

), (2.4)

where

y(vt, t) = vt

∫ T

t

e∫u

tp(s) ds du,

α = −(12− qk

ǫ2

)+

√(12− qk

ǫ2

)2+

2ckǫ2

,

γ = 2(α + 1− qk

ǫ2

),

qk = ρǫik − q,

ck = µ+k2 + ik

2.

The confluent hypergeometric function M(α, γ, z) is defined as

M(α, γ, z) =∞∑

n=0

(α)n(γ)n

zn

n!,

where (α)n = (α)(α− 1) · · · (α+n− 1) is the Pochhammer symbol. Later, we generalize theabove analytic result to the 3/2-model with jumps.

2.2 Pricing formulation of derivatives on realized variance

There are two commonly adopted market conventions of computing the discrete realizedvariance, one is measured based on the actual rate of return while the other is based on thelog return. We consider the tenor of the realized variance to be [0, T ] with monitoring dates0 = t0 < t1 < · · · < tN = T , where T is the maturity date and N is the total number ofmonitoring dates. We use V

(1)d to denote the discrete realized variance over [t0, tN ] in terms

of actual return as defined by

V(1)d =

FA

N

N∑

i=1

(Sti − Sti−1

Sti−1

)2, (2.5a)

where FA is the annualized factor. We take FA = 252 for daily monitoring and FA = 52 forweekly monitoring. Alternatively, the discrete realized variance over [t0, tN ] in terms of logreturn is given by

V(2)d =

FA

N

N∑

i=1

(ln

Sti

Sti−1

)2. (2.5b)

Let △ti denote the time interval between ti−1 and ti, i = 1, 2, · · · , N . We assume the timeintervals to be uniform and use △t to denote this common time interval. For the stochastic

5

volatility model with jumps defined in eq. (2.2), it is known that as ∆t → 0, we have

lim△t→0

N∑

i=1

(ln

Sti

St1−1

)2=

∫ tN

t0

vt dt+

NT∑

k=N0

J2k ,

where∫ tNt0

vt dt is the continuous realized variance over [t0, tN ] and the last term sums allthe square of jumps occurring within [t0, tN ]. The fair strike of the vanilla swap on discreterealized variance (either actual return or log return) is given by

Kn = EQ[V(n)d ], n = 1, 2,

where EQ denotes the expectation under the risk neutral measure Q. By the linear propertyof expectation, the determination of the fair strike amounts to the evaluation of individual

risk neutral expectation of squared return: EQ[(Sti

−Sti−1

Sti−1

)2] or EQ[(lnSti

Sti−1

)2] and summation

over all monitoring instants.We adopt the partial integro-differential equation approach to compute the above expec-

tation of squared return over successive monitoring instants when the dynamics of the assetprice St follows the 3/2-model [see eq. (2.2)]. The challenge in the model formulation isthat the risk neutral expectation is taken conditional on the filtration at time t0 with respectto the two stochastic state variables: Sti−1

and Sti . To resolve the path dependence on theasset price, we follow Little and Pant (2001) to introduce the path dependent state variableIt, where

It =

∫ t

0

δ(u− ti−1)Su du =

{Sti−1

ti−1 ≤ t0 0 ≤ t < ti−1

. (2.6)

Since It takes different forms over [0, ti−1) and [ti−1, ti], we solve the governing equation ina two-step procedure. This is done by solving backward in time from ti to ti−1, applying anappropriate jump condition at ti−1, then solving backward from ti−1 to t0. Subsequently, wewrite Si as Sti , i = 0, 1, · · · , N , for simplicity. Also, similar notations are used for vti andIti .

Consider the pricing of a contingent claim with terminal payoff Fi(Si, vi, Ii) at maturitydate ti, i = 1, 2, · · · , N , the time-t0 value of Ui(S0, v0, I0, t0) under the risk neutral valuationframework is given by

Ui(S0, v0, I0, t0) = e−r(ti−t0)EQ[Fi(Si, vi, Ii)], i = 1, 2, · · · , N. (2.7)

By the Feymann-Kac theorem, the governing partial integro-differential equation (PIDE) forUi(S, v, I, t) is given by

∂Ui

∂t+

1

2vS2∂

2Ui

∂S2+ ρǫv2S

∂2Ui

∂S∂v+

ǫ2

2v3

∂2Ui

∂v2+ (r − d− λm)S

∂Ui

∂S+ [p(t)v − qv2]

∂Ui

∂v

+ δ(t− ti−1)S∂Ui

∂I− rUi + λEJ [Ui(Se

J , v, I, t)− Ui(S, v, I, t)] = 0, (2.8)

where EJ is the expectation with respect to the jump process.In our pricing problem of finding the fair strike value of a swap on discrete realized

variance, the terminal payoff is independent of vi. There is a jump in the value for I acrossti−1 as shown in eq. (2.6), while I assumes constant value across [t0, ti−1) and [ti−1, ti] sothat ∂U

∂I= 0 at t 6= ti−1. Note that there is no jump in the value for Ui across ti−1 since there

6

is no cash flow across the monitoring instant. Since I = 0 at times prior to ti−1 and I = Sat times after ti−1, the no-jump condition is stated as

limt→t+

i−1

Ui(S, v, S, t) = limt→t−

i−1

Ui(S, v, 0, t). (2.9)

The two-step solution procedure for solving the PIDE can be summarized as follows:

(i) t ∈ (ti−1, ti)

∂Ui

∂t+

1

2vS2∂

2Ui

∂S2+ ρǫv2S

∂2Ui

∂S∂v+

ǫ2

2v3

∂2Ui

∂v2+ (r − d− λm)S

∂Ui

∂S

+ [p(t)v − qv2]∂Ui

∂v− rUi + λEJ [Ui(Se

J , v, I, t)− Ui(S, v, I, t)] = 0, (2.10)

with terminal condition: Ui(S, v, I, ti) = Fi(S, Si−1). Here, Si−1 is a known parameterand Fi has no dependence on v.

(ii) t ∈ [t0, ti−1)

∂Ui

∂t+

1

2vS2∂

2Ui

∂S2+ ρǫv2S

∂2Ui

∂S∂v+

ǫ2

2v3

∂2Ui

∂v2+ (r − d− λm)S

∂Ui

∂S

+ [p(t)v − qv2]∂Ui

∂v− rUi + λEJ [Ui(Se

J , v, I, t)− Ui(S, v, I, t)] = 0, (2.11)

with terminal condition specified in eq. (2.9).

It is analytically tractable to solve for Ui over the time interval [ti−1, ti] since the terminalpayoff function Fi(S, Si−1) is independent of v and Si−1 is a known parameter. However, sincethe solution Ui at t

−i−1 has dependence on v, the solution of Ui at t0 cannot be obtained in

analytic closed form. We manage to express Ui at t0 in a quasi-closed form in terms of anintegral with the product of the transition density and the known solution Ui at t

−i−1 as the

integrand.To solve for Ui(x, v, t) over [ti−1, ti] as governed by eq. (2.10), we take the Fourier trans-

form of the governing equation with respect to x, where x = lnS. We let τ = ti − t anddefine the Fourier transform of Ui(x, v, τ) (dropping dependence on I) by

Ui(k, v, τ) =

∫ ∞

−∞

e−ikxUi(x, v, τ) dx. (2.12)

We define Hi(k, v, τ) by

Hi(k, v, τ) = exp(−s(k)τ)Ui(k, v, τ),

where

s(k) = ik(r − d− λm)− (r + λ) + λ exp(ikν − ζ2

2k2).

The PIDE (2.10) is transformed into a partial differential equation in Hi as follows:

∂Hi

∂τ= −ckvHi + [p(t) + qkv

2]∂Hi

∂v+

ǫ2

2v3

∂2Hi

∂v2, (2.13)

with initial condition: F [F (ex, Si−1)], where ck = (k2 + ik)/2 and qk = ρǫik − q. If we canfind the fundamental solution to eq. (2.13), then the solution Ui of the PDE (2.10) can beobtained easily by the following proposition.

7

Proposition 1 Let Hi(k, v, τ) denote the fundamental solution to eq. (2.13) with initial

condition: Hi(k, v, 0) = 1, then

Hi(k, v, τ) =Γ(γ − α)

Γ(γ)

[ 2

ǫ2y(v, t)

]αM(α, γ,− 2

ǫ2y(v, t)

).

The solution for Ui(x, v, t) over [ti−1, ti] is given by

Ui(x, v, τ) = F−1[exp(s(k)τ)Hi(k, v, τ)F [Fi(e

x, Si−1)]].

The proof of Proposition 1 is presented in Appendix A.

Transition density function of the 3/2-variance process

Over the time interval [t0, ti−1), it is cumbersome to use the fundamental solution Hi in theFourier domain to perform the expectation valuation with respect to the variance process.Instead, we employ the transition density function of the variance process vt conditionalon v0 to simplify the expectation calculation. The reciprocal of vt is a CIR process, whosetransition density function is well known. By defining wt = 1/vt, their respective densityfunctions pvt and pwt

are related by (Jeanblanc et al., 2009)

pvt(vt, t|v0, 0)

= pwt

( 1

vt, t∣∣∣ 1v0, 0) 1

v2t

=l(0, t)

2l∗(0, t)v2texp

(− 1

v0− 1

vtl(0, t)

2l∗(0, t)

)(v0 l(0, t)

vt

)ν/2

Iν

( √l(0, t)

l∗(0, t)√v0vt

), (2.14)

where

ν =2

ǫ2(q + ǫ2)− 1, l(s, t) = exp

(∫ t

s

p(u) du), l∗(s, t) =

ǫ2

2

∫ t

s

l(s, u) du.

Note that Iν is the modified Bessel function of the first kind of order ν as defined by

Iν(z) =(z2

)ν ∞∑

k=0

(z2

2

)k

k!Γ(ν + k + 1).

2.3 Fair strike formulas for vanilla variance swaps

The generalized Fourier transform of the terminal payoff Fi(S, Si−1) takes different forms,depending on whether actual return or log return is considered in the squared return.

(i) Actual return

F[F

(1)i (S, Si−1)

]= F

[(S − Si−1

Si−1

)2]

= F

[( ex

Si−1

− 1)2]

= 2π[δ(k + 2i)

S2i−1

− 2δ(k + i)

Si−1

+ δ(k)]. (2.15a)

8

(ii) Log return

F[F

(2)i (S, Si−1)

]= F

[(ln

S

Si−1

)2]

= F [(x− lnSi−1)2]

= 2π[−δ(2)(k)− 2i δ(1)(k) lnSi−1 + δ(k)(lnSi−1)2], (2.15b)

where δ(n)(k) denotes the nth-order derivative of the Dirac function.

Let U(n)i (x, v, τ) denote the solution to eq. (2.10) corresponding to the terminal condition:

F(n)i (ex, Si−1), n = 1, 2. Using Proposition 1, we derive the solutions for U

(1)i and U

(2)i at

τ = ∆t (t = ti−1) as follows:

U(1)i (x, v,∆t)

= F−1[exp(s(k)∆t)Hi(k, v,△t)F

[( ex

Si−1

− 1)2]]

=

∫ ∞

−∞

eikx exp(s(k)∆t)Hi(k, v,∆t)[δ(k + 2i)

S2i−1

− 2δ(k + i)

Si−1

+ δ(k)]dk

= es(−2i)∆tHi(−2i, v,∆t)− 2es(−i)∆t + e−r∆t (2.16a)

and

U(2)i (x, v,∆t)

= F−1[exp(s(k)∆t)Hi(k, v,∆t)F [(x− lnSi−1)

2]]

=

∫ ∞

−∞

eikx exp(s(k)∆t)Hi(k, v,∆t)[− δ(2)(k)− 2iδ(1)(k) lnSi−1 + δ(k)(lnSi−1)

2]dk

= −f (2)(0) + 2if (1)(0) lnSi−1 + f(0)(lnSi−1)2

= −g(2)(0), (2.16b)

where

f(k) = g(k)eikx, g(k) = exp(s(k)∆t)Hi(k, v,∆t),

f (n)(0) =∂nf

∂kn

∣∣∣k=0

, g(n)(0) =∂ng

∂kn

∣∣∣k=0

.

The realized discrete variance consists of N terms [see eqs. (2.5a,b)]. The risk neutral expec-tation of the first term, i = 1, does not require the two-step expectation calculation since S0

is known. For i ≥ 2, it is necessary to implement the two-step backward induction from ti toti−1, then ti−1 to t0. Given the value of the variance swap at time ti−1, the value at time t0can be obtained by taking the expectation of vi−1 conditional on v0 since U

(n)i (x, v,∆t) does

not depend on x. The fair strike can be expressed as a sum of expectation integrals with re-spect to the conditional transition density of vi−1, i = 2, 3, . . . , N and FA

Ner∆tU

(n)1 (x0, v0,∆t),

the details of which are summarized in the following proposition.

Proposition 2 The fair strike of a variance swap with discrete sampling on dates: 0 = t0 <t1 < · · · < tN = T is given by

Kn =FA

Ner∆t

[U

(n)1 (x0, v0,∆t) +

N∑

i=2

∫ ∞

0

U(n)i (x, v,∆t)pv(v, ti−1|v0, 0) dv

], n = 1, 2.

(2.17)

9

3 Third generation variance swaps: Gamma swaps and

corridor swaps

The third generation (generalized) variance swaps are structured so as to provide morespecific exposure to equity variance by adding weights at different monitoring instants inthe evaluation of the accumulated realized variance. The weight adjusted discrete realizedvariance assumes the form

FA

N

N∑

i=1

wi

(Si − Si−1

Si−1

)2or

FA

N

N∑

i=1

wi ln( Si

Si−1

)2

based on actual return or log return, respectively. In this section, we consider the gammaswaps and corridor swaps, where wi takes the forms

wi =Si

S0

and wi = 1{L<Si≤U},

respectively. The motivation for these two generalized variance swaps and their uses havebeen discussed in Lee (2010) and Zheng and Kwok (2014).

3.1 Fair strike formulas for gamma swaps

The derivation of the fair strike formula for the gamma swap, either in actual return or logreturn, amounts to the following respective risk neutral expectation calculation:

EQ

[Si

(Si − Si−1

Si−1

)2]and EQ

[Si

(ln

Si

Si−1

)2].

Similar to the derivation of the fair strike formula for the vanilla variance swap, we computethe generalized Fourier transform of the terminal payoff of the gamma swap as follows:

(i) actual return

F [G(1)i (S, Si−1)] = F

[S(S − Si−1

Si−1

)2]

= 2π[δ(k + 3i)

S2i−1

− 2δ(k + 2i)

Si−1

+ δ(k + i)]; (3.1a)

(ii) log return

F [G(2)i (S, Si−1)] = F

[S(ln

S

Si−1

)2]

= 2π[−δ(2)(k + i)− 2iδ(1)(k + i) lnSi−1 + δ(k + i)(lnSi−1)2]. (3.1b)

We follow a similar two-step solution procedure, where the risk neutral expectation cal-culations are performed over the successive time intervals [ti−1, ti] and [t0, ti−1]. Unlike the

vanilla variance swaps, the time-ti−1 value of U(n)i for a generalized variance swap has depen-

dence on both Si−1 and vi−1. Consequently, this poses the additional technical challenge offinding the joint transition density of lnSt and vt in the risk neutral expectation calculationover [t0, ti−1]. We write xt = lnSt, and let px,v(xt, vt, t|xs, vs, s) denote the joint transitiondensity function of xt and vt from time s to t. For both the gamma swaps and corridor

10

swaps, there is an extra ex term at time ti−1 after the first step evaluation. Therefore, it ismore convenient to have a closed form formula for the xt-transformed density rather thanthe transition density itself.

To proceed further for the derivation of the analytic fair strike formulas of the gammaswaps, we assume p(t) to be a constant value p in all subsequent calculations in orderto achieve analytic closed form representation of the transformed density function. LetG(τ ;−z, vt|xs, vs) be the xt-transformed density, we obtain

G(τ ;−z, vt|xs, vs) =

∫ ∞

−∞

eizypx,v(y, vt, t|xs, vs, s) dy

= G(τ ;−z, vt, vs)eizxs exp(h(z)τ), (3.2)

where

G(τ ;−z, vt, vs) =e(1+µz)pτ

epτ − 1exp

(− 2p

(epτ − 1)vsǫ2− 2pepτ

(epτ − 1)vtǫ2

)

( 2p

ǫ2vt

)(vsvt

)µz

Iνz

(4pepτ/2

(epτ − 1)ǫ2√vsvt

),

and

h(z) = i(r − d− λm)z + [exp(iνz − ζ2z2/2)− 1]λ,

µz =1

2(1 + θz), θz =

2(q + izρǫ)

ǫ2, cz =

(z2 − iz)

2,

cz =2czǫ2

, νz = 2√

µ2z + cz, τ = t− s.

Note that Iνz is the modified Bessel function of the first kind of order νz. The derivation ofeq. (3.2) is presented in Appendix B.

Let U(n)i (x, v, τ) denote the solution to eq. (2.10) corresponding to the terminal condition:

G(n)i (ex, Si−1), n = 1, 2. Using Proposition 1, we derive U

(1)i (x, v,∆t) and U

(2)i (x, v,∆t) as

follows:

U(1)i (x, v,∆t)

= F−1

[exp(s(k)∆t)Hi(k, v,∆t)

[δ(k + 3i)

S2i−1

− 2δ(k + 2i)

Si−1

+ δ(k + i)]]

= ex[es(−3i)∆tHi(−3i, v,∆)− 2es(−2i)∆tHi(−2i, v,∆t) + es(−i)∆t], (3.3a)

and

U(2)i (x, v,∆t)

= F−1[exp(s(k)∆t)Hi(k, v,∆t)[−δ(2)(k + i)− 2iδ(1)(k + i) lnSi−1 + δ(k + i)(lnSi−1)

2]]

= −f (2)(−i) + 2if (1)(−i) lnSi−1 + f(−i)(lnSi−1)2

= ex[−g(2)(−i)], (3.3b)

where

f (n)(α) =∂nf

∂kn

∣∣∣k=α

, g(n)(α) =∂ng

∂kn

∣∣∣k=α

.

In a similar manner as shown in the computation of the fair strike of a variance swap,the risk neutral expectation calculation of the first term in the realized discrete variance

11

[corresponding to i = 1 in the discrete variance formulas (2.5a,b)] does not require the two-step procedure since S0 is known. For i ≥ 2, it is necessary to perform the following doubleintegration procedure corresponding to the risk neutral expectation calculation from ti−1 tot0 in the second step of backward induction:

E[er∆tU

(n)i (x, v,∆t)

]= er∆t

∫ ∞

0

∫ ∞

−∞

U(n)i (x, v,∆t)px,v(x, v, ti−1|x0, v0, 0) dx dv.

For i ≥ 2, let the inner integral be Ψ(n)i (v,∆t|x0, v0), we then obtain

Ψ(1)i (v,∆t|x0, v0)

=

∫ ∞

−∞

U(1)i (x, v,∆t)px,v(x, v, ti−1|x0, v0, 0) dx

=

∫ ∞

−∞

[ex(es(−3i)∆tHi(−3i, v,∆t)− 2es(−2i)∆tHi(−2i, v,∆t)

+ es(−i)∆t)]px,v(x, v, ti−1|x0, v0, 0) dx

=[es(−3i)∆tHi(−3i, v,∆t)− 2es(−2i)∆tHi(−2i, v,∆t) + es(−i)∆t

]G(ti−1; i, v|x0, v0), (3.4a)

and

Ψ(2)i (v,∆t|x0, v0)

=

∫ ∞

−∞

U(2)i (x, v,∆t)px,v(x, v, ti−1|x0, v0, 0) dx

=

∫ ∞

−∞

[ex(− g(2)(−i)

)]px,v(x, v, ti−1|x0, v0, 0) dx

=[− g(2)(−i)

]G(ti−1; i, v|x0, v0). (3.4b)

The formula for the fair strike of the gamma swap is presented in the following proposition.

Proposition 3 The fair strike of a gamma swap with discrete sampling on dates: 0 = t0 <t1 < · · · < tN = T is given by

Kn =er∆t

S0

FA

N

[U

(n)i (x0, v0,∆t) +

N∑

i=2

∫ ∞

0

Ψ(n)i (v,∆t|x0, v0) dv

], n = 1, 2. (3.5)

3.2 Fair strike formulas for corridor swaps

In a corridor swap, the realized squared return monitored at the time ti is added to theaccumulated variance only when Si−1 falls within the corridor (L,U ]. The calculation of thefair strike amounts to the following risk neutral expectation calculations:

EQ

[(Si − Si−1

Si−1

)21{L<Si−1≤U}

]and EQ

[(ln

Si

Si−1

)21{L<Si−1≤U}

].

Alternatively, one may use Si rather than Si−1 as the monitoring asset price of the corridorfeature on the ith sampling date for the calculation of the accumulated realized variance.The corresponding risk neutral expectation terms become

EQ

[(Si − Si−1

Si−1

)21{L<Si≤U}

]and EQ

[(ln

Si

Si−1

)21{L<Si≤U}

].

12

Carr and Lewis (2004) develop an approximate pricing and hedging method for the dis-crete corridor variance swaps by using a portfolio of European style options with a continuumof strikes. Zheng and Kwok (2014) present closed form formula for the fair strike of the corri-dor variance swap under the SVSJ model. Here, we derive fair strike formula for the corridorswap under the 3/2-model. The use of the following relation

1{L<Si−1≤U} = 1{Si−1≤U} − 1{Si−1≤L}

simplifies pricing of the corridor variance swap into that of the downside variance swap withone-sided upper boundary in the monitoring corridor. Therefore, it suffices to compute therespective downside conditional expectation as follows:

EQ

[ln(Si − Si−1

Si−1

)21{Si−1≤U}

]and EQ

[ln( Si

Si−1

)21{Si−1≤U}

].

The indicator function admits the following Fourier integral representation over the realline

1{Si−1≤U} = 1{xi−1≤u} =1

2π

∫ ∞

−∞

eiuω

iωe−iωxi−1dωr,

where u = lnU , ω = ωr + iωi and ωi ∈ (−∞, 0). This Fourier integral representationessentially modifies the terminal payoff function by multiplying a factor of e−i(ωr+iωi)x. Sincethe indicator function is conditioning on time ti−1 rather than ti, in the backward induction

from time ti to ti−1, we obtain the same U(n)i as those in the vanilla variance swaps (with

no dependence on x at time ti−1). For the terms corresponding to i ≥ 2 in the discretevariance formulas (2.5a,b), by changing the order of integration of x with ωr, the expectationcalculation coupled with the Fourier transform representation of the indicator function canbe expressed as

EQ

[1{x≤u}e

r∆tU(n)i (x, v,∆t)

]

= er∆t

∫ ∞

0

∫ ∞

−∞

[ ∫ ∞

−∞

1

2π

eiuω

iωe−iωxU

(n)i (x, v,∆t)px,v(x, v, ti−1|x0, v0, 0) dωr

]dx dv

=er∆t

2π

∫ ∞

0

∫ ∞

−∞

eiuω

iωU

(n)i (x, v,∆t)G(ti−1;ω, v|x0, v0) dωr dv. (3.6)

For i ≥ 2 and τ = ∆t, we define

Φ(1)i (ω, v,∆t|x0, v0) =U

(1)i (x, v,∆t)G(ti−1;ω, v|x0, v0)

=[es(−2i)∆tHi(−2i, v,∆t)− 2es(−i)∆t + er∆t

]G(ti−1;ω, v|x0, v0), (3.7a)

and

Φ(2)i (ω, v,∆t|x0, v0) = U

(2)i (x, v,∆t)G(ti−1;ω, v|x0, v0) =

[− g(2)(0)

]G(ti−1;ω, v|x0, v0).

(3.7b)Combining the above results, the analytic formula for the fair strike of the downside varianceswap is summarized in the following proposition.

Proposition 4 For a discretely sampled downside variance swap on dates: 0 = t0 < t1 <· · · < tN = T , with Si−1 as the monitoring asset price on the ith monitoring date, the fairstrike is given by

Kn =FA

Ner∆t

[1

2π

∫ ∞

−∞

eiuω

iω

(∫ ∞

0

N∑

i=2

Φ(n)i (ω, v,∆t|x0, v0) dv

)dωr

+ 1{x0≤u}U(n)1 (x0, v0,∆t)

], n = 1, 2, (3.8)

13

where u = lnU , and U(n)1 (x0, v0,∆t) are given by eqs. (2.16a,b).

By following a similar approach, we can derive the analytic fair strike formula for thedownside variance swap with Si as the monitoring asset price on the ith monitoring date.Consider the vanilla variance swap with payoff at maturity ti multiplied by e−iωx, the valueat time ti−1 (τ = ∆t) can be obtained as follows:

(i) actual return

U(3)i (ω, x, v,∆t)

=F−1[exp(s(k)∆t)Hi(k, v,∆t)F

[e−iωx

(exI

− 1)2]]

= e−iωx[es(−2i−ω)∆tHi(−2i− ω, v,∆t)− 2es(−i−ω)∆tHi(−i− ω, v,∆t)

+ es(−ω)∆tHi(−ω, v,∆t)]; (3.9a)

(ii) log return

U(4)i (ω, x, v,∆t)

=F−1[exp(s(k)∆t)Hi(k, v,∆t)F [e−iωx(x− ln I)2]]

= e−iωx[− g(2)(−ω)

]. (3.9b)

For the terms corresponding to i ≥ 2 in the discrete variance formulas (2.5a,b), we denote

the integral of U(n)i and px,v over the x variable as

Φ(3)i (ω, v,∆t|x0, v0)

=

∫ ∞

−∞

U(3)i (ω, x, v,∆t) px,v(x, v, ti−1|x0, v0, 0) dx

=[es(−2i−ω)∆tHi(−2i− ω, v,∆t)− 2es(−i−ω)∆tHi(−i− ω, v,∆t)

+ es(−ω)∆tHi(−ω, v,∆t)]G(ti−1;ω, v|x0, v0), (3.10a)

and

Φ(4)i (ω, v,∆t|x0, v0)

=

∫ ∞

−∞

U(4)i (ω, x, v,∆t) px,v(x, v, ti−1|x0, v0, 0) dx

=[− g(2)(−ω)

]G(ti−1;ω, v|x0, v0). (3.10b)

Combining the above results, we obtain the analytic fair strike formula of the alternativedownside variance swap as summarized in Proposition 5. Recall that in order to recover theoriginal fair strike of the downside variance swap, it is necessary to perform integration withrespect to ωr by using the Fourier integral representation formula of the indicator function.

Proposition 5 For a discretely sampled downside variance swap on dates: 0 = t0 < t1 <· · · < tN = T with Si as the monitoring asset price on the ith monitoring date, the fair strikeis given by

Kn =FA

N

[er∆t

2π

∫ ∞

−∞

eiuω

iω

(U

(n)1 (ω, x0, v0,∆t) +

∫ ∞

0

N∑

i=2

Φ(n)i (ω, v,∆t|x0, v0) dv

)dωr

], n = 3, 4.

(3.11)

14

4 Numerical results

In this section, we present the results of our numerical tests on the fair strike formulasand investigate the impact of different parameters on the fair strikes of the various typesof variance swaps. We show the comparison of the values obtained in our method with thebenchmark results obtained from Monte Carlo simulation for assessing the level of accuracyof our analytic formulas. We also show the relation between the sampling frequency Nand the fair strikes for different types of variance swaps. The impact of different modelparameters, including correlation coefficient ρ, volatility of variance ǫ and jump intensity λ,on the fair strikes of the different types of variance swaps are examined.

In our numerical calculations, we adopted the same set of model parameter values fromDrimus (2012). These parameter values are obtained through simultaneous calibration ofthe 3-month and 6-month S&P 500 implied volatilities on July 31, 2009 for the 3/2-model(see Table 1). We assume d = 0, S0 = 1 and T = 1. In order to produce comparable resultswith the no-jump model in Drimus (2012), we also assume λ = 0 when we investigate theimpact of N , ρ and ǫ on the fair strike values of the variance swaps. When we explore theimpact of jump intensity in Figure 4, λ is set to assume varying values. Besides, we take theinterest rate to be flat at r = 0.48% with reference to the Treasury 1-Year Yield Curve onthe calibrated date. Furthermore, we assume U = 1 for the upper barrier in the corridor ofthe downside variance swaps and there are 252 trading days in one year.

Parameter v0 p q ǫ ρValue 0.060025 4.9790 22.84 8.56 -0.99

Table 1: Model parameters of the 3/2-model.

Assessment of numerical accuracy in evaluation of analytic formulas

One may be concerned with accuracy in numerical evaluation of the confluent hypergeometricfunctions in our analytic formulas for the fair strike prices of variance swaps under the 3/2-model. We performed Monte Carlo simulation to obtain benchmark values for the fair strikeprices of the different types of variance swaps and make assessment on accuracy in numericalevaluation of the analytic fair strike formulas. We applied the Euler scheme in the MonteCarlo simulation procedure with 100,000 simulated paths. As pointed out by Drimus (2012),the 3/2-model shows more radical behavior of the volatility dynamics compared to the Hestonmodel. When he calibrated the same set of option data to both the 3/2-model and Hestonmodel parameters, the 3/2-variance process is seen to lead to more volatile sample pathsunder a comparable set of parameters. We also notice that exceedingly small time step isrequired in order to stabilize the Monte Carlo simulated calculations.

On the other hand, our experience in the computation suggests that the range of vi−1 forintegration should be taken to be [0, 10] and that of ωr to be [−100, 100] while ωi = −0.02would be sufficient for convergence of the generalized Fourier transform. The calculationswere performed on an Intel i7 PC. By taking advantage of the multi-cores CPU in parallelcomputing, we have designed parallel codes for the Monte Carlo simulation and numericalintegration in the expectation calculations. The Monte Carlo simulation was coded in Matlabfor its simplicity in programming. Our numerical integration calculations were performed inMathematica to take advantage of its built-in confluent hypergeometric functions and Besselfunctions.

15

In Table 2, we list the numerical results obtained by the Monte Carlo simulation andnumerical integration of the analytic fair strike formulas with the number of sampling datesN = 52. The numerical evaluation of the analytic formulas is seen to provide reasonableaccuracy with significantly less computation time when compared to that of the Monte Carlosimulation. If we increase the number of the sample paths in the Monte Carlo simulationto 1,000,000, we can obtain more accurate results at the expense of significantly highercomputational costs. For the downside variance swaps, our analytic formulas require nu-merical evaluation of a double integral. As a result, the computational times become moredemanding.

swap type variance swap gamma swapconvention actual return log return actual return log returnanalytic 0.080939 0.083874 0.071909 0.073409time (s) (0.742) (5.742) (1.341) (4.232)

MC (0.1M) 0.080853 0.083827 0.071852 0.073355time (s) (211.571) (211.571) (211.418) (211.418)SE (%) 0.014228 0.017914 0.0074019 0.00843187MC (1M) 0.080937 0.083891 0.071890 0.073394time (s) (2859.911) (2859.911) (2839.121) (2839.121)

Table 2: Comparison of the fair strike values obtained from the numerical integration ofthe analytic strike formulas and Monte Carlo simulation. The computational times aremeasured in units of second. Here, “analytic” stands for numerical evaluation of the analyticformulas via numerical integration, MC (0.1M) and MC (1M) for Monte Carlo simulationusing 0.1 million paths and 1 million paths, respectively, and SE for the standard error inthe simulation. For Monte Carlo simulation, computational times quoted are valid for bothactual return and log return since the discrete realized variance under both conventions canbe computed by using the same set of simulation paths.

Pricing properties of discrete variance swaps under 3/2-model

We would like to explore the impact of sampling frequency on the fair strike values of varioustypes of variance swaps. When pricing under the Heston model, Zhu and Lian (2011) andZheng and Kwok (2014) show that the fair strike values of variance swap and gamma swapdecrease with the sampling frequency. In Figure 1, we show the plots of the fair strike valuesof various types of variance swaps against the sampling frequency under the 3/2-model.We found that the fair strike values of the vanilla variance swap and gamma swap using theconvention of log return are always higher than those of actual return, a similar phenomenonthat has been observed by Zhu and Lian (2012) for pricing discrete variance swaps underthe Heston model. For the variance swap based on log return, the fair strike decreases whenthe sampling frequency increases, consistent with similar results obtained under the Hestonmodel. However, when the convention of actual return is used, we note that the fair strikevalues of vanilla variance swap and gamma swap increase steadily when N increases. Forthe less frequently sampled downside variance swaps, the difference in the fair strike valuescan be substantial under different conventions of calculating the discrete realized return. Wealso note that the fair strike values of the two types of downside variance swaps with eitherSi or Si−1 as the monitoring asset price of the corridor feature are quite close. When thesampling frequency increases, the fair strike values of the various variance products underthe two different conventions of realized return converge to the same set of common values.

16

For simplicity, we only show the plots of the fair strike values of the variance swaps basedon the convention of log return in Figures 2, 3 and 4. Figure 2 shows the fair strike valuesagainst correlation coefficient, ρ. The fair strike of the vanilla variance swap is less sensitiveto ρ when compared to the gamma swap. The fair strike of the variance swap decreases slowlywith increasing value of ρ while for the gamma swap, it shows a moderate increasing trend.When ρ is close to zero, the leverage effect is minimal, thus the effectiveness of avoiding theextreme price movement by adding the weights in the gamma swap is reduced. Therefore,the fair strike values of the gamma swap and vanilla swap almost equal to each other at zerovalue of ρ. For the downside variance swap, the fair strike is decreasing when ρ increases.

A higher value of the volatility of variance ǫ normally indicates more volatile behavior ofthe volatility dynamics in the Heston model. However, the reverse relation is found in the3/2-model. Figure 3 demonstrates the relation of the fair strike values with varying valuesof ǫ. As ǫ increases, the fair strike values of all the variance swap products decrease and therate of decrease is more rapid at small value of ǫ.2 In actual market conditions, large valuesof ǫ are commonly found when calibration is performed for real market data [see Drimus(2012) and Baldeaux and Badran (2012)].

To investigate the effect of jump intensity λ and jump size distribution on the fair strikevalues of different variance swap products, we take µ = −0.3 and ζ = 0.4 to represent thescenario of large jump, while µ = −0.03 and ζ = 0.04 for small jump. Figure 4 shows thatthe fair strike values of all variance swaps are very sensitive (almost insensitive) to the jumpintensity parameter for large (small) jump size. When the jump intensity increases, jumpsoccur more frequently with a higher probability. A large mean of jump size gives a moreacute rate of increase, in particular for the vanilla variance swap; while a small one shows lessobvious impact on the fair strike values. The mean of jump size and its standard deviationare seen to be critical in determining the fair strike other than the jump intensity parameter.

5 Conclusion

In this paper, we illustrate the analytic-numerical procedure for finding the fair strike valuesfor various types of discretely sampled variance products under the 3/2-model with jumps.As revealed by various empirical studies, the 3/2-model is a better choice than the Hes-ton model for fitting the volatility structure. The variance swap products considered inthis paper include the third generation swaps like the gamma swaps and corridor swaps.The payoff structures in these discrete variance swaps consist of summation of terms thatare dependent on two time instants ti−1 and ti over successive monitoring dates. We firstcompute the conditional risk neutral expectation of the payoffs at time ti−1 by solving forthe fundamental solution of the governing partial integro-differential equation in the Fourierdomain. The analytic formula for the fair strike value at initiation is derived by summingterms that are obtained by integrating the conditional value at time ti−1 with respect to thetransition density of the 3/2-model. The numerical integration can be effectively performed

2To give a more intuitive explanation of this phenomenon, we consider the continuous monitoring varianceswap since the fair strike values of the continuous and discrete monitoring variance swaps should exhibita similar behavior when ǫ changes. The reciprocal wt of the 3/2 variance process is a CIR process withmean equals (q + ǫ2)/p [see eq. (2.3)]. The expected value of wt changes with this long term mean, andso does ǫ2 in the numerator. When ǫ2 increases, E[wt] is increasing at the leading order of ǫ2. We expandE[1/wt] by Taylor series as: E[1/wt] ≈ 1/E[wt]+var(wt)/(E[wt])

3. Since E[wt] increases at the leading orderof ǫ2, we can estimate that var(wt) = O(ǫ4) as var(wt) involves squared term of wt. Since 1/E[wt] andvar(wt)/(E[wt])

3 are decreasing functions of ǫ2, so E[1/wt] would be a decreasing function of ǫ2 as well. For

the continuous variance swap, the expected value E[∫T

0vs ds] = E[

∫T

01/wt ds] is then a decreasing function

of ǫ2. We are thankful to Wendong Zheng for the above discussion.

17

via numerical quadrature. The same methodology can be applied to find the fair strike valuesof other higher moment swaps under the 3/2-model, though the pricing procedure becomesnotoriously tedious.

We performed numerical valuation of our fair strike formulas of various types of varianceswaps and compared the numerical results with the strike values obtained from the MonteCarlo simulation. The computational times for the vanilla variance swaps and gamma swapsare significantly less than those required by the Monte Carlo simulation runs to achieve asimilar level of numerical accuracy. We analyze the impact of different model parameters onthe fair strike values of various variance swaps. We show that for the less frequently sampledvariance products, the difference of fair strike values between continuously monitored anddiscretely monitored, or between different conventions of calculating the discrete realizedvariance (actual return or log return) cannot be ignored. The effect of jump intensity in theasset price process on the fair strike values depends sensibly on the underlying jump size andits standard deviation. Finally, we also show that the two types of downside variance swapshave very close fair strike values and both converge to the same value when the monitoringfrequency becomes high.

AcknowledgementThis research has benefited from the discussion with Wendong Zheng and an unpublishedmonograph of Alan Lewis referred to us by Peter Carr. We also thank the funding supportof the Hong Kong Grants Council under Project 642110 of the General Research Funds.

18

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Appendix A. Proof of Proposition 1

At τ = 0, we have

Ui(x, v, 0) = F−1[exp(s(k)0)Hi(x, v, 0)F [Fi(e

x, Si−1)]]= Fi(e

x, Si−1),

so the initial condition is easily seen to be satisfied. Next, we take

Hi(k, v, τ) = Hi(k, v, τ)F [Fi(ex, Si−1)],

and substitute into the two sides of eq. (2.13). By canceling out the term F [Fi(ex, Si−1)] on

both sides, Hi is seen to satisfy eq. (2.13). The Fourier transform of Ui is obtained by therelation:

Ui(k, v, τ) = exp(s(k)τ)Hi(k, v, τ) = exp(s(k)τ)Hi(x, v, τ)F [Fi(ex, Si−1)]

].

Subsequently, Ui is obtained by taking the inverse Fourier transform of Ui. In fact, thefundamental solution is closely related to the marginal Fourier-Laplace transform of x givenin eq. (2.4) with µ = 0. One can show this relationship by using the Parseval identity.

We decompose the governing stochastic differential equation of x into the continuous partxC and the jump part xJ as follow:

xC =

∫ t

s

(r − d− λm− 1

2v) du+

∫ t

s

√vu dW

1u and xJ =

Nt∑

i=Ns+1

Ji.

Let pxi(xi, ti|x, v, t) denote the transition density of xi at time ti given v and x at time t.

Suppose we treat eikxHi(k, v, τ) to be the marginal characteristic function given in eq. (2.4)with µ = 0, then

Hi(k, v, τ) =Γ(γ − α)

Γ(γ)

[ 2

ǫ2y(v, t)

]αM(α, γ,− 2

ǫ2y(v, t)

),

where

y(v, t) = v

∫ ti

t

e∫u

tp(s) ds du,

α = −(12− qk

ǫ2

)+

√(12− qk

ǫ2

)2+

2ckǫ2

,

γ = 2(α + 1− qk

ǫ2

), qk = ρǫik − q, ck =

k2 + ik

2.

By performing the risk neutral valuation of the discounted payoff, we obtain

Ui(x, v, τ) =

∫ ∞

−∞

e−rτFi(exi , Si−1)pxi

(xi, ti|x, v, t) dxi

=1

2π

∫ ∞

−∞

e−rτF [Fi(e

xi , Si−1)]F [pxi(xi, ti|x, v, t)] dk

=1

2π

∫ ∞

−∞

e−rτF [Fi(e

xi , Si−1)]

∫ ∞

−∞

eikxipxi(xi, ti|x, v, t) dxi dk

=1

2π

∫ ∞

−∞

e−rτF [Fi(e

xi , Si−1)]E[eikx+ikxC+ikxJ |x, v] dk

=1

2π

∫ ∞

−∞

F [Fi(exi , Si−1)]e

ikxHi(k, v, τ) exp(s(k)τ) dk

= F−1[F [Fi(e

xi , Si−1)]Hi(k, v, τ) exp(s(k)τ)].

21

Appendix B. Deviation of eq. (3.2)

Let px,v(xt, vt, t|xs, vs, s) denote the joint density of xt and vt given xs and vs at time s ≤ t.By observing

xt = xs +

∫ t

s

(r − d− λm− 1

2v) du+

∫ t

s

√vu dW

1u +

Nt∑

i=Ns+1

Ji,

we obtainpx,v(xt, vt, t|xs, vs, s) = px,v(xt − xs, vt, t|0, vs, s).

With the above translation invariant relation, we may assume xs = 0 for simplicity since thejoint transition density px,v can be obtained easily given any value of xs. Unfortunately, wecannot find an explicit formula for the joint transition density. However, the xt-transformeddensity G(τ ;−z, vt|xs, vs) of px,v(xt, vt, t|xs, vs, s) is available in closed form, where τ = t− sand z is the transform variable.

As the first step in our analytic derivation, we consider a pure diffusion process xt bysetting the jump component xJ = 0. Let pd(y, η, t|0, v, s) be the joint transition density ofthe pure diffusion process, where y and η are assumed to be known at time t, then pd satisfiesthe following Kolmogorov backward equation

∂pd∂s

=1

2v∂2pd∂y2

+ ρǫv2∂2pd∂y∂v

+ǫ2

2v3

∂2pd∂v2

+ (r − d− 1

2v)∂pd∂y

+ (pv − qv2)∂pd∂v

,

with initial condition: pd(y, η, t|0, v, t) = δ(y)δ(v−η). The above partial differential equationcan be solved analytically by performing the Fourier transform with respect to the variabley. We define the y-transformed density G(τ ;−z, η, v) as follows:

G(τ ;−z, η, v) =

[ ∫ ∞

−∞

eizypd(y, η, t|0, v, s) dy]e−i(r−d)zτ .

The above backward equation can be simplified into a one-dimensional problem:

∂G

∂τ=

ǫ2

2v3

∂2G

∂v2+ [pv − (q + izρ)v2]

∂G

∂v− czvG,

wherecz = (z2 − iz)/2,

with initial condition: G(0; z, η, v) = δ(v − η). By using the following set of transformedvariables:

y =p

v, Y =

p

η, t = pτ, p =

2p

ǫ2, cz =

2czǫ2

, qz =2(q + izρǫ)

ǫ2,

and assume that G(τ ; z, η, v) takes the form (Lewis, 2000)

G(τ ; z, η, v) =Y 2

p

( y

Y

)R(z)

e[1−R(z)t ]g(t, y, Y, z),

where

R(z) = −µz − δz, µz =1

2(1 + qz), δz =

õ2z + cz,

22

the partial differential equation for G(τ ; z, η, v) can be further transformed to an ordinarydifferential equation in terms of g(t, y, Y, z) with the independent spatial variable y. The so-lution of this ordinary differential equation is readily available by taking the Laplace trans-form in the variable y. The solution procedure is outlined in an unpublished monographof Alan Lewis. By combining all the above results, we obtain the closed form solution forG(τ ;−z, η, v) as shown in eq. (3.19).

The log asset price process xt [see eq. (2.2)] in our model can be decomposed into thecontinuous component xC and the jump component xJ as shown in Appendix A. By applyingthe above result for the continuous process on xC and evaluating the compound Poissonprocess xJ separately, we obtain the xt-transformed density function G(τ ;−z, vt|xs, vs) ofpx,v(xt, vt, t|xs, vs, s) as follows:

G(τ ;−z, vt|xs, vs) =

∫ ∞

−∞

eizypx,v(y, vt, t|xs, vs, s) dy

= E[eizxt |xs, vt, vs

]

= E[eizxs+izxC+izxJ |xs, vt, vs

]

= eizxsE[eizx

C |xs = 0, vt, vs]E[ejzx

J ]

= eizxs

[ ∫ ∞

−∞

eizxC

pd(xC , vt, t|0, vs, s) dxC

]e[exp(iνz−ζ2z2/2)−1]λτ

= G(τ ;−z, vt, vs)ei(r−d−λm)τeizxse[exp(iνz−ζ2z2/2)−1]λτ

= G(τ ;−z, vt, vs)eizxs exp(h(z)τ),

where

h(z) = i(r − d− λm)z + [exp(iνz − ζ2z2/2)− 1]λ.

23

0 50 100 150 200 2500.05

0.055

0.06

0.065

0.07

0.075

0.08

0.085

0.09

0.095

0.1

N (Times/Year)

Fai

r st

rike

VSAVSLGSAGSLDSA1DSL1DSA2DSL2

Figure 1: Plots of the fair strike values of the vanilla variance swaps, gamma swaps anddownside variance swaps against sampling frequency, N . The label VSA stands for thevanilla variance swap defined by actual return, VSL for log return. Similarly, VGA standsfor the gamma swap defined by actual return, and VGL for log return. Also, DSA1 andDSL1 stand for the downside variance swaps with Si−1 as the monitoring asset price on theith monitoring date based on actual and log return, respectively, while DSA2 and DSL2 forSi as the monitoring asset price.

−1 −0.8 −0.6 −0.4 −0.2 00.04

0.05

0.06

0.07

0.08

0.09

0.1

ρ

Fai

r st

rike

VSLGSLDSL1

Figure 2: Plots of the fair strike values of the vanilla variance swap, gamma swap anddownside variance swap against correlation coefficient, ρ. The label VSL stands for thevanilla variance swap, VGL for the gamma swap. Also, DSL1 stands for the downsidevariance swap with Si−1 as the monitoring asset price on the ith monitoring date. All discretevariance calculations are based on the convention of log return.

24

5 10 150

0.05

0.1

0.15

ε

Fai

r st

rike

VSLGSLDSL1

Figure 3: Plots of the fair strike values of the vanilla variance swap, gamma swap anddownside variance swap against volatility of variance, ǫ. The label VSL stands for thevanilla variance swap, VGL for the gamma swap. Also, DSL1 stands for the downsidevariance swap with Si−1 as the monitoring asset price on the ith monitoring date. Alldiscrete variance calculations are based on the convention of log return.

0 0.5 1 1.5 20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

λ

Fai

r st

rike

VSL, LJGSL, LJDSL1, LJVSL, SJGSL, SJDSL1, SJ

Figure 4: Plots of the fair strike values of the vanilla variance swap, gamma swap anddownside variance swap against jump intensity, λ. The label VSL stands for the vanillavariance swap, VGL for the gamma swap. Also, DSL1 stands for the downside varianceswap with Si−1 as the monitoring asset price on the ith monitoring date. Furthermore, LJstands for large jump with ν = −0.3 and ζ = 0.4 while SJ stands for small jump withν = −0.03 and ζ = 0.04. All discrete variance calculations are based on the convention oflog return.

25