ARTICLES

Sci. Rep. Kanazawa Univ.
Vol. 66, pp. 17–27, 2023

Portfolio optimization with conditional Value-at-Risk
under CEV model
Yuri I MAMURA1, * and Benyanee KOSAPONG2,3
1 Faculty of Mathematics and Physics, Institute of Science and Engineering, Kanazawa University

Kanazawa, 920-1192, Japan
2 Division of Mathematical and Physical Sciences, Graduate School of Natural Science and Technology,

Kanazawa University
Kanazawa, 920-1192, Japan
3 Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University
Bangkok, 10330, Thailand(Received October 22, 2022 and accepted in revised form July 27, 2023)

Abstract Portfolio optimization is a well-known and beneﬁcial procedure used by shareholders to select their portfolios. An investor must seek an equilibrium between risk andproﬁt while making investment decisions. The fundamental concern is risk because theresponsibility of the risk for each investor is different. A risk proﬁle of each investor is characterized as a risk measure. In this paper, we focus on Conditional Value-at-Risk (CVaR).
We numerically consider an optimal portfolio which minimizes CVaR under CEV model.
Finally, the numerical results of CVaR and the optimal portfolio are discussed.
Keywords. Portfolio optimization, risk measure, Value-at-Risk (VaR), Conditional Valueat-Risk (CVaR), CEV model, numerical analysis.

1

Introduction

Portfolio optimization is an important process of selecting the optimal portfolio that provides thehighest proﬁt (portfolio returns) for each unit of risk accepted by the investors or minimizes risk
for a given return. In 1952, Markowitz developed portfolio selection theory by using variance asa risk measure [11]. The variance, or equivalently standard deviation, is a famous tool for measuring a risk of portfolio. The axiom of coherent was introduced as properties of measures ofrisk [3, 4]. Artzner et al. [3] proposed the use of conditional Value-at-Risk (CVaR) (which is also
called expected shortfall, average value at risk, tail value at risk, expected tail loss, superquantile,etc.) to alleviate the problems inherent in value-at-risk (VaR). VaR is a loss distribution percentile,and CVaR is the average of loss values which excess a speciﬁc percentage (e.g., 90%, 95%, and99%) of the worst-case loss scenarios. Rockafellar and Uryasev [10] suggested another risk measure used CVaR to optimize portfolios. They showed that risk management with CVaR can be* Corresponding author

Email: imamuray@se.kanazawa-u.ac.jp

18

Portfolio optimization with conditional Value-at-Risk under CEV model

optimized with linear programming methods, whereas optimization with VaR is difﬁcult numerically (see also [3, 4, 9, 12]). Rockafellar and Uryasev [10] applied CVaR for minimizing the riskof an investment portfolio in the stock market. They reformulated them as convex optimization
problems, and a similar approach appears in Krokhmal, Palmquist and Uryasev [8]. Alexander
et al. [2] applied the CVaR model to a portfolio optimization problem and numerically solved it
by Monte Carlo simulation. In those previous results, under the assumption of the existence ofa density function, the optimization problem is reduced to a linear programming problem. Since
the density functions of some stock price models are not known, we aim to remove the density
function assumption. Moreover, since option prices are lower than stock prices and options canmake more proﬁts than stocks, we consider a portfolio which consists of not only stocks but also
options. For simplicity, the target derivatives of investment are restricted to one stock and its calland put options with the same strike. Under Black-Scholes economy (BS model), the analyticalforms of call and put options are known since the price process is expressed by geometric Brownian motion. Then the distribution function of the loss function is obtained which is explained in
Theorem 4.1. Since the behavior of volatility is not stable in the real market, we focus on solving
the optimization problem under the constant elasticity of variance model (CEV model) which was
proposed by [5]. Since the distribution of price process is not known, the motivation of this paper
is numerically to show properties of the optimal portfolio under the CEV model.
This paper is organized as follows. In Section 2, we study the stock and option market. In
Section 3, we explain deﬁnitions of VaR and CVaR and then we deﬁne the loss function of investment that is used to optimize the portfolio. In Section 4, we study optimization problem of
portfolio with options under BS model and CEV model. Moreover, we show numerical results of
the optimal portfolio using approximations of the stock prices and loss functions. In Section 5, we
provide discussions and conclusions.

2

Option market

An option or a derivative is a contract that gives its owner a right to buy or sell assets at a ﬁxed
price on or before a given date. The option value is based on the value of the underlying asset. Call
option and put option are the basic types. European call (or put) options allow the owner to buy (orsell) an asset at a speciﬁed price on a given maturity. For example, in the case of Japan, the Osaka
exchange exists as an options market. Then, we trade call and put options with certain maturity
times and strike prices for Nikkei 225 like stocks. The primary distinction between options andstocks is that stocks are shares of ownership in individual companies, while options are contracts
with other investors. We can purchase a lot of options to make a proﬁt, however it can make aloss at the same time. In this paper, we focus on a portfolio constructed by not only stock but also
options.

3

VaR, CVaR and Optimal Portfolio

VaR is a statistic quantifying the magnitude of a possible bad situation or loss over a speciﬁc time.
CVaR quantiﬁes the amount of tail risk in an investment portfolio, and it is calculated by taking
a weighted average of losses that exceed the VaR cutoff point. Figure 1 shows an example ofhistogram of loss values with a portfolio of investment.
We consider a portfolio of the investment strategy consisting of speciﬁc three claims; the asset,
a call option and a put option, where the options are associated with the same asset and they haveYuri I MAMURA and Benyanee KOSAPONG

19

Figure 1: The graphical representation of VaR, CVaR and max loss (min proﬁt)
the same strike K and the same maturity T . Let x0 , xc and x p be portfolio weights investing in cash,
a call option and a put option, respectively. We assume that the portfolio weights are restrictedwith x0 + xc + x p = 1, and we denote by A (⊂ R3 ) the set of all of the weights. We denote theinitial prices of a call option and a put option by C(K) and P(K), respectively. The returns ofeach contract are determined as the ratio between the gain at the maturity time T and the initial
prices S0 , C(K) and P(K). We consider that the loss function of the portfolio with the weights(x0 , xc , x p ) ∈ A is given byf (x0 , xc , x p ; ST )
[
]
[
]
[
]
(ST − K)1{ST >K} −C(K)(K − ST )1{K>ST } − P(K)ST − S0
= −x0
− xc
− xp
.
S0
C(K)
P(K)

(3.1)

x0 K
+ 1 if −x0C(K) −S0
xc S0 ≥ 0 and −x0 P(K) + x p S0 ≤ 0, and the loss function with a variable ST is bounded above by
x0 K+ 1 if −x0C(K) − xc S0 ≤ 0 and −x0 P(K) + x p S0 ≥ 0. In the other cases, the loss function−S0
is unbounded.
For each (x0 , xc , x p ) ∈ A and α ∈ R, the probability that the loss does not exceed α is denotedby
We note that the loss function with a variable ST is bounded below by −Ψ(x0 , xc , x p ; α ) = P ( f (x0 , xc , x p ; ST ) ≤ α ) .

(3.2)

Then, for (x0 , xc , x p ) ∈ A and conﬁdence level β in (0, 1), β -VaR αβ (x0 , xc , x p ) and β -CVaR
ϕβ (x0 , xc , x p ) are denoted byαβ (x0 , xc , x p ) = inf{ α ∈ R | Ψ(x0 , xc , x p ; α ) ≥ β },(3.3)

ϕβ (x0 , xc , x p ) = (1 − β )−1 E[ f (x0 , xc , x p ; ST )1{ f (x0 ,xc ,x p ;ST )≥αβ (x0 ,xc ,x p )} ].

(3.4)

and

20

Portfolio optimization with conditional Value-at-Risk under CEV model

The aim of this paper is to ﬁnd an investment portfolio weight (x0 , xc , x p ) which is the optimalvalue of the following minimization:minsubject toϕβ (x0 , xc , x p ),(x0 , xc , x p ) ∈ A .

4 Optimal Portfolio under CEV Model
We assume that the asset price process follows the CEV model which is described in [1, 13]. Thatis, the price process is given by a solution of the following stochastic differential equation (SDE):γ

dSt = µ St dt + σ St dWt , S0 > 0,

(4.1)

where µ is a parameter characterising the drift, σ > 0 is a volatility, γ > 0 is the elasticity which
controls the relationship between volatility and price, and {Wt }t≥0 is the Brownian motion. Generally option prices, VaR and CVaR are numerically calculated since the distribution of ST is
unknown except in the case of BS model (γ = 1), i.e. the elasticity is one. Here the price under
BS model is lognormally distributed with constant volatility.
In Section 4.1, we assume that the price process follows BS model. We give the closed formof option prices and the probability that the loss does not exceed a level, i.e. the cumulative
distribution function of the portfolio loss value. Since the distribution of geometric Brownian
motion is given explicitly, the optimal strategy which attains the minimal CVaR is determined. In
Section 4.2, we focus on CEV model. CVaR is numerically calculated; (i) the Euler-Maruyama
method provides the approximation of the discrete-time price process and (ii) Monte Carlo method
gives the approximation of the probabilities.

4.1 Black-Scholes ModelIn this section, we assume that the asset price process is given by the BS model, that is γ = 1
in (4.1). Here, the asset price process is offered by the solution of the following SDE:
dSt = µ St dt + σ St dWt , S0 > 0.

(4.2)

By Itô’s formula, SDE (4.2) is explicitly solved which is given by the following geometric Brownian motion:{}
1 2
St = S0 exp (µ − σ )t + σ Wt .
(4.3)
2
Since WT follows normal distribution with mean 0 and variance T , the prices of call option C(K)and put option P(K) are determined as follows (for example, see [6]);C(K) = S0 Φ(d1 ) − Ke−rT Φ(d2 ),P(K) = Ke−rT Φ(−d2 ) − S0 Φ(−d1 ),where r is the interest rate of the riskless security and[ ( ) (
) ]
1
S0
1 2
d1 = √ log+ r+ σ T ,K2
σ T[ ( ) (
) ]
S0
1 2
1
d2 = √ log+ r− σ T .
K
2
σ T

(4.4)

Yuri I MAMURA and Benyanee KOSAPONG

∫

21

u2

x √1 − 2
Here S0 is the initial price, and Φ(x) = −∞e du is the cumulative distribution function of
2π
standard normal distribution.
Then we get a closed form of the distribution function of the loss function.

Theorem 4.1 Under the BS model, for (x0 , xc , x p ) ∈ A and α ∈ R,
Ψ(x0 , xc , x p ; α )
A− − B +C1 − A+ +C
=

A− −C




1 − A+ + B −C

if − x0C(K) − xc S0 ≥ 0, −x0 P(K) + x p S0 ≥ 0,
if − x0C(K) − xc S0 ≤ 0, −x0 P(K) + x p S0 ≥ 0,
if − x0C(K) − xc S0 ≥ 0, −x0 P(K) + x p S0 ≤ 0,
if − x0C(K) − xc S0 ≤ 0, −x0 P(K) + x p S0 ≤ 0,

where
((


 log
A = Φ
±
±



(



((

)
) (
) 
(α − 1)C(K) − xc K
K
σ2
∨
− µ−
T
−x0C(K) − xc S0
S0
2
,
√

σ T

) (
) 
σ2
K
 log S0 − µ − 2 T ,√
B = Φ


σ T
 log

C = Φ)) () (α − 1)P(K) − x p K
K
σ2
∧
− µ−
T
−x0 P(K) + x p S0
S0
2

√
.

σ T

Here notations are deﬁned by log(z) := −∞ when z ≤ 0, a ∨ b = max{a, b} and a ∧ b = min{a, b}.

4.2

Numerical Examples

We assume that the price process follows CEV model. In this section, we show some numerical1
examples for CEV model. Here we ﬁx parameters as µ = 0, σ = 0.1206, T = 12
, S0 = 295.42
and K = 300. These values of parameters are from analyzing real data of S&P 500 mini index in
Kosapong [7].
The prices of call option and put option are numerically obtained. The discrete-time price(n)process is generated by Euler-Maruyama approximation; S0 = S0 and for 1 ≤ k ≤ n,
(n)

(n)

(n)

n

n

n

S k T = S k−1 T + µ S k−1 T

{
}γ
T
(n)
+ σ S k−1 T Zk ,nnwhere {Zk }nk=1 is a sequence of independent normally distributed random variables with mean 0
and variance T /n. By Monte Carlo method, the arithmetic means for scenarios of the payoff
functions give approximated prices of options. Here the number of time discretization of the pathof the price process by Euler-Maruyama method is 105 and the number of Monte Carlo simulation
is 107 .
In the case γ = 1.0, we can explicitly calculate the prices of the call option and the put option
by using (4.4), and the results are summarized in Table 1. We notice that Table 1 and the prices by22

Portfolio optimization with conditional Value-at-Risk under CEV model

Table 1: The Call/Put options prices under Black-Scholes modelInterest rate
0.0
0.1

Call price (C(K))
2.24183.1563

Put price (P(K))
6.82185.2467

Euler-Maruyama method, in Table 2 at γ = 1.0, are close enough. Therefore, we can understandthat an approximation error of Euler-Maruyama method is negligible.
(n)
By Euler-Maruyama method, we can generate a lot of scenarios of price process ST , and then(n)for each weight (x0 , xc , x p ) ∈ A , we obtain the loss functions f (x0 , xc , x p ; ST ) deﬁned by (3.1).
By Euler-Maruyama method and Monte Carlo method, we can simulate the distribution of the loss(n)function f (x0 , xc , x p ; ST ). Therefore we obtain VaR as the percentile point deﬁned by (3.3), andCVaR given by (3.4). In our simulation with ﬁxed conﬁdence level β (= 0.90, 0.95, 0.99), VaR iscalculated by using “prctile” function in MATLAB. We note that CVaR is a function of weight(x0 , xc , x p ). Hence by using optimization tool “fmincon” in MATLAB, we obtain the optimal
weight which leads to the minimal CVaR. The code of optimization portfolio and the step of
optimization are represented in Listing 1.
Listing 1: MATLAB code of portfolio optimization algorithm
1 loss = @( x_0 , x_c , x_p ) -(( x_0 /( x_0 + x_c + x_p ) ) * returnMatrix (: ,1)
2+ ( x_c /( x_0 + x_c + x_p ) ) * returnMatrix (: ,2)3+ ( x_p /( x_0 + x_c + x_p ) ) * returnMatrix (: ,3) ) ;4 VaR = @( x_0 , x_c , x_p ) prctile ( loss ( x_0 , x_c , x_p ) , beta ) ;5 MeanLoss_tail = @( x_0 , x_c , x_p ) max ( loss ( x_0 , x_c , x_p )6- VaR ( x_0 , x_c , x_p ) ,0) ;7 CVaR = @( x_0 , x_c , x_p ) VaR ( x_0 , x_c , x_p )8+ (1/(1 - beta /100) ) * mean ( MeanLoss_tail ( x_0 , x_c , x_p ) ) ;9 options = optimset ( ' Display ' , ' iter ' , ' PlotFcns ' ,@ optimplotfval);
10 [ x_opt , fval , exitflag , output ] = fmincon (@( x ) CVaR ( x (1) ,x (2) ,x(3) ) ,x , ' constraint ' , options ) ;

1.5

1.25

1.0

0.75

γ

74.7960

73.4303

0.0

0.1

16.1000

14.91390.0

0.1

3.14772.22320.0

0.1

0.27570.03280.0

0.1

Call priceinterest rate

77.000278.100118.2876

19.5677

5.2822

6.82402.39454.6180

Put price

VaR
0.00500.00870.0140-0.0076-0.0072−3.6817 × 10−4
0.07370.07690.0796-0.0072-0.0033
0.01192.4567 × 10−4
0.0071
0.0668−3.3081 × 10−40.00130.01547.1485 × 10−4
0.01230.0319-0.0068-0.00430.0479Conﬁdence level
90%
95%
99%
90%
95%
99%
90%
95%
99%
90%
95%
99%
90%
95%
99%
90%
95%
99%
90%
95%
99%
90%
95%
99%

0.00510.00980.0164-0.0075-0.0068−2.8478 × 10−4
0.07700.07860.0799-0.0068−7.9341 × 10−50.01202.4576 × 10−4
0.0076
0.0695−1.2789 × 10−50.00130.01769.2344 × 10−4
0.01280.0321-0.0067-0.0034
0.0491CVaR
0.9796
0.97370.9841
0.99200.9856
0.98490.98450.9057
0.9531
0.99310.99500.97290.98450.97680.91370.98440.98340.96540.98470.97170.95280.99130.99290.9415

optimal Cash

Table 2: Options, optimal weights and corresponding VaRs and CVaRs
0.02050.02570.01640.00870.01520.01420.02320.0386
0.05410.01780.0176
0.03600.0652
0.06690.10870.0646
0.06810.07210.26010.26010.26740.25880.25900.2440−7.7903 × 10−50.0006−4.5934 × 10−4
−7.6944 × 10−4
−7.3612 × 10−4
−8.3655 × 10−4
-0.0077
0.0557-0.0072-0.0109-0.0126-0.0089-0.0497-0.0437-0.0224-0.0490-0.0514-0.0375-0.2448-0.2448-0.2203-0.2502-0.2519-0.1855

optimal Put

optimal Call

Yuri I MAMURA and Benyanee KOSAPONG

23

24

Portfolio optimization with conditional Value-at-Risk under CEV model

The simulation results are shown in Table 2, which demonstrates the Call and Put option prices,
the optimal weights of cash, call option and put option, and corresponding VaR and CVaR for each
γ value in the CEV model. In Table 2, we observe that VaR and CVaR values are increasing forsigniﬁcance level. This fact is conﬁrmed in the simulation results in Table 2. We also notice thatfor γ = 1.00, VaR and CVaR values are close. However, the CVaR values are always greater thanVaR values. Those facts are mathematically correct by deﬁnitions of VaR and CVaR.
Figures 2, 3 and 4 show the histograms and graphs of loss functions of simulated prices foreach γ = 1.00, 0.75, 1.25 under CEV model with no interest rate. Here the loss function is thevalue at the optimal weights. The optimal portfolios with 95% conﬁdence level are given at thebottom of graphs 2(b), 3(b) and 4(b). The distributions of loss functions for γ = 0.75 and γ = 1.25
are numerically given, unlike the case γ = 1.0. The optimal solutions or the weights of investmentthat are represented in Figure 3 and Figure 4 are also unlike Figure 2. In these cases with γ = 0.75
and γ = 1.0, we notice that we need to sell both of call option and put option because the weights
of investment are positive values. It opposes with the optimal solutions in case of γ = 1.25. We
need to sell a put option and buy a call option since the weight of call option is a negative value.

(a) The histogram of loss values(b) The loss function

Figure 2: At 95% conﬁdence level (γ = 1.00)(a) The histogram of loss values(b) The loss function

Figure 3: At 95% conﬁdence level (γ = 0.75)

In addition, we consider the optimal portfolio with two interest rates, 0.0 and 0.1. The interest
rates of return affect the simulated values of option prices. When we focus on the case r = 0.0, we
conﬁrm that VaR values are smaller than CVaR values and that they are positive values. However,Yuri I MAMURA and Benyanee KOSAPONG(a) The histogram of loss values

25(b) The loss function

Figure 4: At 95% conﬁdence level (γ = 1.25)

it is no trend of changes when we consider the VaR and the CVaR after changing the interest rates
to 0.1. But CVaR values are slightly higher than VaR values. This is shown in Table 2.

(a) Interest rate 0.0(b) Interest rate 0.1

Figure 5: The optimal short position of call option(a) Interest rate 0.0(b) Interest rate 0.1

Figure 6: The optimal long position of put option

Lastly, the x-axis and y-axis of Figure 5 and 6 are γ values and the optimal solutions that weshould invest in call and put options from Table 2. We ﬁnd that there is a strong relationshipbetween γ and selling call options, or between γ and buying put options. As it might be seen, thedifferent interest rate of return makes the distinct graphs. The relative of interest rate return 0.1 is26

Portfolio optimization with conditional Value-at-Risk under CEV model

more than the relative of another one. Figure 5 looks like an exponential curve, and the proportion
for selling call options signiﬁcantly increases when γ value rises up. Nevertheless, the slopes of γ
values and the proportion for buying put options are slightly increased in Figure 6.

5

Discussions and Conclusions

This paper presents an investigation into the optimization of an options portfolio with the objective of minimizing CVaR value. The study employs the CEV model and utilizes numericalmethods, speciﬁcally the Euler-Maruyama method and Monte Carlo simulations, to calculate theloss function, VaR and CVaR. Subsequently, a numerical approach is employed to derive the optimal portfolio strategy. The results, as shown in Table 2, conﬁrm that all CVaR values exceed theirrespective VaR values, with an upward trend observed as the conﬁdence levels increase. Consequently, the proportion of cash decreases, while the proportions of call and put options increase
correspondingly. It is noteworthy that in the absence of an interest rate (i.e., when the interest rate
is zero), all VaR and CVaR values are positive, indicating consistent losses from these investments.
Hence, the interest rate emerges as a parameter that signiﬁcantly inﬂuences VaR and CVaR values.
The ﬁndings suggest that CVaR values are contingent upon the conﬁdence level and the interest
rate of return. Moreover, based on the data presented in Table 2, Figure 5, and Figure 6, it isevident that a higher value of γ correlates with a reduction in the number of call options (indicating increased selling activity) and an increase in the quantity of put options (reﬂecting augmented
buying activity). This connection assumes signiﬁcance from a ﬁnancial standpoint as a higher γ
signiﬁes greater volatility within the CEV model of the market, prompting investors to rely on
options as a risk mitigation strategy. Consequently, the numerical analysis provides evidence thatskillful construction of options portfolios can assist investors in managing volatility-related risks
in the market. Moreover, we observe that CVaR values and the composition of the minimization
portfolio depend not only on the conﬁdence level and interest rate but also on the parameter γ .
In future research, we aim to extend our portfolio optimization approach to other markets suchas Dow Jones and Nikkei 225, giving their status as large datasets that attract global investment
interest. Additionally, we aspire to enhance our model to optimize portfolios in scenarios involving
multiple maturity times, while considering various types of assets, including call and put options
with different strike prices.

Acknowledgements
The authors are grateful to the referee for the careful reading of the paper and for the comments
and detailed suggestions which helped us improve considerably the manuscript. The ﬁrst author
was supported by JSPS KAKENHI Grant Number 20K03731. The second author is grateful for
the Development and Promotion of Science and Technology Talents Project (DPST) scholarship,
which enabled opportunity for this project.

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