Performance of the Delta-Hedging Strategies Based on the Heston
Model and the Black Scholes Model on Meta Platforms
Yehong Li
School of Materials Science and Engineering, South China University of Technology,
Panyu District, Guangdong Province, China
Keywords: Heston Model, Black Scholes Model, Delta-Hedging.
Abstract: This paper researches the performance of the Delta-Hedging strategies and compares the different results
between the Black Scholes (BSM) model and the Heston model. In this study, parameters in two models are
calibrated separately by utilizing data on ten options on the stock of Meta Platforms. Then, after the two
models have been built, the price of the options and options’ delta values can be calculated. The delta hedging
portfolios are composed of one share of a call option and delta shares of the underlying stock. Finally, the
results of different strategies are compared. The sum of squared errors (SSE) of the BSM model is higher than
the SSE of the Heston model. It means that the Heston model prices the options more precisely than the Black
Scholes model. In addition, both two models can transfer the risk of fluctuating prices by implementing delta-
hedging strategies. This paper is helpful for investors to build their portfolios by using different models in
option pricing.
1 INTRODUCTION
Risk management is a significant topic of companies’
investment. Many companies utilize derivatives
instruments to transfer the risks and fix the profit they
could make. For the commodity suppliers,
implementing hedging strategies can avoid the risk of
fluctuating prices, and then the suppliers produce and
sell their products for making a relatively fixed profit.
Hedging strategy is also a popular way in the field of
investment. Investors can compose a portfolio by
purchasing a set of stocks and shorting index futures
to gain excess profits. When the hedging strategies are
implemented, using different models to price the
options may have different results. Fischer Black
derived a theoretical formula to price the options in
1973, and this Black-Scholes model has become one
of the most important models in option pricing. In the
BSM model, the volatility is assumed to be a constant
number, but the volatility in the real options market is
arbitrary (Black, 1973). Then, stochastic volatility
models were researched. Heston built up a stochastic
volatility model and derived a formula for calculating
a call option price. Since then, option pricing and risk
hedging become interesting topics in the financial
field (Heston, 1993).
To illustrate, Xiong used three delta hedging
strategies on Shanghai Stock Exchange 50 ETF and its
options, and they proved that three strategies can
significantly reduce investment losses (Xiong, 2017).
And Jerbi, Y used neural networks (NNs) to model a
European call option price and found that the NNs
method performed fairly well in option pricing (Jerbi,
2020). Similarly, JAG Cervera used artificial neural
networks to calibrate a call option price efficiently
(Cervera, 2019). Furthermore, D Lamberton
researched some properties of the American option
price and extended some results in the BS model to the
Heston model (Lamberton, 2019). Euch, O. E.
composed the portfolios by the forward variance curve
and the asset, which is theoretically perfect hedging
(Euch, 2018). In addition, L Goudenège utilized the
Heston model and the BSM model in valuing
Guaranteed Minimum Withdrawal Benefit and
presented the numerical results (Goudenège, 2019).
With the rapid development of the metaverse, the
stock of Meta Platforms has caused many investors’
concerns, since Meta Platforms is one of the biggest
companies in the field of the metaverse. However, the
performance of pricing options on the Meta Platforms’
stock is rarely discussed. Therefore, this paper focuses
on this matter.
366
Li, Y.
Performance of the Delta-Hedging Strategies Based on the Heston Model and the Black Scholes Model on Meta Platforms.
DOI: 10.5220/0012033200003620
In Proceedings of the 4th International Conference on Economic Management and Model Engineer ing (ICEMME 2022), pages 366-371
ISBN: 978-989-758-636-1
Copyright
c
2023 by SCITEPRESS – Science and Technology Publications, Lda. Under CC license (CC BY-NC-ND 4.0)
This paper is structured as follows: Section 2
shows the data and methods. Section 3 shows the
results of delta-hedging strategies in different models,
and also discusses the results. Section 4 concludes this
paper.
2 DATA AND METHODS
2.1 Data
The data is all from Yahoo Finance
(https://finance.yahoo.com). Meta Platforms’ stock is
chosen as the underlying stock in this paper since its’
options have a huge trading volume, so enough trading
data can be used to calibrate the parameters. The Data
of the underlying stock is collected from June 8th,
2022 to July 11th, 2022. The data of the options are
collected from June 24th, 2022 to July 11th, 2022. In
the BSM model, these data are used for calibrating
implied volatility for the underlying stocks. In the
Heston model, these data are used for calibrating three
parameters. Then, the option prices and delta values
can be calculated by using the two models. Finally,
two delta hedging strategies are constructed for a call
option of Meta Platforms’ stock from June. 27th,
2022, to July. 11th, 2022.
Table 1: The 10 options of the meta platforms
Call options chosen for calibration
Meta
Platfo
rms
META
22072
2C001
70000
META
22072
2C001
75000
META
22072
2C001
80000
META
22072
2C001
85000
META
220722
C0019
0000
Put options chosen for calibration
Meta
Platfo
rms
META
22072
2P001
65000
META
22072
2P001
55000
META
22072
2P001
50000
META
22072
2P001
45000
META
220722
P0014
0000
Table 2: The option of the Meta Platforms’ stock for
hedging.
Call option selected for hedging
Meta Platforms META220722C00165000
In detail, the options contracts that are collected to
calibrate parameters in different models for Meta
Platforms’ stock are shown in the table Ⅰ. And the
contract of the American call option selected for Meta
Platforms for delta hedging is shown in the table Ⅱ.
Also, the stock price trend for Meta Platforms is
shown in the Figure 1.
As shown above, the stock price of Meta Platforms
has been volatile from June 24th, 2022, to July 11th,
2022. In detail, Meta Platforms’ stock price
experienced a slip from June 27th, 2022, to July 5th,
2022, but it bounced back to the previous range on
July 8th.
The descriptive statistics of the rate of return of
Meta Platforms’ stock from June 24th, 2022, to July
11th, 2022, are shown below. As can be seen, Meta
Platforms’ stock has a large size range.
Figure 1: The price trend of Meta.
Performance of the Delta-Hedging Strategies Based on the Heston Model and the Black Scholes Model on Meta Platforms
367
Table 3: Descriptive Statistics.
Mean Median Variance Max Mini
value 0.007 -0.008 0.001 0.064 -0.037
2.2 Methods
In this study, two models are used for options pricing.
There are three steps in researching this project. The
first step is to calibrate the parameters in two models.
For the BSM model, Meta Platforms’ volatility is
calibrated by utilizing the open price of Meta
Platforms’ stock and the open price of ten options on
the underlying stock. For the Heston model, three
parameters are calibrated. Secondly, after the
parameters in models are calculated, the theoretical
price of call options and options’ delta values can be
calibrated in both models separately. Finally, two
delta hedging strategies are composed of one share of
a call option on Meta Platforms’ stock and delta shares
of the underlying stock. In comparison, one trading
strategy without hedging is also composed. The results
are shown and discussed in Section 3.
Some assumptions are proposed as follows: 1. The
dividend rate of underlying stock D=0; 2. The options
are American options, but they are treated as European
Options. Therefore, the options theoretical price can
be calibrated by the formula in the Heston model and
the BS formula; 3. In the Heston model, ρ equals 0,
which is the correlation between two Brownian
Motions.
In the BSM model, the stock price is described by
a geometric Brownian motion, which is shown in the
formula (1), where σ stands for the implied volatility,
W
t
stands for a Brownian motion. Furthermore, the
call and put options prices are calibrated by the Black-
Scholes formula (2) and (3), where N(d
1
) stands for
the normal distribution of d
1
. d
1
and d
2
can be
calculated by the formula (4) and formula (5) (Black,
1973), where T-t is the time to maturity, S
t
is the open
price of the stock at the time t, K is the strike price of
the option, and r is the interest rate, which is assumed
to be 3.13% in these models since the USA ten-years
Treasury bond yield on June 24th, 2022, equals to
3.13%.
𝑑𝑆
=𝜇𝑆
𝑑𝑡 + 𝜎𝑆
𝑑𝑊
(1)
C(t,S) = S
N(d
)−
K
e
()
N(d
)
(2)
P
(
t,S
)
= −S
N
(
−d
)
+
K
e
(
)
N
(
−d
)
(3)
d
=
log(
S
K
) + (r + 0.5σ
)(T − t)
σ
√
T−t
(4)
d
=
log(
S
K
)+(r−0.5σ
)(T − t)
σ
√
T−t
=d
−σ
√
T−t
(5)
In the Heston model, the stock price is also
described by a geometric Brownian motion, but the
volatility is modelled by the Cox-Ingersoll-Ross (CIR)
process (Heston, 1993). The volatility can be
described by the formula (6) (Heston, 1993).
𝑑𝑉
=𝜅
(
𝜃−𝑉
)
𝑑
+ 𝜎
𝑉
𝑑𝑍
∗
(6)
Where 𝑍
∗
is another Brownian motion, and 𝜌 is
the correlation between 𝑍
∗
and W
, which is assumed
to be 0 in this model. Kappa 𝜅, theta 𝜃, and sigma 𝜎
are three parameters that need to be calibrated. Some
different methods are used for calculating the option
price by constructing different characteristic function.
One of the methods is shown in the formula below.
Because the correlation is assumed to be 0, the
characteristic function is given by formula (7)
(Simonsy):
𝜙
(
𝑢,𝑡
)
=𝐸[exp
(
𝑖𝑢log
(
𝑆
)|
𝑆
,𝜎
]
=
=exp (𝑖𝑢(𝑙𝑜𝑔𝑆
+𝑟𝑡) ×exp (𝜂𝜅𝜃
(
(
𝜅−𝑑
)
𝑡
−2log
1−𝑔𝑒
1−𝑔
)
×exp(𝜎
𝜃
(
𝜅−𝑑
)
1−𝑒
1−
𝑔
𝑒
)
(7)
Where
𝑑=𝜅
−𝜃
(
−𝑖𝑢 − 𝑢
)
.
(8)
𝑔=
𝜅−𝑑
𝜅
+
𝑑
(9)
Carr, P utilized the fast Fourier transform in option
valuation. Referring to the method mentioned above,
a European call option price is calculated by the
formula (10) as below (Hunter, 2000):
𝐶
=𝑆
−
𝑆
𝐾𝑒
𝜋
𝑅𝑒[ exp
(
𝑖𝑢𝐾
)
∅
(𝑢
−
𝑖
2
)
1
𝑢
+
0
.
25
𝑑𝑢]
(10)
After the option prices are calculated, the sum of
squared errors is calculated for options on June 24th,
2022, by using the formula (11) below.
𝑆𝑆𝐸=
(𝑃
𝑖
(
𝑎
)
−𝑃
𝑖
𝑚
)
2
𝑃
𝑖
𝑚
𝑀
𝑖
=
1
(11)
Where 𝑃
(
𝑎
)
represents the theoretical option
price in the BSM model or in the Heston model, 𝑃
represents options’ market price, and M is the number
of options, which equals to 10 in these models. In the
ICEMME 2022 - The International Conference on Economic Management and Model Engineering
368
Black-Scholes model, the SSE is minimized to
calibrate the implied volatility. In the Heston model,
the calibration of the SSE is different. Since the
volatility is modelled by the CIR process, the SSE is
minimized to calibrate three parameters, which are
kappa 𝜅, theta 𝜃 and sigma 𝜎.
The data of the stock price from June 8th, 2022, to
June 23th, 2022 is utilized to calculate the daily yields
of the stock. Then the initial volatility of the
underlying stock v
0
can be estimated, which is
assumed to be the standard deviation of the daily
yields of the stock. v
0
equals to 0.2161 after
calibration. After the calibration of the parameters in
different models. Two delta hedging strategies are
composed by one share of a call option on Meta
Platforms’ stock and delta shares of the underlying
stock. Firstly, the stock price data from June 24th,
2022, to July 11th, 2022, and also the market price of
the option contracts chosen are collected. Secondly,
the theoretical prices of the call option can be
calculated on a daily basis by using the BS model and
the Heston model. Thirdly, the hedging portfolios are
composed by one unit of the call option, and delta
shares of the stocks. The delta value equals to N(d
1
)
for the BSM model. For the Heston model, the delta
value can be estimated by formula (12).
𝑑𝑒𝑙𝑡𝑎(𝑠)
=
𝐶
(
𝑠+∆𝑠
)
−𝐶(𝑠−∆𝑠)
2∆𝑠
(12)
Finally, the data of profit or loss is calculated by
implementing the delta-hedging strategies every day
from June 24th, 2022, to July 11th, 2022. The profit or
loss without hedging is also calculated. In detail, to
calculate the implied volatility in the Black Scholes
model, the Solver function in the Microsoft Excel
software is utilized by minimizing the sum of squared
errors. Also, a Python programme is cited (Jace,
2021). The function named optimize in the Python
module is used to minimize the SSE and calibrates the
parameters in the Heston model.
3 RESULTS AND DISCUSSION
3.1 Results
By minimizing the SSE, the parameters in two models
are calibrated. In the Black Scholes model, the
optimize sigma σ=0.498. The SSE equals to 0.799. In
the Heston model, the parameters are shown in the
table IV. The SSE equals to 0.772.
Figure 2: Comparision of profit/loss holding.
-3
-2
-1
0
1
2
3
4
5
6
7
8
Call option of Meta Platforms
without hedging hedging in the BS model
hedging in the heston model
Performance of the Delta-Hedging Strategies Based on the Heston Model and the Black Scholes Model on Meta Platforms
369
Table 4: The parameters of Heston Model.
𝝆 𝝈 𝜽 𝜿 𝑽
𝟎
value 0 0.0248 0.487 0.6839 0.2161
As shown in the Figure 2, the profit without
hedging always exceeds the profit with hedging in two
models. The trend of the profit with hedging in the
Black Scholes model is similar to that in the Heston
model. From June 27th, 2022 to July 11th, 2022, the
profit without hedging is $3.65, while the loss with
hedging by using the Heston model is $0.726. By
using the Black Scholes model, the loss with hedging
is $0.609.
3.2 Discussion
As shown in the Result section, the strategy of selling
the call option without hedging performs very well.
The reason is that the stock price of the Meta
Platforms decreased during that period. The market
price of the underlying stock fluctuates a lot. It
decreased from 171.32 to 167.07, from June 27th,
2022 to July 11th, 2022. Although this strategy makes
some profits, it does not mean that it has effectively
reduced the risk. In comparison, the performance of
hedging strategies performs well. The profit or loss
changes slightly on a daily basis.
In addition, the SSE in the BSM model is higher
than that in the Heston model. It shows that the Heston
model performs better than the BSM model in the
accuracy of option pricing.
There are some shortages in this paper. Firstly,
when calibrating the parameters in the Heston model
and the BSM model, the open price of the stock and
its options on June 24th, 2022 is used. These data are
discrete. If the continuous time series data is used, the
parameters in two models are revised in a period. The
theoretical options’ price and the delta values can also
be calculated with higher accuracy. Secondly, the
method that calculated the delta values in the Heston
model is not the precise way. In the Black Scholes
Model, the delta value equals to N(d
1
) and the delta
values are estimated by the formula (11) above. If the
formula of the delta values in the Heston model is
known, the delta value can be calculated in a more
precise way. Thirdly, some error in calculating the
options price is caused, because the American options
are considered the European Options. The American
option is a kind of path-dependent option. If the
formula to calculate an American option price is
known, the theoretical price and delta values of the
options can be calculated more accurate.
Finally, the correlation between the two Brownian
motions is assumed to be 0 in the Heston model. If the
correlation ρ is not 0, the formula for calibrating the
option price may be more complicated. However, the
option price can be calculated with higher accuracy.
4 CONCLUSION
This paper studies the performance of the delta-
hedging strategies on Meta Platforms by utilizing the
Heston model and the BSM model. Although
researchers have studied hedging strategies in the
Chinese option market. The topic this paper
researched has not been discussed before. Firstly, the
implied volatility is calibrated using the information
from five call options and five put options on the
shares of Meta Platforms in the BSM model. Three
parameters in the Heston model are also calibrated by
utilizing the same data. Then, after the Black Scholes
model and the Heston model are built, the prices of an
option and delta values of the option are calibrated
every day. Two strategies with hedging are composed,
including one share of a call option and delta shares of
the Meta Platforms’ stock. Finally, the performances
of the hedging strategies are compared. Compared
with the strategy without hedging, the hedging
strategies effectively avoid the risk of fluctuating
prices. Furthermore, when the Heston model is used to
calculate the option price and its delta values, the sum
of squared errors is lower than that when the Black
Scholes model is used. This paper proved that the
BSM model and the Heston model perform similarly
on Meta Platforms options. This result may help the
investors choose a proper delta hedging strategy to
avoid the risk and gain profits in the options market.
However, some assumptions are assumed to be
correct in Section 2, while these assumptions may not
perfectly accurate in the real financial market.
Therefore, the results of these two models deserve
more research and discussions in the future.
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