Options Pricing for Two Stocks by Black Sholes Time Fractional Order NonLinear Partial Differential Equation
AAnalytical solution of option pricing for two Stocks by time fractional ordered Black Scholes partial differential equation
Dr. Kamran Zakaria, Muhammad Saeed Hafeez
Department of Mathematics, NED University of Engineering and Technology, Karachi
Abstract
Time fractional order Black scholes partial differential equation for risk free option pricing in financial market yields the better prediction in financial market of the country. In this paper the modified form of Black Schole equation including two stocks is used for evaluations. Samudu Transform approach is utilized for calculating analytical solution . Solution of the equation has been found in form of convergent infinite series.
Keywords:
Options, Samudu Transforms, Caputo Derivative, the time fractional order partial differential equation, Black Scholes Equation.
Introduction:
This paper is the extended version of the paper presented in 3rd International Conference on Computing, Mathematics and Engineering Technologies (iCoMET) 2020 as given in the reference [1]. Option pricing is the worldwide growing field of financial mathematics to solve financial market pricing problems. It is a not very old discipline consists of techniques to predict the prices of assets in financial market. The Black Scholes athematical model with aid of computer science provides the tool to solve complex financial markets, stock exchange and industrial problems. under discussion model is the benchmark imitations in finance, and it is the first mathematical models which predicts the pricing of options (both call and put) and the implied volatility. The Maximum payoff is always the wish of successful businessman and it is possible only when the risk is minimized and gain is maximized. Specifically, the stock exchange is the good example for option pricing where the prices of shares are highly uncertain and unpredictable. But the risk free prices may be obtained by using the famous Black Scholes partial differential equation. Price and share option valuation of options have been a comer stone in financial markets. Financial study of financial derivatives is one of two most growing areas in the corporate business finance . The mathematical models are servings to measure the variation, predict and forecast the behavior of financial markets. The Fractional calculus presents a highly endorsement and latest tool in business finance. Time fractional ordered Business Financial models are expressed in form fractional ordered stochastic PDE to maintain better accuracy and tolerance, to control variations and investigate random -ness in financial markets. Black-Scholes model is the prominent and well known models to evaluate the option prices containing the brief literature review of its empirical developments theme of beginning with Black and Scholes (1973) experimental examinations close predisposition inside the Black-Scholes model as far as moneyness and development. Studies have been growing additionally noted instability inclination operating at a profit Black Scholes model (1973) utilizing S&P 500 choice list information 1966-969 recommend the fluctuation that appertain the choice delivers a cost among the frameworkโs cost and market cost. Black and Scholes (1973) propose proof, instability isn't fixed. Galai (1977) affirm B-S model that the supposition of verifiable momentary instability need to be loose. Their observational outcomes are likewise steady with the consequences of Geske in the year 1979.MacBeth and Merville (1980) look at the Black-Scholes model against the steady flexibility of difference (CEV) model, which expect unpredictability changes when the stock costs changes. MacBeth and Merville (1980) found that the unpredictability of the hidden stock minimized the risk as the stock value rises. Beckers (1980) tried the Black-Scholes suspicion that the chronicled quick unpredictability of the fundamental stock is a component of the stock value, utilizing S&P 500 record alternatives 1972-1977. Beckers (1980) finds the hidden stock is a converse capacity of the stock cost. Geske and Roll (1984) show that at a unique time both in-the-cash and out-of-the-cash choices contain instability inclination. Geske and Roll (1984) finish up, time and cash predisposition might be identified with inappropriate limit conditions, where as the unpredictability inclination issue might be the consequence of measurable mistakes in estimation. Rubinstein (1994) shows that the inferred instability for S&P 500 file choices applies abundance kurtosis. Shimko (1993) exhibits that inferred conveyances of S&P 500 record are contrarily slanted and leptokurtic. Jackwerth and Rubinstein (1996) show the dispersion of the S&P 500 preceding 1987 apply lognormal appropriations, however since have disintegrated to look like leptokurtosis and egative skewness. A few examinations try to expand the tail properties of the lognormal conveyance by consolidating a hop dissemination process or stochastic instability. Das and Sundaram (1999) show hop dispersion and stochastic instability moderate yet don't take out unpredictability predisposition. Das and Sundaram (1999) recognize hop dispersion and stochastic instability forms don't produce skewness and extra kurtosis looked like actually. Buraschi and Jackwerth (2001) create measurable tests dependent on prompt model and stochastic models utilizing S&P 500 list alternatives information from 1986-1995. Buraschi and Jackwerth (2001) close the information is progressively steady with models that contain extra hazard factors, for example, stochastic unpredictability and hop dissemination. Yang (2006) finds suggested volatilities used to esteem trade exchanged call alternatives on the ASX 200 Index are fair-minded and better than chronicled immediate unpredictability in guaging future acknowledged instability. Yang (2006) finds inferred volatilities used to esteem trade exchanged call alternatives on the ASX 2000 Index are fair and better than authentic immediate instability in determining future acknowledged unpredictability. Writing proposes the Black-Scholes model may undervalue alternatives in light of the fact that the tail properties of the hidden lognormal dissemination are excessively little. In 2016, H. Zhang, F. Liu I. Turner, Q. Yang solved time fractional BlackโScholes model governing equation for European options numerically. In 2019, D. Prathumwan and K. Trachoo solved Black Sholes equation y the method of Laplace Homotopy Perturbation Method for two asset option pricing [1]. In this paper, the technique of Samudu transform method is used to demonstrate the analytical solution of 2-Dimensional, Time Fractional-ordered BS-Model, consists of two different assets in Liouville- Caputo Fractional derivative form for the European call options. The Sammudu Transform provides the value of put option in form of explicit solution in convergent series. In Lapace perturbation method, first step the method of Laplace transform is applied then homotopy method is applied. The solution is found by the certain hectic work but in the method of Samudu Transform, the solution may be found without such a hectic working. The solution is similar to the solution obtained by the method of Laplace Homotopy method. The method for solving two assets BS financial model is described in next section.
Methodology
Consider P.D.E. ๐ โ ๐๐ก โ โ ๐ฅ, ๐ฆ, ๐ก + โฟโ ๐ฅ, ๐ฆ, ๐ก + ๐โ ๐ฅ, ๐ฆ, ๐ก = ๐ ๐ฅ, ๐ฆ, ๐ก โ โ โ โ โ โ โ (๐) Where ๐ โ 1 <โโค ๐; ๐ โ ๐ Subject to: โ ๐ฅ, ๐ฆ, 0 = โ (๐ฅ, ๐ฆ) Where
๐ฟ =
Linear Differential operator
๐ =
Non -Linear Differential operator all Sumudu Transform for the caputo Fractional ordered Derivative of โ ๐ฅ, ๐ฆ, ๐ก on because of equation (i)
๐ ๐ โ ๐๐ก โ โ ๐ฅ, ๐ฆ, ๐ก = ๐ [ โ ๐ฅ, ๐ฆ, ๐ก โ โฟโ ๐ฅ, ๐ฆ, ๐ก โ ๐โ ๐ฅ, ๐ฆ, ๐ก + ๐ ๐ฅ, ๐ฆ, ๐ก โ โ โ โ โ โ โ (๐๐) ๐ ๐ โ ๐๐ก โ โ ๐ฅ, ๐ฆ, ๐ก = ๐ข โโ ๐ [โ ๐ฅ, ๐ฆ, ๐ก โ ๐ข โโ+๐๐โ1๐=0 ๐ ๐ ๐๐ก ๐ โ (๐ฅ, ๐ฆ, 0) If โ< 1, setting ๐ = 1 ๐ ๐ โ ๐๐ก โ โ ๐ฅ, ๐ฆ, ๐ก = ๐ข โโ ๐ โ ๐ฅ, ๐ฆ, ๐ก โ ๐ข โโ โ ๐ฅ, ๐ฆ, 0 ๐ข โโ ๐ โ ๐ฅ, ๐ฆ, ๐ก โ ๐ข โโ โ ๐ฅ, ๐ฆ, 0 = ๐ [๐ ๐ฅ, ๐ฆ, ๐ก โ โฟ โ ๐ฅ, ๐ฆ, ๐ก โ ๐โ ๐ฅ, ๐ฆ, ๐ก ๐ โ ๐ฅ, ๐ฆ, ๐ก = โ ๐ฅ, ๐ฆ, 0 + ๐ข โโ ๐ [๐ ๐ฅ, ๐ฆ, ๐ก โ โฟ โ ๐ฅ, ๐ฆ, ๐ก โ ๐โ ๐ฅ, ๐ฆ, ๐ก Apply Inverse Sumudu Transform. ๐ ๐ โ1 โ ๐ฅ, ๐ฆ, ๐ก = ๐ โ1 โ ๐ฅ, ๐ฆ, 0 + ๐ โ1 ๐ข โ ๐ ๐ ๐ฅ, ๐ฆ, ๐ก โ โฟ โ ๐ฅ, ๐ฆ, ๐ก โ ๐โ ๐ฅ, ๐ฆ, ๐ก Call Inverse property of Samudu Transform (S.T) ๐ผ โ ๐ ๐ฅ, ๐ฆ, ๐ก = ๐ โ1 [๐ข โ ๐ (๐ ๐ฅ, ๐ฆ, ๐ก ] Apply Integral Property of S.T on equation (3) โ ๐ฅ, ๐ฆ, ๐ก = โ ๐ฅ, ๐ฆ + ๐ผ โ [๐ ๐ฅ, ๐ฆ, ๐ก โ โฟ โ ๐ฅ, ๐ฆ, ๐ก โ ๐โ ๐ฅ, ๐ฆ, ๐ก โ โ โ โ โ (4) Sumudu Transform Express the so ๐ ๐ of P.D.E in form of Infinite Convergent series as below. (1) โ ๐ฅ, ๐ฆ, ๐ก = โ ๐ฅ, ๐ฆ + ๐ ๐ (๐ฅ, ๐ฆ)๐ก ๐โ ฮ(1 + ๐ โ) โ๐=1
Where โ ๐ฅ, ๐ฆ = ๐ ๐ฅ, ๐ฆ = ๐ = ๐ ๐ฅ, ๐ฆ, ๐ก โ โฟ ๐ โ ๐(๐ )] ๐ = ๐ ๐ฅ, ๐ฆ, ๐ก โ โฟ ๐ โ ๐(๐ )] ๐ = ๐ ๐ฅ, ๐ฆ, ๐ก โ โฟ ๐ โ ๐(๐ )] โฎ โฎ ๐ ๐ = ๐ ๐ฅ, ๐ฆ, ๐ก โ โฟ ๐ (๐) โ ๐[๐ (๐) ] This rearch paper is the application of Sumudu Integral Transform to evaluate option price of two stocks Fractional order Black Sholes Model. Consider below Fractional order European call option pricing P.D.E for two stocks. ๐ โ ๐๐๐ก โ + ๐ ๐ ๐ ๐๐๐ + ๐ ๐ ๐ ๐๐๐ + ๐๐ ๐๐๐๐ + ๐๐ ๐๐๐๐ + ๐๐ ๐๐๐๐ + ๐ ๐ ๐ ๐ ๐ ๐ ๐๐๐ ๐๐ โ ๐๐ = 0 Subject to pay-off to the investor. ๐(๐ , ๐ , ๐ก) = max (๐ค ๐ + ๐ค๐ โ ๐ , ) ( for European Call Option ) P(๐ , ๐ ๐ก) = max (๐ค ๐ + ๐ค๐ โ ๐ , 0) ( for American Put option) Where c = value of European Call Option P = value of
American
Call Option ๐ = Price of share of stock 1 ๐ = Price of share of stock 2 P= correlation coefficient between price of shares of stock 1 and stock 2 ๐ = Price volatility or S.D of stock 1 = Price volatility or S.D of stock 2 K = Strike price or Exercise price for call option ๐ค = properties of investment on stock 1 ๐ค = properties of investment on stock 2 Equation (4) can be simplified by considering the substitution. ๐ข = ๐๐๐ โ ๐ โ ๐ ๐ก ๐ข = ๐๐๐ โ ๐ โ ๐ ๐ก ๐๐๐๐ = ๐๐๐๐ข ๐๐ข๐๐ ๐๐๐๐ = ๐๐๐๐ฃ ๐๐ข๐๐ c ๐ ๐๐๐ = ๐๐๐ ๐๐๐๐ข = ๐ = ๐๐๐ ๐๐๐๐ข + ๐๐๐๐ข ๐๐๐ ๐ = ๐ = ๐๐๐ข ๐๐๐๐ข ๐๐ข๐๐ โ ๐ ๐๐๐๐ข = ๐ ๐ ๐๐๐ข โ ๐ ๐๐๐๐ข ๐๐๐๐ = 1๐ ๐๐๐๐ฃ โฆ . . . . . (2)
1 โฆ โฆ โฆ ๐๐๐๐ = 1๐ ๐๐๐๐ข ๐๐ โ โ โ โ โ โ ๐ ๐๐๐ = 1๐ โ ๐ ๐๐๐ข โ ๐๐๐๐ข imilarly ๐ ๐๐๐ = ๐ ๐ ๐๐๐ฃ โ ๐๐๐๐ฃ โ โ โ โ โ (๐๐๐) ๐๐๐๐ = ๐ ๐๐๐๐ข ๐๐๐๐ ๐๐๐๐ = ๐๐๐ ๐ โ ๐๐๐๐ข = ๐ ๐๐๐ ๐๐๐๐ข = ๐ ๐๐๐ฃ ๐๐๐๐ข ๐๐ฃ๐๐ = ๐ ๐ ๐๐๐ข ๐๐ฃ ๐ Substitute (i) (ii) , (iii) & (iv) in equation โฆ..(4) ๐ โ ๐ ๐๐ก โ + ๐ ๐ ๐๐ ๐ ๐๐๐ข โ ๐๐๐๐ข + ๐ ๐ ๐ = ๐ ๐๐ฃ โ ๐๐๐๐ฃ + ๐๐ . ๐ ๐๐๐๐ข + ๐๐ . ๐ ๐๐๐๐ฃ + ๐ ๐ ๐ ๐ ๐ ๐ ๐ ๐ ๐ ๐ ๐ ๐ ๐๐๐ข ๐๐ฃ โ ๐๐ = ๐ โ ๐ ๐๐ก โ + ๐ ๐ ๐๐๐ข + ๐ ๐ ๐๐๐ฃ โ ๐ ๐๐๐๐ข โ ๐ ๐๐๐๐ข = ๐ ๐๐ฃ โ ๐๐๐๐ฃ ๐๐ฃ โ โ โ โ โ โ ๐ ๐๐๐ ๐๐ ๐ ๐ ๐๐๐ข ๐๐ฃ = 1๐ โ ๐ ๐๐๐ข โ ๐๐๐๐ข ๐๐๐ก = ๐๐๐๐ข ๐ โ 12 ๐ ๐๐๐๐ก = ๐๐๐๐ฃ ๐ โ 12 ๐ + ๐ ๐๐๐๐ข + ๐ ๐๐๐๐ฃ + ๐ ๐ ๐ โ ๐ ๐๐๐ข ๐๐ฃ โ ๐๐ = ๐ โ ๐ ๐๐ก โ + ๐ ๐ ๐๐๐ข + ๐ ๐ ๐๐๐ฃ + ๐ ๐ ๐๐๐๐ข + ๐ โ ๐ ๐๐๐๐ฃ +๐ ๐ ๐ ๐ ๐๐๐ข ๐๐ฃ โ ๐๐ = We have substitution ๐ข = ๐๐๐ โ ๐ โ ๐ ๐ก ๐ฃ = ๐๐๐ + ๐ โ ๐ก ๐๐๐๐ก = ๐๐๐๐ข ๐๐ข๐๐ก โ ๐๐๐๐ก = ๐๐๐๐ข ๐ โ ๐ ๐๐๐๐ก = ๐๐๐๐ฃ ๐๐ฃ๐๐ก ----------(vii) From (vi) & (vii) equation (v) becomes. ๐ โ ๐ ๐๐ก โ + ๐ ๐ ๐๐๐ข + ๐ ๐ ๐๐๐ฃ โ ๐๐๐๐ก + ๐๐๐๐ก + +๐ ๐ ๐ ๐ ๐๐๐ข ๐๐ฃ โ ๐๐ = ๐ โ ๐ ๐๐ก โ + ๐
2 ๐ ๐๐๐ข + ๐
2 ๐ ๐๐๐ฃ + ๐ ๐ ๐ ๐ ๐๐๐ข ๐๐ฃ โ ๐๐ = 0 ๐ ๐ โ ๐ ๐๐ก โ = โ ๐
2 ๐ ๐๐๐ข + ๐
2 ๐ ๐๐๐ฃ + ๐ ๐ ๐ ๐ ๐๐๐ข ๐๐ฃ โ ๐๐ = 0 ----------(A) Subject to: ๐ ๐ข, ๐ฃ, ๐ = max ( ๐ค ๐ ๐ข + ๐ค ๐ ๐ฃ , ) ----------(viii) Hence equation (viii) be the simplified European style ca;; option pricing model for two stocks Fractional order Black-shole P.D.E Apply sumudu transform on (viii) ๐ข โโ ๐ ๐ ๐ข, ๐ฃ, ๐ก โ ๐ข โโ ๐ข, ๐ฃ, ๐ = โ๐ [ ๐ ๐ ๐๐๐ข + ๐ ๐ ๐๐ฃ ๐ + ๐ ๐ ๐ ๐ ๐๐๐ข ๐๐ฃ โ ๐๐ = ๐ ๐ ๐ข, ๐ฃ, ๐ก โ ๐ ๐ข, ๐ฃ, ๐ = โ๐ข โ ๐ [ ๐ ๐ ๐๐๐ข + ๐ ๐ ๐๐๐ฃ + ๐ ๐ ๐ ๐ ๐๐๐ข ๐๐ฃ โ ๐๐ ] ๐ ๐ ๐ข, ๐ฃ, ๐ก = ๐ ๐ข, ๐ฃ, ๐ โ ๐ข โ ๐ [ ๐ ๐ ๐๐๐ข + ๐ ๐ ๐๐๐ฃ + ๐ ๐ ๐ ๐ ๐๐๐ข ๐๐ฃ โ ๐๐ โ โ โ โ โ ( ix ) Apply inverse samudu transform on -----------(ix) ๐ ๐ข, ๐ฃ, ๐ก = ๐ โ ๐ ๐ข, ๐ฃ, โ ๐ โ [ ๐ข โ ๐ [ ๐ ๐ ๐๐๐ข + ๐ ๐ ๐๐๐ฃ + ๐ ๐ ๐ ๐ ๐๐๐ข ๐๐ฃ โ ๐๐ ๐ข, ๐ฃ, ๐ก = ๐ ๐ข, ๐ฃ, ๐ ๐ โ [ ๐ข โ ๐ [ ๐ ๐ ๐๐๐ข + ๐ ๐ ๐๐๐ฃ + ๐ ๐ ๐ ๐ ๐๐๐ข ๐๐ฃ โ ๐๐ โ โ โ โ โ โโ ( x ) By definition ๐ผ โ ๐ ๐ฅ, ๐ฆ, ๐ก = ๐ โ [๐ข โ ๐ ๐ (๐ฅ, ๐ฆ, ๐ก)] Apply Integral property of Sumudu transform on -------------------- ( x ) ๐ ๐ข, ๐ฃ, ๐ก = ๐ ๐ข, ๐ฃ, ๐ ๐ผ โ [ ๐ ๐ ๐๐๐ข + ๐ ๐ ๐๐๐ฃ + ๐ ๐ ๐ ๐ ๐๐๐ข ๐๐ฃ โ ๐๐ โ โ โ โ โ ( ix ) Sumudu transform expresses So ๐ ๐ of P.D.E by using equation (xi) in form of infinite convergent series as below. ๐ ๐ข, ๐ฃ, ๐ก = ๐ ๐ข, ๐ฃ, ๐ ๐๐ ๐ฅ, ๐ฆ ๐ก ๐โ ฮ( + ๐ โ ) โ ๐= Where ๐ ๐ข, ๐ฃ, ๐ก = ๐ ๐ข, ๐ฃ = ๐ (๐ ๐๐ฆ) ๐ ๐+ ๐ข, ๐ฃ, ๐ก = ๐๐ ๐ก ๐โ ฮ( + ๐ โ ) โ ๐= ๐ ๐ข, ๐ฃ, ๐ก = ๐ ๐ข, ๐ฃ, ๐ + ๐๐(๐ฅ, ๐ฆ) ๐ก ๐โ ฮ( + ๐ โ ) โ ๐= be the European call option price solution at time t. where ๐ ๐ข, ๐ฃ, ๐ = ๐ ๐ = โ ๐ ๐ ๐ ๐๐ข + ๐ ๐ ๐ ๐๐ฃ + ๐ ๐ ๐ ๐ ๐ ๐๐ข ๐๐ฃ โ ๐๐ ๐ = โ ๐ ๐ ๐ ๐๐ข + ๐ ๐ ๐ ๐๐ฃ + ๐ ๐ ๐ ๐ ๐ ๐๐ข ๐๐ฃ โ ๐๐ ๐ = โ ๐ ๐ ๐ ๐๐ข + ๐ ๐ ๐ ๐๐ฃ + ๐ ๐ ๐ ๐ ๐ ๐๐ข ๐๐ฃ โ ๐๐ โ ๐ ๐+ = โ ๐ ๐ ๐ ๐ ๐๐ข + ๐ ๐ ๐ ๐ ๐๐ฃ + ๐ ๐ ๐ ๐ ๐ ๐ ๐๐ข ๐๐ฃ โ ๐๐ ๐ Illustrations of option Price For Two Stocks Fractional Ordered Black Sholes Model
Illustrative example 1 : Option type: European call option, consider the following data. ๐ = Price of stock 1 in dollors. ๐ = Price of stock 2 in dollars ๐
20 40 70 100 150 ๐
50 80 120 180 200 Initial Condition : C( ๐ , ๐ , ๐ก) =Max ( ๐ ๐ + 2๐ ๐ โ 80, 0) ๏ท Exercise price of stock 1 = Rs.80 ๏ท Exercise price of stock 2 = Rs.20 ๏ท Maximum Exercise price for = Rs.80 Option type : European Call option. ๏ท Month of Expiration or time for Exercise data = 8 months ๏ท โ = 0.005 ๏ท S.D of stock = 40 % S.D of stock = 25 % ๏ท Proportion of stock 1 = ๐ค = 2 & proportion of stock 2 = ๐ค =2 ๏ท Risk free rate of return = 8% ๏ท Correlation coefficient between Stock 1 and stock 2= 75% solving the above problem by using Matlab programming ,European call option price of the stocks is represented as below C( ๐ , ๐ , ๐ก) = 1.0512 ๐ ๐ + 2.1274 ๐ ๐ - 86.942 CALL OPTION PRICES
Sock S1 20 40 70 100 150 Stock S2 50 42.193 62.744 93.569 124.39 175.77 80 107.13 127.68 158.5 189.33 240.7 120 193.7 214.25 245.08 275.9 327.28 180 323.57 344.12 374.94 405.77 457.15 200 366.86 387.41 418.23 449.06 500.43 llustrative example 2 : Option type : put option consider the following data. ๐ = Price of stock 1 in dollors. ๐ = Price of stock 2 in dollars ๐
20 40 70 100 150 ๐
50 80 120 180 200 Initial Condition: P( ๐ , ๐ , ๐ก) = ๐๐๐ฅ( 60 โ 3sin๐๐ โ 5๐๐๐ ๐๐ , 0 ) ๏ท Exercise price of stock 1 = Rs.60 ๏ท Exercise price of stock 2 = Rs.20 ๏ท Maximum Exercise price for = Rs.60 Option type : put option. ๏ท Month of Expiration or time for Exercise data = 2 months ๏ท โ = 0.755 ๏ท S.D of stock = 45 % ๏ท S.D of stock = 85 % ๏ท Proportion of stock 1 = ๐ค = 3 & proportion of stock 2 = ๐ค =5 Risk free rate of return = 03% ๏ท Correlation coefficient between Stock 1 and stock 2= 65% solving the above problem by using Matlab programming ,put option price of the stocks is represented as below P ( ๐ , ๐ , ๐) =27.459cos(3.1416 ๐ ) - 2.0559cos(3.1416 ๐ ) - 1.4938sin(3.1416 ๐ ) -32.852sin(3.1416 ๐ )+ 60.514 PUT OPTION PRICES
Stock S
20 40 70 100 150 Stock S
50 98.861 96.764 94.274 96.115 98.926 80 43.341 41.244 38.754 40.595 43.406 120 20.404 18.307 15.817 17.658 20.469 180 57.17 55.072 52.583 54.424 57.235 00 71.267 69.17 66.68 68.521 71.332
Illustrative example 3 : Option type : European call option consider the following data. ๐ = Price of stock 1 in dollors. ๐ = Price of stock 2 in dollars ๐
20 40 70 100 150 ๐
50 80 120 180 200 Initial condtion: C( ๐ , ๐ , ๐ก) = ๐๐๐ฅ( 2๐ + 5๐ , 0 ) ๏ท Exercise price of stock 1 = Rs.60 ๏ท Exercise price of stock 2 = Rs.90 ๏ท Maximum Exercise price for = Rs.90 Option type : put option. ๏ท Month of Expiration or time for Exercise data = 2 months ๏ท โ = .125 ๏ท S.D of stock = 40 % ๏ท S.D of stock = 65 % ๏ท Proportion of stock 1 = ๐ค = 2 & proportion of stock 2 = ๐ค =5 ๏ท Risk free rate of return = 07% ๏ท Correlation coefficient between Stock 1 and stock 2= 85% solving the above problem by using matlab programming ,put option price of the stocks is represented as below C ( ๐ , ๐ , ๐) = 2.1517 ๐ + 5.3794 ๐ - 99.777 Stock S1 20 40 70 100 150 Stock S2 50 Illustrative example 4 : Option type : European put option , consider the following data. ๐ = Price of stock 1 in dollors. ๐ = Price of stock 2 in dollars ๐
20 40 70 100 150 ๐
50 80 120 180 200 nitial condtion: C( ๐ , ๐ , ๐ก) = ๐๐๐ฅ( (๐ฅ + ๐ฆ ) - ln(y)+ ln(x)-,0) ๏ท Maximum Exercise price for = Rs. (๐ฅ + ๐ฆ ) Option type : put option. ๏ท Month of Expiration or time for Exercise data = 5 months ๏ท โ = .125 ๏ท S.D of stock = 40 % ๏ท S.D of stock = 20 % ๏ท Proportion of stock 1 = ๐ค = 1 & proportion of stock 2 = ๐ค =1 ๏ท Risk free rate of return = 8% ๏ท Correlation coefficient between Stock 1 and stock 2 = 75 % solving the above problem by using matlab programming ,put option price of the stocks is represented as below: p ( ๐ , ๐ , ๐ก) = ๐ฅ - 1.0868ln(y) - 1.0868ln(x) + 0.073569/ ๐ฅ + 0.14482/ ๐ฅ + 1.2248/ ๐ฅ + 2.1735 ๐ฆ + 0.047084/ ๐ฆ + 0.05932/ ๐ฆ + 0.32106/ ๐ฆ - 48.905 Put -0ption prices
Stock S
20 40 70 100 150 Stock S
50 28.452 46.868 60.826 68.965 77.391 80 31.964 52.15 67.538 76.588 86.049 120 34.994 56.708 73.328 83.164 93.519 180 38.025 61.265 79.119 89.74 100.99 200 38.812 62.45 80.623 91.449 102.93
Illustrative example 5 : Option type: put option , consider the following data. ๐ = Price of stock 1 in dollors. ๐ = Price of stock 2 in dollars ๐
20 40 70 100 150 ๐
50 80 120 180 200
Initial condtion: p( ๐ , ๐ , ๐ก) = ๐๐๐ฅ( -5sin(x)-8y+5xy,0); x)-,0) ๏ท Maximum Exercise price for = Rs.
Option type : put option. ๏ท Month of Expiration or time for Exercise data = 5 months ๏ท โ = .125 ๏ท S.D of stock = 40 % ๏ท S.D of stock = 20 % ๏ท Proportion of stock 1 = ๐ค = 1 & proportion of stock 2 = ๐ค =1 ๏ท Risk free rate of return = 8% ๏ท Correlation coefficient between Stock 1 and stock 2 = 75 % solving the above problem by using matlab programming ,put option price of the stocks is represented as below: ( ๐ , ๐ , ๐) = PUT-OPTION PRICES
Stock S
20 40 70 100 150 Stock S
50 50.18 59.961 69.412 76.156 84.502 80 58.571 68.353 77.803 84.548 92.894 120 66.592 76.374 85.824 92.568 100.91 180 75.335 85.117 94.568 101.31 109.66 200 77.725 87.507 96.958 103.7 112.05
Concluding Remarks : In this paper technique of Samudu Transforms and its derivatives and integral properties are applied to compute analytical solution of time fractional non โ linear two dimensional BS PDE model in form of infinite series to evaluate put options of two stocks asset. Illustrative practical example is also presented to understand the reliability, efficiency, simplicity, and effectiveness of the purposed scheme. Samudu Transforms has many powerful and effective techniques to obtain analytic solution of any type time fractional PDE in least time with less computation.
References K. Zakaria and S. Hafeez, "Options Pricing for Two Stocks by Black โ Sholes Time Fractional Order Non โ Linear Partial Differential Equation," , Sukkur, Pakistan, 2020, pp. 1-13, doi: 10.1109/iCoMET48670.2020.9073866. 2.
Hall John: Options, Future and other Derivatives, 5 th edition prentice Hall. 3. Kwok: Mathematical Models of Financial Derivatives (1998) 62-64 4.
P-Vilmot: The Mathematics of Financial Derivatives University press. 5.
Shedon Ross: An Introduction to Mathematical Finance Cambridge university press. 6.
Bernt: Stochastic D.E; An introduction with application; 5 th edition. 7. Financial Derivatives, Theory concept and problems by S.L. Euptc. 8.
G.K Weatuga, Sumudu Transforms, a new integral Transform to solve differential equations and control engineering problems, Mathematical Engineering 61(1993) 329-329 9.
F-B-M Belagacem, A-A Karbali: and S-L Kala; Analytical investigation of sumudu transform and integral to production equations, Mathematical problems in Engineering,3 (200) 103-118 10.
Sumudu Transform method for Analytical solutions of Fractional Type ordinary Differential Equations . (Hindawi Publication Co-orprations, Mathematical problems for Engineering) volume 2015-1315 690. 11.
H-Eltayeb and Kilichman, A note on Sumudu Transforms and differential equations, Applied Mathematics Science (2010)- 4(22) 1089-1098. 12.
On Sumudu Transforms of Fractional Derivatives and its applications to Fractional Differential Equations International, Knowledge press: ISSN 2395-4205(P) ISSN-2395-4213. 13.
Asira MA Further properties of Sumudu Transforms and its applications. International journal of Mathematical education, Science & Technology 2002: 33(2): 441-449 14.
Sumudu Transform Method for solving Fractional D.E Journal of Mathematics & Computer Science 2013; 6: 79-84 15.
Computational and Analytical solution of Fractional order Linear Partial Differential Equations by using Sumudu Transforms and its properties. IJCNS international Journal of Computer science and Network security Vol 18
Basic Properties of Sumudu Transform to some Partial Differential Equations, Sakarya University Journal of science 23(4), 509-514, 2019. 17.
D-Kumar, J Singh, Sumudu Decomposition method for Non- Linear equations, International Mathematics Forum 7(2012) 515-521. 18.
European option Pricing of Fractional Black Sholes model using Sumudu Transforms and its Derivaties, Refad; general letters in Mathematics Vol1, No.3 Dec 2016 PP:74-80 e-ISNN 2319-9277 19.
Barles, G. and Sooner H.M: Option pricing with Transaction cost and a Non-Linear Black Sholes Equations. Finance stock 2,4 (1998)- 369-397 20.
Black Sholes, M. The pricing of options and co-orprate liabilities 1973, 81,637-654. 21.
Misrana, M lub, Fractional Black sholes Model with application of Fractional option Pricing; International Conference of optimization and control 2010, 17,99-111 (Cross reference) pp-573-588 22.
Tracho; Two dimensional Black Sholes model will European call option. Math compute.2017,2223(eross reference) 3.