bsnotes13 - Lecture Notes Black-Scholes-Merton Model(David...

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Lecture Notes: Black-Scholes-Merton Model (David D. Yao) 1 Itˆ o’s Calculus Throughout, we use B t to denote the standard Brownian motion (BM). Let dB s := B s + ds - B s denote an increment of the BM (with ds > 0). We also use N ( μ, σ 2 ) to denote a normal distribution with mean μ and variance σ 2 . Recall some of the key properties of BM: (i) B 0 = 0; (ii) independent increments, i.e., dB s and dB t are independent, for any s + ds t ; (iii) stationary increments, i.e., dB s follows a normal distribution N (0 , ds ). Note this last distribution depends only on the length of the increment, not on when it starts (hence, “shift invariant”, or stationary). Also note that the variance is proportional to the length of the increment. The BM is a Markov process, with a continuous trajectory over time. Another important property of BM is the following result (see the Appendix for more details): [ dB s ] 2 = ds, as ds 0 . (1) Note that whereas E ([ dB s ] 2 ) = ds , we have, with Z denoting the standard normal variate, Var ([ dB s ] 2 ) = Var ( ds · Z 2 ) = ( ds ) 2 [ E ( Z 4 ) - E 2 ( Z 2 )] = 2( ds ) 2 . (Note E ( Z 4 ) = 3.) Therefore, the relation in (1) appeals to intuition: the random variable on the left had side has a variance that is a higher-order infinitesimal than its mean. Hence, as ds 0, the random variable becomes deterministic. In general, the multiplication rules in Table 1, the so-called “box algebra” 1 is useful when it comes to taking derivatives on functions that involve Brownian motion. (The zero’s in the table should be read as higher-order infinitesimals w.r.t. dt .) And this is essentially what leads to Itˆo’s calculus . To motivate, consider the ordinary calculus. Suppose x t is a deterministic function of time, or, a deterministic “path/trajectory”. Write dx t = ˙ xdt , where ˙ x denotes the derivative of x over time. Consider a smooth function f . We have df ( x t ) = f 0 ( x t ) dx t . 1 J.M. Steele, Stochastic Calculus and Financial Applications, Springer-Verlag, New York, 2001. 1
· dt dB t dt 0 0 dB t 0 dt Table 1: “Box Algebra” More precisely, df ( x t ) = f 0 ( x t ) dx t + 1 2 f 00 ( x t )( dx t ) 2 + · · · , with the higher-order terms vanishing when dt 0. For instance, ( dx t ) 2 = ˙ x 2 ( dt ) 2 . Now, if x t is replaced by the BM, B t , then in view of (1), we must include the second- derivative term, since ( dB t ) 2 = dt , i.e., at the same order as dt . This results in what is known as Itˆo’s formula : df ( B t ) = f 0 ( B t ) dB t + 1 2 f 00 ( B t ) dt. (2) Taking integral on both sides, we have another form of Itˆ o’s formula: f ( B t ) = f (0) + Z t 0 f 0 ( B s ) dB s + 1 2 Z t 0 f 00 ( B s ) ds. (3) As an example, consider f ( X t ), where X t = μt + σB t . (4) ( X t is the generalized Wiener process, or Brownian motion with drift.) Following (2), we have df ( X t ) = f 0 ( X t ) dX t + 1 2 f 00 ( X t )( dX t ) 2 + o (( dX t ) 2 ) . (5) Since ( dX t ) 2 = ( μdt + σdB t ) 2 = μ 2 ( dt ) 2 + 2 μσ ( dt )( dB t ) + σ 2 ( dB t ) 2 = σ 2 dt, (6) where on the last line we have ignored terms that are of higher order w.r.t. dt . Substituting (6) into (5) and omitting higher-order terms, we have df ( X t ) = f 0 ( X t ) dX t + 1 2 σ 2 f 00 ( X t ) dt = [ μf 0 ( X t ) + 1 2 σ 2 f 00 ( X t )] dt + σf 0 ( X t ) dB t . (7) 2
More generally, consider a bivariate function f ( t, x ). The formuals in (2) and (3) extend to: df ( t, B t ) = ∂f ∂t dt + ∂f ∂x dB t + 1 2 2 f ∂x 2 dt, (8) and f ( t, B t ) = f (0 , 0) + Z t 0 f 0 t ( s, B s ) ds + Z t 0 f 0 x ( s, B s ) dB s + 1 2 Z

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