The New Variation Method of Volatility Risk Premium
Yoshihiko Sugihara ∗†First draft: May 2, 2008
This draft: October 17, 2008

Abstract
Integrated implied volatility is an expected quadratic variation that can be derived from European option prices under the risk neutral measure, whereas integrated
realized volatility is a historical quadratic variation that can be derived from highfrequency ex-post stock returns under the physical measure. Earlier studies used thedifference as the volatility risk premium and showed the premium was negative for buyers of options in most periods. Their method, however, has a filtration inconsistency
problem when we evaluate an expected risk premium. This problem results from thedifference of the filtration required for the computation of the two volatility estimates.
We proposes a new method of evaluating the expected risk premium with consistentfiltration. In our method, the premium is identified as the gap between the current
integrated implied volatility and the integrated volatility estimated by a time seriesmodel of realized volatilities. We calculate the premium under the condition that the
one-day realized volatility obeys either the Heston model or the ARFIMAX model.
We found the premium based on our method correlates with market risk indicators,
such as Citi Macro Risk Index or iTraxx Japan, more strongly than those based on theearlier works. This indicates our proposed measure of expected volatility risk premiumreflects the market sentiment well and it is a valid indicator for the market risk aversion. As a byproduct, we can quantify the probability where the realized volatility goes
below the implied volatility. This probability might also be an indicator for market’s
volatility risk aversion.

Keywords: volatility risk premium, model-free implied volatility, realized
volatility, VIX, Heston model, ARFIMAX model
JEL Classification: G12, G13, G14∗

I would like to thank Peter Carr, Sacha Stoikov, Toshiaki Watanabe and seminar participants at
Diawa International Workshop on Financial Engineering 2008 for useful comments and suggestions. Views
expressed in this paper are those of the author and do not necessarily reflect the official views of the Bankof Japan.
†
Institute for Monetary and Economic Studies, Bank of Japan (E-mail: yoshihiko.sugihara@boj.or.jp).

11

Introduction

Volatilities in asset prices often vary roughly. This generates a premium for a volatility
implied in a derivative. The existence of the premium is widely known and that is calleda volatility risk premium, hereafter VRP. Various earlier works studied the VRP in aparametric way under a stochastic volatility model, or SV model.
Scott [1987], Wiggins [1987] and Hull and White [1987] introduced the SV model and
Wiggins [1990] firstly proved the SV model fitted using the option prices with various
strikes and term to maturities. In their works, a VRP was derived by the partial differential
equation of a derivative price under the SV model (see Appendix A.1 and A.2). Various
studies on a VRP also applied parametric models for volatilities. One of the most recent
studies, Bakshi and Kapadia [2003], introduced a parametric method to capture the VRPfrom stock option buyers’ delta-hedged gains. They showed that the VRP was negativefor option buyers and the absolute value was likely to stay large in periods of higher
volatilities. The negative premium for option buyers implies that the option sellers gain
a positive premium, an excess return beyond the risk neutral volatility value. That isfrequently observed in the US stock market. Uchida and Miyazaki [2008] applied thismethod to the Japanese stock market and found similar results.
Recently, some works has examined non-parametric estimation for a VRP defined by
the gap between the model-free integrated implied volatility and the integrated realized
volatility. The model-free integrated implied volatility is an aggregated implied volatility
derived in non-parametric way from put and call European option prices with different
strikes. On the other hand, the integrated realized volatility is a non-parametric consistentestimator of a realized variance computed from high-frequency intraday price data. Carrand Wu [2005] examined this type of VRPs on some stock indices and individual stocks
in the US market. They found that these VRPs were negative for option buyers in mostperiods, and the VRPs in individual stock options can primarily be explained by the VRP
in the stock index options. Bollerslev et al. [2007] estimated the VRP using the realized
volatility of the S&P500 and the VIX index, the model-free integrated implied volatility
index on the S&P500. They assumed the VRP can be explained by macro economic
variables and lagged VRP, and showed that the VRP was negative for option buyers andit well predicted the future returns of the S&P500.
These non-parametric estimation of the VRP owes to the theory and computation
method of the model-free integrated implied volatility and the realized volatility. Neuberger [1990], Dupire [1992], Demeterfi et al. [1999] and Britten-Jones and Neuberger[2000] generated and developed the theory of the model-free implied volatility, and Jiang
and Tian [2007] developed its computation methodology. The VIX index, traded on the
Chicago Board Options Exchange, is based on the theory. Sugihara [2008 to appear] summarised the theory and computation method, and showed the empirical characteristics of2

the model-free integrated implied volatility on the Nikkei 225 Japanese Stock Index, hereafter Nikkei. The availability of high-frequency intraday prices data led to the developmentof the estimation methodology of the realized volatility. Andersen and Bollerslev [1997,
1998] proposed the concept of realized volatility and showed the realized volatility was atheoretically reliable estimator of volatility. McAleer and Mederios [2008] or Shibata [2008]
surveyed recent developments of the realized volatility.
The non-parametric estimation for the VRP is much simpler and easier to implement
than the estimation based on earlier works under the SV model. However, a filtration
inconsistency problem arises in the non-parametric estimation. This problem results fromthe difference of the filtration required for the computation of the two volatility measures.
At a given day, the realized volatility computed from past market prices is an ex post
volatility, whereas the model-free integrated implied volatility on the day is an expected
future volatility. If we try to match their integration periods under the same filtration tocompute an expected VRP instead of a realized VRP, we have to work out future realized
volatilities which cannot be observed at the day.
In order to determine the expected VRP, we propose a new and simple method which
addresses this filtration inconsistency problem. In our method, the expected VRP is identified as the gap between the model-free integrated implied volatility estimated at a day t
and the integrated volatility estimated by a time series model of realized volatilities using
data up to the day t. We calculate the premium under the condition that the realized
volatility obeys either the Heston model or the ARFIMAX model, and obtain different
results from earlier works. The estimated risk premium by our method shows stickier
developments than those by earlier works. It also shows co-movement with risk indices,
proxies of investors’ risk aversion. As a byproduct, we can compute the probability where
the integrated implied volatility stays higher than the integrated realized volatility. This
probability might indicate the risk aversion of the option market.
Note that there are some viewpoints of the VRP in earlier works. According to Lee[2001], those differences mainly came from two matters. One is whether the VRP is basedon the correlated stochastic factor between stock prices and volatilities or uncorrelated
one, and the other is whether the VRP measures risk premiums for derivative sellers or
buyers. In this paper, we assume the VRP reflects the correlation, and is captured from
option buyers’ standpoint. Hence, the sign of our VRP is opposite of VRPs based on option
sellers’ standpoint in some earlier works.
The rest of this paper is organised as follows. Section 2 defines the integrated implied
and realized volatilities. Section 3 describes the four measures of VRP and discusses
the filtration inconsistency problem arisen from the difference in those VRPs. Section
4 formulates our proposing measure of an expected VRP with its calculation procedure.
Section 5 shows the empirical results of the expected VRP and discusses the difference of
conventional VRPs and the expected VRP. Lastly, Section 6 concludes the paper.
3

2

Integrated Implied and Realized Volatilities

Suppose an asset price at time t, St , moves stochastically with i.i.d. instantaneous returns.
The integrated quadratic variation at time t until time T (T > t) is defined as:
1
hSit,T =
T −t

Z Tµ
t

dSu
S u−

¶2
,

(1)

where T − t is an annualizing conversion coefficient and St− is the stock price just before
time t. The expected value of the future quadratic valuation can be estimated by options
prices, and the realized value of the past quadratic variation can be estimated by theunderlying asset processes. The former estimator is called the integrated implied variance,
hereafter IIV, which is an expected variance under the risk neutral measure Q,
2
σIIV(t, T ) = E Q [hSit,T |Ft ],(2)

where Ft is the filtration generated by the asset prices process until time t. The latter iscalled the integrated realized variance, hereafter IRV, which is a consistent estimator of
the realized variance under the physical measure P,
2
σIRV(t, T ) = E P [hSit,T |FT ] = hSit,T .

(3)Note that the IRV is a time-T value, not a time-t value.
The IIV is estimated from out-of-the-money European put and call options prices acrossall strikes. Demeterfi et al. [1999] and Britten-Jones and Neuberger [2000] derived the IIV
by option prices as:22
σIIV(t, T ) =(T − t)B(t, T )ÃZF (t,T )0

P (t, T, K)
dK +
K2

Z ∞

C(t, T, K)
dK
K2
F (t,T )!
,(4)

where B(t, T ), F (t, T ), P (t, T, K) and C(t, T, K) denote a risk free discounted bond price,a forward rate, a put and a call option price at time t with a maturity T and a strike price
K, respectively. The IIV represents the aggregated expectation of the future volatility inthe options market. The volatility index on the S&P500, the VIX index, traded on the
Chicago Board Options Exchange, is based on a discrete formula of eq.(4).
The IRV is estimated from high-frequency ex post intraday stock returns by discretizing
the integral in eq.(1). Assume we have M days from time t to time T and let {ti }M
i=1 denotes
set of time of days between time t to time T where t0 = t and tM = T , the IRV can be
estimated as:

M

2
σIRV
(t, T ) =1 X
RVti ,
T −t
i=1

4

(5)

where RVti is the one-day realized variance (RV) on day i. Further assuming there are N
i.i.d. intraday returns on each day i, {rj,i }N
j=1 , we can estimate RVti as:
RVti =

N
X

2
rj,i
.

(6)

j=1

Theoretically, when N is large enough, the right hand side of eq.(5) converges in probability to eq.(1), i.e.:
2plim σIRV(t, T ) = hSit,T .

(7)

N →∞

However, many earlier studies showed that, when the data frequency, or N , is too large,eq.(5) tends to overestimate the IRV because of an increase in market microstructure noises
such as the price reciprocal development between the bid and offer prices within a shortperiod, so-called “bid ask bounce.”3

Volatility Risk Premium

3.1

Definitions and filtration inconsistency problem

Using the IIV and IRV defined in section 2, the VRP is considered to be the gap between
them, i.e.,
VRP = IRV − IIV.

(8)

The VRP is equivalent with the covariance between pricing kernel of volatility and thequadratic variation. Let ξt denotes the Radon-Nikodým derivative process as
¯ ¸
dQ ¯¯
ξt = E
Ft ,
dP ¯

(9)

Mt,T = ξT /ξt .

(10)

·

and let Mt,T the pricing kernel as

Note that E[Mt,T |Ft ] = 1. Then the IIV and IRV have the following relationship.
2
σIIV(t, T ) = E Q [hSit,T |Ft ]= E P [Mt,T hSit,T |Ft ]= E P [hSit,T |Ft ] + CovP (Mt,T , hSit,T |Ft )2= σ̃IRV(t, T ) − λ̃(t, T ),(11)2
where σ̃IRV(t, T ) = E P [hSit,T |Ft ] and λ̃(t, T ) = −CovP (Mt,T , hSit,T |Ft ) denotes the VRP

at time t integrated from time t to time T . Earlier works showed the VRP is related to5the representative investor’s risk aversion. See Appendix A.5 for the details. In eq.(11),
we add a tilde ˜· on the IRV to emphasise filtration difference from our previous defined22IRV; σIRV(t, T ) defined in eq.(3) is FT -measurable, whereas σ̃IRV(t, T ) in eq.(11) is Ft 2
(t, T ), and hence λ̃(t, T ), cannot be computed from historical market
measurable. σ̃IRV

data.
Alternatively, we define a realized VRP, λ(t, T ), hereafter rVRP, using FT -measurable
variances as:
λ(t, T ) = E P [hSit,T |FT ] − E Q [hSit,T |FT ]22= σIRV(t, T ) − σIIV(t, T ),

(12)

which can be computed from historical market data any time after T . As we show in
Appendix A.4, the rVRP is related to the delta-hedged gain of options. Though we canconduct ex post calculation of eq.(12), the rVRP is still not a market expected VRP attime t.
In order to estimate the market expected VRP, we have to resolve the filtration inconsistency problem which arises from the difference in the measurement periods of IIV and
IRV. Because this problem stems from the prediction of the future physical measure, we
introduce a model to predict the future IRV based on the past time series of the RV. Thisapproach gives a new VRP, namely an expected VRP, or eVRP.
In the estimation for the eVRP, we model RV process rather than IRV process to avoid
the overlapping problem proposed by Christensen et al. [2002], that is the adjacent IRVs
share the same RVs for integration. We simply assume that the model that best fit to thepast RV’s time series is the best estimate for the future physical measure. Let Gt,T denotes
the model-based estimation of the future RV under the physical measure from time t totime T . The eVRP, λ(t, T ), is defined under the filtration Ft ∨ Gt,T as:
2
λ(t, T ) = E P [hSit,T |Ft ∨ Gt,T ] − σIIV(t, T ),(13)λ(t, T ) = −CovP (Mt,T , hSit,T |Ft ∨ Gt,T ).

(14)

or

The concept of the eVRP is displayed in Figure 1.
As a byproduct, the estimation procedure of the eVRP gives the probability where the22 (t, T ). We define the degree of volatility risk
forecasted σIRV(t, T ) is lower than the σIIV

aversion for option sellers: Γ as222
Γ(t, T ) = Prob(σIRV(t, T ) ≤ σIIV(t, T )) = FIRV (σIIV(t, T )),

(15)

where FIRV is the cumulative density function of the model expected IRV. The VRA is

6

given by the shaded area in the Figure 1. It is close to 100% when option sellers are veryrisk averse and require a large premium. Note that, since both option sellers and buyers
are risk averse investors, Γ shows an imbalance of risk aversion between option sellers andbuyers.
Figure 1: The concept of eVRP and VRAMVRP = the gap between
the quantile and the mean

Integrationof RVsPast RV process
2
E t [σ IRV(t , T )]Forecast distribution ofsquare root of IRV
projected on maturity

IIV curve at time t
Expansion of RV distribution
estimated from the past RVprocess

Present

Maturity(day t)(day T)Γ(t,T) = the shaded area

Quantile = σ2IIV(t , T )The simplest estimation of Gt,T is the linear estimation; simply set Gt,T = σ({Su }tu=t−(T −t) )using lagged samples of the daily RV. We call the IRV the trailed IRV denoted as σ 2IRV (t, T )and defined as:
σ 2IRV (t, T ) = hSit−(T −t),t ,(16)

and a trailed VRP, or tVRP, is defined as:2
λ(t, T ) = σ 2IRV (t, T ) − σIIV(t, T ).

(17)

The tVRP is an expected VRP under the assumption that the IRV in the previous monthis the best prediction of the IRV in the following month.
So far, we have introduced four types of VRPs. The definitions and the differences of
VRPs are summarised in Table 1.
Note that the filtration inconsistency problem has not been recognised in earlier studies
that uses parametric estimation, because most of those discussed on the instantaneous VRP.

7

Table 1: The definition, notation and difference of VRPs
term

notation

explanation

true VRPλ̃(t, T )True expected VRP, unobservable at any time.

realized VRP (rVRP)

λ(t, T )Ex-post VRP, related to delta-hedged gain of
options, observable after time T .

trailed VRP (tVRP)

λ(t, T )Expected VRP under static expectation assumption on future IRV, observable at time t.

expected VRP (eVRP)

λ(t, T )Our proposed expected VRP using model-baseforcast of IRV, observable at time t.

Since λ(t, T ) = T 1−t
limT →t λ(t, T ) =RTt λ(t,¯s)ds, the instantaneous VRP, λ(t, t), is defined as λ(t, t) =∂¯∂T λ(t, T ) T =t for any VRPs, and those converges to the same value

whichever VRPs we use, i.e.:λ̃(t, t) = λ(t, t) = λ(t, t) = λ(t, t).

(18)

See Appendix A.3 for the instantaneous VRP defined under the SV model.

3.2

Volatility risk premiums in the options market

Figure 2 shows the movements of the one-month IIV computed using option prices with
one-month maturity and the one-month IRV on Nikkei from April 2003 to September 2007.
They are plotted in the volatility dimension, i.e., the square root of the integrated variancein eq.(4) and eq.(5). Two types of IRV are shown: future one-month IRV and past onemonth IRV. The former covers the same period as the IIV, whereas the latter uses lagged
samples of the daily RV. In other words, the future IRV is σIRV (t, t + 1month), and thepast IRV is σIRV (t − 1month, t).
The IRVs and IIV normally move parallel to each other, and the level of the IIV isgreater than the IRVs in most of the periods. The gap is corresponds to the VRP 1 . We
consider the gap between the future IRV and the IIV to be the rVRP, and that between
the past IRV and the IIV to be the tVRP. Since IIV moves over IRV in most periods, thesigns on the VRPs are negative.
The negative sign on the VRPs indicates that option sellers require a premium for
uncertainty of the future volatility, or the volatility risk. We find that the past IRV seems
to move more consistently with the IIV rather than the future IRV, and the future IRV
sometimes exceeds the IIV. These observations lead to the following hypotheses: (a) option
sellers require a premium by observing past IRV to compensate for the future volatility
1

These values are computed in the volatility dimension and may be little smaller than the square root of
the VRP. We neglect those difference in the discussion here.

8Figure 2: Time series of IRV and IIV on Nikkei from April 2003 to September 2007
volatility [%]60

50
Future IRV
Past IRV
IIV

40

30

20

10

0
03/4

03/8 03/12 04/4

04/8 04/12 05/4

05/8 05/12 06/4

06/8 06/12 07/4

07/8
YY/M

Notes: The values are one-month term to maturity in the volatility dimension. The IIV iscalculated using the method proposed by Jiang and Tian [2007].

risk, and option buyers are willing to pay the premium, and (b) the premium covers thevolatility risk for option sellers in most periods, but sometimes the premium is undervalued.
These results show the risk aversion of the financial markets, and reveal how the market
participants form their expectation. The hypothesis (a) supports our assumption on thecalculation of the eVRP; the expectation on the future RV seems to be based on the pastdevelopment of RV. We guess the model-base eVRP is a better estimate of expected VRP
than the tVRP that based on a static expectation on the future RV.
Under SV model, the risk premium on a derivative is composed of those on the stochastic
factors in the process of underlying asset and the volatility, as we discussed in eq.(A-17)
in Appendix A.2. Decomposition of the risk premium gives(Risk premium on a derivative) = w1 ·(Risk premium on the underlying)+w2 λ(t, t), (19)where w1 , w2 are weights which are proportional to the delta and the vega of the derivative
respectively. Since the vega is positive for European options, the negative VRP reduces
the risk premium on an option itself.

9

4

Variation Method of Expected Volatility Risk Premium

In this section, We show two models to forecast future RV process, which is used in theestimation of the eVRP, and describe the calculation procedures of the eVRP and Γ definedin Section 3.1.

4.1

Two Models for RV forecast

The first model is the Heston type model defined as:p

RVt St dWt1 ,dSt = µSt dt +d(RVt ) = κ(θ − RVt )dt + σV³´p
RVt ρdWt1 + 1 − ρ2 dWt2 ,

p

(20)

where dWt1 , dWt2 are independent Weiner processes and κ, θ, σV , ρ (σV > 0, |ρ| ≤ 1) areparameters. The original model of eq.(20) was proposed by Heston [1993], which is thestochastic volatility model with mean-reverting square root volatility process. This model
substitutes the volatility process in the original Heston model for RV. So, we call thismodel the Heston type model for RV hereafter. The Heston model is commonly usedamong practitioners, and this Heston type model for RV may well represent forcastingmodels which major market participants use.
The second model for RV forecast is the ARFIMAX model proposed by Giot andLaurent [2004] as:
Rt =p
RVt zt ,(1 − L)d {ln(RVt ) − µ0 − µ1 |Rt−1 | − µ2 1{Rt−1 <0} |Rt−1 |} = (1 + δL)ut ,zt v N (0, σz2 ),(21)

ut v N (0, σu2 ),where L is a lag operator2 , Rt = ln(St /St−1 ) is the one-day return on day t, and µ0 , µ1 , µ2 , δ, σz
and σu (σz , σu > 0) are parameters. According to Watanabe and Sasaki [2007] and Shibata [2008], the ARFIMAX model provides a good fit of the past movement of the RV for
Nikkei.
Note that both models have mean reverting property of RV. The Heston type model has
fixed and deterministic mean of RV, whereas the ARFIMAX model has stochastic mean
which changes based on the previous day’s return. The ARFIMAX model expressed thevolatility movement’s asymmetric feature, that is the volatility is normally higher when theprevious day’s return was negative than when positive. This feature is expressed by theµ3 parameter and 1{Rt−1 <0} in eq.(21) which takes non-zero value only when the previous
day’s return was negative. The “X” in ARFIMAX is added to address this asymmetry. We
assume the mean reverting property of RV in those models better expresses the market’s2

Lxt = xt−1 .

10expectation for the future RV than the simple static expectation assumption in tVRP.

4.2

The calculation procedure

The eVRP and Γ can be calculated by the following procedure.
First, estimate the IIV in eq.(4) from the option prices data, following the methodology
proposed by Jiang and Tian [2007]. The one- to six-month IIVs are estimated by interpolating market-traded terms of IIV using a cubic spline function. For the risk free discountbond price B(t, T ), the interpolated rate of LIBOR and swap rates is used.
Second, compute the RV from intraday stock price data by eq.(6). The appropriate
choice of N , or the choice of a time interval to measure returns, is an ongoing issue for RV
estimation. Because five-minute returns are widely used in earlier studies including thoseon Nikkei, we follow this choice of N . Note that the return during lunch break and thereturn from closing to the next morning’s opening are included in the daily RV. The firstsample, j = 1 in eq.(6), corresponds to the nighttime return from the last day’s closing to
the next morning’s opening.
Third, estimate the parameters of the models for the RV. The parameters are estimatedat each day ti by the quasi-maximum likelihood method using the previous one-year of ex
post RV data including the day t0 itself 3 . Since estimation of IIV and IRV in Section
3.2 suggests that market volatility expectation relies on the recent development of RV, theparameters are updated daily for the better tracking of the development.
Forth, simulate RV process from the day t0 to day tM using the estimated parameters,
and set the estimated future physical measure Gt,T using the simulated RV process. Thencompute the model expected value of future IRV under the filtration Ft ∨ Gt,T by ten
thousand times of Monte Carlo simulation each day.
Lastly, compute the eVRP, λ(t, T ), and the Γ(t, T ) from the IIV on day t0 and thesimulated IRV. The Monte Carlo simulation is used for the calculation of Γ(t, T ). We
repeat the above process for each day to get the time series of eVRP and Γ.

5

Empirical Analysis

5.1

Volatility risk premium

5.1.1

Differences in three VRPs: eVRP, tVRP and rVRP

The upper and middle panels in Figure 3 display the eVRPs on Nikkei under the Hestonand ARFIMAX models respectively. The VRPs from one, three and six months term tomaturity are shown in square percent point scale. The lower panel shows one month rVRP
3

The reason for the inclusion of the day t0 is that the options market for the Nikkei closes at 15:10 whereas
the stock market closes at 15:00. By comparing the closing IIV with the IRV, which are computed withpast data including the current day, an unbiased VRP can be computed.

11

and tVRP for comparison. Comparing the eVRPs with tVRP and rVRP, we observe thefollowing differences.
First, we focus the difference of RV forecast models. The eVRPs based on the Heston
model and that on ARFIMAX model move parallel to each other. The directions of thosechanges are almost the same, but the levels differ occasionally.
Second, the eVRPs are less volatile and stickier than the tVRP and rVRP. The eVRPs
suggest that the risk premium may primarily fluctuate with inertia. The eVRPs exhibits
the cyclical development of the risk premium, compared with the tVRP and rVRP, andthe eVRP also seems to detect the turning points of the long-term risk premium trends
which are hard to be detected by the tVRP or rVRP.
Third, the eVRPs stay stably around zero while the realized volatility stays low. Thatmay indicate the bargaining power balances between options buyers and sellers while thevolatility stays low. In periods from October 2004 to July 2005 and from December 2006
to June 2007 when the IRV stayed below 15% as seen in Figure 2, the eVRPs stayed stably
around zero. In the same periods, however, the tVRP moved above zero and the rVRP
fluctuates. The tVRP and rVRP did not detect the bargaining power balance.
Forth, comparing eVRPs with different terms, we see the shorter term eVRPs are morevolatile than the longer term eVRPs. This indicates that the shorter term VRP is moresensitive to changes in market sentiments. This is consistent with the characteristic of theimplied volatility term structure showed in Sugihara [2008 to appear].
5.1.2

Consistency with other market risk indicators

In order to determine whether the eVRP is better estimate of the risk premium than thetVRP and rVRP, we compare those VRPs with market risk indicators to be regarded as aproxy of risk preference. The VRPs are expected to move consistent with the risk indicators
derived from the other markets, because investors in the options market normally invest
in other asset classes such as government bonds or credit default swaps, and their risk
aversion in the other market affects the options market.
The Citi Macro Risk Index, hereafter CMRI 4 , and iTraxx Japan 5 are chosen for therisk indicators. The former measures global investors’ overall risk aversion. Because morethan half of investment into the Japanese stock market comes from overseas, the VRP in
Nikkei is expected to be consistent with the global investors’ risk preference such as CMRI.
The iTraxx Japan measures credit risk in the referenced Japanese enterprises. It reflects
4

The Citi Macro Risk Index is a equally weighted index of emerging market sovereign spreads, US credit
spreads, US swap spreads and implied FX, equity and swap rate volatilities. The index is expressed ina rolling historical percentile and ranges between 0 (low risk aversion) to 1 (high risk aversion). Data
source is Citibank Ltd.
5
The iTraxx Japan refers 50 most liquid investment grade credit default swap premiums on Japaneseentities. The indication is arithmetic average of index CDS premiums collected from broker dealers inTokyo. It starts to be computed from July 2004. Data source is Markit Group Ltd.

12

Figure 3: eVRP, tVRP and rVRP
(%2)
-2000

(a) eVRP (Heston)
6m

-1500

3m
-1000

1m

-500
0
500
03/4
-2000

03/8 03/12 04/4

04/8 04/12 05/4

05/8 05/12 06/4

06/8 06/12 07/4

07/8

06/8 06/12 07/4

07/8(b) eVRP (ARFIMAX)
6m

-1500

3m
-1000

1m

-500
0
500
03/4
-2000

03/8 03/12 04/4

04/8 04/12 05/4

05/8 05/12 06/4

(c) tVRP and rVRP (one month)

-1500
rVRP

tVRP

-1000
-500
0
500
03/4

03/8 03/12 04/4

04/8 04/12 05/4

05/8 05/12 06/4

06/8 06/12 07/4

07/8
(YY/M)

Notes: (a) The eVRP on Nikkei where RV obeys the Heston model, (b) the eVRP on Nikkei
where RV obeys the ARFIMAX model, (c) the rVRP and tVRP. One-, three-, sixmonths-terms of eVRPs, and one-month-term rVRP and tVRP are shown. The
ordinates are inverted to show the negative risk premium.

13

the downside risk aversion of the Japanese credit markets. Since the risk preference in thecredit market is generally shared with the stock market, the VRP is expected to have apositive correlation with iTraxx Japan.
Table 2 displays the correlation between the VRPs and the risk indicators. Every VRP
has a negative correlation with risk indicators and the eVRPs show stronger correlation
than the rVRP and tVRP. The negative correlation indicates when the option sellers’ risk
aversion agrees with overall investors’ risk aversion. And those stronger correlations resultsfrom coherent cycle of eVRPs and the risk indicator. Figure 4 displays the time series of
eVRPs, tVRP and risk indicators. The upper panel displays the CMRI and VRPs, andthe lower panel displays the iTraxx Japan and VRPs. During the periods from August to
December 2004 and from July 2006 to January 2007, the eVRPs co-move well with thedecrease in the risk indicators. The eVRP goes down in the periods while the tVRP almost
unchanges. The eVRP also detects the rise of the risk aversion better than the tVRP in
the period from May 2005 to February 2006. The same feature also holds when compared
iTraxx Japan. In the periods from October to November 2004, from January to July 2006,
from July to December 2006 and from April to June 2007, the eVRP tracks uprise anddownturn of the risk aversion while the tVRP almost unchanges. These results suggest theeVRP is the better indicator of the risk aversion in the stock market rather than the tVRP
or rVRP.
Table 2: Correlation between the VRPs and risk indicators

eVRP (Heston)eVRP (ARFIMAX)
tVRP
rVRP

CMRI
−0.5935−0.5680−0.4008−0.2283

iTraxx Japan−0.5045−0.4424−0.3078−0.2057

Notes: The correlations with the CMRI are computed using data from April 2003 to September 2007, whereas those with iTraxx use data from July 2004 to September 2007.
VRPs with one month term to maturity are used.

5.2

Prediction for the future

We further investigate the prediction accuracy of the eVRP and tVRP for the realized
VRP, the future IRV and the future increase in IRV. We will look into the root meansquare error, or RMSE, for the prediction accuracy of VRP for the rVRP, the future IRV

14

Figure 4: Time series of eVRPs, tVRP and the risk indicators
VRP [ %2 ]
-1600
-1400
-1200
-1000
-800
-600
-400
-200
0
200
400
03/4

CMRI

eVRP (Heston)

03/10

04/4

04/10

tVRP

05/4

CMRI (right axis)

05/10

06/4

06/10

07/4

VRP [ %2 ]

1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
07/10
YY/M

iTraxx [bps]

-1600
-1400

45

eVRP (Heston)tVRP

iTraxx Japan (right axis)
40

-1200
-1000

35

-800
-600

30

-400
25

-200
0

20

200
400
04/7

15
05/1

05/7

06/1

06/7

07/1

07/7
YY/M

Notes: The upper and lower panel displays the eVRP, tVRP with the CMRI and the iTraxx
Japan respectively. Those are plotted weekly to show the direction of the move. For
the eVRP, the Heston model case is shown and for every VRP, one-month-term tomaturity VRP are shown. Note that the periods in horizontal axes differ between
upper and lower panels due to the data availability. The dotted circles indicate
the periods when the movements of eVRPs move more consistently with the risk
indicators.

15

Table 3: Prediction error of the eVRPs and tVRP for the rVRP measured by RMSE
defined in eqs.(22) to (24)eVRP(Heston)eVRP(ARFIMAX)tVRPrealized VRP
0.43940.55900.3464future IRV
0.75480.79770.6015future increase in IRV
0.81380.95750.7411and the increase in IRV, defined respectively as:
v
u
m
u1 X
¡
¢2
t
RMSE(realized VRP) =λ(ti , Ti ) − λ(ti , Ti ) ,
m

(22)

i=1

v
u
m
u1 X
¡
¢2
2
−λ(ti , Ti ) − σIRV(ti , Ti ) ,RMSE(future IRV) = t
m

(23)

i=1

v
u
m
u1 X
¢2
¡
2
2
(2ti−1 − Ti , ti ) , (24)
RMSE(future ∆IRV) = t(ti , Ti ) + σIRV−λ(ti , Ti ) − σIRVmi=1

where m is a number of observations. In eq.(23) and eq.(24), the sign of λ is minus
as we assume negative VRP predicts the future level or future increase in IRV. Notethat the prediction accuracy does not matter for the evaluation of expected VRP, since
unpredictable shocks in future volatility may diverse realized RV from forecasted RV.
Table 3 displays the RMSE of the eVRPs with one to six months term to maturity.
As seen, the eVRP with Heston model has a higher prediction accuracy than that with
ARFIMAX for all cases. Table 3 also displays the tVRP’s prediction accuracy. The tVRP
has a smaller error and more accurately predicts the rVRP than the eVRPs for each case.
Though those difference in error is relatively small, especially between eVRP in Heston’s
and tVRP, this result indicates that the market expectation of future volatility risk does
not always correctly predict the future, and the simple linear estimation gives the better
prediction. Since the tVRP simply extends the present level of RV for the future andneglect the mean reverting property of RV process, the tVRP fluctuates largely enoughto cover the wide range of realized RV (see Figure 5). On the other hand, the forecasted
RV stays around middle range of realized one in Figure 5, which may cause the worse
performance in RMSE.
Now, we face a question that why the eVRPs show stronger correlation with risk indicators in spite of their low prediction power. Possible hypotheses are the following: (a)the formation of the market expectation cannot be measured only by ex post performanceof prediction. Because there are some future unpredictable shocks, the prediction error of
16

Figure 5: Realized IRV and forecasted IRV: comparison between the Heston model forecasts
and static forecasts in the tVRP

800
linear approximation of
static forecast

Heston Model forecast
static forecast2

R = 0.4386

2

forecasted IRV [% ]

600

400

200

linear approximation ofmodel forecast

R2 = 0.1323
0
0

100

200

300
400
realized IRV [%2]

500

600

700

800

the expected VRP become high. (b) The mean reverting property of RV models in theeVRP might explain the formation of the market expectation better than the static expectation assumption on the tVRP, and (c) even though the models has better prediction
accuracy for one day ahead, the prediction accuracy up to maturity become worsen due tothe accumulation of uncertainty in the model on each day up to maturity.

5.3

Volatility risk aversion

In Section 4, we define another volatility risk aversion indicator: Γ(t, T ). Figure 6 displays
the estimation results of Γ. The upper and lower panels display the Γ computed under the
Heston model and the ARFIMAX model respectively.
First, low Γ period is much shorter than high Γ period. This indicates that a risk
seeking market vanishes easier than a risk averse market and once the market goes intorisk averse the state tends to continue.
Second, the shorter-term-to-maturity Γ precedes the longer-term Γ when the Γ is low,
whereas the longer-term Γ precede the shorter-term Γ when the Γ is high. This indicatesthat the longer the term of the Γ the more risk sensitive the market.
Third, the ARFIMAX-Γ moves more drastically than Heston-Γ. That indicates the the17distribution of model projected IRV is narrower in ARFIMAX than that in Heston.

6

Conclusion

This paper addresses the filtration inconsistency problem regarding the estimation of expected volatility risk premium, and proposes a new evaluation method which resolves the
problem. We find that the volatility risk premium computed by our method has stronger
correlation with risk indicators for general market risk preferences than those computedby the conventional methods. This suggests that our measure is a better indicator of markets’ aversion of volatility risk. We also propose a new measure of markets’ volatility risk
aversion.
Our method depends on RV models to describe how market participants form their
expectation on the future volatility. We do not directly examine the relevance of the RV
models due to difficulty in obtaining the expectation. This issue is remained for our futurechallenge.
Although our volatility risk measures are assessed to be the better risk aversion indicator, we also find that the tVRP, the VRP computed under the simple linear assumption of
RV process, has more accurately predicts the future than the eVRP. Those results indicatesthat even the model-base expectation for future risk has difficulty in predicting the future
correctly, and much simpler linear estimation provides a better prediction accuracy.

References
Andersen, T.G. and T. Bollerslev, “Intraday Periodicity and Volatility Persistence in Financial Markets,” Journal of Empirical Finance, 4, pp. 115–158, 1997.
“Answering the Skeptics: Yes, Standard Volatility Models Do Provide Accurate
Forecasts,” International Economic Review, 39, pp. 885–905, 1998.
Bakshi, G. and N. Kapadia, “Delta-Hedged Gains and the Negative Market Volatility Risk
Premium,” The Review of Financial Studies, 16(2), pp. 527–566, 2003.
Bollerslev, T., M. Gibson, and H. Zhou, “Dynamic Estimation of Volatility Risk Premia
and Investor Risk Aversion from Option-Implied and Realized Volatilities,” Finance andEconomics Discussion Series, 2007.
Britten-Jones, M. and A. Neuberger, “Option Prices, Implied Price Processes and Stochastic Volatility,” Journal of Finance, 55, pp. 839–866, 2000.
Carr, P. and L. Wu, “Variance Risk Premia,” Working Paper, 2005.

18

Figure 6: Time series of Γ(a) Heston
100%
90%
80%
70%
60%
50%

1M

40%

3M

30%

6M

20%
10%
0%

(b) ARFIMAX
100%
90%
80%
70%
60%
50%
40%
30%
20%
10%
0%
03/4

1M
3M
6M

03/10

04/4

04/10

05/4

05/10

06/4

06/10

07/4

07/10
YY/M

Notes: (a) displays the Γ with Heston model and (b) displays Γ with the ARFIMAX model.
One, three, six months term of Γ are shown.

19Christensen, B.J., C.S. Hansen, and N.R. Prabhala, “The Telescoping Overlapping Problem
in Options Data,” Working Paper, University of Maryland, 2002.
Demeterfi, K., E. Derman, M. Kamal, and J. Zou, “More Than You Ever Wanted To
Know About Volatility Swaps,” Quantitative Strategies Research Notes, Goldman Sachs,
March, 1999.
Derman, E., Volatility Smile Lecture Notes, Master of Financial Engineering Program,
Columbia University, Lecture 10, pp.10–14, 2007.
Dupire, B., “Arbitrage Pricing with Stochastic Volatility,” Societe Generale, 1992.
Giot, P. and S. Laurent, “Modelling Daily Value-at-Risk Using Realized Volatility and
ARCH Type Models,” Journal of Empirical Finance, 11(3), pp. 379–398, 2004.
Heston, S., “A Closed Form Solutiob for Options with Stochastic Volatility with Applications to Bond and Currency Options,” The Review of Financial Studies, 6, pp. 327–343,
1993.
Hull, J. and A. White, “The Pricing of Options on Assets with Stochastic Volatilities,”
Journal of Finance, 42(2), pp. 281–300, 1987.
Jiang, G.J. and Y.S. Tian, “Extracting Model-Free Volatility from Option Prices: An
Examination of the VIX Index,” Journal of Derivatives, Spring, pp. 35–60, 2007.
Lee, R.W., “Implied and Local Volatility under Stochastic Volatility,” International Journal of Theoretical and Applied Finance, 4(1), pp. 45–89, 2001.
McAleer, M. and M.C. Mederios, “Realized Volatility: A Review,” Econometric Review,27(1-3), pp. 10–45, 2008.
Neuberger, A., “Volatility Trading,” Working Paper, London Business School, 1990.
Scott, L.O., “Option Pricing when the Variance Changes Randomly: Theory, Estimation,
and an Application,” Journal of Financial and Quantitative Analysis, 22(4), pp. 419–438,
1987.
Shibata, M., “Estimation for Volatility using Intraday Market Data: Survey for Realized
Volatility and Application for Japanese Stock Index and Stock Index Futures,” Kin’yu
kenkyu (Monetary and Economic Studies), 27(1), pp. 1–54, 2008. in Japanese.
Sugihara, Y., “Model Free Implied Volatility in Nikkei Index Options Market,” IMES
Discussion Paper Series, Institute for Monetary Economic Studies, Bank of Japan, 2008
to appear.

20

Uchida, Y. and K. Miyazaki, “Volatility Risk Premium in Nikkei 225 Index Options Market,” Gendai Finance, 23, pp. 35–59, 2008. in Japanese.
Watanabe, T. and K. Sasaki, “The Volatility Estimation by an ARCH howpublished Modeland the Realized Volatility, and the Value at Risk,” Kin’yu kenkyu (Monetary and Economic Studies), pp. 39–74, 2007. in Japanese.
Wiggins, J.B., “Option Values under Stochastic Volatilities,” Journal of Financial Economics, 19, pp. 351–372, 1987.
“The Pricing of Foreign Currency Options with Stochastic Volatility,” Journal of
Econometrics, 45, pp. 239–265, 1990.

21

A

Volatility Risk Premium under Stochastic Volatility ModelIn this section, we consider the volatility risk premium using SV model. Many of earlier
works captured VRP under the SV model and we show how their works. We also show our
VRP is equivalent with the volatility risk premium under SV model.

A.1

Derivation of PDE of derivatives on SV modelFirst, we derive the partial differential equation of a derivative on an asset under SV model,
following Scott [1987]. Suppose an asset St follows a general SV model,(S)dSt = µSt dt + σt St dWt,(σ)

dσt2 = α(St , σt2 , t)dt + β(St , σt2 , t)dWt ,(S)(σ)
dWt dWt(S)

where dWt(σ)

and dWt(A-1)= ρt dt,

denote correlated Weiner processes, and µ, α, β and ρt denoteparameters. We assume the asset is not a dividend paying asset.
Considering two derivatives on the asset, which has the value f1 (St , σt2 , t) and f2 (St , σt2 , t).
Both depend on the asset price St and volatility σt , but has different parameters.
Suppose an investor holding one of the derivatives, who hedge off the risk of the volatility
and the volatility’s volatility by the asset and the other derivative. The investor longs one
contract of the derivative f1 and shorts δ1 of the underlying asset and δ2 contracts of thederivative f2 . The value of the portfolio Πt is
Πt = f1 (St , σt2 , t) − δ1 St − δ2 f2 (St , σt2 , t).

(A-2)

By Ito’s lemma, the growth of the portfolio value become
µ

dΠt

¶
∂f1
1 2 ∂ 2 f1
∂f1
∂f1 2 1 2 2 ∂ 2 f1
∂ 2 f1=dt +dSt +dσ + σt St
dt + βdt + σt St βρt 2
dt
∂t
∂St
2
2 (∂σt2 )2∂σt2 t∂St2∂σt ∂St−δ1 dStµ¶∂f2∂f2 2 1 2 2 ∂ 2 f2
∂ 2 f2∂f2
1 2 ∂ 2 f2
dSt +dt + σt St βρt 2
dt
−δ2
dt +
dσ + σt St
dt + β∂t∂St22 (∂σt2 )2∂σt2 t∂St2∂σt ∂St·µ¶∂f1 1 2 2 ∂ 2 f1 1 2 ∂ 2 f1
∂ 2 f1=+ σ t St+ β
+ σt St βρt 2∂t22 (∂σt2 )2∂St2∂σt ∂Stµ¶¸2
2∂f2 1 2 2 ∂ f2 1 2 ∂ f2
∂ 2 f2− δ2+ σ t Stdt
+ β
+ σt St βρt 2∂t22 (∂σt2 )2∂St2∂σt ∂St
¶
µ
¶
µ
∂f1
∂f2
∂f2
∂f1
− δ2
− δ1 dSt +− δ2 2 dσt2 .
(A-3)
+
2
∂St
∂St
∂σt
∂σt
In order to hedge the risk in the dSt and dσt , the investor should chooses δ1 and δ2 so22

as to the last two terms in eq.(A-3) become zero, i.e.,
∂f1
∂f2
− δ2
− δ1 = 0,
∂St
∂St

∂f1
∂f2
− δ2 2 = 0.
∂σt2
∂σt

(A-4)

then, the δ1 and δ2 are computed as
δ1 =

∂f1
∂f2 ∂f1 ∂f2−/,∂St ∂St ∂σt2 ∂σt2

δ2 =∂f1 ∂f2/.
∂σt2 ∂σt2(A-5)

This makes the portfolio value Πt riskless. Since the return of the risk free portfolio
should be equal to the risk free rate,dΠt = rt Πt dt,(A-6)

where rt is the risk free rate. Applying eq.(A-2), eq.(A-3) and eq.(A-5) to eq.(A-6) yieldsµ¶∂f1∂f2 ∂f1 ∂f2∂f1 ∂f2
rt f1 −−
/ 2 rt St −
/
rt f2
2
∂St ∂St ∂σt ∂σt∂σt2 ∂σt2µ¶∂ 2 f1∂f1 1 2 2 ∂ 2 f1 1 2 ∂ 2 f1
+ σ t St
+
β
+
σ
S
βρ
=
t t
t
∂t
2
2 (∂σt2 )2∂St2∂σt2 ∂Stµ¶∂f2 1 2 2 ∂ 2 f2 1 2 ∂ 2 f2
∂ 2 f2∂f1 ∂f2+ σ t St+ β
+ σt St βρt 2− 2/ 2 ·. (A-7)∂t
2
2 (∂σt2 )2∂σt ∂σt∂St2∂σt ∂St
Defining an operator V as
µ
V=

∂
∂σt2

¶−1 Ã

∂
∂
1
∂2
1
+ rt S t+ σt2 St2 2 + β 2
∂t
∂St 2
2
∂St

µ

∂
∂σt2

¶2

!
∂2
+ σt St βρt 2− rt ,∂σt ∂St(A-8)

eq.(A-7) can be rewritten as
Vf1 = Vf2

(A-9)

Since the left and right hand side of eq.(A-9) are function of derivative f1 and f2 respectively, the value should be completely independent from the variables of the derivative,
such as strike price or maturity. We can write the value as Vf = −ϕ(St , σt2 , t).
Then any derivative f on the asset S satisfy the following PDE.
∂f
∂f
∂2f
∂2f
∂2f
∂f
1
1
+ rt S t+ ϕ 2 + σt2 St2 2 + β 2= rt f.
+
σ
S
βρ
t
t
t
∂t
∂St
2
2 (∂σt2 )2∂σt∂St∂σt2 ∂St(A-10)

That is the PDE for an derivative value with stochastic volatility, hereafter SV-PDE, which
considered to be the SV version of the Black-Scholes-Merton’s PDE.

23

A.2

Definition of VRP

The VRP is defined under SV model with same analogy with the market price of risk.
We follow the Derman [2007] in the following discussion. The change of the value of anderivative f can be written by Ito’s lemma as:
·
df

=

¸
∂f
∂f
1 2 2 ∂2f
1 2 ∂2f
∂2f
∂f
+ β
dt
+ µSt
+ α 2 + σ t St
+ St σt βρt
∂t
∂St
2
∂σt
∂St2 2 (∂σt2 )2∂St ∂σt2
∂f
∂f
(S)
(σ)
+σt StdWt + β 2 dWt .
(A-11)
∂St
∂σt
(S)

(σ)

Letting µf , σf , σf(S)denote f ’s drift, volatility for dWt(σ)

and for dWt

respectively,

we view the eq.(A-11) as the two dimensional geometric Brownian motion as
df
(S)
(S)
(σ)
(σ)
= µf dt + σf dWt + σf dWt ,f(A-12)where·¸
1 2 2 ∂2f
1 2 ∂2f
∂f
1 ∂f
∂f
∂2f
µf =
+ β
+ µSt
+ α 2 + σ t St
+ St σt βρ
,
f ∂t
∂St
2
∂σt
∂St2 2 (∂σt2 )2∂St ∂σt2(A-13)

St σt ∂f,
f ∂St

(A-14)β ∂f.
f ∂σt2(A-15)·¸∂f∂f1(µ − rt )St+ (α − ϕ) 2 + rt f .
f
∂St
∂σt

(A-16)

(S)

σf

=

(σ)

σf

=

Applying eq.(A-10) to eq.(A-13), we getµf =Eq.(A-16) can be written using eq.(A-14) and eq.(A-15) as:(S)µf − rt =

σf

σt

(σ)

(µ − rt ) +

σf

β
(S)

We consider µ − rt to be a risk premium of dWt(α − ϕ).

(A-17)or that on the asset price itself. By(σ)

the same logic, α − ϕ is considered to be a risk premium of dWt

or the volatility risk

premium. In consequence, ϕ is considered to be a kind of “risk free rate” of volatility. The
eq.(A-17) shows the risk premium on the derivative is the weighted sum of the risk premiumof the asset and the volatility. Eq.(A-17) is called “the market price of risk equation”.
We guess our VRP, λ, equals to the volatility risk premium instantaneously. Thatmeans λ(t, t) has the form
λ(t, t) = α − ϕ.
24

(A-18)

We have showed λ is negative in most periods. The overall risk premium in a derivativewith positive vega is smaller than the risk premium of the underlying asset itself, since theoverall risk premium of a derivative is composed of the risk premium of underlying asset
and its volatility as in eq.(A-17) and α − ϕ in eq.(A-17) is negative.
Further setting

r³

´
(S) 2

σf

σf =

³
´
(σ) 2
(S) (σ)+ σf+ 2ρt σf σf ,(A-19)

we get the Sharpe ratio version of eq.(A-17) as
(S)

(σ)

σ f µ − rt σ f α − ϕµf − rt
=
+
.
σf
σf
σt
σf
β

(A-20)The Sharpe ratio, or the market value of risk, of the derivative is weighted sum of those(S)of the underlying asset and its volatility. Since σf ≤ σf

(σ)

+ σf , the Sharpe ratio of thederivative is smaller than the weighted average of the Sharpe ratios of the asset and thevolatility unless the asset and the volatility is perfectly correlated.

A.3

Relation between α − ϕ and λThe risk neutral representation of the stochastic volatility model, eq.(A-2), changes to thefollowing:(S̃)

dSt = rt St dt + σt St dWt,(σ̃)

dσt2 = ϕ(St , σt2 , t)dt + β(St , σt2 , t)dWt ,
(S̃)
(σ̃)
dWt dWt(A-21)= ρt dt,

where
(S̃)

dWt

(σ̃)

dWt

µ − rt
dt,
σt
α−ϕ
(σ)
dt.
= dWt +
β
(S)

= dWt

+

(A-22)For simplicity, we assume α and β in eq.(A-2) and ϕ in eq.(A-22) are function of t andSt , and do not depend on σt2 from now. The integrated volatility under the risk neutral
measure Q can be computed asQ

E [hSit,T |Ft ] =
=

Z T
1
Q
σu2 du
E
T −t
t
¸
Z u
Z T·
1
2
ϕdv du,σt +T −t tt

25(A-23)

whereas the integrated volatility under the physical measure P isP

E [hSit,T |Ft ] =
=

Z T
1
P
E
σu2 du
T −t
t
¸
Z T·
Z u
1
2
σt +
αdv du.
T −t tt(A-24)

These difference (eq.(A-23)−eq.(A-24)) is considered to be the true VRP, λ̃(t, T ), as definedin eq.(11), i.e.:λ̃(t, T ) = E P [hSit,T |Ft ] − E Q [hSit,T |Ft ]Z TZ u1=(αv − ϕv )dvdu.
T −t t t(A-25)

Instantaneously,
λ(t, t) =
=
=
=

¯
¯
∂
λ̃(t, T )¯¯
∂T
T →t
·
¸
Z TZ u∂1lim(αv − ϕv )dvdu∂T T →t T − t t t
¯
Z u
¯
∂
(αv − ϕv )dv ¯¯∂u tu→tαt − ϕt ,(A-26)

as we guess in eq.(A-18).

A.4

Relation between dynamic delta hedge and VRP

From eq.(A-17), we can compute the value of the delta hedged portfolio. In Section A.1,
we assume a portfolio that consists of two derivatives and one underlying asset to hedge off
the risk of both the asset and the volatility. Here, we assume a portfolio that hedges therisk of asset only. We call this a delta-hedged portfolio. In the stochastic volatility setting,the delta of a derivative is also stochastic, therefore we assume the portfolio manager
dynamically rebalances the position every time the delta changes. Let Πt denotes the valueof the delta-hedged portfolio at time t. The change of the portfolio value is computed as:
dΠt∂ft(St + dSt ) + (1 + rt dt)= ft + dft −
∂St
µ
¶
∂ft
∂ft
= dft −
dSt − rt ft − St
dt.
∂St
∂St

26

µ

∂ft
St − ft∂St¶(A-27)

By eq.(A-12), eq.(A-13), eq.(A-14) and eq.(A-15),
(S)

(S)

(σ)

(σ)

dft = µf ft dt + σf ft dWt + σf ft dWt
·
¸
∂f
∂f
= (µ − rt )St+ (α − ϕ) 2 + rt f dt
∂St
∂σt
∂f
∂f
(σ)
(S)
+St σtdWt + β 2 dWt .
∂St∂σt(A-28)

Eq.(A-27) can be rewitten using eq.(A-28) and eq.(A-2) as:dΠt = (αt − ϕt )

∂f
∂f
(σ)
dt + β 2 dWt
2
∂σt
∂σt

(A-29)

Hence, the value of the portfolio hedged from time t to time T is given by integrating
both sides of eq.(A-29) as:
Z T
ΠT =

t

∂f
(αu − ϕu ) 2 du +
∂σu

Z T
β
t

∂f
dWu(σ) .
∂σu2(A-30)The expected value of eq.(A-30) become·Z T
E t ΠT = E t

t

¸
·Z T
¸
∂f
∂f
(αu − ϕu ) 2 du = Et
λ(t, u) 2 du .
∂σu
∂σu
t

(A-31)

Eq.(A-31) indicates that the growth of the value of the delta-hedged portfolio is proportional to the volatility risk premium instantaneously, when we assume the portfolio manager
can perfectly and dynamically hedge off the risk of the asset. When considering an equity
option for the derivative f , the delta hedge gain may be negative, because the option has
a positive vega and we know α − ϕ = λ is negative in most periods.

A.5

Relation between investors’ risk aversion and VRP

Many earlier works, such as Heston [1993], Bakshi and Kapadia [2003] and Bollerslev et al.
[2007], discussed the VRP related to the investors’ utility. In their works, the pricing
kernel in eq.(10) is supposed to be equal to the marginal utility of the representative
investor whose utility function is assumed to be a power function of the investor’s wealthwt , which is defined as:
U (wt ) = e−rt

wt1−γ
1−γ

(A-32)

where the parameter γ indicates the investor’s risk aversion; the larger the γ the morerisk averse the investor. In wealth based asset pricing model, the pricing kernel defined in27

eq.(10) is equal to the inter-temporal marginal rate of substitutions, i.e.:
U 0 (wT )Mt,T = 0= e−r(T −t)
U (wt )

µ

wT
wt

¶−γ
(A-33)

Suppose the representative investor invests in one share of a stock whose price obeys
standard SV model in eq.(A-2). Then, the pricing kernel become Mt,T = e−r(T −t) (ST /St )−γ .
The instantaneous risk premium for the investor can be computed from eq.(1), eq.(11) andIto’s lemma as:¯¯∂Cov (Mt,T , hSit,T )¯¯
λ(t, t) = −
∂T
T =t
¯
¯ ¶
µ
¯
¯
∂
∂
¯
= −Cov
Mt,T ¯,
hSit,T ¯¯∂T
T =t ∂T
T =t
Ã
µ
¶2 !
dSt dSt= γCov,StSt= γβt ρt dt.

(A-34)

Integrating both sides yields

Z T
λ(t, T ) = γtβs ρs ds.

That shows the VRP is proportional to the investors’ risk aversion γ.

28(A-35)