=Paper=
{{Paper
|id=Vol-2927/paper7
|storemode=property
|title=Derivative Pricing: Predictive Analytics Methods and Models
|pdfUrl=https://ceur-ws.org/Vol-2927/paper7.pdf
|volume=Vol-2927
|authors=Ivan Burtnyak,Anna Malytska
}}
==Derivative Pricing: Predictive Analytics Methods and Models==
https://ceur-ws.org/Vol-2927/paper7.pdf
83
Derivative Pricing: Predictive Analytics Methods and
Models
Ivan Burtnyak 1[0000-0002-9440-1467], Anna Malytska2[0000-0003-5811-9288]
1,2
Vasyl Stefanyk Precarpathian National University, Ukraine, bvanya@meta.ua,
hanna.malytska@pnu.edu.ua,
Abstract. The article deals with the study of pricing and calculating the
volatility of European options with general local-stochastic volatility applying
Taylor series methods for degenerate diffusion processes, in particular for
diffusion with inertia. The application of this idea requires new approaches
caused by degradation difficulties. Price approximation is obtained by solving
the Cauchy problem of partial differential equations diffusion with inertia, and
the volatility approximation is completely explicit, that is, it does not require
special functions. If the payoff of options is a function of only x, then the
Taylor series expansion does not depend on and an analytical expression of
the fundamental solution is considerably simplified. We have applied an
approach to the pricing of derivative securities on the basis of classical Taylor
series expansion, when the stochastic process is described by the diffusion
equation with inertia (degenerate parabolic equation). Thus, the approximate
value of options can be calculated as effectively as the Black-Scholes pricing of
derivative securities.
Keywords: stochastic volatility, European options, degenerate diffusion
processes, Kolmogorov equation.
1. Introduction
Derivative prices tend to change and predict their behavior is becoming more
and more complicated. Stochastic processes that are described by the equation of
diffusion with inertia are widely used in the theory of mass service, in particular in the
theory of queues. Often, such processes occur on financial markets with the pricing of
European and Asian options. the theory of pricing derivatives and the study of the
behavior of volatility for the analysis of profitability are necessary for flexibility in
the adoption of managerial strategic decisions by managers. The validity of strategic
decisions allows managers to make step-by-step additional investments in order to
maintain strategic positions of the company in the stock market. Typically, a high
level of volatility gives the manager more opportunities to change their decisions in
the future. Volatility is important for traders when pricing several different series of
options with different execution rates and maturities. Since is stable in the Black-
Copyright © 2021 for this paper by its authors. Use permitted under Creative
Commons License Attribution 4.0 International (CC BY 4.0).
84
Scholes model [6], speculative price changes can occur in these cases, which can not
be verified without the presence of a certain kind of specialized knowledge. This is
one example of how volatility pricing options are related to the principles of modeling
and risk management in the financial market. Therefore, it is necessary to use models
with variable volatility.
The purpose of this article is to introduce a unified approach to pricing and find the
implied volatility of European options, which are described by degenerate diffusion
processes of diffusion type with inertia, based on classical Taylor series
approximation. Therefore, the approximate value of options can be calculated as
effectively as the price of Black-Scholes [6] for European options.
2. Literature Review
Modeling and forecasting of pricing dynamics, using financial instrument is
an important element of investment activity, as they have a high level of financial
leverage and fulfillment of obligations is realized in the future. Derivative prices tend
to change and predict their behavior is becoming more and more complicated.
Various operations with financial products suitable for selling and buying are carried
out using derivative financial instruments [1]. The effective investment of decision
making while operating in the Ukrainian stock market is the major factor of investors’
interest in it. Securities market participants should have a good understanding of
derivative pricing to achieve successful financial results. Derivative securities
transactions occupy an important place in the stock market financial activity, since
each participant must hedge the risks to obtain extra profit based on stock market
speculations [2]. Therefore, the derivatives are one of the major instruments in the
securities market. An important task is to study the state and dynamics of the
domestic stock market in close interrelation with other countries’ stock markets and
analyze the volatility of financial instruments to increase the efficiency of investment
operations. Nowadays many approaches have been developed to calculate local and
stochastic volatility that describe the overall dynamics of underlying price using CEV
models [4, 12], JDCEV [7]., Heston-model [11], SABR-model [16], but the
application of these models requires the use of special functions and numerous
integrations of complex functions. Hence, this may lead to miscalculations, but
considering the time law (dependency), a considerable number of the models become
unstable. Other methods for pricing derivative securities are required to get direct
calculations [9]. The use of Taylor series expansion depends on the model structure,
specifically, on function properties which are used in the model and model ability to
maintain the stability at time change.
We consider models without default of diffusion process with inertia, with
coefficients depending on the variables ( ), for derivatives pricing we use Taylor
series expansion for degenerate diffusion processes. In particular, complete
correlation matrices are non-degenerate in the following works [4, 13, 14], and in our
work they are degenerate, so the application of this idea requires new approaches
caused by degradation difficulties
The purpose of this article is to introduce a unified approach to pricing and find the
implied volatility based on classical Taylor series approximation. Therefore, the
85
approximate value of options can be calculated as effectively as the price of Black-
Scholes [6] for derivative securities.
3. Methodology and Data
We consider market without arbitration, zero interest rate and no dividends.
Without losing generality these considerations may be extended to deterministic
interest rate. We consider probabilistic space with martingale measure , with
filtering { } defining market history. Let asset S represent such phenomena as
stocks, price index, a reliable short-term interest rate i.e. and where { }
and processes and are set by such system of equations
( ) ( )
( ) (1)
where is the stopping time { ∫ ( ) }, with the
exponential distribution , which does not depend on X, the drift function μ has the
form
( ) ( ) ( ),
Let be a non-arbitrary price of the European option, which at time T is a
gain ( ) [3], that
∫ ( )
{ } { ( ( ) )| }
( ) ( ) ( ) To calculate the price of the European option, we
∫ ( )
must calculate the mathematical expectation from ( ( ) ) in
particular
∫ ( )
( ) { ( ( ) )| } (2)
The function ( ) satisfies the Kolmogorov equation of diffusion with
inertia
( ) ( )| ( )
(3)
where operator has the form
( )( ) ( )( )
(4)
where ( ) is equal to
( ) ( )
86
If we consider deterministic interest rates then we need to calculate the
mathematical expectation of this form
( ̃
̃( ̃ ) { ∫ )
( (̃ ) )| ̃ ̃ }
̃ ( )
A direct verification proves that ( ( ̃) ) satisfies (3).
An approximate solution to the Cauchy problem for diffusion equation with
inertia (3) is obtained by adapting the ideas of [5,10] which are applied to non-
degenerate diffusion processes. We consider degenerate diffusion processes with
inertia on which singular integro-differential pricing operators of Levy type are
applied. We have introduced a unified approach to pricing and estimation of implicit
volatility using Taylor series expansion. We consider that and are infinitely
differentiable functions of variables ( ), continuous on and bounded ( )
[ ]
Let ( ̅ ̅) R2 be a fixed point then any analytical function ( ) we
may write down Taylor series expansion
( ) ∑∑ ( )( ̅) ( ̅)
( ) ( ̅ ̅)
( )
Applying the above considerations to the coefficients and , we obtain
that operator in (4) has the form
∑ ∑( ̅) ( ̅) ( )
where { } are differential operators of the
( )( ) ( )( )
Operator of a parabolic type and has a degenerate parabolicity on
(since there is no second order derivative on ), which usually operates in financial
spaces, that is, in the financial markets. Based on Taylor series expansion for , the
pricing function has the form [8]
∑ ( )
Substituting (5) and (6) into (3), we obtain the Cauchy problems for diffusion
equations with inertia.
( ) ( ) ( ) (7)
87
( ) ∑ ( ) ( )
We construct a fundamental solution of the homogeneous Cauchy problem
(7) and use the properties of the fundamental solution (7) to find as a solution
of the problem (8), while we use only the properties of the distribution function for
the diffusion equation with inertia expressed by the classical Chapman-Kolmogorov-
Planck equations and the Duhamel’s principle are applied to partial derivatives
equations.
Let’s consider the Cauchy problem (7). Operator is a degenerate parabolic
diffusion operator with inertia, or a Kolmogorov operator with time-dependent
coefficient . Thus, the solution has the form
∫ ( )
( ) ∫ ( ) ( ) ( )
where ( ) is a two-dimensional Gaussian density of diffusion
process with inertia.
We find ( ), so we solve the Cauchy problem for the diffusion
equation with inertia and variable coefficients depending on
( )( ) ( )( ) ( )
( )| ( ) ( )
We apply Fourier transform to solve the Cauchy problem (10), (11)
assuming that are completely integrable on ( ).
( ( )) ∫ { } ( ) ( )
( )
Since
( ( )) ( )
( ( )) ( )
( ( )) ( )
( ( )) ( )
then we will have
( ) ( ) { ( )[( ) ] ( ) } ( ) ( )
( )| ̃( ) ( )
where ̃ ( ) ( ( )).
The problems (10), (11) are reduced to (12), (13) so these are the Cauchy
problem for a linear differential equation with partial first order derivatives [10].
We formulate the corresponding characteristic equation
88
( ( ) ( ) ( ) ( ))
Equation of characteristics
( ( ) ( ( ) ( )) ( ))
So, we have the first integral from
( ( ) ( ) ( ) ( ))
we have
∫ [( ) ( ) ( )( )]
taking into consideration that , we have
( ) {∫ {[ ( ) ( )] ( )
( )( ( ) )} },
providing that
( ) ̃( ) ̃( )
( ) ̃( ) {∫ {[ ( ) (
)] ( ) ( )( ( ) )} },
since , then
( ) ̃( ( ) ) {∫ {[ ( ( ) ) ( ( )
)] ( ) ( )( ( ) )} },
We take an inverse Fourier transform from v (t, ξ, η)
( ( )) ∫ { } ( ) ( )
( ) ∫ ̃( ( ) ) {∫ {[ ( ( ) )
( ( ) )] ( ) ( )[ ( ) ]} { } .
(14)
In (14) we will substitute the variables
89
( ) ( )
{
then
( ) ∫ ̃( ) { ( ( )) ∫ {[ ( (
) ) ( ) ] ( ) ( )[ ( ) ]} } (15)
by substituting the value ̃ ( ) ( ) in (15) we have
∫ ( )
( ) ∫ ( ) ( ) .
(16)
where ( )is the density of the two-dimensional diffusion
process with inertia and with coefficients dependent on a time variable.
( )
( ) (∫ ( ) ) { ( ∫ ( ) ) ( ∫ ( ( )
( )) )
( ) [ ( ) ( ∫ ( ( ) ( )) )∫ (
) ( ) (∫ ( ) ) ∫ ( )( ( ) ( )) ] }
( ) ( )
( ) is the initial point, ( ) is the end point (current),
∫ ( ) ( ) (∫ ( ) ( ) ) (∫ ( ) )
First, we solve the Cauchy problem (8) with with payoff ( )
( ) ( ). We have
( )
( ) ∫
∫ ∫ ( ) ( )
( )
∫ ( ) ( )
∫ ( )
We multiply both sides by and using (9) we have
( ) ( ) ∫ ( )
Using method of mathematical induction and taking into consideration
properties of the fundamental solution, we have
( ) ( )
90
∑ ∫ ∫ ∑ ( )( ( ) ( )
( ) ( ) ( )) ( ( ) ( ) (
) ( ) ( )) ,
{ ( ) | }
( ) ( ̅ ̅), ( )|( ) ( ̅ ̅) ( ),
( ) ∑ ( ) ( ) ( ( )) ( ( ))
where
( ) ( ∫( ( ) ( )) ∫ ( )
(∫ ( ) ( ) ( )∫ ( ) ) )
( ) ( ∫( ) ( ) (∫ ( ) ) ∫( ( )
( )) ∫( )( ( ) ( )) ∫( ) ( )
( ∫( ) ( ) (∫ ( ) ) ) (∫ ( ) ( )
( )∫ ( ) ) ) ,
The results of the asymptotic approximation are proved in [13,15]. These
formulas are considerably simplified when the coefficients do not depend on time.
4. Results and analysis
The degenerate diffusion process that describes price dynamics and implicit
volatility, depending on time and ( ), indexes of base assets, stocks, options of
financial flows in the method of calculating the company's rating based on the
available quotations of securities, is considered in the article. The research was
conducted on the basis of Taylor series. The density distribution of the probabilities of
passing this process is constructed. With density distribution, you can find , you
can find the price ( ) at any given time. The method of successive finding of
the price is developed when the coefficients depend on ( ). Using approximate
approximations using the Black-Scholes price function as an initial approximation, we
obtain explicit formulas for finding the initial approximations of implicit volatility.
We note that they are in the form of the record match the formulas [13], but in this
case they are more complicated by the degeneration of the equation and the formula
for the approximation of the price.
The obtained implications of implicit volatility and consistent price
approximations make it possible to analyze the process of passage in the financial
91
market. Make corrections and concrete steps to improve the situation for optimizing
financial strategies.
A Taylor series expansion method for the Kolmogorov equation (degenerate
Heston model) is used.
( )
with the initial function, which is the Black-Scholes value-dependent volatility based
on S&P500 options data in the time interval from November 1 to December 31, 2020.
The exact values of changes in volatility and profitability are found, which allows
managers to predict the process of forming a portfolio, financial flows, and changes in
pricing.
Figure 1: The implied volatility obtained by the Ordinary Least Squares and our
second order approximation the Taylor series for the degenerate Heston model.
.
Figure 2: The yield curve first and second order approximation the Taylor series for
the degenerate Heston model. .
In the example of the degenerate Heston model, which describes the dynamics
of pricing and the development of implicit volatility, the initial approximation of the
Black-Scholes price function made volatility and yield calculations based on the
second Taylor approximation and the the Ordinary Least Squares. The results
obtained are almost identical, indicating a high accuracy of approximation.
Knowledge of the approximate price and implied volatility at each step at a
fixed time gives an opportunity to develop a strategy for managing the dynamics of
derivative prices in financial markets and to avoid speculative changes in pricing.
92
The density of distribution for degenerate diffusion processes is constructed
when the coefficients depend only on time (7).
The method of constructing the density of distribution in the case of
coefficients dependence on time and spatial variables is developed (3), when the
correlation matrix is not strictly positive definite.
Knowing the density of distribution, one can always find the price by the formula
(13) where the role is played by the density of distribution of probabilities of the
investigated economic process.
5. Conclusions
This paper expands the methodology of approximate pricing for a wide range of
derivative assets. Price approximation is obtained by solving the Cauchy problem for
differential equations in partial derivatives of diffusions with inertia. If the payoff of
options is a function of only x, then the Taylor series expansion does not depend on
and an analytical expression of the fundamental solution is considerably simplified.
We have applied an approach to the pricing of derivative securities on the basis of
classical Taylor series expansion, when the stochastic process is described by the
diffusion equation with inertia (degenerate parabolic equation). For a degenerate
parabolic equation, the approximate price of options is fairly simple since it uses only
estimates of derivatives of the distribution density of diffusion with inertia, we
obtained explicit formulas for derivatives price approximations.
References
1. Aboulaich, R., Baghery, F. & Jrai A. Option Pricing for a Stochastic Volatility
Jump-Diffusion Model, International Journal of Mathematics and Statistics, 13 (1),
pp. 1-19 (2013).
2. Andersen, L. “Option pricing with quadratic volatility: a revisit”, Finance and
Stochastics, Volume 15, Issue 2, pp. 191–219 (2011).
3. Burtnyak, І.V. & Malytska A. On the Fundamental Solution of the Cauchy Problem
for Kolmogorov Systems of the Second Order.Ukrainian Mathematical
Journal.Volume 70, Issue 8, 1 January, 1275-1287 (2019).
4. Burtnyak, І.V. & Malytska A., Gvozdytskyi V. Modelling of the derivatives pricing
with multifactor volatility. // Proceedings of the Workshop on the XII International
Scientific Practical Conference Modern problems of social and economic systems
modelling (MPSESM-W 2020). 92-108, (2020).
5. Burtnyak, І.V. & Malytska H.P. Construction of the fundamental solution of a class
of degenerate parabolic equations of high order. // Carpathian Math. Publ. 12 (1), 79-
87 (2020)
6. Black, F & Scholes, M. The Pricing of Options and Other Corporate Liabilities.
Jounal of Political Economy. Vol. 81, No. 3, pp. 637-654 (1973).
7. Carr, P. & V. Linetsky. A jump to default extended CEV model: An application of
Bessel processes. Finance and Stochastics 10 (3), pp. 303–330 (2006).
93
8. Corielli, F., Foschi, P. & Pascucci A. “Parametrix approximation of diffusion
transition densities”, SIAM Journal on Financial Mathematics, Volume 1, pp. 833–
867 (2010).
9. Forde M., Jacquier, A. & Lee, R. “The small-time smile and term structure of
implied volatility under the heston model”, SIAM Journal on Financial Mathematics,
Volume 3, Issue 1, pp. 690–708 (2012).
10. Gatheral, J., Hsu, E. P., Laurence, P., Ouyang, C. & Wang, T.-H. “Asymptotics of
implied volatility in local volatility models”, Mathematical Finance, Volume 22, Issue
4, pp. 591–620 (2012).
11. Heston, S. (1993). A Closed-Form Solution for Options with Stochastic Volatility
with Applications to Bond and Currency Options, Review of Financial Studies, 6 (2),
pp. 327–343 (1993).
12. Lindsay, A.E. & Brecher, D.R. Simulation of the CEV Process and the Local
Martingale Property, Mathematics and Computers in Simulation, 82, pp. 868-878
(2012).
13. Lorig, M. The exact smile of certain local volatility models. Quantitative Finance
13 (6), pp. 897–905 (2013).
14. Mendoza-Arriaga, R., Carr, P., & Linetsky V. Time-changed Markov processes in
unified creditequity modeling. Mathematical Finance 20, pp. 527–569 (2010).
15. Pagliarani, S., Pascucci A., & Riga C. Adjoint expansions in local Levy models.
SIAM J. Financial Math. 4, pp. 265–296 (2013).
16. Ren, Y, Madan, D. & Qian, M.Q. “Calibrating and pricing with embedded local
volatility models”, Risk, pp. 138–143 (2007).