Comparing of some sensitivities for nonlinear models comparing of some sensitivities (Greeks) for nonlinear models of option pricing with market illiquidity Текст научной статьи по специальности «Математика»

Математические заметки СВФУ Апрель—июнь, 2019. Том 26, №2

UDC 517.957+336.76

COMPARING OF SOME SENSITIVITIES (GREEKS) FOR NONLINEAR MODELS OF OPTION PRICING WITH MARKET ILLIQUIDITY M. M. Dyshaev and V. E. Fedorov

Abstract. We discuss the numerical solving of nonlinear options pricing models to a market with the insufficient liquidity. Also for these models the sensitivity coefficients of the option price (Greeks) were found numerically. These nonlinear models were selected by us on the basis of our group classification of a general model and were previously obtained in the works of Frey and Stremme, Sircar and Papanicolaou, and Schonbucher and Wilmott. The behavior of the price and its sensitivity coefficients in the nonlinear models and in the linear Black—Scholes model is compared. The results of the comparing presents in the form of graphs, a brief comparative analysis of them was made.

DOI: 10.25587/SVFU.2019.102.31514 Keywords: options pricing, nonlinear Black—Scholes type model, illiquid market, sensitivities (Greeks), numerical solution.

Introduction

The classical options pricing models of [1,2] or [3] are based on the perfect market hypothesis, in this case participants use only the prices, presented at the market. It is assumed, that traders with their operations can not influence on the prices. These models describes the pricing of options with a sufficient liquidity, absent transaction costs and a constant volatility of underlying assets. In the case of insufficient liquidity, it is necessary to take into account the effects of deals on the prices.

Recently, many authors have been investigating the issues of accounting for the effect of hedging deals on the prices of the underlying asset. [4] presented a model which assumes the dividing on reference and program traders. Traders of the first type are investing to make a profit due to the growth of the assets. A second type trader performs operations on insuring its option portfolios, following dynamic hedging strategies based on the Black-Scholes model. Model of R. Frey and A. Stremme has the form

2

a2x2Cxx + r(xCx - C) = 0, (1)

The work was supported by the Russian Foundation of Basic Research, under grant 2019-01— 00244.

© 2019 M. M. Dyshaev and V. E. Fedorov

Ct + т:

1 — рф(Ь, x)

1 - p<t>{t,x) - px^(t, x)

where <(t, x) is the normalized function of the program traders' hedge strategy, p is the fraction of the program traders, r is the risk-free interest rate, t is time in the model, x is the price of the underlying asset, C is the price of the call option.

In the model of [5] there are two types of the market participants also: a large trader and many small traders. The large trader uses a trading strategy that is not necessarily aimed at options portfolio hedging. The trading strategy f (x,t) is the sum of the underlying assets, which the large trader wants to hold in the portfolio at any time. The Schonbucher-Wilmott model equation is written (with the preservation of the authors' notations), as

1

dX{x,W,t) \ 2

ptnhwm7î7îT pxx + r{xpx-p) = o, (2)

2 \ dX(x,W,t) df(x,t) dx dx

where x(S, W, t) is the excess demand of the reference traders, W is a stochastic process of entering information on the market, P is the price of a put option.

To describe the pricing of options, [6] obtained a family of nonlinear partial differential equations

V(1 - pCx)U'(V(1 - pCx))

|_V(1 - pCx)U'(V(1 - pCx)) - pxCx

a2x2Cxx + r(xCx - C) = 0, (3)

where V(•) = U-1(-) (the inverse function), and U(•) is the relative demand function of the reference traders. In the absence of the program traders (p ^ 0+), the model reduces to the classical Black-Scholes model.

Remark 1. If in equation (3) we make the substitutions pC = u, V'/V = v and use the derivative formula of the inverse function, then the equation can be written in the more compact form

2 2

@ Uxx / \ / \

ut + —--,-7-—+r{xux-u)= 0. (4)

(1 - xv(ux)uxx)2

Remark 2. Equation (2) also reduces to equation (4), see [5,7] for details. This model corresponds to the case v(ux) = fi/ux in (4).

Using the group classification of equation (4), obtained in the works [7-10], authors single out several cases of the free functional parameter v, which are nonequi-valent in the group structure sense. These cases correspond to the all widest symmetry groups of the equation. For the completeness of the comparison, authors also consider the case v (ux) = 0, which is not included in the classification, it corresponds to the Black-Scholes linear model:

0, if the model does not take into account the demand, it is the Black-Scholes model; v(ux) = < /3, if the demand function is logarithmic: U(z) = jj lnz + A; (5) P/ux, if the demand is described by the power function U(z) = Az , A is a constant.

2

Further, we shall consider the logarithmic and the power functions of the demand, and LGR model and PWR model respectively. In this paper, for these models calculations and a comparative analysis with the linear Black-Scholes model (BSM model) will be performed.

Black-Scholes type equations are actively investigated with using various approaches, including numerical methods. In the article [11], explicit and implicit schemes have been constructed for finding a numerical solution of the linear Black-Scholes equation. In the works [12-18], difference schemes for various nonlinear Black-Scholes equations are obtained. [19, 20] investigated the convergence of numerical solutions for some nonlinear Black-Scholes type equations.

In addition to searching for a numerical solution of the equation, it is of interest to calculate the dynamics of the main indicators of the option price sensitivities to changes in the basic market parameters. This indicators widely used in the practice of the options trading and the portfolio insurance. Their common name in the market community is Greeks or sensitivities. They reflect the dependence of the option price (and these indicators themselves) on changes in the market parameters, such as time, the volatility or the price of the underlying asset. About 30 such indicators are used, and the most detailed, apparently, their description and analysis for the linear model of Black — Scholes are given in the book of [21, table 2.1, p. 22].

In most studies on the numerical integration of nonlinear Black-Scholes equations, only some price sensitivities are searched and compared, often just so-called Delta and Gamma. This work aims to partially fill the gap for other sensitivities also,

i. e. to find numerical values of Greeks for the nonlinear models, mentioned above. Here numerical solutions of the initial-boundary value problem, corresponding to European call option, for equation (4) with v from (5) are found, several different Greeks are calculated for the nonlinear models and compared with the values of the same sensitivities for the linear Black-Scholes model, which are also calculated here. For other Greeks, one can see a preliminary results in the preprint [22].

This article is organized as follows. Section 2 is devoted to the construction of a difference scheme for the search of numerical solutions for the initial-boundary value problem, which corresponds to the call option in our nonlinear models. The values of the numerical solutions are used for the finding of Greeks values. In Section 3 details of the sensitivities calculation are described. Section 4 gives brief sense descriptions of Greeks, which are calculated in this work. Also the results of the calculations are presented here in the form of the comparative graphs.

2. Construction of the difference scheme

The first aim in this work is to find a numerical solution of the initial-boundary value problem for European call option a2 x2u

ut + 7TTi-F~T-^+r(xux-u) = 0, (x,t) £ [0,+oo) x [0,T], (6)

2(1 - xv(ux)uxx)2

u( x t)

u(0, t) = 0, lim --——— = 1, u(x, T) = max{i — K, 0}, (7)

x^to x — Ke

where K > 0, a > 0, r > 0.

In this article, only the replacement of time direction and the truncation of domain on the right are performed. Thus, in equation (6) time was replaced by t' = T — t, so the time in the problem becomes reversed. Note that this truncation is done according to the method described in [23]. As result, we obtained the initial-boundary value problem (again marking the variable t' as t) 2 2

ut-on a x Uxx-- - r{xux - u) = 0, (x, t) £ [0,1] x [0,1], (8)

2(1 — xv (ux) uxx)2

u(0, t) = 0, u(1,t) = 1 — Ke-rt, u(x, 0) = max{x — K, 0}, (9)

where K > 0, a > 0, r > 0.

For numerical solution of problem (8), (9), the stencil for six-point two-layer implicit-explicit scheme with weights was used. The corresponding difference scheme in the case 0 = 1/2 is called the Crank-Nicolson scheme (see [24]). In our scheme, the values of the sought function utt+1 on the layer m + 1 have the weight 0, and the values from the previous layer are considered with the weight 1 — 0.

To find an approximate solution of problem (8), (9) the difference representations of the function u(x,t) and its derivatives

um+1 _ u m

1 i il o\ „,m un un

u - eum+1 + (1 -

Ut

T

um+1 _ um+1 m _

Ux~en+1hn +(i-e)n+1hUn, (io)

u m+1 _ 2um+1 + um+1 um _2um + um aan+1 2un + un-1 . /, n\an+1 2un + un-1 UXX ~ 0--2- + (1 - 0)--2-,

were used, where t is the grid spacing for t and h is the constant grid spacing for x.

For the discretization of boundary and initial conditions, h and t were specified. The following algorithm is proposed. In the domain [0,1] x [0,1] the uniform grid xn = nh is defined with the constant grid spacing h and n = 0,1,... ,N for the price x, and tm = mT with a variable grid spacing t for time t. Hence the number M = M (t) of the time steps depends on the choice of t .

The way for specifying t is to vary the value of t from layer to layer:

h2 h2 \ Atn_ *2(xmf

Tm+1 = min —-r—T , All =

m _

2 ' 2 max (A™) J ' ~ 2(l - v™x™{uxx)™)2 '

where At is calculated for each node k in the layer m, and 0, for the Black-Scholes model;

= ^ P, for the model with the logarithmic demand function; (11)

for the model with the power demand function.

0h

The boundary conditions for the difference equation have the following form. The left boundary condition u(0, t) = 0 corresponds to

um = 0, m = 1, 2,..., M.

n

The right boundary condition lim = 1 is transformed to the condition

u^ = 1 - Ke-

m = 1, 2,. .., M ; 0 <K< 1.

The initial condition u(x, 0) = max{x — K, 0} corresponds to the condition

u°n = max{nh — K, 0}, n = 1, 2,...,N.

Substituting difference representations (10) into the equation (8), we get a system of linear equations

n um+1 — h um+1 + c um+1 = d nn un-l hnun + cnun+1 - dn

1, 2.....N - 1,

1, 2.....M - 1,

where

O-n =--

2( 1 — vmnh(uxx)m)

1 Oa2n2

bn = - - +

t (1 — <nh(ux*)m)2

+ 6rn + r6

— vmnh(uxx)m)

— Orn,

dn =

(1-

(1 — vmnh(uxx)m)

— (um —2um + um \

2 \an+1 2un + un-l)

(12)

- r(i - e)um+r(i - e)n(um+1 - um),

The tridiagonal matrix algorithm is applicable in this case, since the condition |bn| > |an| + |cn| for the predominance of diagonal elements is satisfied (no division by zero).

Note that for the denominator of the fractions in (12) and for the free functional parameter v(ux) in (11), the differential representations of the derivatives on the layer m (not m +1) are used. These values are already calculated at the previous time step. But, if we shall use the differential representation of the second derivative (and of v(ux)) on the layer m + 1 in the denominator of the fraction, the tridiagonal matrix algorithm will not be applicable. For each step of the computation, we shall obtain a nonlinear system of equations.

n

m

Oa2 n2

2

Oa2n2

cn

2

a2n2

1

un +

3. Calculation of the sensitivities

The sensitivity coefficients (Greeks) are partial derivatives of the function u(x, t) by various parameters of the model, which describe market indicators. The names, formulas and brief descriptions of each of Greeks will be given below. Here we describe the general procedure for their calculation.

To calculate Greeks, characterizing the change in the function u(x, t) depending on the change in x, a or r, the constant step of the argument is taken. When calculating the time derivative, the current value of t is used as the argument step.

The approximate value of the derivative is found as the value of the corresponding difference relation. For the calculation of Greeks, which are partial derivatives of the higher order, arrays of the approximate values of the previous order derivatives are used. In the case, when one of the arguments is time t, for definiteness, the derivative by time is considered the first, then the derivatives with respect to other variables are found.

The presented algorithm is implemented in the programming language C+—h For the construction of graphs, the free GNUplot package (http://gnuplot.info/) is used.

To get closer to a market situation, the following parameters of model were used for the all calculations:

— the volatility a = 0.3;

— the step of the volatility change Aa = 0.01;

— the interest rate r = 0.04;

— the step of the interest rate change Ar = 0.001;

— the strike price K = 0.4;

— the parameter ft was adopted as ft =1.0 for LGR model for definiteness;

— the share of program traders p = 0.1.

Remark 3. Note that ft defines the power in the demand function in PWR model. In this case the equation is nonlinear, but homogeneous and has the same form for u(x,t) and for C(x,t) (see Remark 1). But for LGR model the value of u(x,t) for ft corresponds to the value of C(x,t) at pft. In other words, if one need have to calculate values C(x, t) for LGR model with some constant ft, then it must be used constant ft/p for calculation of values u(x,t) for PWR model.

Note that the linear BSM model is stable at most of the parameters, so the preliminary numerical experiments were mainly aimed at ensuring computational stability for LGR and PWR models. It should also be noted that the stability of the solution was taken into account not only for the unknown function u(x,t), but also for the sensitivity coefficients (Greeks). Thus, it was decided to use for the final calculation following parameters of the difference scheme:

— the weighting factor of the six-point scheme: © = 0.9;

— the number of the nodes along the x axis: N = 120.

4. Results of calculations

Below we give the graphical results of finding the approximate solutions u(x, t), as well as of the solutions comparison for the different models (BSM model is the Black-Scholes model, LGR model is the model with the logarithmic demand of the reference traders, PWR model is the model with the power demand), which are considered in this paper.

Fig. 1 presents graphs, showing the difference in the values between the numerical solutions of the linear model (BSM) and of the nonlinear LGR and PWR models.

Even more clearly this difference at time t = 0, i.e. one year before the option is exercised, is visible on Fig. 2.

In addition, we compared the values of various sensitivities of the option price for the models, studied in the paper. The most widely used Greek is Delta: Delta = characterizing the change in the value of an option, when the price of the underlying asset changes.

As in the case with the call option price, Delta is less smoothed near the strike price in the models with the insufficient liquidity compared to the linear model. The graphs for comparing Delta in the linear model and the models with the different demand functions are shown on Fig.3. The difference in the Delta values of the call option for the different models at the moment t = 0 is shown on Fig. 4.

Gamma shows, how Delta changes at changing of the underlying asset price: Gamma = 9DQxta = f^nf • Gamma is one of the most important Greeks for the options traders. Comparing for Gamma between BSM and nonlinear models is presented at Fig. 5. As can be seen on Fig. 6, Gamma increases sharply in the nonlinear models near the strike price (in our calculations the strike price is taken as K = 0.4).

Theta =—■§!, shows, how the price of option will change over time, if all other parameters will be constant. For different models comparing is presented at Fig. 7. It should be noted, that for large x the decreasing rate of the call option price in all the models under consideration tends to the same value -rKe-rT (see Fig. 8), which is fully consistent with theory and practice. This happens because in this case the call option price reduces daily only by the value, corresponding to the risk-free interest rate.

Rho = (Fig- 9 and 10), shows the change in the price of an option, when the risk-free interest rate on the market changes. When the Black-Scholes model was obtained, it was assumed that the risk-free interest rate would remain constant until the option is expires, and there are no any payments on the underlying asset. However, with long periods of the options expiration, when the probability of the risk-free interest rate change is high, Rho should be taken into account, when forming a portfolio or hedging.

On Fig 11 and 12 comparing for Vega = is presented. Vega shows the changing of the option price value with small changes in the volatility of the underlying asset. Despite the fact, that Black and Scholes assumed a constant volatility, when building their model, in practice the change of volatility plays a key role in the option pricing.

5. Conclusion

We obtained numerical solutions of some nonlinear Black-Scholes type equations, describing the option pricing in illiquid market. It allowed to calculate numerical values for the series of Greeks, used in practice of the options trading and the portfolio insurance. The graphs of the sensitivities are constructed, that demonstrate the differences between the linear Black-Scholes model and the models, which take

Fig. 1. The difference of u(x,t) values between Black-Scholes model and models with logarithmic (BSM-LGR) and power (BSM-PWR) demand function

Fig. 2. Price of the European call option u(x,t) at the moment t = 0 (r = 0.04, a = 0.30), for different considered models, for one year before execution of option. The line BSM correspondent Black- Scholes model (BSM model), line PWR — the model with power function of demand (PWR model). The line LGR is corresponding to the model with logarithmic demand function (LGR model)

Fig. 3. The difference of values of Delta between Black- Scholes model and models with logarithmic (BSM-LGR) and power (BSM-PWR) demand function

Fig. 4. Delta of the European call option at the moment t = 0 (r = 0.04, a = 0.30), for different considered models. The line BSM correspondent Delta from Black-Scholes model, line PWR — Delta from model with power function of demand. The line LGR is corresponding to the values of Delta for model with logarithmic demand function

Fig. 5. The difference of values of Gamma between Black- Scholes model and models with logarithmic (BSM-LGR) and power (BSM-PWR) demand function

Fig. 6. Gamma of the European call option at the moment t = 0 (r = 0.04, a = 0.30), for different considered models. The BSM line correspondent Black-Scholes model (BSM model), the PWR line is the model with power function of demand (PWR model). The LGR is corresponding to the model with logarithmic demand function (LGR model)

Fig. 7. The difference of values of Theta between Black- Scholes model and models with logarithmic (BSM-LGR) and power (BSM-PWR) demand function

Fig. 8. Theta of the European call option at the moment t = 0 (r = 0.04, a = 0.30), for different considered models. The BSM line correspondent Black-Scholes model (BSM model), the PWR line — the model with power function of demand (PWR model). The LGR line is corresponding to the model with logarithmic demand function (LGR model)

Fig. 9. The difference in values of Rho between Black-Scholes model and models with logarithmic (BSM-LGR) and power (BSM-PWR) demand function

Fig. 10. Rho of the European call option at r = 0.2 (t = 0.50, a = 0.30), for different considered models. The BSM line correspondent Black-Scholes model (BSM model), the PWR line — the model with power function of demand (PWR model). The LGR line is corresponding to the model with logarithmic demand function (LGR model)

Fig. 11. The difference in values of Vega between Black-Scholes model and models with logarithmic (BSM-LGR) and power (BSM-PWR) demand function

Fig. 12. Vega of the European call option at a = 1.0 (t = 0.50, r = 0.04), for different considered models. The BSM line correspondent Black-Scholes model (BSM model), the PWR line — the model with power function of demand (PWR model). The LGR line is corresponding to the model with logarithmic demand function (LGR model)

into account the effects of feedback between the option price and the price of the underlying asset.

It can be seen that the both nonlinear models considered in the work, with the power and with the logarithmic demand functions, have one common feature. Namely, unlike the linear Black-Scholes model, there is a dramatic change in the state of the system, when the price of the underlying asset passes through the exercise price (strike price). This imposes certain restrictions and changes the nature of the actions of traders to hedge their own portfolio of options.

REFERENCES

1. Black, F., and Scholes, M. "The pricing of options and corporate liabilities." Journal of Political Economy 81, 3 (1973), 637-654.

2. Merton, R. C. "Theory of rational option pricing." The Bell Journal of Economics and Management Science 4, 1 (1973), 141-183.

3. Cox, J. C., Ross, S. A., and Rubinstein, M. "Option pricing: a simplified approach." Journal of Financial Economics 7, 3 (1979), 229-263.

4. Frey, R., and Stremme, A. "Market volatility and feedback effects from dynamic hedging." Mathematical Finance 7, 4 (1997), 351-374.

5. Schonbucher, P. J., and Wilmott, P. "The feedback effect of hedging in illiquid markets." SIAM J. Appl. Math. 61, 1 (2000), 232-272.

6. Sircar, R. K., and Papanicolaou, G. "General Black-Scholes models accounting for increased market volatility from hedging strategies." Appl. Math. Finance 5, 1 (1998), 45-82.

7. Dyshaev, M. M., and Fedorov, V. E. "Symmetries and exact solutions of a nonlinear pricing options equation." Ufa Mathematical Journal 9 (2017), 29-40.

8. Fedorov, V. E., and Dyshaev, M. M. "Invariant solutions for nonlinear models of illiquid markets." Math Meth Appl Sci 41, 18 (2018), 8963-8972.

9. Dyshaev, M. M., and Fedorov, V. E. "Symmetry analysis and exact solutions for a nonlinear model of the financial markets theory." Mathematical Notes of North-Eastern Federal University 23, 1 (2016), 28-45. In Russ.

10. Dyshaev, M. M. "Group analysis of a nonlinear generalization for Black-Scholes equation." Chelyabinsk Physical and Mathematical Journal 1, 3 (2016), 7-14. In Russ.

11. Brennan, M. J., and Schwartz, E. S. "Finite Difference Methods and Jump Processes Arising in the Pricing of Contingent Claims: A Synthesis." Journal of Financial and Quantitative Analysis 13, 3 (1978), 461-474.

12. Ankudinova, J., and Ehrhardt, M. "On the numerical solution of nonlinear Black-Scholes equations." Computers & Mathematics with Applications 56, 3 (2008), 799-812.

13. Company, R., Jodar, L., Ponsoda, E., and Ballester, C. "Numerical analysis and simulation of option pricing problems modeling illiquid markets." Computers & Mathematics with Applications 59, 8 (2010), 2964- 2975.

14. Company, R., Navarro, E., Pintos, J. R., and Ponsoda, E. "Numerical solution of linear and nonlinear Black-Scholes option pricing equations." Computers & Mathematics with Applications 56, 3 (2008), 813-821.

15. During, B., Fournie, M., and Jungel, A. "High order compact finite difference schemes for a nonlinear Black-Scholes equation." International Journal of Theoretical and Applied Finance 06, 07 (2003), 767-789.

16. Heider, P. "Numerical methods for non-linear Black-Scholes equations." Applied Mathematical Finance 17 (2010), 59-81.

17. Guo, J., and Wang, W. "On the numerical solution of nonlinear option pricing equation in illiquid markets." Computers & Mathematics with Applications 69, 2 (2015), 117-133.

18. Arenas, A. J., Gonzalez-Parra, G., and Caraballo, B. M. "A nonstandard finite difference scheme for a nonlinear Black- Scholes equation." Mathematical and Computer Modelling 57, 7 (2013), 1663-1670.

19. During, B., Fournie, M., and Jiingel, A. "Convergence of a high-order compact finite difference scheme for a nonlinear Black-Scholes equation." M2AN, Math. Model. Numer. Anal. 38, 2 (2004), 359-369.

20. Pooley, D. M., Vetzal, K. R., and Forsyth, P. A. "Numerical convergence properties of option pricing PDEs with uncertain volatility." IMA Journal of Numerical Analysis 23, 2 (2003), 241-267.

21. Haug, E. G. " The complete guide to option pricing formulas, second ed." McGraw-Hill, New York, 2007.

22. Dyshaev, M. M., and Fedorov, V. E. "The Sensitivities (Greeks) for some models of option pricing with market illiquidity." DOI: https://doi.org/10.13140/RG.2.2.27157.58083

23. Kangro, R., and Nicolaides, R. "Far field boundary conditions for Black-Scholes equations." SIAM Journal on Numerical Analysis 38, 4 (2000), 1357-1368.

24. Crank, J., and Nicolson, P. "A practical method for numerical evaluation of solutions of partial differential equations of the heat-conduction type." Mathematical Proceedings of the Cambridge Philosophical Society 43, 1 (1947), 50-67.

Submitted May 17, 2019 Revised May 23, 2019 Accepted June 3, 2019

Mikhail M. Dyshaev

Chelyabinsk State University,

Scientific Research Department,

129, Br.Kashirins str., Chelyabinsk 454021, Russia

Mikhail.DyshaevSgmail.com

Vladimir E. Fedorov

Chelyabinsk State University,

Mathematical Analysis Department,

129, Br.Kashirins str., Chelyabinsk 454021, Russia;

South Ural State University (National Research University),

Laboratory of Functional Materials,

76, Lenin Av., Chelyabinsk 454080, Russia

kar@csu.ru