Asymmetric Jumps, Sampling, and Variance Swap Rates

ABSTRACT

We investigate the effect of asymmetric jumps and the number of sampling dates on variance

swap rates in the Heston’s (1993) stochastic volatility framework. Analytic formulae for fair

continuous and discrete variance strikes are both derived and verified by Monte Carlo

simulations. Numerical results show that asymmetric jumps could strongly affect the

discretely sampled variance and the option replication strategy through underlying prices

even though the fair continuous variance strike remains unchanged. With S&P 500 index

market data, our empirical analysis tends to support the superiority of asymmetric jumps

modelled by Kou (2002) to normal distribution jumps.

JEL Classification: G12, G13, G17

Keywords: Asymmetric jump; Discretely sampling; Option replication strategy; Stochastic

volatility; Variance swap;

1

1. Introduction

Variance swaps are forward contracts on future-realized underlying variance and provide an

easy means for investors to gain exposure to the future level of return variances.1 Although

there are some popularly traded exchange-listed volatility-based products, most variance

swaps are over-the-counter (OTC) contracts.2 Variance swaps that have a payout equal to the

difference between the realized variance and a pre-agreed strike level multiplied by the

notional amount have been popular since 2004.3 Recent market turmoil due to the subprime

crisis in the United States may possibly enhance the trading of variance-based financial

derivatives, and greatly promote research in this area. For example, Broadie and Jain (2008a)

present a partial differential equation approach to price volatility derivatives in Heston’s

(1993) stochastic volatility model and an approach to hedge volatility derivatives using

variance swaps. Furthermore, Broadie and Jain (2008b) integrate Merton’s (1976) normal

distribution jumps into Heston’s (1993) stochastic volatility model (the MertonSV model)

and discuss the effect of discrete sampling and asset price jumps on fair variance strikes.4 In

1 Dew-Becker et al. (2017) argue that, due to believing that realized variance will be high in

future market declines, large institutional asset managers, such as Man Group, Deutsche

Bank, and JP Morgan, and their investors use variance swaps to protect against tail risk. 2 For example, VIX is the volatility index of the Chicago Board Options Exchange (CBOE),

which shows the market's expectation of 30-day volatility, and it is often referred to as the

"investor fear gauge." From (but excluding) October 3, 2014, VIX measures the 30-day

volatility of the S&P 500 implied by the out-of-the–money put and call options (SPX options)

using both monthly SPX options as well as weekly SPX options. In addition, the CBOE

launched three-month variance futures on the S&P 500 in May, 2004, and 12-month variance

futures in March, 2006. 3 The market for variance swaps has grown sharply over the past few years. Deutsche Bank

estimates that volumes grew five-fold in 2005 and were up 50% year-on-year for the first half

of 2006. 4 According to Barndorff-Nielsen and Shephard (2004, 2006), continuous- and jump-part

contributions can be separated by comparing the difference between the quadratic variation

and the bipower variation. In Figure 1, we used intra-daily S&P500 index data obtained from

Price-Data.com to distribute realized volatility into continuous and jump components, and we

calculated the difference between the quadratic variation and the bipower variation as a

measure of jumps. Figure 1 shows the daily realized jumps of the S&P500 index from 2007

to 2009 and justifies the incorporation of jumps into pricing models. Because Lemon

Brothers collapsed in 2008, the downward jump size observed in 2008 is larger than those

2

this paper, we extend the work of Broadie and Jain (2008b) to incorporate investors’

sentiment through the inclusion of the double exponential jump model of Kou (2002) and

investigate the effect of asymmetric jumps on variance swaps.

Although Merton’s (1976) normal distribution jump model is the first jump-diffusion

model including the leptokurtic feature, Kou’s (2002) double exponential jump model has

significantly more pronounced kurtosis and is known for its asymmetric leptokurtic feature,

which includes a psychological interpretation of markets tending to exhibit both overreaction

and underreaction to good or bad news.5 The empirical phenomenon that the daily return

distribution tends to have more kurtosis than the distribution of monthly returns tends to

support Kou’s (2002) argument that “the double exponential jump-diffusion model and the

stochastic volatility model can complement each other.” The reason is that the kurtosis in

stochastic volatility models decreases as the sampling frequency increases, while the

instantaneous jumps are independent of the sampling frequency. Our study also provides

some evidence to show the superiority of Kou’s (2002) double exponential jumps over

Merton’s (1976) normal distribution jumps.

In this paper, we first derive the fair continuous variance strike of variance swaps in the

hybrid model of Kou’s (2002) double exponential jumps and Heston’s (1993) stochastic

volatility (the DEJSV model) and then go on to derive the fair discrete variance strikes using

respective sampling dates. We further analyze the effect of asymmetric price jumps on fair

continuous and discrete variance strikes respectively. Subsequently, we follow Bakshi and

Cao (2003) and Floc′h (2018) to use the data of market options6 to calibrate the model

occurring in 2007 or 2009. 5 Kou (2002) asserts that “Because the double exponential distribution has a both high peak

and heavy tails, it can be used to model both the overreaction (attributed to the heavy tails)

and underreaction (attributed to the high peak) to outside news.” 6 Dew-Becker et al. (2017) also examine claims to variance constructed using option prices

and confirm their main results by showing that the term structure and returns obtained from

investments in options are similar to those obtained from variance swaps.

3

parameters and investigate whether the DEJSV model is superior to the MertonSV model in

predicting the future realized volatility.

The remainder of this paper is organized as follows. The concept and definition of

variance swaps is introduced in Section 2. In Section 3, the DEJSV model is described, and

both fair variance strikes, continuous and discrete, are derived and further justified by some

numerical experiments. In Section 4, we conduct an empirical study to compare the DEJSV

model with some competing models. The conclusion is presented in Section 5.

2. Variance Swaps

A variance swap is a contract that gives an investor direct exposure to the realized variance of

an underlying asset such as an index or a stock. The pre-agreed level is called the strike of the

swap. The payoff of a variance swap is defined as

𝑁 × (𝑉𝑑(0, 𝑛, 𝑇) − 𝐾𝑉𝑎𝑟(𝑛)) (1)

where 𝑁 is the notional amount of the swap contract, 𝑉𝑑(0, 𝑛, 𝑇) is the floating leg of the

variance swap, which is the realized variance over the contracted period [0, 𝑇], and 𝐾𝑉𝑎𝑟(𝑛)

is the delivery price of the variance swap (the fixed leg of the variance swap). The life of the

contract is partitioned into 𝑛 equal segments of length Δ𝑡 for sampling, i.e., 0 = 𝑡0 <

𝑡1 < ⋯ < 𝑡𝑛 = 𝑇, and 𝑡𝑖 = 𝑖 𝑇 𝑛⁄ for each 𝑖 = 0, 1, 2,⋯ , 𝑛 ; 𝑉𝑑(0, 𝑛, 𝑇) is calculated

using the second moment of log returns of the underlying asset

𝑅𝑖 = 𝑙𝑜𝑔 (𝑆𝑡𝑖

𝑆𝑡𝑖−1

) , 𝑖 = 1, 2,⋯ , 𝑛. (2)

At expiration, the holder of a variance swap receives 𝑁 dollars for every point by which the

underlying realized variance 𝑉𝑑(0, 𝑛, 𝑇) has exceeded the variance delivery price 𝐾𝑉𝑎𝑟(𝑛).

The realized variance in most traded contracts is defined by

4

𝑉𝑑(0, 𝑛, 𝑇) =𝑛

(𝑛 − 1)𝑇∑𝑅𝑖

2

𝑛

𝑖=1

. (3)

This contract is important because it is easier to replicate than volatility swaps, and it can also

serve as the building block for the construction of other variance derivatives. One of the most

significant applications for variance swaps is in the area of volatility trading. For investors

who have traditionally employed delta-neutral option strategies to implement views on

volatility, the variance swap or its corresponding volatility swap offers a more exact method

for formulating views on future volatility.7

3. The Model and Numerical Experiments

The DEJSV model describes the dynamics of the underlying return and the variance

processes under the risk-neutral measure as

𝑑𝑆𝑡

𝑆𝑡− = (𝑟 − 𝜆𝑚) 𝑑𝑡 + √𝑉𝑡 (𝜌𝑑𝐵𝑡

𝜈 + √1 − 𝜌2𝑑𝐵𝑡𝑠) + 𝑑 (∑(𝑌𝑖 − 1)

𝑁𝑡

𝑖=1

) (4)

where 𝐵𝑡𝑠 and 𝐵𝑡

𝜈 are two Brownian motion processes, and 𝜌 represents the instantaneous

correlation between the return process and the variance process. 𝐵𝑡𝑠 , 𝐵𝑡

𝜈, 𝑁𝑡, and {𝑌𝑖} are

assumed to be independent. Eq. (4) gives the dynamics of the stock price: 𝑆𝑡 denotes the

stock price at time t, 𝑟 is the risk free rate, 𝑚 is the mean proportional size of the jump,

7 For Instance, delta-neutral long-option strategies are based on buying options that carry an

implied volatility that is less than the volatility that will ultimately be realized. Conversely,

data-neutral short-option strategies are based on selling options at an implied volatility that is

higher than the anticipated realized volatility. The profitability of these strategies, however,

depends on a complicated interaction of factors, such as movements in the underlying asset

and the passage of time. Instead, variance swaps provide a clear way of speculating on

realized versus implied volatility. Variance swap is also potentially useful as a defensive

strategy that seeks to protect a portfolio from a market sell-off. Because a significant market

decline is usually accompanied by an increase in volatility, a long variance swap position will

help offset portfolio losses that result from a steep market decline. Egloff, Leippold, and Wu

(2010) further provide an analysis on the optimal investment decision on the variance swaps

and the stock index.

5

𝐸(𝑌𝑖 − 1) = 𝑚 = 𝑝𝜂1

𝜂1−1+ 𝑞

𝜂2

𝜂2+1− 1, √𝑉𝑡 is Heston’s (1993) stochastic volatility, 𝑁𝑡 is a

Poisson process with rate 𝜆, and {𝑌𝑖} is a sequence of independent, identically distributed

nonnegative random variables such that 𝑋 = 𝑙𝑜𝑔(𝑌) has an asymmetric double exponential

distribution with the density8

𝑓𝑋(𝑥) = 𝑝 ∙ 𝜂1𝑒−𝜂1𝑥1{𝑥≥0} + 𝑞 ∙ 𝜂2𝑒

𝜂2𝑥1{𝑥<0}, 𝜂1 > 1 , 𝜂2 > 0 (5)

where ≥ 0, 𝑞 ≥ 0, and 𝑝 + 𝑞 = 1 represent the probabilities of upward and downward

jumps. 1 𝜂1⁄ and 1 𝜂2⁄ are means of two exponential random variables standing for upward

and downward jump sizes, respectively.

𝑑𝑉𝑡 = 𝜅(𝜃 − 𝑉𝑡) 𝑑𝑡 + 𝜎𝑣√𝑉𝑡𝑑𝐵𝑡𝜈 . (6)

Eq. (6) gives the evolution of the variance that follows the square root process: 𝜃 is the

long-run mean variance, 𝜅 represents the speed of mean reversion, and 𝜎𝜈 is a parameter

that determines the volatility of the variance process. In the case of continuous sampling, the

continuously sampled realized variance, 𝑉𝑐(𝑡, 𝑇), is defined by Eq. (7),

𝑉𝑐(𝑡, 𝑇) ≡ lim𝑛→∞

𝑉𝑑(𝑡, 𝑛, 𝑇) (7)

and consists of two components. The first is the accumulated variance contributed by the

diffusive component of the underlying asset price process, and the second is the contribution

from jumps. The contribution to the accumulated variance from diffusion between time 𝑡

and 𝑇 is given by

�̅�𝐷(𝑡, 𝑇) = ∫ 𝑉𝑠 𝑑𝑠𝑇

𝑡

. (8)

8 Unlike Kou’s (2002) double exponential jumps, the distribution of log(𝑌𝑖) in the Merton (1976) model is

normal; 𝑌𝑖 is specified according to 𝐿𝑁(𝑎, 𝑏2) and 𝐸(𝑌𝑖 − 1) = 𝑒𝑎+1

2𝑏2

− 1 = 𝑚. The variance strike derived

by Broadie and Jain (2008b) for the MertonSV model is given in Appendix B for comparison.

6

If there are 𝑁𝑡𝑇 price jumps in [𝑡, 𝑇], the contribution to the accumulated variance from

jumps is

�̅�𝐽(𝑡, 𝑇) = ∑(𝑙𝑜𝑔(𝑌𝑖)2)

𝑁𝑡𝑇

𝑖=1

. (9)

Thus, the continuously sampled realized variance in the DEJSV model, 𝑉𝑐, can be expressed

as

𝑉𝑐(𝑡, 𝑇) = �̅�𝐷(𝑡, 𝑇) + �̅�𝐽(𝑡, 𝑇). (10)

Define 𝑃𝑐(𝑡, 𝑇, 𝐾, 𝑉𝑅) to be the expected present value at time 𝑡 of the payoff of a

continuous variance swap with variance strike, 𝐾𝑣𝑎𝑟, i.e.,

𝑃𝑐(𝑡, 𝑇, 𝐾𝑉𝑎𝑟 , 𝑉𝑅) ≡ 𝐸𝑡𝑄 (𝑒−𝑟(𝑇−𝑡) (

1

𝑇(𝑉𝑅 + 𝑉𝑐(𝑡, 𝑇)) − 𝐾𝑉𝑎𝑟)) (11)

where the superscript 𝑄 denotes the risk-neutral measure, the subscript 𝑡 denotes

expectation at time 𝑡, and 𝑉𝑅 = 𝑉𝑅(0, 𝑡) is the continuous realized accumulated variance

from the start of the contract (time 0) until time 𝑡. As illustrated by Eq. (3.9), the fair

continuous variance strike, 𝐾𝑉𝑎𝑟, is often determined by setting the variance swap value to

be zero at time 0, i.e.,

𝑃𝑐(0, 𝑇, 𝐾𝑉𝑎𝑟 , 0) = 𝐸0𝑄 (𝑒−𝑟𝑇 (

1

𝑇𝑉𝑐(0, 𝑇) − 𝐾𝑉𝑎𝑟)) = 0. (12)

Solving Eq. (12) for 𝐾𝑉𝑎𝑟 yields Eq. (13),

𝐾𝑉𝑎𝑟 = 𝐸0𝑄 (

1

𝑇𝑉𝑐(0, 𝑇)) = 𝐸0

𝑄 (1

𝑇�̅�𝐷(0, 𝑇)) + 𝐸0

𝑄 (1

𝑇�̅�𝐽(0, 𝑇)) = 𝐾𝑆𝑉 + 𝐾𝐽 (13)

where 𝐾𝑆𝑉 = 𝐸0𝑄 (

1

𝑇�̅�𝐷(0, 𝑇)) denotes the fair variance strike in the Heston (1993)

7

stochastic volatility model, and 𝐾𝐽 = 𝐸0𝑄 (

1

𝑇�̅�𝐽(0, 𝑇)) denotes the fair variance strike in the

Kou (2002) double exponential jump model. Because 𝑉𝑇 − 𝑉0 = ∫ 𝑑𝑉𝑠 𝑇

0= ∫ 𝜅(𝜃 −

𝑇

0

𝑉𝑠)𝑑𝑠 +𝜎𝜈 ∫ √𝑉𝑠𝑑𝐵𝑠𝜈𝑇

0, we have

𝐸0𝑄(𝑉𝑇) − 𝑉0 = 𝜅𝜃𝑇 − 𝜅𝐸0

𝑄 (∫ 𝑉𝑠𝑑𝑠𝑇

0

) + 𝜎𝜈𝐸0𝑄 (∫ √𝑉𝑠𝑑𝐵𝑠

𝜈𝑇

0

)

= 𝜅𝜃𝑇 − 𝜅 ∫ 𝐸0𝑄(𝑉𝑠)𝑑𝑠

𝑇

0

. (14)

Substituting 𝜇(𝑠) = 𝐸0𝑄(𝑉𝑠) into Eq. (14) results in

𝜇(𝑇) − 𝑉0 = 𝜅𝜃𝑇 − 𝜅 ∫ 𝜇(𝑠)𝑑𝑠𝑇

0

. (15)

Differentiating Eq. (15) with respect to 𝑇 gives

𝜇′(𝑇) = 𝜅(𝜃 − 𝜇(𝑇)). (16)

The solution to this linear differential equation is

𝜇(𝑇) = 𝜃 + 𝑒−𝜅𝑇 𝑐 (17)

where 𝑐 is a constant. Because 𝜇(0) = 𝑉0, then 𝑐 = 𝑉0 − 𝜃. We have

𝐸0𝑄(𝑉𝑇) = 𝜃 + 𝑒−𝜅𝑇 (𝑉0 − 𝜃). (18)

Hence, Eq. (19) is derived as

𝐸0𝑄(�̅�𝐷(0, 𝑇)) = ∫ (𝜃 + 𝑒−𝜅𝑠 (𝑉0 − 𝜃))𝑑𝑠

𝑇

0

= 𝜃𝑇 +(𝑉0 − 𝜃)(1 − 𝑒−𝜅𝑇)

𝜅. (19)

Therefore, the fair variance strike in the Heston (1993) stochastic volatility model is given by

8

𝐾𝑆𝑉 = 𝐸0𝑄 (

1

𝑇�̅�𝐷(0, 𝑇)) = 𝜃 +

𝑉0 − 𝜃

𝜅𝑇(1 − 𝑒−𝜅𝑇) (20)

which is the same result as Broadie and Jain (2008b). Because 𝐸(𝑋) =𝑝

𝜂1−

𝑞

𝜂2 and

𝑉𝑎𝑟(𝑋) = 𝑝𝑞 (1

𝜂1+

1

𝜂2)2

+𝑝

𝜂12 +

𝑞

𝜂22, we have

9

𝐸(𝑋2) = 𝑉𝑎𝑟(𝑋) + 𝐸(𝑋)2 =2𝑝

𝜂12 +

2𝑞

𝜂22 (21)

and

𝐸0𝑄 (�̅�𝐽(0, 𝑇)) = 𝐸0

𝑄 (∑(𝑙𝑜𝑔(𝑌𝑖)2)

𝑁0𝑇

𝑖=1

) = 𝜆𝑇 (2𝑝

𝜂12 +

2𝑞

𝜂22). (22)

Therefore, the fair variance strike in the Kou double exponential jump model is given by

𝐾𝐽 = 𝐸0𝑄 (

1

𝑇�̅�𝐽(0, 𝑇)) = 𝜆 (

2𝑝

𝜂12 +

2𝑞

𝜂22). (23)

Thus, the fair continuous variance strike in the DEJSV model is given by

𝐾𝑉𝑎𝑟 = 𝐾𝑆𝑉 + 𝐾𝐽 = 𝜃 +𝑉0 − 𝜃

𝜅𝑇(1 − 𝑒−𝜅𝑇) + 𝜆 (

2𝑝

𝜂12 +

2𝑞

𝜂22). (24)

Note that the fair continuous variance strike does not depend on the volatility of the variance

process, 𝜎𝜈.

Nevertheless, sampling is performed daily or weekly and sometimes monthly in most

variance swap contracts. Hence, the following is the derivation of the discrete variance strike

in the DEJSV model. Applying Itô’s lemma to ln(𝑆𝑡) in Eq. (4) and integrating from 𝑡𝑖−1

to 𝑡𝑖 gives

9 Note that 𝑝 + 𝑞 = 1.

9

ln (𝑆𝑡𝑖

𝑆𝑡𝑖−1

) = ∫ (𝑟 − 𝜆𝑚 −1

2𝑉𝑡)𝑑𝑡

𝑡𝑖

𝑡𝑖−1

+ ∫ √𝑉𝑡 (𝜌𝑑𝐵𝑡𝑣 + √1 − 𝜌2𝑑𝐵𝑡

𝑠)𝑡𝑖

𝑡𝑖−1

+ 𝑙𝑛 (∏𝑌𝑗

𝑛𝑖

𝑗=1

) (25)

where 𝑛𝑖 is the number of jumps in the stock price during time 𝑡𝑖 − 𝑡𝑖−1. The result after

squaring Eq. (25), summing from 0 to T, dividing both sides by (n-1)Δ𝑡, and accounting for

expectation in the risk-neutral measure is

𝐾𝑉𝑎𝑟(𝑛) ≡ 𝐸0𝑄 [∑

1

(𝑛 − 1)∆𝑡(𝑙𝑛 (

𝑆𝑡𝑖

𝑆𝑡𝑖−1

))

2𝑛

𝑖=1

]

=1

𝑇𝐸0

𝑄 [∫ 𝑉𝑡𝑑𝑡𝑇

0

] + 𝑔(𝑟 − 𝜆𝑚, 𝜌, 𝜎𝑣, 𝜅, 𝜃, 𝑛) + 𝜆 (2𝑝

𝜂12 +

2𝑞

𝜂22)

+𝜆 (

2𝑝

𝜂12 +

2𝑞

𝜂22) + 𝜆2 (

𝑝

𝜂1−

𝑞

𝜂2)2

𝑇 + 𝜆 (𝑝

𝜂1−

𝑞

𝜂2) (2(𝑟 − 𝜆𝑚)𝑇 − 𝐸0

𝑄 [∫ 𝑉𝑡𝑑𝑡𝑇

0])

(𝑛 − 1)

= 𝐾𝑉𝑎𝑟 + 𝑔(𝑟 − 𝜆𝑚, 𝜌, 𝜎𝑣, 𝜅, 𝜃, 𝑛)

+𝜆 (

2𝑝

𝜂12 +

2𝑞

𝜂22) + 𝜆2 (

𝑝

𝜂1−

𝑞

𝜂2)2

𝑇 + 𝜆 (𝑝

𝜂1−

𝑞

𝜂2) (2(𝑟 − 𝜆𝑚)𝑇 − 𝐸0

𝑄 [∫ 𝑉𝑡𝑑𝑡𝑇

0])

(𝑛 − 1) (26)

where 𝐸0𝑄 [∫ 𝑉𝑡𝑑𝑡

𝑇

0] is calculated according to Eq. (19) and the function 𝑔(∙) is given

explicitly (see Appendix A). Note that 𝑔(∙) is 𝑂 (1

𝑛), and the last term in Eq. (26) is also

𝑂 (1

𝑛). Thus, lim𝑛→∞ 𝐾𝑉𝑎𝑟(𝑛) = 𝐾𝑉𝑎𝑟.

We further verify our fair continuous and discrete variance strike formulae using Monte

Carlo simulations. Model parameters used in Monte Carlo simulations are similar to those of

10

Broadie and Jain (2008b) and Kou (2002). Table 1 shows these parameters. Table 2 shows

that the daily sampled variance strike formula of Eq. (26) for the DEJSV model is consistent

with the results of Monte Carlo simulations. When the simulation path number is 10,000, the

relative difference between the simulation and the formula of Eq. (26) is 0.269%. When the

simulation path number increases to 100,000, the relative difference further decreases to

0.055%. Therefore, our derived formulae are successfully justified by Monte Carlo

simulations.

Then, we investigate the fair variance strikes of one-year maturity based on monthly,

weekly, and daily sampling, i.e., n=12, 52, 252, respectively. For each sampling size, n, we

compute fair variance strikes using our analytical formulae. Table 3 shows that, in the DEJSV

model, the fair discrete variance strike in the case of monthly sampling (n=12) is 28.355%2

and the fair continuous variance strike is 27.129%2. This corresponds to a relative difference

of 4.519% higher than the fair continuous variance strike. Table 3 shows that discrete

sampling increases the fair variance strike. To illustrate, the fair discrete variance strike in the

case of monthly sampling is larger than that of weekly sampling, which is also larger than

that of daily sampling. Nevertheless, in the case of daily sampling, the relative difference

between the discrete variance and the fair continuous variance strike is reduced to 0.543%.

Hence, the effect of discrete sampling on variance swaps is typically small in the case of

daily sampling.

Subsequently, we investigate the effect of jump arrival rates on fair variance strikes.

Previous studies such as Dupire (1993), Carr and Madan (1998), and Demeterfi et al. (1999)

show that when the price process for the underlying asset is continuous, a continuously

sampled variance swap can be replicated by a static portfolio of out-of-the-money put and

call options. In the DEJSV model, jumps in the underlying asset can be turned off with the

specification of 𝜆 = 0, and then it reduces to Heston’s (1993) stochastic volatility model.

11

After turning off jumps, the option replication strategy to the fair continuous variance strike

is

𝐾𝑆𝑉 = 2𝑟 −2

𝑇(𝑙𝑛

𝑆∗

𝑆0+

𝑆0𝑒𝑟𝑇

𝑆∗− 1 − ∫

1

𝐾2𝑒𝑟𝑇𝑝(𝐾)𝑑𝐾 −

𝑆∗

𝐾=0

∫1

𝐾2𝑒𝑟𝑇𝑐(𝐾)𝑑𝐾

∞

𝐾=𝑆∗

) (27)

for all values of 𝑆∗ where 𝑐(𝐾) and 𝑝(𝐾) are the prices of European call and put options

with strike 𝐾 and maturity 𝑇. Table 4 shows that if there are no jumps (λ = 0) in the

underlying asset price, the option replication strategy could give the same variance value as

the formula of Eq. (24). However, if jumps occur in the underlying asset price and are priced

by market participants, the option replication strategy may deviate from the fair variance

strike.10

The result is consistent with claims of Carr and Lee (2009) that market participants

have often observed discrepancies between market variance swap quotes and the option

replication value. In this numerical example, the probability and size of negative jumps are

larger than those of positive jumps. Hence, the option replication strategy tends to be smaller

than the fair variance strike.11

Because the discretely sampled variance is higher than the

continuous variance strike, the option replication strategy could be even smaller than the

discretely sampled variance. Table 4 shows that discretely sampling and the increase of jump

frequency both tend to deteriorate the discrepancy problem.

Next, we investigate the effect of asymmetric jumps on fair variance strikes. Two types

of jump asymmetry are analyzed. One could arise from the probabilities of upward jumps and

downward jumps. The other could appear in the amplitudes of upward jumps and downward

jumps. Table 5 shows that if the two exponential random variables, standing for upward

jumps and downward jumps respectively, have the same mean, an increase in the upward

10

Given model parameters of DEJSV, we made a set of options using the formulae of (C.1)

and (C.6) and then used those options to price variance swaps with the option replication

approach. 11

Broadie and Jain (2008b) also find that the strike from the portfolio options is smaller than

the fair variance strike when there are negative jumps.

12

jump probability might have no influence on the fair continuous variance strike, but it could

increase the discretely sampled variance strike. Table 5 also shows that the variance strike

underestimate problem arising from the option replication strategy tends to disappear as the

upward jump probability increases. This result seems to justify the argument that negative

jumps cause the strike from the portfolio options smaller than the fair variance strike.

However, when 𝜂1 = 𝜂2, 𝑝=0.5 and 𝜆 = 10 (in the top panel of Table 5), the expected

overall jump size mean (𝑝 𝜂1⁄ − 𝑞 𝜂2⁄ ) is equal to zero, and the strike from the option

replication strategy could be larger than the fair continuous variance strike by 1.873%,

whereas the strike from the option replication strategy is still smaller than the monthly

sampling variance strike by 3.301%.12

In the bottom panel of Table 5, the jump arrival rate is

increased to 20 and we find that the underestimate problem of the option replication strategy

incurred by negative jumps actually deteriorates but the overestimate problem of the option

replication strategy induced by positive jumps seems to mitigate.

Table 6 shows if the upward jump and the downward jump have the same probability

(i.e., 𝑝 = 0.5), an increase in jump size mean could increase the variance value calculated

from the option replication strategy and the discretely sampled variance strikes without

changes in the fair continuous variance strike. In the bottom panel of Table 6, the jump

arrival rate is increased to 20 and we also find that the underestimate problem of the option

replication strategy incurred by negative jumps actually deteriorates but the overestimate

problem of the option replication strategy induced by positive jumps seems to mitigate.

Therefore, asymmetric jump sizes as asymmetric jump probabilities could have the similar

effect on variance strikes. Both types of jump asymmetry in the DEJSV model could strongly

affect the fair discrete variance strike and the variance value calculated from the option

12

If 𝜂1 = 𝜂2 and 𝑝=0.5, then the double exponential distribution is called “the first law of

Laplace” by Laplace in 1774, while “the second law of Laplace” means the normal density

(see Kou, 2002).

13

replication strategy even though the fair continuous variance strike remains unchanged.

4. Market Data Calibration and Empirical Results

We follow Floc′h (2018)13

by applying the daily S&P500 index option market data to

calibrate the parameters of the DEJSV model and other comparable models, and investigate

whether the DEJSV model is superior to other competing models in predicting the future

realized variance. In the spirit of Dew-Becker et al. (2017) to explore the behavior of realized

variance during financial crises, we discuss three different in-sample/out-of-sample tests on

the basis of whether the VIX is greater than 30%. The three sample periods used in market

data calibration are (i) in-sample: from Oct 22, 2007 to June 20, 2008 (a low volatility period);

out-of-sample: from Jun 23, 2008 to Sept 12, 2008 (a low volatility period), (ii) in-sample:

from Jan 14, 2008 to Sept 12, 2008 (a low volatility period); out-of-sample: from Sept 15,

2008 to Dec 5, 2008 (a high volatility period), (iii) in-sample: from Sept 15, 2008 to May 15,

2009 (a high volatility period); out-of-sample: from May 18, 2009 to Aug 7, 2009 (a low

volatility period). We used the intraday S&P500 index price data (every five minutes) to

calculate the realized variance. All option data were obtained from WRDS (OptionMetrics),

and all S&P500 index data were obtained from Price-Data.com.

Consider the DEJSV model whose implementation requires the estimation of volatility

𝑉𝑡 and the parameter vector 𝜗 ≡ {𝜃, 𝜅, 𝜎𝑣, 𝜆, 𝜂1, 𝜂2, 𝑝, 𝜌}. For 𝑚 = 1,… ,𝑀, let 𝑂𝑡̅̅ ̅(𝜏𝑚, 𝐾𝑚)

be the market price of the option with time-to-maturity 𝜏𝑚 and strike price 𝐾𝑚 and

𝑂𝑡(𝜏𝑚, 𝐾𝑚; 𝜗) be the corresponding model price. Similar to Bakshi and Cao (2003), we

search for 𝑉𝑡 and 𝜗 to solve

13

Floc′h (2018) studies the variance swap replication problem by calibrating the Heston’s

stochastic volatility model (1993) against the mid-price of the market SPX500 option quotes,

computing the vanilla option prices under the calibrated Heston’s stochastic volatility model

(1993) for each market strike, and using those as the basis of the various variance swap

replication.

14

𝛺𝑡 ≡ min𝑉𝑡,𝜗

∑(𝑂𝑡̅̅ ̅(𝜏𝑚, 𝐾𝑚) − 𝑂𝑡(𝜏𝑚, 𝐾𝑚; 𝜗))

2𝑀

𝑚=1

+ 𝑝𝑒𝑛𝑎𝑙𝑡𝑦 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛14. (28)

European option values in the DEJSV model are obtained from the computation of analytic

formulae in Appendix C. A similar procedure is also applied for its competing model.

European option values in the MertonSV model are obtained from the computation of

analytic formulae in Appendix D. Table 7 presents these parameters for these two competing

models in three different sub-samples.

Next, we verify the fair discrete variance strike formula calculated by model parameters

calibrated against the market SPX option quotes in predicting the future realized variance,

and compare the DEJSV model with the BS model and the MertonSV model. We use the

mean square error (MSE) to determine which model is better based on the realized variance.

Daily realized variance is calculated using the S&P500 index intraday data (every five

minutes).

The upper half of Table 8 shows the in-sample MSE between the realized variance15

and the fair one-month variance strike of competing models with daily sampling during Oct

22, 2007–Jun 20, 2008 (a low volatility period) for: (a) the DEJSV model, (b) the MertonSV

model, and (c) the BS model. The “Total” denotes that the sample includes all the

out-of-the-money call/put options and all the at-the-money call/put options. In the upper half

of Table 8, the MSE for the “Total” in the DEJSV model (0.0221%) is less than that in the BS

model (0.0906%), and the MertonSV model (0.0641%). The left section of Figure 2

illustrates realized variance, and the fair discrete variance strike in (a) the DEJSV model, (b)

14

The 𝑝𝑒𝑛𝑎𝑙𝑡𝑦 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛 can be (i) |𝜗 − 𝜗0|2 given by Mikhailov and Nogel (2004) or (ii)

|𝜗 − 𝜗0| given by Deryabin (2010). Nevertheless, the penalty function in this research is

specified by 𝑝𝑒𝑛𝑎𝑙𝑡𝑦 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛 = 𝑤𝑒𝑖𝑔ℎ𝑡 ∗ |𝜗 − 𝜗0|, where 𝑤𝑒𝑖𝑔ℎ𝑡 increases with the

volume of the traded option. 15

Because the variance strike calculated by model parameters calibrated against the SPX

option quotes is forward-looking for a future period, the realized variance is calculated with

the market data during the same corresponding period for comparison.

15

the MertonSV model, and (c) the BS model. The fair discrete variance strike in the DEJSV

model is closer to the realized variance than is the fair discrete variance strike in the

MertonSV model. The upper half of Table 8 and the left section of Figure 2 illustrates that the

estimation error in the DEJSV model is smaller than that in the MertonSV model in the

in-sample period of (i).

The lower half of Table 8 shows that the out-of-sample MSE for the “Total” in the

DEJSV model (0.0173%) is less than that in the BS model (0. 0770%), and also less than that

in the MertonSV model (0.0518%) during Jun 23, 2008–Sept 12, 2008 (a low volatility

period). The right section of Figure 2 illustrates the realized variance and the fair discrete

variance strike in the competing models. The lower half of Table 8 and the right section of

Figure 2 show that the MSE in the DEJSV model is smaller than that in the MertonSV model

in the out-of-sample period of (i).

The upper half of Table 9 shows that the in-sample MSE for the “Total” in the DEJSV

model (0.0200%) is less than that in the BS model (0.0815%), and also less than that in the

MertonSV model (0.0553%) during Jan 14, 2008–Sept 12, 2008 (a low volatility period). The

left section of Figure 3 illustrates the realized variance and the fair discrete variance strike in

competing models. The upper half of Table 9 and the left section of Figure 3 show that the

MSE in the DEJSV model is less than that in the MertonSV model in the in-sample period of

(ii).

The lower half of Table 9 shows the out-of-sample MSE for the “Total” in the DEJSV

model (0.4945%) is less than that in the BS model (2.5547%), and also less than that in the

MertonSV model (0.5198%) during Sept 15, 2008–Dec 5, 2008 (a high volatility period). The

right section of Figure 3 illustrates the realized variance and the fair discrete variance strike

in competing models. The lower half of Table 9 and the right section of Figure 3 illustrates

that the MSE in the DEJSV model is less than that in the MertonSV model in the

16

out-of-sample period of (ii).

The upper half of Table 10 shows that the in-sample MSE for the “Total” in the DEJSV

model (0.3145%) is less than that in the BS model (1.5754%), and also less than that in the

MertonSV model (0.4802%) during Sept 15, 2008–May 15, 2009 (a high volatility period).

The left section of Figure 4 illustrates the realized variance and the fair discrete variance

strike in competing models. The upper half of Table 10 and the left section of Figure 4

illustrates that the MSE in the DEJSV model is less than that in the MertonSV model in the

in-sample period of (iii).

The lower half of Table 10 shows that the out-of-sample MSE for the “Total” in the

DEJSV model (0.0721%) is less than that in the BS model (0.2334%), and also smaller than

that in the MertonSV model (0.0841%) during May 18, 2009–Aug 7, 2009 (a low volatility

period). The right section of Figure 4 illustrates the realized variance and the fair discrete

variance strike in competing models. The lower half of Table 10 and the right section of

Figure 4 illustrate that the MSE in the DEJSV model is less than that in the MertonSV model

in the out-of-sample period of (iii).

Therefore, our empirical results show that the DEJSV model performs better in

predicting the future realized variance in almost all cases than the MertonSV model. In

addition, both of the DEJSV model and the MertonSV model perform far better than the BS

model. These may shed light on the importance of incorporating asymmetric jumps into the

model for pricing variance derivatives.

5. Conclusion

In this paper, we investigate the effect of asymmetric jumps and the number of sampling

dates on variance swap rates and derive analytic formulae for both of continuous and discrete

variance strikes in the DEJSV model. Similar works such as Broadie and Jain (2008) and

17

Floc′h (2018) often discuss the effect of price jumps on variance swaps according to the

Merton (1976) jump model specification in which log stock price jump size follows normal

distribution. Our numerical experiments show that it is possible for the fair discrete variance

strike to increase with the upward jump probability whereas the fair continuous variance

strike could remain unchanged and for the option replication strategy to increase with the

upward jump probability faster than the fair discrete variance strike. Moreover, increasing the

upward jump size mean could have similar effects as increasing the upward jump probability.

Furthermore, we use market data to investigate whether asymmetric jumps would

perform better than normal distribution jumps in predicting future realized variance in the

framework of Heston’s (1993) stochastic volatility framework. We use three different

in-sample/ out-of-sample testing periods to analyze the estimation error and the prediction

error and find that both errors in the DEJSV model are much less than those in the MertonSV

model in all testing periods. Therefore, the superiority of asymmetric jumps to normal

distribution jumps in describing market data seems to be a robust phenomenon.

18

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21

Appendix A

Broadie and Jain (2008b) showed that the discrete variance strike in Heston’s stochastic

volatility model is 1

𝑇𝐸0

𝑄 [∫ 𝑉𝑡𝑑𝑡𝑇

0] + 𝑔(𝑟, 𝜌, 𝜎𝑣, 𝜅, 𝜃, 𝑛) where

𝑔(𝑟, 𝜌, 𝜎𝑣, 𝜅, 𝜃, 𝑛)

=𝑟2𝑇

𝑛 − 1+

1

𝑇𝐸0

𝑄 [∫ 𝑉𝑡𝑑𝑡𝑇

0

] (1

𝑛 − 1−

𝑟𝑇

𝑛 − 1+

𝜌𝜅𝜃𝑇

(𝑛 − 1)𝜎𝑣)

+∑ 𝐸0

𝑄[∫ ∫ 𝑉𝑡𝑉𝑠𝑑𝑡𝑑𝑠

𝑡𝑖𝑡𝑖−1

𝑡𝑖𝑡𝑖−1

]𝑛𝑖=1

(𝑛 − 1)Δ𝑡(1

4−

𝜌𝜅

𝜎𝑣)

−∑ 𝐸0

𝑄 [𝜌 (∫ 𝑉𝑡𝑑𝑡𝑡𝑖𝑡𝑖−1

) (𝑉𝑡𝑖− 𝑉𝑡𝑖−1

)]𝑛𝑖=1

𝜎𝑣(𝑛 − 1)Δ𝑡 (A. 1)

with

∑ 𝐸0𝑄 [(∫ 𝑉𝑡𝑑𝑡

𝑡𝑖

𝑡𝑖−1

) (𝑉𝑡𝑖− 𝑉𝑡𝑖−1

)]𝑛

𝑖=1

=(𝑉0 − 𝜃)𝜎𝑣

2

𝜅2(1 + 𝜅Δ𝑡 − 𝑒𝑥𝑝(𝜅Δ𝑡)) (

1 − 𝑒𝑥𝑝(−𝜅𝑇)

−1 + 𝑒𝑥𝑝(𝜅𝑇 𝑛⁄ ))

− ((𝜃 − 2𝑉0)𝜎𝑣

2

2𝜅2+

(𝑉0 − 𝜃)2

𝜅)(

1 − 𝑒𝑥𝑝(−2𝜅𝑇)

1 − 𝑒𝑥𝑝(−2𝜅𝑇 𝑛⁄ )(1 − 𝑒𝑥𝑝(−𝜅Δ𝑡))

2)

+ (𝑉0 − 𝜃)𝜃Δ𝑡(−1 + 𝑒𝑥𝑝(−𝜅𝑇)) (A. 2)

∑ 𝐸0𝑄 [∫ ∫ 𝑉𝑡𝑉𝑠𝑑𝑡𝑑𝑠

𝑡𝑖

𝑡𝑖−1

𝑡𝑖

𝑡𝑖−1

]𝑛

𝑖=1= ∑ (𝐸0

𝑄 [∫ 𝑉𝑡𝑑𝑡𝑡𝑖

𝑡𝑖−1

])

2𝑛

𝑖=1+ ∑ 𝑉𝑎𝑟 (∫ 𝑉𝑡𝑑𝑡

𝑡𝑖

𝑡𝑖−1

)𝑛

𝑖=1(A. 3)

∑ 𝑉𝑎𝑟 (∫ 𝑉𝑡𝑑𝑡𝑡𝑖

𝑡𝑖−1

)𝑛

𝑖=1

=𝜎𝑣

2𝑒−2𝜅Δ𝑡

𝜅3(𝑒2𝜅Δ𝑡 − 2𝑒𝜅Δ𝑡𝜅Δ𝑡 − 1)(𝑉0 − 𝜃) (

1 − 𝑒𝑥𝑝(−𝜅𝑇)

−1 + 𝑒𝑥𝑝(𝜅𝑇 𝑛⁄ ))

+𝜎𝑣

2𝑒−2𝜅Δ𝑡

2𝜅3(4𝑒𝜅Δ𝑡 − 3𝑒2𝜅Δ𝑡 + 2𝑒2𝜅Δ𝑡𝜅Δ𝑡 − 1)𝜃𝑛 (A. 4)

22

∑ (𝐸0𝑄 [∫ 𝑉𝑡𝑑𝑡

𝑡𝑖

𝑡𝑖−1

])

2𝑛

𝑖=1

= 𝑛(𝜃Δ𝑡)2 + (1 − 𝑒𝑥𝑝(−𝜅Δ𝑡))2𝜃Δ𝑡(𝑉0 − 𝜃)

𝜅∑𝑒𝑥𝑝 (

−(𝑖 − 1)𝜅𝑇

𝑛)

𝑛

𝑖=1

+(1 − 𝑒𝑥𝑝(−𝜅Δ𝑡))

2

𝜅2(𝑛

𝜎𝑣2𝜃

2𝜅+

(𝑉0 − 𝜃)𝜎𝑣2

𝜅∑𝑒𝑥𝑝(

−(𝑖 − 1)𝜅𝑇

𝑛)

𝑛

𝑖=1

+ ((𝑉0 − 𝜃)2 −(𝑉0 − 𝜃)𝜎𝑣

2

𝜅−

𝜎𝑣2𝜃

2𝜅)∑𝑒𝑥𝑝 (

−(𝑖 − 1)2𝜅𝑇

𝑛)

𝑛

𝑖=1

) (A. 5)

Therefore, the discrete variance strike in the DEJSV model is

𝐾𝑉𝑎𝑟(𝑛) = 𝐸0𝑄 [∑

1

(𝑛 − 1)∆𝑡(𝑙𝑛 (

𝑆𝑡𝑖

𝑆𝑡𝑖−1

))

2𝑛

𝑖=1

]

= ∑1

(𝑛 − 1)∆𝑡

𝑛

𝑖=1

𝐸0𝑄 [(∫ (𝑟 − 𝜆𝑚 −

1

2𝑉𝑡)𝑑𝑡

𝑡𝑖

𝑡𝑖−1

)

2

]

+ ∑1

(𝑛 − 1)∆𝑡

𝑛

𝑖=1

𝐸0𝑄 [(∫ √𝑉𝑡 (𝜌𝑑𝐵𝑡

𝑣 + √1 − 𝜌2𝑑𝐵𝑡𝑠)

𝑡𝑖

𝑡𝑖−1

)

2

]

+ ∑2

(𝑛 − 1)∆𝑡

𝑛

𝑖=1

𝐸0𝑄 [(∫ (𝑟 − 𝜆𝑚 −

1

2𝑉𝑡) 𝑑𝑡

𝑡𝑖

𝑡𝑖−1

)(∫ √𝑉𝑡 (𝜌𝑑𝐵𝑡𝑣

𝑡𝑖

𝑡𝑖−1

+ √1 − 𝜌2𝑑𝐵𝑡𝑠))] + ∑

1

(𝑛 − 1)∆𝑡

𝑛

𝑖=1

𝐸0𝑄

[

(𝑙𝑛 (∏𝑌𝑗

𝑛𝑖

𝑗=1

))

2

]

+ ∑2

(𝑛 − 1)∆𝑡

𝑛

𝑖=1

𝐸0𝑄 [(∫ (𝑟 − 𝜆𝑚 −

1

2𝑉𝑡) 𝑑𝑡

𝑡𝑖

𝑡𝑖−1

)(𝑙𝑛 (∏𝑌𝑗

𝑛𝑖

𝑗=1

))]

+ ∑2

(𝑛 − 1)∆𝑡

𝑛

𝑖=1

𝐸0𝑄 [(∫ √𝑉𝑡 (𝜌𝑑𝐵𝑡

𝑣 + √1 − 𝜌2𝑑𝐵𝑡𝑠)

𝑡𝑖

𝑡𝑖−1

)(𝑙𝑛 (∏𝑌𝑗

𝑛𝑖

𝑗=1

))]

23

=1

𝑇𝐸0

𝑄 [∫ 𝑉𝑡𝑑𝑡𝑇

0

] + 𝑔(𝑟 − 𝜆𝑚, 𝜌, 𝜎𝑣, 𝜅, 𝜃, 𝑛) + ∑1

(𝑛 − 1)∆𝑡

𝑛

𝑖=1

𝐸0𝑄

[

(𝑙𝑛 (∏𝑌𝑗

𝑛𝑖

𝑗=1

))

2

]

+∑2

(𝑛 − 1)∆𝑡

𝑛

𝑖=1

𝐸0𝑄 [(∫ (𝑟 − 𝜆𝑚 −

1

2𝑉𝑡)𝑑𝑡

𝑡𝑖

𝑡𝑖−1

)(𝑙𝑛 (∏𝑌𝑗

𝑛𝑖

𝑗=1

))]

+∑2

(𝑛 − 1)∆𝑡

𝑛

𝑖=1

𝐸0𝑄

[(∫ √𝑉𝑡 (𝜌𝑑𝐵𝑡𝑣 + √1 − 𝜌2𝑑𝐵𝑡

𝑠)𝑡𝑖

𝑡𝑖−1

)(𝑙𝑛 (∏𝑌𝑗

𝑛𝑖

𝑗=1

))]

=1

𝑇𝐸0

𝑄 [∫ 𝑉𝑡𝑑𝑡𝑇

0

] + 𝑔(𝑟 − 𝜆𝑚, 𝜌, 𝜎𝑣, 𝜅, 𝜃, 𝑛) + ∑1

(𝑛 − 1)∆𝑡

𝑛

𝑖=1

𝐸0𝑄

[

(𝑙𝑛 (∏𝑌𝑗

𝑛𝑖

𝑗=1

))

2

]

+∑2

(𝑛 − 1)∆𝑡

𝑛

𝑖=1

𝐸0𝑄 [(∫ (𝑟 − 𝜆𝑚 −

1

2𝑉𝑡)𝑑𝑡

𝑡𝑖

𝑡𝑖−1

)(𝑙𝑛 (∏𝑌𝑗

𝑛𝑖

𝑗=1

))] (A. 6)

𝐸[∑ 𝑙𝑛(𝑌𝑗)𝑛𝑖𝑗=1 ] = (

𝑝

𝜂1−

𝑞

𝜂2) 𝑛𝑖 and 𝑉𝑎𝑟(∑ 𝑙𝑛(𝑌𝑗)

𝑛𝑖𝑗=1 ) = (𝑝𝑞 (

1

𝜂1+

1

𝜂2)2

+p

η1

2 +q

η2

2)𝑛𝑖

given 𝑛𝑖 . In addition, 𝐸[𝑛𝑖] = 𝜆Δ𝑡 and 𝐸[𝑛𝑖2] = 𝑉𝑎𝑟(𝑛𝑖) + (𝐸[𝑛𝑖])

2 = 𝜆Δ𝑡 + (𝜆Δ𝑡)2 .

Therefore, 𝐸0𝑄 [(∑ 𝑙𝑛(𝑌𝑗)

𝑛𝑖𝑗=1 )

2

] = 𝑉𝑎𝑟(∑ 𝑙𝑛(𝑌𝑗)𝑛𝑖𝑗=1 ) + (𝐸0

𝑄[∑ 𝑙𝑛(𝑌𝑗)𝑛𝑖𝑗=1 ])

2

=

(𝑝𝑞 (1

𝜂1+

1

𝜂2)2

+𝑝

𝜂12 +

𝑞

𝜂22)𝐸0

𝑄[𝑛𝑖] + (𝑝

𝜂1−

𝑞

𝜂2)2

𝐸0𝑄[𝑛𝑖

2] = (2𝑝

𝜂12 +

2𝑞

𝜂22) 𝜆Δ𝑡 + (

𝑝

𝜂1−

𝑞

𝜂2)2(𝜆Δ𝑡)2 .

Hence,

𝐾𝑉𝑎𝑟(𝑛) =1

𝑇𝐸0

𝑄 [∫ 𝑉𝑡𝑑𝑡𝑇

0

] + 𝑔(𝑟 − 𝜆𝑚, 𝜌, 𝜎𝑣, 𝜅, 𝜃, 𝑛) + 𝜆 (2𝑝

𝜂12 +

2𝑞

𝜂22)

+𝜆 (

2𝑝

𝜂12 +

2𝑞

𝜂22) + 𝜆2 (

𝑝

𝜂1−

𝑞

𝜂2)2

𝑇 + 𝜆 (𝑝

𝜂1−

𝑞

𝜂2) (2(𝑟 − 𝜆𝑚)𝑇 − 𝐸0

𝑄 [∫ 𝑉𝑡𝑑𝑡𝑇

0])

(𝑛 − 1) (A. 7)

24

Appexdix B

Broadie and Jain (2008b) showed the discrete variance strike in the MertonSV model is given

by:

𝐾𝑣𝑎𝑟∗ (𝑛) = 𝐾𝑣𝑎𝑟

∗ + ℎ(𝑟, 𝜌, 𝜎𝑣, 𝜅, 𝜃,𝑚, 𝑏, 𝑛) (B. 1)

where

𝐾𝑣𝑎𝑟∗ = 𝜃 +

𝑣0 − 𝜃

𝜅𝑇(1 − 𝑒−𝜅𝑇) + 𝜆(𝑎2 + 𝑏2) (B. 2)

and

ℎ(𝑟, 𝜌, 𝜎𝑣, 𝜅, 𝜃, 𝑚, 𝑏, 𝑛) = 𝑔(𝑟 − 𝜆𝑚, 𝜌, 𝜎𝑣, 𝜅, 𝜃, 𝑛)

+𝜆(𝑎2 + 𝑏2) + 𝜆2𝑎2𝑇 + 𝜆𝑎 (2(𝑟 − 𝜆𝑚)𝑇 − 𝐸 [∫ 𝑉𝑡𝑑𝑡

𝑇

0])

𝑛 − 1 (B. 3)

with 𝑔(𝑟, 𝜌, 𝜎𝑣, 𝜅, 𝜃, 𝑛) is given in (A. 1).

25

Appendix C

The present value of a European call option in the DEJSV model can be formulated as

𝐶(𝐾) = 𝐸0𝑄[𝑒−𝑟𝑇𝑚𝑎𝑥{𝑆𝑇 − 𝐾, 0}] and given by

𝐶(𝐾) = 1

2𝐽(𝑡, 𝑇; −𝑖) −

1

𝜋∫

𝐼𝑚[𝑒𝑖𝑣𝑙𝑜𝑔(𝐾)𝐽(𝑡, 𝑇; −𝑖 − 𝑣)]

𝑣𝑑𝑣

∞

0

− 𝐾 (1

2𝐽(𝑡, 𝑇; 0)

−1

𝜋∫

𝐼𝑚[𝑒𝑖𝑣𝑙𝑜𝑔(𝐾)𝐽(𝑡, 𝑇; −𝑣)]

𝑣𝑑𝑣

∞

0

) (C. 1)

where 𝐽(𝑡, 𝑇; 𝜙) ≡ 𝐸𝑡[𝑆𝑇𝑖𝜙

] is the characteristic function of the state density. 𝐼𝑚(𝑐) denotes

the imaginary part of 𝑐 ∈ 𝐶. The characteristic function is given by

𝐽(𝑡, 𝑇; 𝜙) = 𝑒𝑥𝑝(𝐴(𝑇; 𝜙) + 𝐵(𝑇; 𝜙)𝑉)𝑆𝑖𝜙 (C. 2)

where 𝐴(𝑇; 𝜙) and 𝐵(𝑇; 𝜙) are

𝐴(𝑇; 𝜙) = (𝑖𝜙) (𝑟 − 𝜆 (𝑝𝜂1

𝜂1 − 1+ 𝑞

𝜂2

𝜂2 + 1− 1))𝑇 − 𝑟𝑇

+ 𝜆 (𝑝−𝜂1

−𝜂1 + 𝑖𝜙+ 𝑞

𝜂2

𝜂2 + 𝑖𝜙− 1) 𝑇

−𝜅𝜃

𝜎𝑣2((𝜀 + 𝑖𝜙𝜎𝑣𝜌 − 𝜅)𝑇

+ 2𝑙𝑜𝑔 (1 −(𝜀 + 𝑖𝜙𝜎𝑣𝜌 − 𝜅)(1 − 𝑒𝑥𝑝(−𝜀𝑇))

2𝜀)) (C. 3)

𝐵(𝑇; 𝜙) =𝑖𝜙(𝑖𝜙 − 1)(1 − 𝑒𝑥𝑝(−𝜀𝑇))

2𝜀 − (𝜀 + 𝑖𝜙𝜎𝑣𝜌 − 𝜅)(1 − 𝑒𝑥𝑝(−𝜀𝑇)) (C. 4)

𝜀 = √(𝑖𝜙𝜎𝑣𝜌 − 𝜅)2 − 𝑖𝜙(𝑖𝜙 − 1)𝜎𝑣2 (C. 5)

The proof is similar to the work of Duffie, Pan, and Singleton (2000) and is therefore omitted.

Once we have the solution for European calls, the formula for put, 𝑃(𝐾), can be obtained by

26

the put-to-call conversion equation of Grabbe (1983):

𝑃(𝐾) = 𝐶(𝐾) − 𝑆 + 𝐾𝑒−𝑟𝑇 (C. 6)

27

Appendix D

Yedder (2012) showed the present value of a European option in the hybrid model of

Heston’s stochastic volatility and Merton’s jump-diffusion is given by

𝐹(𝑥𝑡, 𝑡) =1 + 𝜑

2𝑒𝑥𝑡−𝜏𝜆𝑚 +

1 − 𝜑

2𝑒1−𝜏𝑟𝐾 − 𝑒−𝜏𝑟𝑓(𝑥, 𝜆, 𝜏) (D. 1)

where

𝜑 = +1 for a call and 𝜑 = −1 for a put; 𝜏 = 𝑇 − 𝑡; 𝑥𝑡 = ln(𝑆𝑡) and 𝑋 = ln(𝑆𝑡/𝐾) +

(𝑟 − 𝜆𝑚)𝜏

and

𝑓(𝑥, 𝜆, 𝜏) =𝐾

𝜋∫ ℜ[

𝑄(𝑘, 𝑥, 𝜆, 𝜏)

𝑘2 + 1/4]

∞

0

𝑑𝑘 (D. 2)

with

𝑄(𝑘, 𝑥𝜆, 𝜏) = 𝑒(−𝑖𝑘+1/2)𝑋+𝐴(𝑘,𝜏)+𝐵(𝑘,𝜏)𝑉0+𝐶(𝑘,𝜏)+𝐷(𝑘,𝜏)𝜆 (D. 3)

The coefficients 𝐴(𝑘, 𝜏), 𝐵(𝑘, 𝜏), 𝐶(𝑘, 𝜏) and 𝐷(𝑘, 𝜏) are specified as

𝐴(𝑘, 𝜏) = −𝜅𝜃

𝜎𝑣2[𝜓+𝜏 + 2 ln (

𝜓− + 𝜓+𝑒−𝜏𝜍

2𝜍)] (D. 4)

𝐵(𝑘, 𝜏) = −(𝑘2 +1

4)

1 − 𝑒−𝜏𝜍

𝜓− + 𝜓+𝑒−𝜏𝜍 (D. 5)

𝐶(𝑘, 𝜏) = 0 (D. 6)

𝐷(𝑘, 𝜏) = 𝜏Λ(𝑘) (D. 7)

where

𝜓± = ∓(𝑢 + 𝑖𝑘𝜌𝜎𝑣) + 𝜍 (D. 8)

28

ς = √𝑘2𝜎𝑣2(1 − 𝜌2) + 2𝑖𝑘𝜌𝜎𝑣𝑢 + 𝑢2 + 𝜎𝑣

2/4 (D. 9)

𝑢 = 𝜅 −𝜌𝜎𝑣

2 (D. 10)

Λ(𝑘) = 𝑒−𝑖𝑘(𝑎+

𝑏2

2)−(𝑘2−

1

4)𝑏2

2+

1

2𝑎 − 1 − (𝑖𝑘 +

1

2) (𝑒𝑎+

𝑏2

2 − 1) (D. 11)

29

Table 1. Model parameters used in numerical experiments.

Parameters Values

Risk-free rate 𝑟 3.19%

Stochastic Volatility Model Initial volatility √𝑉0 10.11%

Correlation 𝜌 -0.70

Long-run mean variance 𝜃 0.019

Speed of mean reversion 𝜅 6.21

Volatility of variance 𝜎𝑣 0.31

Double Exponential Jump Model Jump arrival rate 𝜆 1

Upward jump probability 𝑝 0.4

Upward jump size mean 1 𝜂1⁄ 1 10⁄

Downward jump size mean 1 𝜂2⁄ 1 5⁄

30

Table 2. Fair variance strikes with different trial numbers of Monte Carlo simulations in the

DEJSV model: a daily-sampling case.

Daily Sampling (𝑛 = 252)

Monte Carlo Simulation Analytical Formula

Paths 𝐾𝑉𝑎𝑟(𝑛) SD (%) Diff (%) 𝐾𝑉𝑎𝑟(𝑛)

5,000 27.313 0.200 0.471

27.185 10,000 27.258 0.110 0.269

20,000 27.166 0.072 0.070

100,000 27.200 0.042 0.055

The first column shows the trial number in computing the fair variance strike of one-year

maturity swap in the DEJSV model using the Monte Carlo simulation. The second column

shows the fair daily-sampled variance strike (quoted in volatility percentage points)16

for the

respective trial number computed using the simulation. The third column shows the standard

deviation in the estimate of the fair variance strike computed using simulation. The fourth

column shows the absolute difference between the simulation and the analytical formula

divided by the analytical formula. The last column shows the fair variance strike values

computed using the analytical formula in Eq. (26).

16

The variance strike is often quoted as the square root of variance (e.g., a 739.024 variance

strike would be denoted by 27.1852) to allow investors to easily related the quantity back to

volatility terms.

31

Table 3. Fair variance strikes with different sampling dates in the DEJSV model.

The first column shows the sampling dates in computing the realized variance of one-year

maturity swap in the DEJSV model. The second column shows the fair variance strike values

(quoted in volatility percentage points) computed using the analytical formula in Eq. (26).

n Analytical Formula 𝐾𝑉𝑎𝑟(𝑛)

Monthly (n=12) 28.355

Weekly (n=52) 27.404

Daily (n=252) 27.185

Cont. (n=∞) 27.129

32

Table 4. Effects of price jumps and sampling on the discrepancy problem of variance strikes

arising from the option replication strategy in the DEJSV model.

𝜆

Option

Replication

Strategy

Sampling Dates

Cont.

n=∞

Daily

n=252

Weekly

n=52

Monthly

n=12

𝐾𝑉𝑎𝑟(n)

0 13.266 13.266

(0%)

13.304

(-0.286%)

13.416

(-1.131%)

13.928

(-4.990%)

0.25 17.234 17.776

(-3.145%)

17.804

(-3.307%)

17.972

(-4.282%)

18.601

(-7.932%)

0.5 20.469 21.354%

(-4.324%)

21.401

(-4.553%)

21.564

(-5.350%)

22.338

(-9.131%)

0.75 23.259 24.413

(-4.962%)

24.454

(-5.138%)

24.658

(-6.015%)

25.534

(-9.781%)

1 25.749 27.129

(-5.359%)

27.185

(-5.577%)

27.404

(-6.427%)

28.355

(-10.121%)

The first column shows the jump arrival rate 𝜆, and the second column shows the variance

strike of one-year maturity swap computed using the option replication strategy with Eq. (27).

The third column shows the fair continuous variance strike. The fourth column shows the fair

discrete variance strike values for daily sampling computed using the analytical formula in

Eq. (26). The fifth column shows the fair discrete variance strike values for weekly sampling,

and the last column shows the fair discrete variance strike values for monthly sampling. The

number in the parenthesis is the difference between the option replication value and the fair

variance strike divided by the fair variance strike for respective sampling dates.

33

Table 5. Effects of asymmetric jump probability on fair variance strikes in the DEJSV model.

p

Option

Replication

Strategy

Sampling Dates

Cont.

n=∞

Daily

n=252

Weekly

n=52

Monthly

n=12

𝐾𝑉𝑎𝑟(n)

0.1 84.605 90.421

(-6.874%)

90.626

(-7.117%)

91.422

(-8.057%)

94.963

(-12.243%)

0.3 88.470 90.421

(-2.205%)

90.631

(-2.443%)

91.444

(-3.362%)

95.074

(-7.465%)

0.5 92.147 90.421

(1.873%)

90.637

(1.639%)

91.471

(0.734%)

95.189

(-3.301%)

0.7 95.708 90.421

(5.524%)

90.642

(5.293%)

91.499

(4.398%)

95.321

(0.404%)

0.9 99.151 90.421

(8.805%)

90.648

(8.576%)

91.531

(7.685%)

95.462

(3.721%)

𝜂1 = 𝜂2 = 5, 𝜆 = 10

0.1 118.059 127.185

(-7.730%)

127.511

(-8.006%)

128.775

(-9.077%)

134.402

(-13.843%)

0.3 123.628 127.185

(-2.877%)

127.523

(-3.151%)

128.849

(-4.223%)

134.726

(-8.977%)

0.5 128.969 127.185

(1.383%)

127.542

(1.106%)

128.926

(0.033%)

135.074

(-4.734%)

0.7 134.112 127.185

(5.165%)

127.558

(4.887%)

129.012

(3.803%)

135.451

(-0.998%)

0.9 139.104 127.185 127.577 129.105 135.857

34

(8.568%) (8.287%) (7.188%) (2.334%)

𝜂1 = 𝜂2 = 5, 𝜆 = 20

The first column shows the upward jump probabilities, and the second column shows the

variance strike of one-year maturity swap computed using the option replication strategy with

Eq. (27). The third column shows the fair continuous variance strike. The fourth column

shows the fair discrete variance strike for daily sampling computed using Eq. (26). The fifth

column shows the fair discrete variance strike for weekly sampling, and the last column

shows the fair discrete variance strike for monthly sampling. The number in the parenthesis is

the difference between the option replication value and the fair variance strike divided by the

fair variance strike for respective sampling dates.

35

Table 6. Effects of asymmetric jump sizes on fair variance strikes in the DEJSV model.

1 𝜂1⁄ 1 𝜂2⁄

Option

Replication

Strategy

Sampling Dates

Cont.

n=∞

Daily

n=252

Weekly

n=52

Monthly

n=12

𝐾𝑉𝑎𝑟(n)

√2

10

√6

10

85.440 90.421

(-5.830%)

90.626

(-6.070%)

91.422

(-7.001%)

94.984

(-11.170%)

√3

10

√5

10

88.707 90.421

(-1.932%)

90.631

(-2.169%)

91.444

(-3.085%)

95.079

(-7.183%)

√4

10

√4

10

92.147 90.421

(1.873%)

90.637

(1.639%)

91.471

(0.734%)

95.189

(-3.301%)

√5

10

√3

10

95.697 90.421

(5.513%)

90.642

(5.282%)

91.499

(4.387%)

95.321

(0.393%)

√6

10

√2

10

99.303 90.421

(8.944%)

90.648

(8.716%)

91.531

(7.827%)

95.473

(3.857%)

𝑝 = 0.5, 𝜆 = 10

√2

10

√6

10

119.281 127.185

(-6.626%)

127.511

(-6.900%)

128.791

(-7.973%)

134.473

(-12.736%)

√3

10

√5

10

124.012 127.185

(-2.559%)

127.526

(-2.834%)

128.853

(-3.904%)

134.748

(-8.657%)

√4

10

√4

10

128.969 127.185

(1.383%)

127.542

(1.106%)

128.926

(0.033%)

135.074

(-4.734%)

√5

10

√3

10

134.067 127.185

(5.133%)

127.558

(4.855%)

129.012

(3.771%)

135.451

(-1.032%)

36

√6

10

√2

10

139.241 127.185

(8.658%)

127.577

(8.377%)

129.108

(7.277%)

135.879

(2.415%)

𝑝 = 0.5, 𝜆 = 20

The first column shows the upward jump sizes, the second column shows the downward

jump sizes, and the third column shows the variance strike of one-year maturity swap

computed using the option replication strategy with Eq. (27). The fourth column shows the

fair continuous variance strike. The fifth column shows the fair discrete variance strike for

daily sampling computed using the analytical formula in Eq. (26). The sixth column shows

the fair discrete variance strike for weekly sampling, and the last column shows the fair

discrete variance strike for monthly sampling. The number in parenthesis is the difference

between the option replication value and the fair variance strike divided by the fair variance

strike for respective sampling dates.

37

Table 7. Model parameters used in market data calibration.

Parameters Initial

Values

Sample (i) Sample (ii) Sample (iii)

Risk-free rate 𝑟 5.19%

Panel A: Model parameters of the DEJSV model

Stochastic

Volatility

Model

Correlation 𝜌 -0.70 -0.8376 -0.8294 -0.9513

Long-run mean variance 𝜃 0.019 0.0811 0.0836 0.1431

Speed of mean reversion 𝜅 6.21 1.2589 1.2611 1.3054

Volatility of variance 𝜎𝑣 0.31 0.7549 0.7219 0.7194

Double

Exponential

Jump Model

Jump arrival rate 𝜆 1 0.0113 0.0139 0.0982

Upward jump probability 𝑝 0.4 0.1078 0.1029 0.1941

Upward jump size mean 1 𝜂1⁄ 1/10 0.1031 0.0999 0.3927

Downward jump size mean 1 𝜂2⁄ 1/5 0.1941 0.1931 0.6831

Panel B: Model parameters of the MertonSV model

Stochastic

Volatility

Model

Correlation 𝜌 -0.70 -0.4840 -0.4746 -0.4465

Long-run mean variance 𝜃 0.019 0.0731 0.0710 0.1263

Speed of mean reversion 𝜅 6.21 1.4865 1.4448 0.9519

Volatility of variance 𝜎𝑣 0.31 0.0184 0.0180 0.0169

Merton

Jump-Diffusion

Model

Jump arrival rate 𝜆 0.089 0.5082 0.5170 0.6450

Jump mean 𝑎 -0.89 -0.8114 -0.8101 -0.9999

Jump standard deviation 𝑏 0.45 0.1195 0.1194 0.1638

Three sub-sample periods used in market data calibration are (i) in-sample: from Oct 22,

2007 to Jun 20, 2008 (a low volatility period); out-of-sample: from Jun 23, 2008 to Sept 12,

2008 (a low volatility period), (ii) in-sample: from Jan 14, 2008 to Sept 12, 2008 (a low

volatility period); out-of-sample: from Sept 15, 2008 to Dec 5, 2008 (a high volatility period),

38

(iii) in-sample: from Sept 15, 2008 to May 15, 2009 (a high volatility period); out-of-sample:

from May 18, 2009 to Aug 7, 2009 (a low volatility period).

39

Table 8. In-sample test during Oct 22, 2007–Jun 20, 2008 (a low volatility period) in the

upper half of the table; out-of-sample test during Jun 23, 2008–Sept 12, 2008 (a low volatility

period) in the lower half of the table. MSE between the realized variance and the fair discrete

variance strike in different models: (a) the DEJSV model, (b) the MertonSV model, and (c)

the BS model.

Moneyness

Mean Square Error (%)

DEJSV MertonSV BS

In-sample test during Oct 22, 2007–Jun 20, 2008 (a low volatility period)

0.8625~0.8875 0.0223 0.0481 0.0813

0.8875~0.9125 0.0239 0.0508 0.0809

0.9125~0.9375 0.0215 0.0459 0.0997

0.9375~0.9625 0.0212 0.0561 0.0884

0.9625~0.9875 0.0209 0.0655 0.0972

0.9875~1.0125 0.0272 0.0849 0.0897

Total 0.0221 0.0641 0.090617

Out-of-sample test during Jun 23, 2008–Sept 12, 2008 (a low volatility period)

0.8625~0.8875 0.0127 0.0284 0.0760

0.8875~0.9125 0.0136 0.0274 0.0765

0.9125~0.9375 0.0181 0.0389 0.0720

0.9375~0.9625 0.0183 0.0475 0.0756

0.9625~0.9875 0.0163 0.0540 0.0835

0.9875~1.0125 0.0238 0.0765 0.0746

Total 0.0173 0.0518 0.077018

17

The MSE between realized variance and VIX2 is 0.0976%.

18 The MSE between realized variance and VIX

2 is 0.0815%.

40

The first column shows the moneyness computed using the S&P500 index closing price

divided by the strike price for calls (the strike price divided by the S&P500 index closing

price for puts). The second, third, and fourth columns, respectively, show the MSE between

the realized variance and the variance strike in the DEJSV model, the MSE between the

realized variance and the variance strike in the MertonSV model, and the MSE between the

realized variance and the BS model.

41

Table 9. In-sample test during Jan 14, 2008–Sept 12, 2008 (a low volatility period) in the

upper half of the table; out-of-sample test during Sept 15, 2008–Dec 15, 2008 (a high

volatility period) in the lower half of the table. MSE between the realized variance and the

fair discrete variance strike in different models: (a) the DEJSV model, (b) the MertonSV

model, and (c) the BS model.

Moneyness

Mean Square Error (%)

DEJSV MertonSV BS

In-sample test during Jan 14, 2008–Sept 12, 2008 (a low volatility period)

0.8625~0.8875 0.0172 0.0347 0.0776

0.8875~0.9125 0.0192 0.0385 0.0762

0.9125~0.9375 0.0203 0.0418 0.0808

0.9375~0.9625 0.0190 0.0483 0.0838

0.9625~0.9875 0.0196 0.0594 0.0740

0.9875~1.0125 0.0253 0.0768 0.0862

Total 0.0200 0.0553 0.081519

Out-of-sample test during Sept 15, 2008–Dec 5, 2008 (a high volatility period)

0.8625~0.8875 0.6447 0.6919 2.5605

0.8875~0.9125 0.6151 0.5776 2.5494

0.9125~0.9375 0.6797 0.8622 2.5571

0.9375~0.9625 0.6039 0.6419 2.5472

0.9625~0.9875 0.5208 0.5892 2.5504

0.9875~1.0125 0.5213 0.6358 2.5529

Total 0.4945 0.5198 2.554720

19

The MSE between realized variance and VIX2 is 0.0856%.

20 The MSE between realized variance and VIX

2 is 2.6096%.

42

The first column shows the moneyness computed using the S&P500 index closing price

divided by the strike price for calls (the strike price divided by the S&P500 index closing

price for puts). The second, third, and fourth columns, respectively, show the MSE between

the realized variance and the variance strike in the DEJSV model, the MSE between the

realized variance and the variance strike in the MertonSV model, and the MSE between the

realized variance and the BS model.

43

Table 10. In-sample test during Sept 15, 2008–May 15, 2009 (a high volatility period) in the

upper half of the table; out-of-sample test during May 18, 2009–Aug 7, 2009 (a low volatility

period) in the lower half of the table. MSE between the realized variance and the fair discrete

variance strike in different models: (a) the DEJSV model, (b) the MertonSV model, and (c)

the BS model.

Moneyness

Mean Square Error (%)

DEJSV MertonSV BS

In-sample test during Sept 15, 2008–May 15, 2009 (a high volatility period)

0.8625~0.8875 0.3537 0.5028 1.6749

0.8875~0.9125 0.3452 0.4754 1.5840

0.9125~0.9375 0.3612 0.5794 1.6573

0.9375~0.9625 0.3527 0.5328 1.6512

0.9625~0.9875 0.3260 0.5276 1.4989

0.9875~1.0125 0.3214 0.5495 1.5092

Total 0.3145 0.4802 1.575421

Out-of-sample test during May 18, 2009–Aug 7, 2009 (a low volatility period)

0.8625~0.8875 0.0574 0.0565 0.2499

0.8875~0.9125 0.0459 0.0429 0.2777

0.9125~0.9375 0.0597 0.0692 0.2482

0.9375~0.9625 0.0634 0.0849 0.1901

0.9625~0.9875 0.0633 0.1011 0.2331

0.9875~1.0125 0.0489 0.0786 0.1895

Total 0.0721 0.0843 0.233422

21

The MSE between realized variance and VIX2 is 1.6271%.

22 The MSE between realized variance and VIX

2 is 0.2500%.

44

The first column shows the moneyness computed using the S&P500 index closing price

divided by the strike price for calls (the strike price divided by the S&P500 index closing

price for puts). The second, third, and fourth columns, respectively, show the MSE between

the realized variance and the variance strike in the DEJSV model, the MSE between the

realized variance and the variance strike in the MertonSV model, and the MSE between the

realized variance and the BS model.

45

2007.1 2007.4 2007.7 2007.10 2008.1 2008.4 2008.7 2008.10 2009.1 2009.4 2009.7 2009.10

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

Date

Ju

mp

Siz

e (

%)

Figure 1. The daily realized jumps of S&P500 index from 2007 to 2009.

Jump information: The number of positive jump is 90, positive jump size mean is 0.49%; the

number of negative jump is 92, and negative jump size mean is -0.52%.

46

2007.11 2007.12 2008.1 2008.2 2008.3 2008.4 2008.5 2008.6 2008.7 2008.8 2008.90

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Date

Va

ria

nce

Realized Var

Total (DEJSV)

Total (MertonSV)

Total (BS)

Figure 2. In-sample test during Oct 22, 2007–Jun 20, 2008 (a low volatility period) in the left

section of the figure; out-of-sample test during Jun 23, 2008–Sept 12, 2008 (a low volatility

period) in the right section of the figure. The realized variance and the fair discrete variance

strike in (a) the DEJSV model, (b) the MertonSV model, and (c) the BS model. are

respectively denoted by “Realized Var”, “Total (DEJSV)”, “Total (MertonSV)”, and “Total

(BS)”.

47

2008.2 2008.3 2008.4 2008.5 2008.6 2008.7 2008.8 2008.9 2008.10 2008.11 2008.120

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Date

Va

ria

nce

Realized Var

Total (DEJSV)

Total (MertonSV)

Total (BS)

Figure 3. In-sample test during Jan 14, 2008–Sept 12, 2008 (a low volatility period) in the left

section of the figure; out-of-sample test during Sept 15, 2008–Dec 15, 2008 (a high volatility

period) in the right section of the figure. The realized variance and the fair discrete variance

strike in (a) the DEJSV model, (b) the MertonSV model, and (c) the BS model are

respectively denoted by “Realized Var”, “Total (DEJSV)”, “Total (MertonSV)”, and “Total

(BS)”.

48

2008.10 2008.11 2008.12 2009.1 2009.2 2009.3 2009.4 2009.5 2009.6 2009.7 2009.80

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Date

Va

ria

nce

Realized Var

Total (DEJSV)

Total (MertonSV)

Total (BS)

Figure 4. In-sample test during Sept 15, 2008–May 15, 2009 (a high volatility period) in the

left section of the figure; out-of-sample test during May 18, 2009–Aug 7, 2009 (a low

volatility period) in the right section of the figure. The realized variance and the fair discrete

variance strike in (a) the DEJSV model, (b) the MertonSV model, and (c) the BS model are

respectively denoted by “Realized Var”, “Total (DEJSV)”, “Total (MertonSV)”, and “Total

(BS)”.