Доклади на Българската академия на науките Comptes rendus de l’Acad´emie bulgare des Sciences Tome 67, No 8, 2014
MATHEMATIQUES Math´ematiques appliqu´ees
NUMERICAL IMPLEMENTATIONS OF GENERALISATIONS OF BLACK–SCHOLES MODEL FOR ESTIMATION OF CALL- AND PUT-OPTION Angela Slavova, Nikolay Kyurkchiev (Submitted by Academician I. Popchev on May 14, 2014)
Abstract In this paper we propose new modules in MATHEMATICA programming environment for the generalisations of Black–Scholes (BS) model taking into account the market price and coefficient of variation. First we derive generalisation of Black–Scholes PDE and present its explicit solution. Then we derive the Garman–Kohlhagen’s model as generalisation of BS model. The proposed modules give the possibility for visualisation and hypersensitive analysis. Key words: Black–Scholes model, market price, coefficient of variation, Garman–Kohlhagen model, programming environment MATHEMATICA 2000 Mathematics Subject Classification: 65M12, 65Y20
1. Introduction. The area of derivative securities has been one of the fastest growing areas of finance as well as one of the most active areas of research on stochastic analysis, stochastic control and computations. The fundamental problem of derivative valuation amounts to determining the derivative’s fair value and in specifying the hedging policy which eliminates the risk inherent to the contract. The Black and Scholes valuation approach [2 ] brought to modern finance the powerful methodologies of martingale theory and stochastic calculus. Today, numerous different kinds of derivative instruments are traded all around the world and various new contracts are being created every day. The valuation of these contracts gives rise to a number of challenging problems in the areas of stochastic analysis, martingale theory, stochastic control and partial differential equations. In their seminal paper Black and Scholes [1 ] developed a theory for the valuation of derivative securities in frictionless markets. They considered the 1053
problem of determining the value of European call which is written on an underlying stock whose price Ss follows the diffusion process, as described in Z t Z t σ2 σSτ dWτ = S0 exp µ− µSτ dτ + t + σWt , (1) St = S0 + 2 0 0 where µ is the mean rate of return and σ is the volatility; µ and σ are constants such that µ > r and σ 6= 0, St is a stock price, Wt is one-dimensional standard Brownian motion. The market is also endowed with a riskless security whose price is given by: Bt = ert B0 ,
(2)
t ≥ 0,
where r is the constant rate of interest. The investor holds ut dollars of the bond and vt dollars of the stock at date t. We consider a pair of right-continuous with left limits (CADLAG), nondecreasing processes Lt , Mt such that Lt represents the cumulative dollar amount transferred into stock account and Mt the cumulative dollar amount transferred out of the stock account. By convention, L0 = M0 = 0. The stock account process is: Z t Z t σvτ dWτ + Lt − Mt µvτ dτ + (3) vt = v + 0
0
with v0 = v. The European claim is written at the time t > 0 and expires at maturity time T . Its payoff at expiration is given by (ST − K)+ , where K is the (prespecified) exercise price. The valuation problem amounts to specifying the fair value of the security at its birth time t. Black and Scholes had the novel idea of constructing a dynamic portfolio whose value coincides with the terminal payoff, (ST − K)+ , of the call. Then they argued that the amount needed to set up this hedging portfolio, at time t, yielded the correct price of the European call. Moreover, the components of this portfolio, across time, give the perfectly replicating (hedging) strategies which reproduce the value of the security. Black and Scholes [1 ] postulate that the call price is a smooth function of the current stock price and time. Therefore, there exists a smooth function C : [0, +∞) × [0, T ) → R+ such that the call price process hs , t ≤ s ≤ T can be represented as hs = C(Ss , s). Applying Ito’s formula to hs yields (4) ∂C 1 2 2 ∂2C ∂C ∂C dhs = (Ss , s) + σ xs 2 (Ss , s) + µxs (Ss , s) ds + σSs (Ss , s) dWs . ∂t 2 ∂S ∂S ∂S Next we assume that the riskless interest rate is r > 0 and that the components of the replicating portfolio are βs and δs . In other words, at any time s, 1054
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we would have to purchase βs shares of the riskless security and δs shares of the underlying stock. According to the perfect replication idea of Black and Scholes, the following equalities must hold (5)
βs Bs + δs Ss = hs ,
a.e. t ≤ s < T,
βT BT + δT ST = (ST − K)+ .
(6)
Taking into account the price equations (2) and (3), (5) yields: (7)
dhs = [(µ + r)δs Ss − rhs ] ds + σδs Ss dWs .
We recall that the processes βs and δs satisfy certain “self-financing” assumptions which in turn justify the above differential forms. Equating formally the coefficients in (4) and (7) yields ∂C (Ss , s), δs = ∂S (8) h − δs Ss βs = s , Bs as long as the following condition holds a.e. ∂C 1 ∂2C ∂C (Ss , s) + σ 2 Ss2 2 (Ss , s) + rSs (Ss , s). ∂t 2 ∂S ∂S Therefore, in order to specify the components (βs , δs ) of the replicating portfolio, it suffices for C = C(S, t) to solve the second order nonlinear partial differential equation
(9)
rhs =
∂C 1 ∂2C ∂C + σ 2 S 2 2 + rx ∂t 2 ∂S ∂S together with the boundary and terminal conditions, for 0 ≤ t ≤ T , S ≥ 0, (10)
(11)
rC =
C(0, t) = 0
and
C(S, T ) = (S − K)+ .
The solution of (10) and (11) is given by C(S, t) = SN (d1 ) − e−r(T −t) KN (d2 ), where N is the cumulative standard normal distribution and the quantities d1 and d2 are defined as S σ2 + r+ (T − t) ln K 2 √ d1 = σ T −t and S σ2 ln + r− (T − t) √ K 2 √ d2 = = d1 − σ T − t. σ T −t Equation (10) is the Black–Scholes equation for European type claims written on a stock with constant volatility and when the riskless interest rate is r > 0. Compt. rend. Acad. bulg. Sci., 67, No 8, 2014
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2. Modified Black–Scholes equation with coefficient of market price of risk and coefficient of variation. Black–Scholes (BS) model can be modified when we take into account the following coefficients: λ is the market price per standardised unit volatility, or, simply, the market price of risk; α is the coefficient of variation. Then the solution of the so modified BS model in order to find the call-options is the following: CM = SM N (d1 ) − Ke−r(T −t) N (d2 ),
(12) where
(λ · α)2 ln + r+ (T − t) 2 √ d1 = , λ·α T −t √ d2 = d1 − λ · α T − t.
(13)
SM K
Modulus I. The Black–Scholes model with market-price and coefficient of variation. The code is shown in Fig. 1. Remark 1. The above developed modulus in the program environment MATHEMATICA can be successfully upgraded if we include the mentioned coefficients λ and α. Modulus I is modified based on the extended model (12), (13). 3. Other moduli in programming environment MATHEMATICA. 3.1. Modulus II. The Garman–Kohlhagen’s model. In 1983 Garman and Kohlhagen extended the Black–Scholes model to cope with the presence of two interest rates (one for each currency). Let us suppose that rd is domestic risk free simple interest rate and rf is foreign risk free simple interest rate. Then the domestic currency value of a call option into the foreign currency is:
(14)
C0 = S0 e−rf (T −t) N (d1 ) − Xe−rd (T −t) N (d2 ) = e−rd (T −t) S0 e(rd −rf )(T −t) N (d1 ) − XN (d2 ) . The value of a put option is:
(15)
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P0 = Xe−rd (T −t) N (−d2 ) − S0 e−rf (T −t) N (−d1 ) −rd (T −t) (rd −rf )(T −t) =e XN (−d2 ) − S0 e N (−d1 ) , A. Slavova, N. Kyurkchiev
Fig. 1
where (16) σ2 ln ln + rd − rf + (T − t) 2 √ d1 = = σ T −t √ d2 = d1 − σ T − t,
S0 X
! S0 e(rd −rf )(T −t) σ2 + (T − t) X 2 √ , σ T −t
S0 is the current spot rate; N (d) is the cumulative normal distribution function; X is the strike price; T is the time to maturity; σ is the volatility. The code is shown in Fig. 2. 3.2. Modulus III. Black–Scholes model with interest of dividends. By investments, the investors are trying to obtain income – dividend or increase of the market price of the stocks. Compt. rend. Acad. bulg. Sci., 67, No 8, 2014
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Fig. 2.
Black–Scholes model can be modified in the case, when we take into consideration the coefficient q – interest of the dividend in the period [t + ∆t). In this case the formulas for the calculation of call-put options can be modified in the following way: (17) C0 = e−r(T −t) S0 e(r−q)(T −t) N (d1 ) − XN (d2 ) , (18) P0 = e−r(T −t) XN (−d2 ) − S0 e(r−q)(T −t) N (−d1 ) , 1058
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where
(19)
! S0 e(r−q)(T −t) + ln X √ d1 = σ T −t √ d2 = d1 − σ T − t.
σ2 2 (T
− t) ,
The code is shown in Fig. 3. Remark 2. The same formula is used to price options on foreign rates, expecting that now q plays the role of the foreign risk-free interest rate in Garman– Kohlhagen’s model.
Fig. 3 Compt. rend. Acad. bulg. Sci., 67, No 8, 2014
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4. Conclusions. Let us mention that in the field of industrial insurance, the Black–Scholes model and its modifications have application in forming insurance bill of exchange, considered as put-option. In financial crises the problem of imitation modelling, simulation and detailed investigation of the functions of call- and put-options is very modern. The modules presented in this paper (see also [6 ]) are components of web-based application, realised in the program environmental with central mathematical kernel and in some sense realises the problem proposed above, and the build software instruments can be used for research investigations, as well as for training. We want to point out more general problems connected with the detailed investigation of the models studied in the paper such as models of reporting of return of discrete dividends with discrete taxes and their modifications, construction of portfolios and minimising of the risk, as objectives of our other investigations and usage of software platform with provided access to WEB-based servers for scientific calculations, interval calculations and graphical design. For other results see [2–9 ].
REFERENCES [1 ] Black F., M. Scholes. J. Pol. Econ., 81, 1973, 637–659. [2 ] Brandimarte P. Numerical Methods in Finance and Economics. A MATLABBased Introduction, 2nd ed. Hoboken, New Jersey, John Willey & Sons, Inc., 2006. [3 ] Levy G. Computational Finance. Numerical Methods for Pricing Financial Instruments. Oxford, U.K., Butterworth-Heinemann Elsevier Ltd, 2004. [4 ] Popchev I., N. Velinova. Cybernetics and Information Technologies, 5, 2005. [5 ] Slavova A. In: IEEE Proc. CNNA, 2008, art. No 4588674, 181–185. [6 ] Slavova A., N. Kyurkchiev. Compt. rend. Acad. bulg. Sci., 66, 2013, No 5, 643–650. [7 ] Milev M., A. Tagliani. J. of Comput. and Appl. Math., 233, 2013, No 10, 2468– 2480. [8 ] Chernogorova T., R. Valkov. Finite-Volume Difference Scheme for the Black– Scholes Equation in Stochastic Volatility Models. In: Proc. 7th Int. Conf. NMA. Springer LNCS, 6046, 2011, 377–385. [9 ] Kyurkchiev N. Selected Topics in Applied Mathematics of Finance. Sofia, Prof. Marin Drinov Academic Publishing House, 2012. Institute of Mathematics and Informatics Bulgarian Academy of Sciences Acad. G. Bonchev St, Bl. 8 1113 Sofia, Bulgaria e-mails:
[email protected] [email protected]
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