5. Itô Calculus. Partial derivative are abstractions. Usually they are called multipliers or marginal effects (cf. the Greeks in option theory).

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1 5. Itô Calculus Types of derivatives Consider a function F (S t,t) depending on two variables S t (say, price) time t, where variable S t itself varies with time t. In stard calculus there are three types of derivatives: Partial derivative: F s = F(S t,t), F t = F(S t,t) (). S t t Total derivative: () Chain rule: df (S t,t) (3) dt df = F s ds t + F t dt. = F s ds t dt + F t. Partial derivative are abstractions. Usually they are called multipliers or marginal effects (cf. the Greeks in option theory). Total derivative describes the total change or response in F as time t S t change The chain rule indicates the chain effects in the change of the price S t F as time changes. We will consider in this section stochastic counterparts for the total differential chain rule. Essentially as we will see the major differences are that we have to interpret the differential in stochastic processes via the stochastic integral that the second order term (ds t ) is not negligible as in stard calculus. Itô s stochastic differential equation Itô s lemma gives the stochastic version for the chain rule. Let (4) ds t = a(s t,t) dt + σ(s t,t) dw t. where a(s t,t) σ(s t,t) are nonanticipating W t is the stard Brownian motion. We interpret ds t via the stochastic integral such that (5) ds u = S t S, so that (6) S t = S + ds u = a(s u,u) du + σ(s u,u) dw u, where the first integral is the usual Riemann integral the second one is the Itô integral. 3 4

2 Consider a function F (S t,t) (e.g. derivative of a stock) As time t changes by dt what is the total effect on F (S t,t). The change is (7) t W t S t F (S t,t). The interest is df (S t,t). Suppose the observation interval of S t is,t]. Let = t <t < <t n = T be a partition with (8) h = t k t k = T n so that T = nh. k =,...,n, Consider the finite difference representation of ds t (9) ΔS k = a k h + σ k ΔW k, k =,...,n, with ΔS k = S tk S tk, a k = a(s tk,t k ), σ k = σ(s k,t k ), ΔW k = W tk W tk. 5 6 Itô formula is derived using the Taylor expansion of a smooth function. If f(x) is such a function the Taylor expansion around x becomes For a function with two variables the Taylor expansion is F (S tk,t k ) = F (S tk,t k )+F sδs k + F th + Fss (ΔS k) () + Ftt h + F st h ΔS k + R. f(x) =f(x )+f (x )(x x )+ f (x )(x x ) + R, () where R is the remainder. Arranging terms ΔF k = F s ΔS k + F t h + F ss (ΔS k ) () + F tt h + F st h ΔS k + R, where (3) ΔF k = F (S tk,t k ) F (S tk,t k ). 7 8

3 As n, h = t k t k dt, ΔS k ds, ΔF k df, R because it consists of terms (Δt k ) m (ΔW k ) m with m 3. So we get df (S t,t) (4) = F s ds t + F t dt + Fss (dst) + Ftt (dt) + F st dt ds t. Using the calculation rules for the differentials, we obtain (5) dt ds t = dt (a(s t,t)dt + σ(s t,t) dw t ) = a(s t,t)(dt) + σ(s t,t) dt dw t =, because (dt) = dt dw t =. So we get Remark 5.: If S t is non-stochastic then (ds t) = the above formula is just the total derivative df = F s ds + F t dt. Replacing ds t with its Itô representation, we have df (S t,t) = F s a(s t,t) dt + σ(s t,t) dw t] + F t dt (7) + Fss a(st,t) dt + σ(st,t) dwt]. Using the infinitesimal calculation rules again yields (8) (ds t ) = σ (S t,t) dt. Arranging terms, we obtain finally the famous Itô s differential formula df = F t + a(s t,t)f t + ] σ (S t,t)f ss dt + σ(s t,t) dw t. (9) (6) df (S t,t)=f s ds t + F t dt + Fss(dSt) 9 The result is summarized as Itô s Lemma: Lemma 5.: (Itô s Lemma) Let F (S t,t) be a twicedifferentiable function of t of the rom process S t with Itô differential equation ds t = a t dt + σ t dw t, t, with a t = a(s t,t) σ t = σ(s t,t) continuously twicedifferentiable (real valued) functions. Then The major usage of the Itô formula in finance is to find the (Itô) stochastic differential equation (SDE) for the financial derivative once the (Itô) SDE of the underlying asset is given. () df = F s ds t + F t dt + Fssσ t dt, or, after substituting for the right h side of ds t above df = (F s a t + F t + ) () Fssσ dt + F sσ t dw t, where () F s = F,F t = F S t t, Fss = F. S t

4 Itô s formula can be used also in some cases to find the stochastic integral itself. Example 5.: (7) F (W t,t)=3+t + e Wt. Example 5.: Let (3) F (W t,t)=wt. Using formula (6) with S t = W t we obtain (4) F w = W / W =W, (5) F ww = F/ W =. Then because F t =, Using again Itô s formula (6) with S t = W t (8) df (W t,t) = F t dt + F w dw t + Fww (dwt) = dt + e Wt dw t + ewt dt = ( + ewt ) dt + e Wt dwt. df (W t,t) = F w dw t + Fww(dWt) (6) = dt +W t dw t. Thus the drift of F is a(f, t) = diffusion parameter is σ(f, t) =W t. 3 4 Example 5.3: Consider the geometric Brownian motion (9) S t = S e {(μ σ )t+σwt}, where S is a constant. Then using again Itô with formula (), noting that σ(w t,t) =, we get ds t = St dw Wt t + St (3) or (3) or (3) t dt + St W t dt = S σe {(μ σ )t+σwt} dw t +(μ σ )S e (μ σ )t+σwt dt + σ S e (μ σ )t+σwt dt = S tσdw t +(μ σ )S t dt + σ S t dt = S t(μdt+ σdw t), ds t S t = μdt+ σdw t, ds t = μs tdt + σs tdw t. Remark 5.: Comparing to the general formula ds t = a(s t,t)dt + σ(s t,t)dw t we find that in (3) a(s t,t)= μs t, σ(s t,t)=σs t. Itô s formula as an integration tool Suppose our task is to evaluate (33) W s dw s. Make a guess (34) Then using Itô (35) F (W t,t)= W t. df (W t,t)=w t dw t + dt. The integral form is t W t = F (W t,t)= df (W s,s)= W s dw s + ds. (36) So (37) W s dw s = W t t. The start off point here is to make a good guess. 5 6

5 Example 5.4: Consider Itô integral (38) Make a start off guess sdw s. (39) F (W t,t)=tw t. Then (4) df (W t,t)=w t dt + tdw t. (4) So (4) tw t = df (W s,s)= W s ds + sdw s = tw t W s ds. sdw s. Example 5.5: Consider ds t (43) = μdt+ σdw t. S t Let (44) F (S t,t) = log S t. Then df (S t,t) = F t dt + F s ds t + Fss(dSt) (45) = StdS t S t (ds t) = μdt+ σdw t σ S S t dt t = (μ σ ) dt + σdw t. We get log S t = log S + df (Su,u) (46) So (47) = log S + (μ σ )du + σdwu = log S +(μ σ )t + σw t. S t = S e (μ σ )t+σwt. 7 8 Integral form of Itô s Lemma Integrating both sides of () yields Itô s formula in integral form: F (S t,t) = F (S, ) + F (S ] u,u)+ F (S u,u)σu du (48) where (49) + F (x, y) = F s ds u. F(x, y),f (x, y) = x F(x, y), y F (x, y) = F (x, y) (5) x, we have used (5) df (S u,u)=f (S t,t) F (S, ). Remark 5.3: Rearranging terms in the Itô s integral form yields (5) F sds u = F (S t,t) F (S, )] F ] (S u,u)+ F(Su,u)σ u du, which is a representation of the stochastic integral as a function of integrals with respect to time. 9

6 Multivariate Itô formula (53) ( ) ( ) ( ds (t) a (t) σ (t) σ = dt+ (t) ds (t) a (t) σ (t) σ (t) or )( ) dw (t) dw (t) (54) ds (t) = a (t) dt + σ (t) dw (t)+σ (t) dw (t) ds (t) = a (t) dt + σ (t) dw (t)+σ (t) dw (t), where it is assumed that Wiener processes W (t) W (t) are independent. Suppose F (S (t),s (t),t) is a twice differentiable real valued function. Use of the Taylor expansion taking limit in the same manner as in the univariate case, yields (with (dt) =,dt ds =, dt ds = ) (55) df = F t dt + F s ds + F s ds + Fs,s(dS ) + F s,s(ds ) +F s,sds ds ]. The independence of W W implies that dw dw = (otherwise if they were correlated with correlation ρ, then dw dw = ρdt). Then (56) (57) (58) (ds ) =(σ + σ ) dt, (ds ) =(σ + σ ) dt, ds ds =(σ σ + σ σ )dt. Using these in the multivariate Itô gives a formula as a function of dw dw. Example 5.6: Interest rate derivatives. Assume that the yield curve depends on two state variables, short rate r t, long rate R t. Denote the price of the derivative as F (r t,r t,t). Assume (59) dr t = a (t) dt + σ (t) dw (t)+σ (t) dw (t), (6)dR t = a (t) dt + σ (t) dw (t)+σ (t)dw (t). Straightforward application of the Itô formula gives (6) df = F t dt + F r dr t + F R dr t + Frr(σ + σ )+FRR(σ + σ )+FrR(σσ + σσ)] dt, which indicates how the price of an interest rate derivative will change during a small interval dt. Remark 5.4: (6) Cov(dr t,dr t)=σ (t)σ (t)+σ (t)σ (t)] dt. 3 4

7 Example 5.7: Total value of wealth (63) Y (t) = N i (t)p i (t), where N i (t) is units of the ith asset P i (t) the price. Increment of wealth as time passes dy (t) = Y t dt + Y Y dn i (t)+ dp i (t) N i P i Y Y = + N (dn i (t)) + i Y + dn i (t) dp i (t) N i P i P i P i (t)dn i (t)+ N i (t)dp i (t) + dn i (t) dp i (t). (dp i (t)) (64) In the stard calculations the last term would not be present. 5