STA 2502 / ACT 460 :Stochastic Calculus Main Results© 2006 Prof. S. JaimungalDepartment of Statistics andMathematical Finance ProgramUniversity of Toronto© S. Jaimungal, 2006 2Brownian Motion| A stochastic process Wtis called a Brownian motion (or Wiener process) if| Starts at 0 with Probability 1z Wt= 0 almost surely| Has independent increments :z Wt–Wsis independent of Wv–Wuwhenever [t,s] ∩ [u,v] = ∅| Has stationary increments as follows:z Ws–Ws+t~ N( 0; σ2t)© S. Jaimungal, 2006 3Stochastic Integration| A diffusion or Ito process Xtcan be “approximated” by its local dynamics through a driving Brownian motion Wt:| FtXdenotes the information generated by the process Xson the interval [0,t] (i.e. its entire path up to that time)Adapted drift processAdapted volatility processfluctuations© S. Jaimungal, 2006 4Stochastic Integration| Stochastic differential equations are generated by “taking the limit” as Δt → 0| A natural integral representation of this expression is| The first integral can be interpreted as an ordinary Riemann-Stieltjes integral| The second term cannot be treated as such, since path-wise Wtis nowhere differentiable!© S. Jaimungal, 2006 5Stochastic Integration| Instead define the integral as the limit of approximating sums| Given a simple process g(s) [ piecewise-constant with jumps at a < t0< t1< … < tn< b] the stochastic integral is defined as| Idea…z Create a sequence of approximating simple processes which converge to the given process in the L2sensez Define the stochastic integral as the limit of the approximatingprocessesLeft end valuation© S. Jaimungal, 2006 6Martingales| Two important properties of conditional expectations| Iterated expectations: for s < tz double expectation where the inner expectation is on a larger information set reduces to conditioning on the smaller information set| Factorization of measurable random variables : if Z ∈ FtXz If Z is known given the information set, it factors out of the expectation© S. Jaimungal, 2006 7Martingales| A stochastic process Xtis called an Ft-martingale if 1. Xtis adapted to the filtration {Ft}t ≥0  given information to time t X_t is “known”2. For every t ≥ 03. For every s and t such that 0 ≤ t < sThis last condition is most important:the expected future value is its value now© S. Jaimungal, 2006 8Martingales : Examples| Standard Brownian motion is a Martingale| Stochastic integrals are Martingales© S. Jaimungal, 2006 9Martingales : Examples| A stochastic process satisfying an SDE with no drift term is a Martingale| A class of Geometric Brownian motions are Martingales:© S. Jaimungal, 2006 10Ito’s Lemma| Ito’s Lemma: If a stochastic variable Xtsatisfies the SDEthen given any function f(Xt, t) of the stochastic variable Xtwhich is twice differentiable in its first argument and once in its second,© S. Jaimungal, 2006 11Ito’s Lemma…| Can be obtained heuristically by performing a Taylor expansion in Xtand t, keeping terms of order dt and (dWt)2and replacing| Quadratic variation of the pure diffusion is O(dt)!| Cross variation of dt and dWtis O(dt3/2) | Quadratic variation of dt terms is O(dt2)© S. Jaimungal, 2006 12Ito’s Lemma…© S. Jaimungal, 2006 13Ito’s Lemma : Examples| Suppose Stsatisfies the geometric Brownian motion SDEthen Ito’s lemma givesTherefore ln Stsatisfies a Brownian motion SDE and we have© S. Jaimungal, 2006 14Ito’s Lemma : Examples| Suppose Stsatisfies the geometric Brownian motion SDE| What SDE does Stβsatisfy?| Therefore, Sβ(t) is also a GBM with new parameters:© S. Jaimungal, 2006 15Dolean-Dade’s exponential| The Dolean-Dade’s exponential E(Yt) of a stochastic process Ytis the solution to the SDE:| If we write| Then,© S. Jaimungal, 2006 16Quadratic Variation| In Finance we often encounter relative changes| The quadratic co-variation of the increments of X and Y can be computed by calculating the expected value of the productof course, this is just a fudge, and to compute it correctly youmust show that© S. Jaimungal, 2006 17Ito’s Product and Quotient Rules| Ito’s product rule is the analog of the Leibniz product rule for standard calculus| Ito’s quotient rule is the analog of the Leibniz quotient rule for standard calculus© S. Jaimungal, 2006 18Multidimensional Ito Processes| Given a set of diffusion processes Xt(i)( i = 1,…,n)© S. Jaimungal, 2006 19Multidimensional Ito Processeswhere μ(i)and σ(i)are Ft– adapted processes andIn this representation the Wiener processes Wt(i)are all independent© S. Jaimungal, 2006 20Multidimensional Ito Processes| Notice that the quadratic variation of between any pair of X’s is:| Consequently, the correlation coefficient ρijbetween the two processes and the volatilities of the processes are© S. Jaimungal, 2006 21Multidimensional Ito Processes| The diffusions X(i)may also be written in terms of correlated diffusions as follows:© S. Jaimungal, 2006 22Multidimensional Ito’s Lemma| Given any function f(Xt(1), … , Xt(n), t) that is twice differentiable in its first n-arguments and once in its last,© S. Jaimungal, 2006 23Multidimensional Ito’s Lemma| Alternatively, one can write,© S. Jaimungal, 2006 24Multidimensional Ito rules| Product rule:© S. Jaimungal, 2006 25Multidimensional Ito rules| Quotient rule:© S. Jaimungal, 2006 26Ito’s Isometry| The expectation of the square of a stochastic integral can be computed via Ito’s Isometry| Where htis any Ft-adapted process| For example,© S. Jaimungal, 2006 27Feynman – Kac Formula| Suppose that a function f({X1,…,Xn}, t) which is twice differentiable in all first n-arguments and once in t satisfiesthen the Feynman-Kac formula provides a solution as :© S. Jaimungal, 2006 28Feynman – Kac Formula| In the previous equation the stochastic process Y1(t),…, Y2(t) satisfy the SDE’sand W’(t) are Q – Wiener processes.© S. Jaimungal, 2006 29Self-Financing Strategies| A self-financing strategy is one in which no money is added or removed