The pricing of options generated by diffusion processes, where diffusion depends on two groups of variables, was carried out. An algorithm for calculating the approximate price of derivatives and the accuracy of valuations has been developed, which allows to perform the analysis and to make precautionary to minimize the risk of derivatives pricing arising on the stock market. The method of finding the indicative price for a wide class of derivatives has been expanded. Using the spectral theory of self-adjoint operators in Hilbert space and the wave theory of singular and regular perturbations, an analytical formula of the approximate asset price was set, which was described by models with stochastic volatility dependent on l-fast variable and n-slow variable factors, and on local variable.
been used by many scientists, namely to forecast options prices [
Short-term interest rate dynamics models were considered in Vasicek's work [
Using spectral analysis, Linetsky [
. In particular, this theory applies to the short-term interest rates described by Vasicek's model. The approximate price of the bonds and their profitabilty are calculated. Applying the Sturm-Liouville theory, Fredholm alternatives, and analyzing singular and regular perturbations at different time scales, we obtained explicit formulas for the approximation of bond prices and profitability.
The goal of the article is to develop an algorithm for finding the approximate price of derivatives and to find explicit formulas for finding their value based on the development of eigen functions and eigenvalues of self-adjoint operators using boundary tasks for singular and regular perturbations. To set the theorem of estimating the accuracy of option prices approximation.
The main advantage over other developed methods is that finding the price of derivatives is reduced to solving the problem of finding the eigenvalues and eigenfunctions of a particular equation that fits this model.
II Methodology and Data
Let represent short interest rates. One of the most widely known models of short interest rates is the Vasicek model, in which is modeled as an Ornstein-Uhlenbeck process with multidimensional stochastic volatility. i.e.
The Ornstein – Uhlenbeck process is described by a second-order differential equation ща parabolic type
Let’s calculate the density of distribution of this process. To do this, consider the Cauchy problem for (1). With the initial condition ( ) ( ) ( ) where
( )– smooth finite function.
Let’s apply the Fourier transform. In particular, [
| √ ( ∫ ( ) ∫ ( ) )]
The Cauchy problem (3), (4) for a linear non-uniform differential equation in partial first-order derivatives is solved by the method of characteristics ( ) ) }) ) )}
)} ( ) ) (
√ ( ( ) ))
} ( ( { )} (
) ( ( ∫ ( ) ) √ ∫ ( ) ∫ { ( ( ) )}} { ( ) Let’s distinguish the complete square by ( ) (
( (
Indeed ∮( ) – closed contour, integral theorem, the integral is zero. √
( {( (
( ( ( ) ) ( ( ∫ ( )) ( √ √ )) ( ) (
( ) ) ∫ )) ( )) ( )) ( ) ) ) ) ) )} √ ) √ √ ) ) √
( ) )) ( ) – analytical function, so, by Cauchy's
( ) , the contour is arranged symmetrically will transform into ∫ , and ∫
∫ ( ) | || | with ( (
)) along the axis with if ∫ ∫ with therefore equality (7) holds.
| | With fixed |∫ |
| ∫ |
| ( )) as ( ), where ( ( (
))√ ) ∫ | then because Similarly, ∫
. Will get so fundamental solution or the Green's function has the form
On the other hand, on the probabilistic side, the Green's function is the density of distribution.
Using the methods of spectral theory and the theory of singular and regular
perturbations, we can find the approximate price of Ornstein-Uhlenbeck two-barrier
options with multivariate volatility, as a self-function decomposition using infinitesimal
generators of (l + n + 1)-dimensioned diffusions, that is,
diffusion depends on one local variable, the l-dimensional fast-variable factor and the
ndimensional slow-variable factor. This work is an extension of [
Process can represent many economic phenomena and processes. For example, inventory value, index price, reliable short interest, etc. More broadly, is an external factor that characterizes the cost of any of the above processes. By physical measure of process , we understand process , which has an instant drift ( ) and stochastic volatility ( ) ( ) , which contains both components: local ( )and non-local ( ). It should be noted, that infinitesimal generators for and have a form ( ( ) ( ) ) ( ( ) ( ) ) are characterized by the values and respectively. Thus, and an internal timeline and
Let’s consider and , to make the inner time scale small and the inner time scale – large. Therefore, fast variables, and are slowly variable factors. Note that and the form of the Ornstein-Uhlenbeck process [20] have
are have ̅ the right part of which has a form
̅ (9) Let’s reduce (9) to the equation , where , so that there is no first derivative in the obtained equation, that is, we look for a solution in the form ( ) ( ) where ( ) new unknown function, and let’s choose ( )in such way to have ( ) . (10)
Having reduced into (10), we will have ̅ , herefrom will have
( ) ( ) ( ̅ { ∫
̅ ) ( ̅ ) } ̅ ̅ ) For reasons of solution, let's make a substitution we will have therefrom
̅ √ ̅ ) (√̅ √ ∫ ) ( ( ̅ √ ̅ ̅ ) ( (
̅ ̅ ( ( ̅ ( ̅
) √ ̅ ̅ √
) ̅ )
) )
̅ ) √ ̅ ( ) ( √ ̅ ̅ √
) √ (
̅ √ ̅
) then √ ̅ (
̅ need solution to be equal 0 at points picture, so we will replace the variables ̅ and
, and , we’ll have ̅ ) ) √ ( ̅ ) ̅ and do not give such a ( ̅ or then on eigen values and eigen functions we will explore ̅ ̅
)
√ ( ̅ Let’s check the fulfilling of boundary conditions, if , with ( ) ( ̅
) (
( ( ) ( )) ∫
̅ ∫ ∫ { ̅
} ( ( ) )
| ∫( √ √ (
̅ ̅
̅ ̅ ) { ∫ ̅ ( ∫[ ( ̅ √ ) ) √ √
√ ( ) }
( ) ) | { ̅ ) ( ( | } ( √ ) ( ) ( )) { III Results and analysis
Let be the securities without paying dividends on an asset (for example, stock,
index, etc.). Often, is modelled as a geometric Brownian motion with constant volatility
(e.g. Black-Scholes formula) [
̅ For a double barrier option with and ( ) ( )
̅ ̅ barrier values, the payout is: ( )
( ) { } ( ) { }, ( ), ,
To calculate the value of this parameter, at first, it is needed to find the eigenvalues of the operator 〈 〉 with boundary conditions
( ) , ( ) .
It should be noted that a regular killing of boundary conditions at the ends and are entered. Equation
〈 〉 (〈 〉)
with the above boundary conditions can be found in [
̅√
,
and
On the basis [
) ( ( )
) ( ( ̃ ̅ ( )
̅ ( ( ( )
( ) ( ) ̅
( ) ( ( ) ) (( )( ( ̅ ) ( )) And for let’s find ( ̅
( ( ̅ ( ( ) ) ̅ (
) ( ) ( ) ̅ ) ) ( ( )) ̅ ) ̅ )) ̅ ( ̅ ( ( ) ̅ ( ) ( )) ) ( ) ( ) ̃ Calculation (
( ) ̅ ( ̅ ( )( ( ) ( ) ( ) ( )) ( ) ( ) ( ) ̅ ( ̅ ( ) ( ) ( ) ( ))
( ) ( ) (√ { ( ̅
) } (
) ̅ (
{ ( ) ) ̅̅̅̅ { ̅ ̅ ) ̅ ) } (
̅ ( ̅
) } ̅ ) ̅ ) ) ( ) (√̅ ( ̅ ) √ √ ̅ )
̅ ) ̅ { (̅ ( ( ̅ ) ̅( ̅ ) (̅ ( ̅ ) } ̅ ) ̅) ) After replacing the variable √ (
̅ ( ( )
( ))
Because therefrom it follows that
( ) ) ( ) ∫ (
) ( ) ) √ { ∫ ( ) and :
By integrating by the parts, given that such an integral contains pair degrees decreases and reduces to ∫ and integral which contains – not pair equals zero, times, taking by parts we will get ( ( ) ( ))
To find the price of the bond with payments ( ) { } , it’s needed to solve
the equation (15) on finding the eigenvalues at the segment ( ) with 〈 〉,
according to (15). As both ends and are natural boundaries, then the solution has a
form [
̅ ̅̅̅̅ {( ) ) ( ( ̅̅̅̅ {( ) ̅ )
̃ are written on the basis of recurrence ratios:
̅
For zero coupon bonds, the profitability curve is considered more often rather than the price of the bond itself. Return in zero-coupon bonds, for which one dollar is paid at time is determined by the ratio:
Let’s get an approximation for a zero coupon bond, sorting it out both bond prices and return by degrees √ and √ : ( ( √ ̅ √ ̅ ) ) √ ̅ )
( )
( )( ̅
̅ √ ̅ ̅ ( ( ) ) ̅ )] ̅ ( ( ̅√ )
( ) √ ( ) ) ) ) )] ̅
) )
} (
) ̅ ( ( ) ̅ (
} } ̅ ) ) } } } } } ∑ √
∑ √ { ( ∑ √ ∑ √ ) } Grouped by degrees√ and √
we will get: ( ) ∑ √ ∑ √
Note: the drawings are built by component in each corresponding timeline, similarly
for the two components as in [
IV Conclusions
Thus, the studies conducted in the work allow us to draw the following conclusions.
An algorithm for finding the approximate price of derivatives has been developed and explicit formulas have been found for finding their value based on the decomposition of eigen functions and eigenvalues of self-adjoint operators using boundary tasks for singular and regular perturbations. The theorem of estimating the accuracy of derivatives prices approximation is established, on the scales of systems of slow and fast variable factors on which volatility of derivative financial instruments depends.
The general method of finding the approximate price for a wide range of derivatives has been obtained. It has been established that derivative payments can be pathdependent, and the underlying process may exhibit a jump whose intensity depends on multidimensional volatility. The price of options depends on the stochastic multidimensional volatility, which is described by a path-dependent process. Finding the price of derivatives comes down to the task of finding the eigenvalues and eigenfunctions of a particular equation that fits this model.
The approximate price of bonds and their profitability are determined by the methods of spectral theory and wave perturbation theory. The spectral theory and the theory of singular and regular perturbations have been applied to short-term interest rates described by the Vasicek model. The approximate price of the bonds and their profitability are calculated.
The main advantage of the reviewed pricing methodology is that, by combining methods with spectral theory, regular perturbation theory and singular perturbation theory, it reduces to solving equations on finding eigenfunctions and eigenvalues.