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- 1. STA 2502 / ACT 460 : Stochastic Calculus Main Results © 2006 Prof. S. Jaimungal Prof S Department of Statistics and Mathematical Finance Program University of Toronto
- 2. Brownian Motion A stochastic process Wt is called a Brownian motion (or Wiener process) if Starts at 0 with Probability 1 Wt = 0 almost surely Has independent increments : Wt – Ws is independent of Wv– Wu whenever [t s] Å [u v] = ∅ [t,s] [u,v] Has stationary increments as follows: Ws – Ws+t ~ N( 0; σ2 t) © S. Jaimungal, 2006 2
- 3. Stochastic Integration A diffusion or Ito process Xt can be “approximated” by its local dynamics through a driving Brownian motion Wt: Adapted drift process fluctuations Adapted volatility process FtX denotes the information generated by the p g y process Xs on the interval [0,t] (i.e. its entire path up to that time) © S. Jaimungal, 2006 3
- 4. Stochastic Integration Stochastic differential equations are generated by “taking the limit” as ∆t → 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 Wt is nowhere differentiable! ff © S. Jaimungal, 2006 4
- 5. Stochastic 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 Left end valuation Idea… Create a sequence of approximating simple processes which converge t the given process i th L2 sense to th i in the Define the stochastic integral as the limit of the approximating processes © S. Jaimungal, 2006 5
- 6. Martingales Two important properties of conditional expectations Iterated expectations: for s < t p double d bl expectation where th i t ti h the inner expectation i on a l t ti is larger information set reduces to conditioning on the smaller information set Factorization of measurable random variables : if Z ∈ FtX If Z is known given the information set, it factors out of the expectation © S. Jaimungal, 2006 6
- 7. Martingales A stochastic process Xt is called an Ft-martingale if 1. Xt is adapted to the filtration {Ft }t ≥0 given information to time t X_t is “known” 2. 2 For every t ≥ 0 3. For every s and t such that 0 ≤ t < s This last condition is most important: the expected future value is its value now © S. Jaimungal, 2006 7
- 8. Martingales : Examples Standard Brownian motion is a Martingale Stochastic integrals are Martingales © S. Jaimungal, 2006 8
- 9. Martingales : 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 9
- 10. Ito’s Lemma Ito’s Lemma: If a stochastic variable Xt satisfies the SDE then given any function f(Xt, t) of the stochastic variable Xt which is twice differentiable in its first argument and once in its second, © S. Jaimungal, 2006 10
- 11. Ito’s Lemma… Can be obtained heuristically by performing a Taylor expansion in Xt and t, keeping terms of order dt and (dWt)2 and replacing Quadratic variation of the pure diffusion is O(dt)! Cross variation of dt and dWt is O(dt3/2) Quadratic variation of dt terms is O(dt2) © S. Jaimungal, 2006 11
- 12. Ito’s Lemma… © S. Jaimungal, 2006 12
- 13. Ito’s Lemma : Examples Suppose St satisfies the geometric Brownian motion SDE then Ito’s lemma gives Therefore ln St satisfies a Brownian motion SDE and we have © S. Jaimungal, 2006 13
- 14. Ito’s Lemma : Examples Suppose St satisfies the geometric Brownian motion SDE What SDE does Stβ satisfy? Therefore, Sβ(t) is also a GBM with new parameters: © S. Jaimungal, 2006 14
- 15. Dolean-Dade’s exponential The Dolean-Dade’s exponential E(Yt) of a stochastic process Yt is the solution to the SDE: If we write Then, , © S. Jaimungal, 2006 15
- 16. Quadratic 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 product of course, this is just a fudge, and to compute it correctly you must show that © S. Jaimungal, 2006 16
- 17. Ito’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 q q g quotient rule for standard calculus © S. Jaimungal, 2006 17
- 18. Multidimensional Ito Processes Given a set of diffusion processes Xt(i) ( i = 1,…,n) © S. Jaimungal, 2006 18
- 19. Multidimensional Ito Processes where µ(i) and σ(i) are Ft – adapted processes and In thi I this representation the Wiener processes Wt(i) are all t ti th Wi ll independent © S. Jaimungal, 2006 19
- 20. Multidimensional Ito Processes Notice that the quadratic variation of between any pair of X’s is: Consequently, th correlation coefficient ρij b t C tl the l ti ffi i t between the two th t processes and the volatilities of the processes are © S. Jaimungal, 2006 20
- 21. Multidimensional Ito Processes The diffusions X(i) may also be written in terms of correlated diffusions as follows: © S. Jaimungal, 2006 21
- 22. Multidimensional 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 22
- 23. Multidimensional Ito’s Lemma Alternatively, one can write, © S. Jaimungal, 2006 23
- 24. Multidimensional Ito rules Product rule: © S. Jaimungal, 2006 24
- 25. Multidimensional Ito rules Quotient rule: © S. Jaimungal, 2006 25
- 26. Ito’s Isometry The expectation of the square of a stochastic integral can be computed via Ito’s Isometry Where ht is any Ft-adapted process For example, © S. Jaimungal, 2006 26
- 27. Feynman – Kac Formula Suppose that a function f({X1,…,Xn}, t) which is twice differentiable in all first n-arguments and once in t satisfies then the Feynman-Kac formula provides a solution as : © S. Jaimungal, 2006 27
- 28. Feynman – Kac Formula In the previous equation the stochastic process Y1(t),…, Y2(t) satisfy the SDE’s and W’(t) are Q – Wiener processes. © S. Jaimungal, 2006 28
- 29. Self-Financing Strategies A self-financing strategy is one in which no money is added or removed from the portfolio at any point in time. All changes in the weights of the portfolio must net to zero value Given a portfolio with ∆(i) units of asset X(i), the Value process is The total change is: © S. Jaimungal, 2006 29
- 30. Self-Financing Strategies If the strategy is self-financing then, So that self-financing requires, © S. Jaimungal, 2006 30
- 31. Measure Changes The Radon-Nikodym derivative connects probabilities in one measure P to probabilities in an equivalent measure Q This random variable has P-expected value of 1 Its It conditional expectation i a martingale process diti l t ti is ti l © S. Jaimungal, 2006 31
- 32. Measure Changes Given an event A which is FT-measurable, then, For Ito processes there exists an Ft-adapted process s.t. the Radon Nikodym Radon-Nikodym derivative can be written © S. Jaimungal, 2006 32
- 33. Girsanov’s Theorem Girsanov’s Theorem says that if Wt(i) are standard Brownian processes under P then the Wt(i)* are standard Brownian processes under Q where © S. Jaimungal, 2006 33

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