Using of stochastic Ito and Stratonovich integrals derived security pricing Laura Pânzar and Elena Corina Cipu Abstract We seek for good numerical approximations of solutions for stochastic differential equation (SDE) of stock price of underlying asset of an European call option. Using Ito and Stratonovich integrals and SDE approximate solution we predict the stock price on market. Mathematics Subject Classification: 65C3, 65C, 65C5. Key words: random variable, Wiener process, Ito integral, Stratonovich integral, SDE, numerical simulation. 1 Introduction In the most important problem on the financial market is to estimate the price of underlying asset of an European call option (see [5], [6]). In the central form of SDE for evolution of a firm stock price is: (1.1.1) ds = Sµdt + SσdW where µ, σ and W (t) express respectively the drift, volatility and the Wiener process. Generaly (see [1]), a differential equation (1.1.) dx(t) dt = b(x(t), t) + B(X(t), t)ξ(t), X() = X with X( ) : [, ) R n random function, b : R n [, T ] R n, B : R n [, T ] M n m (R), ξ : R R m, m - dimensional white noise, defines a stochastic differential equation (SDE): (1.1.3) dx(t) = b(x(t), t)dt + B(X(t), t)dw (t), X() = X if the white noise is solution of a m - dimensional Wiener process. Proceedings of The 3-rd International Colloquium Mathematics in Engineering and Numerical Physics October 7-9, 4, Bucharest, Romania, pp. -9. c Balkan Society of Geometers, Geometry Balkan Press 5.
Using of stochastic Ito and Stratonovich integrals derived security pricing 3 Remark 1: For m=1, the random { variable ξ is 1-dimensional white noise if E(ξ(t)) =, t s, t R and E(ξ(t) ξ(s)) = σ. The m-dimensional Wiener process (see, t = s [6]) is solution of differential equation: Ẋ(t) = ξ(t), X() =. In one dimensional case Wiener process has the following characteristics: W () = a.s. W (t) W (s) N(, t s), t s < t 1 < t <... < t n random variables W (t 1 ), W (t ) W (t 1 ),..., W (t n ) W (t n 1 ) are independent W (t) has continuous path. Stochastic integrals Integral form of solution of SDE (1.3) is: (..1) X(t) = X + b(x(s), s)ds + B(X(s), s)dw (s), t. The central problem is to calculate the third term of the right side of equation (.1). We shall give then the general form of a stochastic integral..1 Discretization and properties of Ito integral Definition 1: Let be [, T ] R and = {t = < t 1 <... < t m = T } partition of [, T ], with norm = max (t k+1 t k ) and intermediary points τ k = k {,1,...,} (1 λ)t k λt k+1, k {, 1,..., m 1}, λ [, 1] then (..) B(X(s), s)dw (s) = lim k= B(X(τ k ), τ k )(W (t k+1 ) W (t k )). Particular case: i) Ito integral definition: taking in (.) λ = τ k = t k, ii)stratonovich integral definition: taking in (.) λ = 1/ τ k = (t k + t k+1 )/. Let be S(t) express the stock price at moment t satisfying (1.1) and f(t k) number of shares in a portofolio, buyed at time t k, immediately after the announcing the price of underlying asset and before change the portofolio at time t k+1, f(t) = f(t k ), t [o, T ] represent the total gain at moment t, from trading shares. (..3) n 1 I(t) = f(t k )[S(t k ) S(t k )]. k= Definition : Let be = {t = < t 1 <... < t m = T } partition of [, T ], with norm = max (t k+1 t k ) k {,1,...,}
4 L. Pânzar and E. C. Cipu i) F V [,T ] (f) = lim ii) < f > (T ) = lim k= f(t k+1 ) f(t k ) is first variation of function f, (f(t k+1 ) f(t k )) is quadratic variation of f. k= The most important Ito integral properties (see [6], and [1]), used to solve stochastic differential equations, could be enumerated as follows: i) linearity: a, b R if I(t) = ii) martingale: iii) continuity: iv) Ito isometry: if I(t) = then: ai(t) + bj(t) = f(s) dw (s), J(t) = (af(s) + bg(s)) dw (s), g(s) dw (s) {I(t)} t [,T ] is a martingale with respect to the filtration generated by standard Brownian motion, E(I(t)) =, t [, T ], I(t) is continuous, v) < I > (T ) = T or di(t)di(t) = f (t)dt. f(s) dw (s) then E(I(t) ) = f (s)ds. Existence and uniqueness of solution for SDE (Ito approach) For the following stochastic differential equation: (..4) dx(t) = b(x(t), t)dt + B(X(t), t)dw (t), X() = X where X : [, T ] R n, b : R n [, T ] R n, B : R n [, T ] M n m (R), continuous with the property: C > such that: b(x, t) b(x, t) C x x, x, x R n, B(x, t) B(x, t) C x x, x, x R n, b(x, t) C(1 + x ), x R n, t [, T ], B(x, t) C(1 + x ), x R n, t [, T ], X a R n valued random variable such that: E( X ) < and independent of W + () = σ(w (s) W (t)) s t future of Brownian motion beyond time t =, there exist a unique solution X L (, T ) of (.4). Remark: The uniqueness is in the sense that if we find two such solutions for SDE (.4), X(t) and X(t ), then it is necessary that P (X(t) = X (t), t [, T ]) = a.s. The solution will be construct using a sequence of random variable: X := X ; X n+1 (t) = X (t) + b(x n (s), s)ds + B(X n (s), s)dw (s).
Using of stochastic Ito and Stratonovich integrals derived security pricing 5.3 The relationship between Ito and Stratonovich integrals As defined in (.) we can write in general case, for any λ [, 1] and partition of [, T ] the stochastic integral: (..5) W (s)dw (s) = lim k= W (τ k )(W (t k+1 ) W (t k )) with τ k = (1 λ)t k λt k+1, k {, 1,..., m 1}. If we denote: R m = k= W (τ k )(W (t k+1 ) W (t k )) then (..6) lim R m = W (t) + (λ 1 m )T From (.5) and (.6) one could find the relations: i) for Ito stochastic integral: ii) for Stratonovich stochastic integral (denoted with ): W (s)dw (s) = 1 W (t) 1 T, W (s) dw (s) = 1 W (t), that express the relationship between the two integrals. More generaly, one dimensional Ito-Stratonovich conversion formula is given by: (..7) b(w, t) dw = b(w, t)dw + 1 b (W, t)dt. x 3 Application for stock prices We shall use a sample set of data that represents the evolution of a firm stock prices precised in Table 1., in order to approximate the solution of (1.1). Table 1. Evolution of a firm stock prices and rentabilities estimated:
6 L. Pânzar and E. C. Cipu Day of Data Stock R i Day of Data Stock R i the week S i the week S i We 1.3.95.11 Fr 4.3.95.73.76 Th.3.95 1.9 -.995 Mo 7.3.95.91.659 Fr 3.3.95.18.1474 Tu 8.3.95.9.34 Mo 6.3.95.16 -.9 We 9.3.95.9. Tu 7.3.95 1.91 -.1157 Th 3.3.95 3.1.685 We 8.3.95 1.86 -.6 Fr 31.3.95 3.14.64 Th 9.3.95 1.97.591 Mo 3.4.95 3.13 -.3 Fr 1.3.95.7.153 Tu 4.4.95 3.4.351 Mo 13.3.95.49.969 We 5.4.95 3.5.31 Tu 14.3.95.76.184 Th 6.4.95 3.8.31 We 15.3.95.61 -.543 Fr 7.4.95 3.1 -.13 Th 16.3.95.67.3 Mo 1.4.95 3. -.59 Fr 17.3.95.64 -.11 Tu 11.4.95 3.8.199 Mo.3.95.6 -.15 We 1.4.95 3.19.357 Tu 1.3.95.59 -.38 Mo 17.4.95 3.1.63 We.3.95.59. Tu 18.4.95 3.17 -.15 Th 3.3.95.55 -.154 We 19.4.95 3.4.1 We estimate the drift µ and volatility σ using unbiased estimators. In discrete time the rentability of stock S over an time interval (t k 1, t k ) is: R(t k ) = S(t k) S(t k 1 ), S(t k 1 ) k 1, and in continuous time the rentability stock at time t is: R(t) = ds(t)/s(t). We use a Kolmogorov - Smirnov test to verify if values of R follow a normal probability distribution. We find: E(R) =.1474; V AR(R) =.3584; σ = V AR(R) =.593196. So the rentability folow a normal distribution with the estimated mean µ =.1474 expressing the drift and square root variance σ =.593 expressing the volatility. Equation (1.1) could be written in a first case: (3.3.1) with (3.3.) ds S = µdt + σdw (t), S() = S µ = E(R) =.1474, σ = V AR(R) =.593, S =.11, t = T = 33, W (t) N(, t). Using Ito stochastic integral the solution of (3.1) - (3.) has the form: (3.3.3) σ σw (t)+(µ S(t) = S e )t. In a second case we consider in (3.1) that the drift is a function of t and we estimate the values µ k = µ(t k ), k {1,..., m}. In this case the solution of (3.1) is: (3.3.4) σw (t)+ S(t) = S e (µ(s) σ (s) )ds.
Using of stochastic Ito and Stratonovich integrals derived security pricing 7 Because of the difficulty to estimate the values σ k = σ(t k ), k {1,..., m} we shall consider only the first and second case. But also the difficulty increase when the volatility is considered as random variable. In order to obtain approximative values of solution (3.3) we must generate the Wiener process. We can do that either using 1 independent uniform random variables U(, 1), using the algorithm RJNORM, using the polar method (from Box - Muller theorem, see [[3]]), or using randn MATLAB function for variables N(, 1). The polar method is convenient to generate the Wiener process, as it could be seen in Fig. 1 rentability values are mean of the Wiener process generated values. Then we shall use the average of the generated values of N(, 1) for Wiener process such that finally to find a stable generation process. 8 6 4 4 3.5 3 S.5 4 6 8 5 1 15 5 3 35 1.5 5 1 15 5 3 35 4 t Figure 1: Generation of Wiener process and estimation of stock prices using polar method, µ = const, σ = const The calculus for integral i) Simpson composed formula: k k µ(s)ds = 1 3 [µ k + 4 µ(s)ds is made by using Particular case: k k+1 µ(s)ds = 1 3 [µ k + 4 µ i 1 + k 1 µ i 1 + µ i ], k even, k µ i ],h = 1, k odd, ii)trapezoidal method: k µ(s)ds = 1 k 1 [µ k + µ i ],h = 1, the obtained values are presented in Fig.. In case with a variable drift we consider also for values of S(t) a fit with a five
8 L. Pânzar and E. C. Cipu 4 3.5 observed data trapezoidal method Simpsin method Stock prices S 3.5 1.5 5 1 15 5 3 t 35 Figure : Estimation of stock prices using polar method for Wiener process, for µ = µ(t), σ = const degrees polynom that has the form: y =.3e 7x 5 + 1.9e 5x 4.6x 3 +.84x.18x +.1 The approximate and predicted values of stock prices are depicted in Fig. 3. 4.5 Estimation S 5th degree fit polynom Observed S 4 3.5 Stock prices S 3.5 1.5 5 1 15 5 3 35 4 t Figure 3: Estimated and predicted values of stock prices We conclude that with the mean values obtained by the polar method for generating the Wiener process and trapezoidal composed formula for the integral we obtain a good approximation of given data and a local good prediction (for one or two days only). The goodness of approximation is given by the coefficient R (Sest S obs ) = 1 (Sobs S)
Using of stochastic Ito and Stratonovich integrals derived security pricing 9. For the values of stock prices estimated in case µ = const, σ = const in Fig. 1 we find for the coefficient R=.87, respectively R=.841. In case when µ = µ(t), σ = const in Fig. 3 we obtain for the coefficient R=.954 for the trapezoidal method in order to calculate the integral, respectively R=.8811 for the case of using Simpson method. References [1] L.C. Evans, Introduction to stochastic differential equations, U.S.Berkely, 3. [] P. Wilmott, Derivative. Inginerie financiara. Teorie si practica, Ed. Economica, Bucuresti,. [3] I. Vaduva, Modele de simulare,ed. Univ. Buc., Bucharest, 4. [4] T. Cocs, A. Murphy, An introduction to the mathematics of financial derivatives, (course online),. [5] T. Bjork, Arbitrage theory in continuous time,oxford University Press, Oxford, 1998. [6] J. C. Hull, Options, futures and other derivatives, Prentice Hall, London,. [7] T. Mikosch, Elementary stochastic calculus with finance in view, World Scientific Publishing, Singapore, 1998. [8] P. Wilmott, Derivatives: The theory and practice of financial engineering, John Wiley and Sons, Chichester, 1998. Laura Pânzar and Elena Corina Cipu University Politehnica of Bucharest, Department Mathematics III, Splaiul Independenţei 313, RO-64, Bucharest, Romania e-mail: corinac@math.pub.ro, corinac@math.math.unibuc.ro