Expected Shortfall computation with multiple control variates1
Luis Ortiz-Gracia2
Department of Econometrics, University of Barcelona, Spain3
E-mail: luis.ortiz-gracia@ub.edu4
Abstract5
In this work we derive an exact formula to calculate the Expected Shortfall (ES) value for the one-6
factor delta-gamma approach which, to the best of our knowledge, was still missing in the literature.7
We then use the one-factor delta-gamma as a control variate to estimate the ES of the multi-factor8
delta-gamma approach. A one-factor delta-gamma approximation is used for each risk factor appear-9
ing in the problem. Since the expected values of control variates are computed by means of an exact10
formula, the additional effort of computation with respect to the naive estimator of the multi-factor11
delta-gamma can be neglected. With this method, we achieve a considerable reduction of the variance.12
We have established a theorem to prove that the variance is further reduced when we use all the risk13
factors instead of just some of them. We show that one of the main potential applications takes place14
in the insurance industry regulation within the Swiss solvency test framework. We perform a model15
risk analysis and illustrate these results with numerical experiments.16
Key words. Swiss solvency test, market risk, nonlinear portfolio, delta-gamma approximation,17
Expected Shortfall, exact formula, control variates18
AMS subject classifications. 62P05, 65C05, 65C5019
1 Introduction20
A classical but still important problem in market risk management is the estimation of a profit and21
loss distribution of a portfolio over a specified time horizon and the associated risk measures. Value-22
at-Risk (VaR) has become an important measure for estimating and managing portfolio market risk.23
VaR is defined as a certain quantile of the change in value of a portfolio during a specified holding24
period. While the basic concept of VaR is simple, many complications can arise in practical use. An25
important complication is caused by nonlinearity in the portfolio payoff structure. This problem arises26
for all portfolios that include assets with nonlinear payoffs, such as option positions. For such nonlinear27
portfolios, VaR cannot be computed directly from a risk factor distribution. Instead, the risk factor28
distribution first needs to be converted into a profit and loss distribution for the portfolio. VaR is then29
computed from this profit and loss distribution. The Basel Committee of Banking Supervision initiated a30
fundamental review of the trading book regime (see [2, 3]), beginning with an assessment of those things31
that went wrong during the financial turmoil. The revised standards for minimum capital requirements32
for market risk were established in [4]. The Committee has focused, among other things, on moving33
from VaR to the ES for measuring the portfolio risk. ES is a coherent measure (in the sense of [1]) that34
takes into account the tail of the portfolio loss distribution beyond the VaR value.35
Monte Carlo (MC) simulation is typically used to calculate these risk measures, first simulating36
changes in the risk factors of a portfolio, then the portfolio is evaluated at each new price and the change37
in value of the portfolio is estimated. This method is known as full revaluation. However, accurate VaR38
and ES estimates with a full revaluation method, are obtained at the cost of a considerable computational39
effort, since there might be a large number of instruments in the portfolio and when the confidence level is40
high, a large number of simulations may be required to obtain accurate estimates of the tail probability.41
In the present work, we adopt the delta-gamma approximation [6, 11, 18] as an alternative approach42
to the full revaluation method. This approach is based on the assumption that the change in portfolio43
1
value is a quadratic function of the changes in the risk factors. This method is sometimes called partial44
MC, since the only part of the procedure that involves simulation is the one related to the price change.45
This fact makes the delta-gamma method much more competitive from the computational standpoint.46
The boost in terms of CPU time is at the cost of less accurate values when measuring the risk, since it47
is a second order Taylor approximation to the true change in portfolio value (the suitability of the delta-48
gamma approach is studied in [21]). Typically, the changes in the risk factors are assumed to be normally49
distributed. The assumption of normality is found, for instance, in the insurance industry. As pointed50
out by [15, 16, 19], the standard market model of the Swiss solvency test (SST), in force since 2011,51
defines the capital required by a Swiss insurance company to absorb negative financial scenarios. The52
capital requirement imposed by the Swiss financial market supervisory authority (FINMA) corresponds53
to the ES calculated at 99% confidence level by means of a delta-gamma approach where the risk factors54
follow a multivariate normal distribution, and this computation is generally done by means of Monte55
Carlo simulation. Thus, the possibility of rare events such as financial crises are ignored under the56
multivariate normal framework. As stated in [19], the FINMA has amended the model to take into57
account the possibility of exceptionally high losses in financial markets, so we consider this model as well58
in the present work. Other potential applications of the delta-gamma approach appear in counterparty59
risk, and more precisely, to speed up the calculation of initial margin payments as suggested by [20]. It60
has also been employed to compute the VaR value of straddles, strangles and spread options in [7].61
The delta-gamma method has also been studied in [8, 19] and [21] within the context of Fourier62
inversion methods, avoiding this way MC simulation. The probability density function (PDF) and the63
cumulative distribution function (CDF) are recovered through Fourier inversion from the characteristic64
function of the random variable representing the change in value of the portfolio. However, as pointed65
out in [8], for certain holding periods the risk measures are difficult to estimate by means of Fourier66
inversion, since numerical errors may hamper the accurate estimation of the PDF. Further, when a67
numerical method is employed, then some parameters have to be fixed, and that is generally a trial and68
error problem. The implementation of a MC method is usually a more simple task, and this is why69
simulation is often preferred by financial companies.70
The main contributions of this work within the delta-gamma framework are the following.71
• We derive an exact formula to calculate the ES value for the one-factor delta-gamma approach72
under the assumption of normal changes in the risk factor. While the density function of the one-73
factor delta-gamma approach was already known in closed-form (see for instance [13] or [21]), an74
exact formula for estimating the ES value was still missing. We therefore overcome the numerical75
problems stated before in the presence of a single risk factor when using Fourier inversion.76
• We extend the one-factor exact formula to the one-factor SST framework. We provide an exact77
formula for the ES where extreme scenarios are taken into account, and we give the mathematical78
expression that relates the VaR value with and without extreme scenarios.79
• We propose a conservative estimation of the ES for separable and multivariate portfolios. This80
is achieved by means of the exact formula and the subadditivity property of the ES. For those81
portfolios involving only one asset, like for instance the trading strategies described in [17], the82
exact formula can be applied straightforwardly.83
• We use the one-factor delta-gamma as a control variate to estimate the ES of the multi-factor delta-84
gamma approach in the normal case as well as when extreme scenarios are considered. A one-factor85
delta-gamma approximation is used for each risk factor appearing in the financial context. Since86
the expected values of control variates are computed by means of an exact formula, the additional87
effort of computation with respect to the naive estimator of the multi-factor delta-gamma can be88
neglected. We achieve a considerable variance reduction factor (VRF). This fact will be shown89
in the numerical experiments. It is worth remarking that the closed-form formula given for the90
univariate case is a key aspect for the successful application of the variance reduction method put91
forward in this work.92
• When we use a control variates technique in the context of multi-factor delta-gamma, we can93
potentially include as many control variates as risk factors appearing in the problem. So we have94
2
to make a decision on which factor(s) should be used. We prove in Theorem 1 that the variance is95
reduced (or at least equal) when we use all the risk factors as control variates. To the best of our96
knowledge, this is a new contribution to the literature. We illustrate this result in the numerical97
experiments part.98
Other authors have used control variates with a rather different approach in the context of portfolio99
losses for computing the VaR value (see for instance [11]) or valuation of derivatives (like [9] for basket100
and Asian options).101
The paper is organized as follows. We introduce the delta-gamma approach in Section 1.1. We derive102
an exact formula for a non-central chi-squared distribution with one degree of freedom in Section 2. That103
formula will be used in Section 3 to obtain an exact formula for the ES under the one-factor delta-gamma104
approach. We extend the computation of the ES with an exact formula to the SST model in Section105
4. The problem of multiple control variates is put forward in Section 5, and the numerical experiments,106
along with an analysis of model risk, are presented in Section 6. Finally, Section 7 concludes.107
1.1 The delta-gamma approach108
Suppose the current value of a portfolio is V (t), the holding period is ∆t, and the value of the portfolio109
at time t+ ∆t is V (t+ ∆t). The change in the portfolio value during the holding period is ∆V , where110
∆V = V (t+ ∆t)− V (t). The VaR value of ∆V at a confidence level α, is given by qα, where,111
P(∆V < qα) = α.
In practice, ∆t ranges from one day to two weeks and α ∈ (0, 1) is close to 1. By definition, the ES risk112
measure at confidence level α is given by,113
ESα(∆V ) = E (∆V |∆V > qα) ,
or, alternatively,114
ESα(∆V ) =
1
1− α
∫ +∞
qα
xf∆V (x)dx, (1)
where f∆V is the PDF of ∆V .115
We assume that there are p risk factors and that S(t) = (S1(t), . . . , Sp(t))
T denotes the value of these116
factors at time t. Define ∆S = S(t+ ∆t)− S(t) to be the change in the risk factors during the interval117
[t, t+ ∆t]. Then, the delta-gamma approximation is given by,118
∆V ' ∆Vγ = Θ∆t+ δT∆S + 1
2
∆STΓ∆S, (2)
where Θ = ∂V∂t , δi =
∂V
∂Si
, Γi,j =
∂2V
∂Si∂Sj
, are called the Greeks, and all partial derivatives are being119
evaluated at S(t). Further, ∆S is governed by a normal distribution with mean zero and covariance120
matrix Σ. Observe that in the univariate case (p = 1), we have,121
∆Vγ =
n∑
i=1
xi
∂vi
∂t
∆t+
n∑
i=1
xi
∂vi
∂S
∆S +
1
2
n∑
i=1
xi
∂2vi
∂S2
(∆S)2, (3)
where n represents the number of assets in the portfolio, xi is the amount of asset i and vi the value of122
asset i. In this particular case,123
Θ =
n∑
i=1
xi
∂vi
∂t
, δ =
n∑
i=1
xi
∂vi
∂S
, Γ =
n∑
i=1
xi
∂2vi
∂S2
.
3
2 Non-central chi-squared distribution with one degree of freedom124
We devote this section to the study of the non-central chi-squared distributions with one degree of125
freedom, since as we will see in the next section, they are the building blocks of the delta-gamma126
approach. More precisely, if X is a random variable that follows a non-central chi-squared distribution127
with one degree of freedom and non-centrality parameter ζ, we will compute its CDF, its VaR value128
and give a closed-form expression for the ES. The starting point is the PDF of X, given by (see [14] for129
details),130
fζ(x) =
e−
1
2
(x+ζ)
√
2pix
cosh
(√
ζx
)
, (4)
where,131
cosh
(√
ζx
)
=
e
√
ζx + e−
√
ζx
2
, x ∈ (0,+∞), ζ > 0.
We can derive the CDF of X,132
Fζ(x) =
∫ x
0
fζ(y)dy, x ∈ (0,+∞), ζ > 0,
by integrating,133
Fζ(x) =
∫ x
0
e−
1
2
(y+ζ)
√
2piy
· e
√
ζy + e−
√
ζy
2
dy, x ∈ (0,+∞), ζ > 0.
If we make the change of variables y = t2 we obtain,134
Fζ(x) = Φ
(√
x−
√
ζ
)
+ Φ
(√
x+
√
ζ
)
− 1, x ∈ (0,+∞), ζ > 0,
or, alternatively,135
Fζ(x) = Φ
(√
x−
√
ζ
)
− Φ
(
−√x−
√
ζ
)
, x ∈ (0,+∞), ζ > 0,
where Φ is the CDF of the standard normal distribution.136
Let qζα be the VaR value of X calculated at the confidence level α ∈ (0, 1), this is Fζ(qζα) = α. We137
derive a closed-form formula for the ES of X by means of its density (4) and expression (1),138
ESα(X) =
1
1− α
∫ +∞
qζα
xfζ(x)dx.
Proposition 1. The ES at confidence level α of a non-central chi-squared random variable X with one139
degree of freedom and non-centrality parameter ζ is,140
ESα(X) =
1
1− α
[
(ζ + 1)(1− α) + φ
(√
qζα +
√
ζ
)((√
qζα +
√
ζ
)
e2
√
qζαζ +
√
qζα −
√
ζ
)]
,
where φ denotes the PDF of the standard normal distribution.141
Proof. If we make the change of variables x = t2 then,142
ESα(X) =
1
1− α
∫ +∞
√
qζα
t2
(
φ
(
t−
√
ζ
)
+ φ
(
−t−
√
ζ
))
dt.
If we define,143
g(t) = (ζ + 1)
(
Φ
(
t−
√
ζ
)
+ Φ
(
t+
√
ζ
)
− 1
)
− φ
(
t+
√
ζ
)
·
((
t+
√
ζ
)
e2
√
ζt + t−
√
ζ
)
,
then,144
4
ESα(X) =
1
1− α
(
g (+∞)− g
(√
qζα
))
.
Finally, we conclude by using that,145
Fζ(q
ζ
α) = Φ
(√
qζα −
√
ζ
)
+ Φ
(√
qζα +
√
ζ
)
− 1 = α.
146
3 Closed-form Expected Shortfall in the delta-gamma framework147
In this section we derive a closed formula to calculate the ES of ∆Vγ for a single risk factor. We will148
use that formula for each risk factor to reduce the variance of the multi-factor delta-gamma approach149
estimator. The details will be given in Section 5 and Section 6. We follow the results provided in [21],150
where it is established the link between the density function of ∆Vγ and the density function of a non-151
central chi-squared distribution with one degree of freedom, called Q, with non-centrality parameter ζ152
and density function fQ. It is worth mentioning that the density function of the one-factor delta-gamma153
approach was also given in [13]. More precisely,154
f∆Vγ (x) =
2
|λ|fQ
(
2
λ
(x−Θ∆t− µ¯Q)
)
, (5)
where λ = ΓC2, d = δC,C = σ
√
∆tS(t), µ¯Q = −12 d
2
λ and ζ =
(
d
λ
)2
. For completeness, we give at the155
beginning of Section 6 a brief explanation on the selection of the variance Σ = C2 considered to simulate156
the price change of individual risk factors ∆Sj , j = 1, . . . , p. Further, the VaR value of ∆Vγ is given by,157
qα =
{
λ
2 q
ζ
α + µ¯Q + Θ∆t, if λ > 0,
λ
2 q
ζ
1−α + µ¯Q + Θ∆t, if λ < 0,
(6)
where the quantiles qζα and q
ζ
1−α represent the VaR value of Q at confidence level α and 1−α respectively.158
In order to derive the expression for the ES, we differentiate between positive and negative λ. We159
start by assuming that λ > 0. In this case, it is shown in [21] that f∆Vγ is either unimodal or bimodal160
in its domain of definition (µ¯Q + Θ∆t,+∞). If we take into account (5) the ES is given by,161
ESα(∆Vγ) =
1
1− α
∫ +∞
qα
xf∆Vγ (x)dx =
1
1− α ·
2
λ
∫ +∞
qα
xfQ
(
2
λ
(x−Θ∆t− µ¯Q)
)
dx.
If we make the change of variables y = 2λ (x−Θ∆t− µ¯Q), then by (6) we get,162
ESα(∆Vγ) =
1
1− α
∫ +∞
qζα
(
λ
2
y + Θ∆t+ µ¯Q
)
fQ(y)dy =
λ
2
ESα(Q) +
1
1− α · (Θ∆t+ µ¯Q)
∫ +∞
qζα
fQ(y)dy,
where ESα(Q) denotes the Expected Shortfall of the random variable Q. Finally, taking into account163
that,164 ∫ +∞
qζα
fQ(y)dy = 1− α,
we conclude that,165
ESα(∆Vγ) =
λ
2
ESα(Q) + Θ∆t+ µ¯Q.
We now consider λ < 0. In this case, it is shown in [21] that f∆Vγ is either unimodal or bimodal in166
its domain of definition (−∞, µ¯Q + Θ∆t), where x = µ¯Q + Θ∆t is a vertical asymptote for f∆Vγ . If we167
take that observation into account then by (5) the ES is given by,168
5
ESα(∆Vγ) =
1
1− α
∫ µ¯Q+Θ∆t
qα
xf∆Vγ (x)dx =
−1
1− α ·
2
λ
∫ µ¯Q+Θ∆t
qα
xfQ
(
2
λ
(x−Θ∆t− µ¯Q)
)
dx.
If we make the change of variables y = 2λ (x−Θ∆t− µ¯Q), then by (6) we get,169
ESα(∆Vγ) =
−1
1− α
∫ 0
qζ1−α
(
λ
2
y + Θ∆t+ µ¯Q
)
fQ(y)dy
=
1
1− α
[
λ
2
∫ qζ1−α
0
yfQ(y)dy + (Θ∆t+ µ¯Q)
∫ qζ1−α
0
fQ(y)dy
]
.
Taking into account that,170 ∫ qζ1−α
0
fQ(y)dy = 1− α,
then,171
ESα(∆Vγ) =
1
1− α ·
λ
2
∫ qζ1−α
0
yfQ(y)dy + (Θ∆t+ µ¯Q) . (7)
Since Q is a non-central chi-squared distribution with one degree of freedom and non-centrality parameter172
ζ, we follow the same steps as in Section 2 to obtain,173
∫ qζ1−α
0
yfQ(y)dy = g
(√
qζ1−α
)
− g(0)
= (ζ + 1)
(
Φ
(√
qζ1−α −
√
ζ
)
+ Φ
(√
qζ1−α +
√
ζ
)
− 1
)
− φ
(√
qζ1−α +
√
ζ
)
·
·
((√
qζ1−α +
√
ζ
)
e
2
√
qζ1−αζ +
√
qζ1−α −
√
ζ
)
= (ζ + 1) (1− α)− φ
(√
qζ1−α +
√
ζ
)
·
((√
qζ1−α +
√
ζ
)
e
2
√
qζ1−αζ +
√
qζ1−α −
√
ζ
)
.
(8)
Then, by (7) and (8),174
ESα(∆Vγ) =
λ
2
· 1
1− α
[
(ζ + 1) (1− α)− φ
(√
qζ1−α +
√
ζ
)
·
((√
qζ1−α +
√
ζ
)
e
2
√
qζ1−αζ +
√
qζ1−α −
√
ζ
)]
+ Θ∆t+ µ¯Q.
We can summarize in the following proposition the value of the ES for all values of λ.175
Proposition 2. The ES at confidence level α of the delta-gamma approximation ∆Vγ reads,176
ESα(∆Vγ) =
λ
2
· 1
1− α
[
(ζ + 1) (1− α) + sign(λ)φ
(√
qζ +
√
ζ
)
·
((√
qζ +
√
ζ
)
e2
√
qζζ +
√
qζ −
√
ζ
)]
+
+ Θ∆t+ µ¯Q,
(9)
where sign(λ) is the sign function (takes the value 1 for positive λ and −1 for negative λ) and qζ = qζα177
for positive λ and qζ = qζ1−α for negative λ.178
Proof. The result follows from the expressions above.179
6
3.1 Quantile computation180
Looking at formula (9), the only required computation for obtaining the ES value is the quantile qζ .
The quantile qζ satisfies Fζ
(
qζ
)
= η, where η = α for positive λ, η = 1 − α for negative λ, and Fζ is
the distribution function of a non-central chi-squared random variable with one degree of freedom and
non-centrality parameter ζ. We define,
Gζ(x) = Fζ(x)− η,
where,181
Fζ(x) = Φ
(√
x−
√
ζ
)
− Φ
(
−√x−
√
ζ
)
, x ∈ (0,+∞), ζ > 0,
as seen in Section 2. We observe that,182
Gζ(0) = Fζ(0)− η = −η < 0, and,
Gζ(ζ) = Fζ(ζ)− η = Φ(0)− Φ
(
−2
√
ζ
)
− η = Φ
(
2
√
ζ
)
− η − 1
2
.
Since G′ζ(x) > 0 for all x ∈ (0,+∞), there is a unique solution of Gζ(x) = 0 in the interval [0, ζ] provided183
that Φ
(
2
√
ζ
)−η− 12 > 0. In that case, we can safely apply a bisection method to the function Gζ(x) with184
initial interval [0, ζ]. When Φ
(
2
√
ζ
)−η− 12 < 0, then the unique root is located at some point beyond ζ185
and we apply a Newton-Raphson method with initial seed ζ (we prefer not to use the Newton-Raphson186
method in the first case to avoid negative values in subsequent iterations).187
4 Shocks in the risk factors: the SST model188
In this section we derive a closed formula to calculate the ES of the change in value of the portfolio for a189
single risk factor under the SST model. To do this, we consider multiple scenarios, which occur with small190
probabilities and are mutually exclusive. To be more precise, the new model considers l + 1 scenarios191
with associated probabilities of occurrence p0, p1, . . . , pl, where p0 stands for the normal scenario and,192
l∑
i=0
pi = 1.
For scenarios i ≥ 1 the change in value of the portfolio V is modified by the additive term,193
si := ∆S
T
i Γ∆Si + δ
T∆Si,
where ∆Si represents the change in value of the risk factors corresponding to the scenario i ≥ 1. In194
summary, the scenario-adjusted value of ∆V is given by,195
∆V sγ := ∆Vγ + S, (10)
where ∆Vγ is given in (2), S =
∑l
i=0 Iisi and the indicator random variables Ii select which scenario196
occurs, i.e., with probability pi, Ii = 1 and Ik = 0 for k 6= i (this is, the random variables are mutually197
exclusive) and the indicator variables Ii are independent of the risk factors. Let fS be the density198
function of S, then,199
fS(x) = p0δ(x) +
l∑
i=1
piδ(x− si),
where δ stands for the Dirac delta function. Since ∆Vγ and S are assumed to be independent, then the200
density of ∆V sγ is the convolution product between both densities,201
7
f∆V sγ (x) =
(
f∆Vγ ∗ fS
)
(x) =
∫
R
f∆Vγ (y)fS(x− y)dy =
∫
R
f∆Vγ (y)
[
p0δ(x− y) +
l∑
i=1
piδ(x− y − si)
]
dy,
where f∆Vγ is considered to be zero outside its domain. Finally, if we take into account that δ(x) = δ(−x)202
and, if we define s0 = 0, then,203
f∆V sγ (x) =
l∑
i=0
pif∆Vγ (x− si) =
2
|λ|
l∑
i=0
pifQ
(
2
λ
(x− si −Θ∆t− µ¯Q)
)
, (11)
where, for the second equality in (11), we have used the expression (5). It is worth remarking that the204
first equality in (11) is also given in [19]. If we restrict ourselves to the single risk factor case, then we205
will show that an exact formula for the ES can be developed also in this case.206
Let us assume that λ > 0 in (11). In this case, since the domain of definition of f∆Vγ (x − si) is207
(si + µ¯Q + Θ∆t,+∞), then the distribution function F∆V sγ of ∆V sγ reads,208
F∆V sγ (x) =
l∑
i=0
pi
∫ x
si+Θ∆t+µ¯Q
f∆Vγ (y − si)dy =
2
λ
l∑
i=0
pi
∫ x
si+Θ∆t+µ¯Q
fQ
(
2
λ
(y − si −Θ∆t− µ¯Q)
)
dy.
If we make the change of variables z = 2λ(y − si −Θ∆t− µ¯Q), then,209
F∆V sγ (x) =
l∑
i=0
pi
∫ 2
λ
(x−si−Θ∆t−µ¯Q)
0
fQ(z)dz =
l∑
i=0
piFζ
(
2
λ
(x− si −Θ∆t− µ¯Q)
)
.
If qsα represents the VaR of ∆V
s
γ at confidence level α then it can be obtained by solving the equation210
F∆V sγ (q
s
α) = α, this is,211
l∑
i=0
piFζ
(
2
λ
(qsα − si −Θ∆t− µ¯Q)
)
= α.
We now use the VaR value to compute the ES,212
ESα(∆V
s
γ ) =
1
1− α
l∑
i=0
pi
∫ +∞
qsα
xf∆Vγ (x− si)dx
=
2
λ
· 1
1− α
l∑
i=0
pi
∫ +∞
qsα
xfQ(
2
λ
(x− si −Θ∆t− µ¯Q))dx.
If we make the change of variables z = 2λ(x− si −Θ∆t− µ¯Q), then,213
ESα(∆V
s
γ ) =
1
1− α
l∑
i=0
pi
[
λ
2
∫ ue
le
zfQ(z)dz + (si + Θ∆t+ µ¯Q)
∫ ue
le
fQ(z)dz
]
,
where le =
2
λ(q
s
α − si −Θ∆t− µ¯Q) and ue = +∞. Finally, if we use the function g defined in Section 2214
we end up with,215
ESα(∆V
s
γ ) =
1
1− α
l∑
i=0
pi
[
λ
2
(
ζ + 1− g
(√
le
))
+ (si + Θ∆t+ µ¯Q) (1− Fζ(le))
]
.
Let us assume that λ < 0 in (11). In this case, since the domain of definition of f∆Vγ (x − si) is216
(−∞, si + µ¯Q + Θ∆t), then the distribution function F∆V sγ of ∆V sγ reads,217
8
F∆V sγ (x) =
∫ x
−∞
f∆V sγ (y)dy =
l∑
i=0
pi
∫ x
−∞
f∆Vγ (y−si)dy = −
2
λ
l∑
i=0
pi
∫ x
−∞
fQ
(
2
λ
(y − si −Θ∆t− µ¯Q)
)
dy.
If we make the change of variables z = 2λ(y − si −Θ∆t− µ¯Q), then,218
F∆V sγ (x) =
l∑
i=0
pi
∫ +∞
2
λ
(x−si−Θ∆t−µ¯Q)
fQ(z)dz =
l∑
i=0
pi
[
1− Fζ
(
2
λ
(x− si −Θ∆t− µ¯Q)
)]
= 1−
l∑
i=0
piFζ
(
2
λ
(x− si −Θ∆t− µ¯Q)
)
.
If qsα represents the VaR of ∆V
s
γ at confidence level α then it can be obtained by solving the equation219
F∆V sγ (q
s
α) = α, this is,220
l∑
i=0
piFζ
(
2
λ
(qsα − si −Θ∆t− µ¯Q)
)
= 1− α. (12)
We now use the VaR value to compute the ES,221
ESα(∆V
s
γ ) =
1
1− α
l∑
i=0
pi
∫ si+Θ∆t+µ¯Q
qsα
xf∆Vγ (x− si)dx
= − 2
λ
· 1
1− α
l∑
i=0
pi
∫ si+Θ∆t+µ¯Q
qsα
xfQ(
2
λ
(x− si −Θ∆t− µ¯Q))dx.
If we make the change of variables z = 2λ(x− si −Θ∆t− µ¯Q), then,222
ESα(∆V
s
γ ) =
1
1− α
l∑
i=0
pi
[
λ
2
∫ u¯e
l¯e
zfQ(z)dz + (si + Θ∆t+ µ¯Q)
∫ u¯e
l¯e
fQ(z)dz
]
,
where l¯e = 0 and u¯e =
2
λ(q
s
α − si −Θ∆t− µ¯Q). Finally, if we use the function g defined in Section 2 we223
end up with,224
ESα(∆V
s
γ ) =
1
1− α
l∑
i=0
pi
[
λ
2
g
(√
u¯e
)
+ (si + Θ∆t+ µ¯Q)Fζ(u¯e)
]
.
5 Multiple control variates225
As pointed out by [10], the method of control variates is one of the most effective methods for improving226
the efficiency of MC simulation. The method takes advantage of the information about the errors in227
estimates of known quantities to reduce the error in an estimate of an unknown quantity. For the time228
being, we consider a unique control variate and the new estimator,229
E∗ = E − c1(E1 − τ1),
where E stands for the naive MC estimator, E1 is the control variate with known expected value τ1, and230
c1 is the optimal coefficient minimizing the variance of E
∗, this is,231
c1 =
Cov(E,E1)
Var(E1)
, (13)
where Cov and Var are the covariance and variance, respectively. If we use the optimal c1 in (13) then,232
9
Var(E∗) =
(
1− Corr(E,E1)2
)
Var(E), (14)
where Corr(E,E1) denotes the correlation between E and E1. Thus, the variance of the new estimator233
is dramatically reduced with respect to the variance of the naive estimator when the correlation between234
E and E1 is close to one (in absolute value). As pointed out in [9], the coefficient c1 can be estimated235
by using a pilot run with a smaller sample size or by using the full sample of the simulation. The236
former approach leads to an unbiased estimate while the second one has a bias which is negligible when237
the sample size is large (the bias is of order O(1/n)). In this work, we will use the full sample of the238
simulation for estimating c1, since the sample size will be large.239
Let us now consider multiple control variates. The general formulae for an arbitrary number d of240
control variates are taken from [12]. The estimator in this case reads,241
E∗ = E − cT (E − τ),
where cT = (c1, . . . , cd) is the vector of coefficients minimizing the variance of E
∗ and E = (E1, . . . , Ed)242
is the vector of control variates with known expected value τ = (τ1, . . . , τd). Then, c is selected as the243
optimal value of the problem,244
min Var(E∗) = Var(E)− 2cTC + cTDc, (15)
where C is the d-dimensional vector of covariances of E with each of the components of E , and D is245
the covariance matrix of E . If we assume that D is a non-singular matrix, then the first and second246
order optimality conditions of the minimization problem (15) imply that there is an optimal and unique247
solution given by c = D−1C with optimal value,248
Var(E∗) = (1−R2)Var(E),
where,249
R2 =
CTD−1C
Var(E)
. (16)
In particular, we can easily calculate the variance reduction factor achieved when using two control250
variates (this is, for d = 2) and compare it with the variance reduction factor obtained if we use a unique251
control variate. Thus,252
C =
(
Cov(E,E1)
Cov(E,E2)
)
, D =
(
Var(E1) Cov(E1, E2)
Cov(E1, E2) Var(E2)
)
,
and,253
c =
1
|D| ·
(
Var(E2) −Cov(E1, E2)
−Cov(E1, E2) Var(E1)
)
C. (17)
With the optimal solution c = (c1, c2)
T in (17), we have,254
R2 =
1
|D|VaR(E) · C
T
(
Var(E2) −Cov(E1, E2)
−Cov(E1, E2) Var(E1)
)
C
=
Var(E)Var(E1)Var(E2)
|D|VaR(E) ·
(
Corr(E,E1)
2 + Corr(E,E2)
2 − 2Corr(E,E1)Corr(E,E2)Corr(E1, E2)
)
=
Corr(E,E1)
2 + Corr(E,E2)
2 − 2Corr(E,E1)Corr(E,E2)Corr(E1, E2)
1− Corr(E1, E2)2 ,
(18)
since |D| = 1− Corr(E1, E2)2Var(E1)Var(E2).255
Within the delta-gamma framework, we can choose as many control variates as risk factors, so the256
natural and first question arising at this stage is whether it would be better to use one or two control257
10
variates for measuring the risk with a greater reduction of variance. We answer this question in the258
following lemma.259
Lemma 1. If we define R21 = Corr(E,E1)
2 as in expression (14) and,
R22 =
Corr(E,E1)
2 + Corr(E,E2)
2 − 2Corr(E,E1)Corr(E,E2)Corr(E1, E2)
1− Corr(E1, E2)2 ,
as in (18) then R22 ≥ R21, and the equality holds if and only if Corr(E,E1)Corr(E1, E2)−Corr(E,E2) = 0.260
Proof. Indeed, R22 ≥ R21 if and only if,261
Corr(E,E1)
2 +Corr(E,E2)
2−2Corr(E,E1)Corr(E,E2)Corr(E1, E2) ≥ (1−Corr(E1, E2)2)Corr(E,E1)2,
this is,262
(Corr(E,E1)Corr(E1, E2)− Corr(E,E2))2 ≥ 0,
and this completes the proof.263
Note that the same result applies if we select E2 instead of E1. We generalize the result of Lemma264
1 to an arbitrary dimension.265
Theorem 1. Let Dl and Cl be the matrix D and vector C, respectively, of expression (16) corre-266
sponding to the first l control variates. Let bTl−1 = (Cov(E1, El),Cov(E2, El), . . . ,Cov(El−1, El)) and267
C(l) = Cov(E,El) the last component of vector Cl. If we define,268
R2l =
CTl D−1l Cl
Var(E)
,
then, R2k ≥ R2k−1 for all k ≥ 2, and the equality holds if and only if bTk−1D−1k−1Ck−1 − C(k) = 0.269
Proof. We can write matrix Dk in block form,270
Dk =
(Dk−1 bk−1
bTk−1 Var(Ek)
)
,
with inverse matrix (see for instance [22] for details),271
D−1k =
(D−1k−1 + 1FD−1k−1bk−1bTk−1D−1k−1 − 1FD−1k−1bk−1
− 1F bTk−1D−1k−1 1F
)
,
where F = Var(Ek)− bTk−1D−1k−1bk−1.272
We have that R2k ≥ R2k−1 if and only if,273
CTk D−1k Ck ≥ CTk−1D−1k−1Ck−1. (19)
After some basic algebraic manipulation we get,274
CTk D−1k Ck = CTk−1D−1k−1Ck−1 +
1
F C
T
k−1D−1k−1bk−1bTk−1D−1k−1Ck−1 −
1
F C
T
k−1D−1k−1bk−1C(k)
− 1F b
T
k−1D−1k−1Ck−1C(k) +
1
F C(k)
2.
Expression (19) is satisfied if and only if,275
1
F C
T
k−1D−1k−1bk−1bTk−1D−1k−1Ck−1 −
1
F C
T
k−1D−1k−1bk−1C(k)−
1
F b
T
k−1D−1k−1Ck−1C(k) +
1
F C(k)
2 ≥ 0.
11
If we use that (D−1k−1)T = D−1k−1 then we derive an equivalent expression,276
1
F
(
bTk−1D−1k−1Ck−1
)2 − 2F bTk−1D−1k−1Ck−1C(k) + 1F C(k)2 ≥ 0,
which holds if and only if,277 (
bTk−1D−1k−1Ck−1 − C(k)
)2 ≥ 0,
provided that F > 0. Indeed, F , called the Schur complement of Dk−1 in Dk, is positive semi-definite,278
since every covariance matrix is positive semi-definite and by [23] Dk is positive semi-definite if and only279
if Dk−1 and F are positive semi-definite.280
Note that Lemma 1 is a particular case of Theorem 1 for k = 2.281
6 Expected Shortfall and control variates282
In this section we focus our attention in the efficient computation of the ES value under the multi-factor283
delta-gamma approach given in (2). We aim at calculating the aforementioned risk measure by means284
of an improved version of crude MC simulation. For this purpose, we consider each risk factor of the285
delta-gamma approximation as a control variate to reduce the variance. We will perform a consistency286
check in Section 6.1 for the one-factor delta-gamma approach. In Section 6.2 we will consider a European287
call option where the asset and the interest rate are the risk factors and only one control variate will be288
used. Finally, we calculate the ES in Section 6.3 for a basket put option, which has a payoff depending289
on two assets. In this last case, we illustrate the results obtained in Theorem 1 by using a different290
number of control variates in separated examples.291
We will assume the geometric Brownian motion (GBM), also called Black-Scholes model, for option292
valuation. The GBM model assumes that log
(
Sj(t+∆t)
Sj(t)
)
is normally distributed with mean µj∆t and293
standard deviation σj
√
∆t, for j = 1 . . . , p and S(t) = (S1(t), . . . , Sp(t)). Thus, there seems to be an294
inconsistency between the valuation model (this is GBM) and the model used for path simulation (this295
is the normal distribution introduced in Section 1.1). However, for small ∆t (as the holding period is),296
Sj(t+ ∆t)
Sj(t)
= 1 +
∆Sj
Sj(t)
' exp
(
∆Sj
Sj(t)
)
,
which is log-normally distributed if ∆Sj is normally distributed. In that case, ∆Sj follows a normal297
distribution with mean µj∆t and standard deviation Sj(t) · σj
√
∆t. A common assumption within the298
delta-gamma framework consists of approximating the mean by zero and we therefore have that ∆Sj299
follows a normal distribution with mean zero and variance Σ =
(
σj
√
∆tSj(t)
)2
.300
6.1 Consistency check of the one-factor delta-gamma approach301
We consider in this section the one-factor delta-gamma approach given in (3). Under this assumption,302
we can compute the ES value by means of the exact formula of Proposition 2, once the quantile value qζ303
has been obtained following the steps given in Section 3.1. In all of our experiments we fix a tolerance304
error of 10−6 for obtaining the quantile. As pointed out in Section 3.1, when the solution qζ is located305
within the interval [0, ζ], we can apply a bisection method. We use a faster version of the classical306
bisection method called Brent’s method (see [5]) which combines the bisection procedure with linear or307
quadratic inverse interpolation1.308
Our first test portfolio is taken from [8]. It consists of one short European call and half a short309
European put with maturity 60 days (T = 60/365). The underlying asset at time t is S(t) = 100310
with volatility level σ = 0.3, interest rate r = 0.1 and strike price K = 101 for each option. The311
pricing formula and the Greeks are detailed in Appendix A. We consider different holding periods ∆t,312
ranging from one to thirty days. We present in Table 1 the VaR and ES values obtained by means of313
1Computations were carried out in R code, and Brent’s method is implemented in the function uniroot.
12
MC simulation at confidence level α = 0.99, as well as the exact values calculated with the formula314
of Proposition 2. We show in the last column of the table, the value reported in [8]. This value was315
obtained by means of a numerical method based on the inversion of the characteristic function of ∆Vγ316
with wavelets. As it was explained in [8], for the two last cases ∆t = 10/365, 30/365 there are some317
numerical difficulties that hamper the computation of the VaR value and it is replaced by the (known)318
loss upper bound. The density plots in Figure 1 illustrate this fact (for concrete details see [8]).319
∆t VaR (MC) VaR (Exact) ES (MC) ES (Exact) VaR (reported in [8])
1/365 0.90311421 0.90307268 0.96464937 0.96460523 0.9038
10/365 1.70443274 1.70443156 1.70478356 1.70478331 1.7050
30/365 3.04335253 3.04335308 3.04368835 3.04306448 3.0439
Table 1: VaR and ES values corresponding to different holding periods ∆t and α = 0.99. MC values are
calculated with 108 simulations of price change ∆S.
We report in Table 2 the MC and exact values for VaR and ES at very high confidence levels. We320
know from [8] and [21] that the loss upper bound in this case is 1.102455.321
α VaR (MC) VaR (Exact) ES (MC) ES (Exact)
0.999 1.03507382 1.03536925 1.06162291 1.06187675
0.9999 1.09139676 1.09120022 1.09765512 1.09757439
Table 2: VaR and ES values corresponding to holding period ∆t = 1/365 and different confidence levels
α = 0.999, 0.9999. MC values are calculated with 108 simulations of price change ∆S.
−2.0 −1.5 −1.0 −0.5 0.0 0.5 1.0
0.
0
0.
2
0.
4
0.
6
0.
8
Loss
D
en
si
ty
(a) ∆t = 1/365.
1.40 1.45 1.50 1.55 1.60 1.65 1.70
1
2
3
4
5
6
7
Loss
D
en
si
ty
(b) ∆t = 10/365.
1.5 2.0 2.5 3.0
0
2
4
6
Loss
D
en
si
ty
(c) ∆t = 30/365.
Figure 1: Density plots of ∆Vγ .
In the following sections we use this exact formula within the control variates technique to provide322
a solution for the multi-factor delta-gamma approach.323
6.2 Delta-gamma approach with two risk factors and one control variate324
In this section we address the efficient computation of ES within a two-factor delta-gamma framework325
by using the one-factor delta-gamma as the unique control variate. For the one-factor delta-gamma,326
the risk factor considered is the underlying asset. We therefore apply the formula of Proposition 2 to327
the one-factor delta-gamma approach with a unique risk factor (the underlying asset). We consider a328
short European call option with strike K = 101, maturity T = 60/365, S(t) = 100, r(t) = 0.1, and the329
volatility of the underlying asset is σ = 0.3, being the underlying asset and the interest rate the risk330
factors. In this case, the two-factor version of (2) reads,331
∆Vγ = Θ∆t+ δS∆S + δr∆r +
1
2
(
ΓSS(∆S)
2 + Γrr(∆r)
2
)
+ ΓSr∆S∆r,
13
where δS , δr,ΓSS ,Γrr,ΓSr are the corresponding delta and gamma Greeks of the option (observe that we332
have replaced the notation δi,Γi,j of Section 1.1 by δS , δr,ΓSS ,Γrr,ΓSr to emphasize the risk factors).333
The pricing formula and the Greeks are detailed in Appendix A. If we assume that,334
log
(
r(t+ ∆t)
r(t)
)
∼ N (0, σ¯2),
where N (0, σ¯2) denotes a normal distribution with mean zero and variance σ¯2, then following the ar-335
gument given at the beginning of Section 6, we can assume that ∆r ∼ N (0, (σ¯r(t))2) and ∆S ∼336
N (0, (σ√∆tS(t))2). Further, we assume correlated normals with correlation parameter ρ = 0.5. Then,337
Σ =
(
σ2S σSσrρ
σSσrρ σ
2
r
)
,
is the covariance matrix, with σS = σ
√
∆tS(t), σr = σ¯r(t). We set σ¯ = 0.1. We calculate the ES with338
the naive estimator and the control variates estimator and present the results in the second and third339
column of Table 3, respectively. Two different confidence levels are considered and specified in the first340
column of the table.The holding period is set to ∆t = 1/365. The VRF achieved at confidence level341
α = 0.99 is 7.7, where it has been computed as the ratio between the variance of the naive estimator342
and the variance of the control variate estimator. The VRF is almost the double for α = 0.9.343
α ES (naive) ES (CV) VRF
0.9 1.487314 1.487373 14.1
0.99 2.129052 2.129427 7.7
Table 3: ES values for ∆t = 1/365. MC values are calculated with 108 simulations of price change ∆S.
It is worth mentioning that the additional computational cost of carrying out the estimation with344
the control variate can be neglected, since the expected value of the control variate is obtained with an345
exact formula (the only numerical part involved is the quantile computation, which is done efficiently).346
6.3 Delta-gamma approach with two risk factors and two control variates347
In this section we calculate the ES within a two-factor delta-gamma framework by using the one-factor348
delta-gamma as the control variate for each risk factor. For this purpose, we consider a geometric basket349
put option with pricing formula and Greeks detailed in Appendix B. In this case, the risk factors are350
the underlying assets S1 and S2 and the parameters employed are as follows. Strike K = 100, maturity351
T = 1, S1(t) = 90, S2(t) = 110, r = 0.04, and the volatilities of the underlying assets are σ1 = 0.2 and352
σ2 = 0.3. We assume a correlation ρ = 0.75 between the assets with normal distributions for the price353
change,354
∆S1 ∼ N (0, (σ1
√
∆tS1(t))
2), ∆S2 ∼ N (0, (σ2
√
∆tS2(t))
2).
The two-factor delta-gamma approach (2) reads,355
∆Vγ = Θ∆t+ δS1∆S1 + δS2∆S2 +
1
2
(
ΓS1S1(∆S1)
2 + ΓS2S2(∆S2)
2
)
+ ΓS1S2∆S1∆S2.
We consider a time horizon of ∆t = 10/365 and two different confidence levels α = 0.9 and α = 0.99.356
We present in Table 4 the VRF obtained in three different situations. We consider first a single control357
variate corresponding to the risk factor S1, in second place we perform a similar experiment with S2 and358
finally we take the two risk factors, this is, we use two control variates. The outcome is in line with the359
statement of Theorem 1, and a greater reduction variance is achieved when we employ the one-factor360
delta-gamma approach for each risk factor as a control variate.361
14
α VRF(S1) VRF (S2) VRF (S1, S2)
0.9 2.2 3.3 5.6
0.99 1.5 2.0 2.9
Table 4: VRF for different factors and confidence levels.
6.4 The SST model362
We devote this section to show how the variance reduction technique developed in this work can be363
used within the SST framework. Extreme scenarios definitions and probabilities of occurrence currently364
used in practice can be found in [15]. Further, the standard estimation of risk capital by means of the365
delta-gamma model takes into account 96 risk factors (82 market risk factors plus 14 life risk factors).366
For sake of clarity and brevity, we will consider an arbitrary set of probabilities associated to the extreme367
scenarios as well as the two-risk factor portfolio of Section 6.3 with two control variates.368
In order to appreciate the difference in risk between the normal case of Section 6.3 and the SST369
model, we use the same set of parameters as before. The extreme scenarios considered in this section370
are defined in Table 5,371
Scenario pi ∆S1 ∆S2
0 0.4 0 0
1 0.3 3 0
2 0.2 0 4
3 0.1 5 5
Table 5: Extreme scenarios.
The ES values calculated at confidence levels α = 0.9, 0.99 are shown in Table 6 with the correspond-372
ing VRF presented in Table 7.373
α = 0.9 α = 0.99
ES without shocks ES with shocks ES without shocks ES with shocks
Naive 2.340301 3.062148 3.236419 4.216899
CV 2.340422 3.062348 3.236460 4.217128
Table 6: ES values for ∆t = 10/365. MC values are calculated with 108 simulations of price change ∆S.
α VRF (S1, S2)
0.9 4.0
0.99 2.2
Table 7: VRF for different confidence levels with shocks.
We observe a considerable reduction of the variance under the SST model, although the reduction374
factor is lower with shocks than without shocks.375
6.5 Model risk376
We start by analyzing the model risk in the univariate case by comparing the VaR value qα with the377
VaR value qsα corresponding to ∆Vγ and ∆V
s
γ , respectively. We consider the case λ < 0 as it is the378
situation in Section 6.2 and Section 6.4. The case λ > 0 can be treated analogously. From expression379
(6) we know that,380
15
qα =
λ
2
qζ1−α + µ¯Q + Θ∆t,
this is,381
Fζ
(
2
λ
(qα −Θ∆t− µ¯Q)
)
= 1− α. (20)
Then, from expression (20) and (12) we have,382
l∑
i=0
piFζ
(
2
λ
(qsα − si −Θ∆t− µ¯Q)
)
= Fζ
(
2
λ
(qα −Θ∆t− µ¯Q)
)
. (21)
Finally, we can isolate qα from (21) and we end up with,383
qα =
λ
2
F−1ζ
(
l∑
i=0
piFζ
(
2
λ
(qsα − si −Θ∆t− µ¯Q)
))
+ Θ∆t+ µ¯Q. (22)
The expression (22) shows the relation between the risk measured by the VaR value within the SST model384
and the delta-gamma without shocks in the underlying factor. We consider the example in Section 6.4.385
The VaR value at confidence level α = 0.9 of the univariate version with risk factor S1 is qα = 0.859797386
without shocks and qsα = 1.294235 with shocks. We illustrate in Figure 2 the relation given in formula387
(22) for this particular example. The upper extreme of the horizontal axis corresponds to the value388
Θ∆t + µ¯Q, which is the maximum level of losses for the model without shocks. We can observe that389
under the SST model, the univariate risk grows (almost) linearly with respect to the model without390
shocks.391
0.0 0.5 1.0 1.5 2.0 2.5 3.0
0
1
2
3
VaR
Va
R
SS
T
Figure 2: Relation given by formula (22) for a range of quantiles.
6.6 Separable portfolios392
If we assume that portfolio V is separable, this is, it can be decomposed into a sum of one-dimensional393
subportfolios,394
V (S1(t), . . . , Sp(t)) = V1(S1) + · · ·+ Vp(Sp), (23)
where each Vj(Sj) depends only on the risk factor Sj , then we can then decompose the p-dimensional395
delta-gamma approach (2) with normal ∆S as,396
∆Vγ =
p∑
j=1
∆V jγ ,
16
where ∆V jγ := Θj∆t + δj∆Sj +
1
2Γ
j∆S2j , and Θ
j =
∂Vj
∂t , δ
j =
∂Vj
∂Sj
, Γj =
∂2Vj
∂S2j
. Since the ES enjoys the397
subadditivity property of a coherent risk measure then,398
ESα(∆Vγ) ≤
p∑
j=1
ESα(∆V
j
γ ), (24)
where ESα(∆V
j
γ ) can be readily computed with the exact formula of Section 3. The same argument399
applies if we consider the scenario-adjusted ∆V sγ in (10). Thus, the right-hand-side of (24) gives us a400
conservative but fast alternative of computing the ES value avoiding MC simulation. When V is not401
separable, we can combine MC simulation for those financial instruments which depend on more than402
one risk factor, with the exact ES for the instruments written on a unique risk factor.403
7 Conclusions404
In this work we have further investigated the well-known delta-gamma approach for computing the ES405
of the change in portfolio value. We have derived an exact formula to calculate the ES value for the406
one-factor delta-gamma approach which was still missing in the literature. We then use the one-factor407
delta-gamma as a control variate to estimate the ES of the multi-factor delta-gamma approach. A408
one-factor delta-gamma approximation is used for each risk factor appearing in the problem. Since the409
expected values of control variates are computed by means of an exact formula, the additional effort410
of computation with respect to the naive estimator of the multi-factor delta-gamma can be neglected.411
With this method, we achieve a considerable VRF. We have established a theorem to prove that the412
variance is further reduced when we use all the risk factors and we have illustrated these results with413
numerical experiments. Two models have been presented for driving the dynamics of the risk factors,414
the normal model and the SST model, and we have included an analysis of model risk. Finally, we415
consider the case of separable portfolios, and we provide an upper bound of the ES by using the exact416
formula of the univariate case. The possibility of either combining control variates with other variance417
reduction techniques or using nonlinear controls has not been explored in this work, and we leave it for418
future research.419
Appendix A. Greeks for European calls and puts420
The Black-Scholes formula for pricing a European call reads,421
v(S(t), σ, T, r,K) = S(t)Φ(d1)− e−r(T−t)KΦ(d2),
where,422
d1 =
ln(S(t)/K) + (r + 12σ
2)(T − t)
σ
√
T − t , d2 = d1 − σ
√
T − t.
The price of the corresponding put option is,423
v(S(t), σ, T, r,K) = e−r(T−t)KΦ(−d2)− S(t)Φ(−d1).
The Greeks used in Section 6.1 are,424
• theta (call),
∂v
∂t
= −S(t)φ(d1)σ
2
√
T − t − rKe
−r(T−t)Φ(d2),
• theta (put),
∂v
∂t
= −S(t)φ(d1)σ
2
√
T − t + rKe
−r(T−t)Φ(−d2),
17
• delta (call),
∂v
∂S
= Φ(d1),
• delta (put),
∂v
∂S
= −Φ(−d1),
• gamma (call and put),
∂2v
∂S2
=
φ(d1)
S(t)σ
√
T − t .
The Greeks used in Section 6.2 (evaluated at t = 0) are, theta (call), delta (call), gamma (call) and,425
• rho (call),
∂v
∂r
= KTe−rTΦ(d2),
• Γrr (call),
∂2v
∂r2
= KTe−rT
(
−TΦ(d2) +
√
T
σ
φ(d2)
)
,
• ΓSr (call),
∂v2
∂S∂r
=
√
T
σ
φ(d1).
Appendix B. Greeks for the geometric basket put option426
The formula for pricing a geometric basket put option under the Black-Scholes dynamics for assets S1427
and S2 with maturity T , strike K and payoff,428
max
(
K −
√
S1(T )S2(T ), 0
)
,
reads,429
v(Sˆ(t), σˆ, T, r,K) = e−r(T−t)KΦ(−dˆ2)− Sˆ(t)Φ(−dˆ1),
where,430
dˆ1 =
ln(Sˆ(t)/K) + (r + 12 σˆ
2)(T − t)
σˆ
√
T − t , dˆ2 = dˆ1 − σˆ
√
T − t,
and Sˆ(t) =
√
S1(t)S2(t), σˆ =
√
σ21+σ
2
2+2σ1σ2ρ
2 , being σ1 and σ2 the volatility of asset S1 and S2, respec-431
tively, ρ their correlation and r the risk-free rate.432
The Greeks used in Section 6.3 (evaluated at t = 0) are,433
• theta (put),
∂v
∂t
= − Sˆ(0)φ(dˆ1)σˆ
2
√
T
+ rKe−rTΦ(−dˆ2),
• δS1 (put),
∂v
∂S1
= −S2(0)
2Sˆ(0)
Φ(−dˆ1),
• δS2 (put),
∂v
∂S2
= −S1(0)
2Sˆ(0)
Φ(−dˆ1),
18
• ΓS1S1 (put),
∂2v
∂S21
=
S2(0)
4S1(0)Sˆ(0)
(
Φ(−dˆ1) + φ(dˆ1)
σˆ
√
T
)
,
• ΓS2S2 (put),
∂2v
∂S22
=
S1(0)
4S2(0)Sˆ(0)
(
Φ(−dˆ1) + φ(dˆ1)
σˆ
√
T
)
,
• ΓS1S2 = ΓS2S1 (put),
∂2v
∂S1∂S2
=
∂2v
∂S2∂S1
=
1
4Sˆ(0)
(
−Φ(−dˆ1) + φ(dˆ1)
σˆ
√
T
)
.
Acknowledgements434
The author acknowledges the Spanish Ministry of Economy and Competitiveness for funding under435
grants ECO2016-76203-C2-2 and MTM2016-76420-P (MINECO/FEDER, UE).436
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