## O

A path of a simple process. We shall think of the interplay between the simple process A(t) and the Brownian motion W(t) in (4.2.1) in the following way. Regard W(t) as the price per share of an asset at time i. (Since Brownian motion can take negative as well as positive values, it is not a good model of the price of a limited-liability asset such as a stock. For the sake of this illustration, we ignore that issue.) Think of to,ti, ,tn x as the trading dates in the asset, and think...

## Xtoiyfattf Jf4tjxtjxtjlxt

+ Y ftt t ,X(tJ))(tJ+l-tj)2+ higher-order terms. (4.4.21) j o The last two sums on the right-hand side have zero limits as i7 > 0 for the same reasons the analogous terms have zero limits in the sketch of the proof of Theorem 4.4.1 (see (4.4.10) and (4.4.11) . The higher-order terms likewise have limit zero. The limit of the first term on the right-hand side of (4.4.21) is Jq ft. (t, X(t))dt. The limit of the second term is T fx t,X(t))dX(t) i fx(t,X(t))A(t)dW(t)+ T fx t,X(t))0(t)dt. Jo Jo Jo...

## F40

Use this to obtain (4.10.33) from (4.10.34).) (iv) Show that defined by Dijie) E (Xi(t0 + e) X4(to)) ( o + e) - X,(io)) (to) -MtieWjie) 0 -(e) pjj-(i0)cri(i0)< 7j(t0)e + o(e), (4.10.35) where we set > n( ) p22(t) 1 and pi2(t) p2i(t) p(t). (Hint You should define the martingales (jiito + ) - Vf(to) + jf Oi u) du j Then expand the expression on the right-hand side of (4.10.36). You should use Ito's product rule to show that the first term in the expansion is E (Yi(to K(io))(Yj(t0 + )- (<...

## F XuPu V yfcPX yk Jnko

(ii) (Integrability) The random variable X is integrable if and only if Now let Y be another random variable on (O, T, P). (iii) (Comparison) If X < Y almost surely (i.e., W X < V 1), and if fnX(u> )dP(uj) and fn F(o)) dP(w) are defined, then X w)dP j) < f y(u)dP(w). Jn Jn In particular, if X Y almost surely and one of the integrals is defined, then they are both defined and f X(oj)d IV) I Y w)dP(ui). Jn Jn (iv) (Linearity) If a and are real constants and X and Y are integrable, or if a...

## ZtT cexp ioWT r A

Where W is a Brownian motion under the risk-neutral measure P. In Exercise 4.9 h , the delta of this option is computed to be c fO, x N d T,x . This problem provides an alternate way to compute CsfO, . i We begin with the observation that if h s s K , then If a K, then h' s is undefined, but that will not matter in what follows because S T has zero probability of taking the value K. Using the formula for h' s , differentiate inside the expected value in 5.9.2 to obtain a formula for c fO, . ii...

## Forward Swap Measure

- f 0,v dv ci u, t, T du d u,t,T dW u . Jt Jo Jo In one case, this is a reversal of the order of two Riemann integrals, a step that uses only the theory of ordinary calculus. In the other case, the order of a Riemann and an Ito integral are being reversed. This step is justified in the appendix of 83 . You may assume without proof that this step is legitimate. ii Take the differential with respect to t in 10.7.21 , remembering to get two terms from each of the integrals fg a u, t, T du and J a...

## X

Although one cannot simply substitute x 0 into 8.5.14 , we have c , 0 0 see equation 4.5.17 and Exercise 4.9. Formula 8.5.14 can be determined by computing the conditional expectation in 8.5.13 under the condition S t x. In the case t tn, using 8.5.12 , this leads to The function cn t, x also satisfies the Black-Scholes-Merton differential equation en t,x rx- cn t,x -a2x2- cn t,x rcn t,x , tn lt t lt T, x gt 0, The function Cn tn,x is convex in x. This is well-known, but we establish it here...

## Info

-rT 1- N -6- T,z 1 -z, 0 lt t lt T, 0 lt z lt l, satisfying the boundary conditions 7.4.19 - 7.4.21 . This function was shown to satisfy boundary condition 7.4.20 in Exercise 7.5 v . Here we verify by direct computation that the limit of u t,z as z I 0 satisfies 7.4.19 and the limit of u t, z as 11 T r J. 0 satisfies 7.4.21 . i If you have not worked Exercise 7.2, then verify 7.8.11 , the second equality in 7.8.14 and 7.8.16 . ii Use 7.8.11 and the second part of 7.8.14 to show that lim o u t,...

## Y

Where 0 gt 0, cr2 gt 0 gt p lt 1 gt an i Mi, M2 are real numbers. More generally, a random column vector X Xi, ,Xn tr, where the superscript tr denotes transpose, is jointly normal if it has joint density Wx - Tmkmr B x C' X ,r 2218 In equation 2.2.18 , x xi, ,xn is a row vector of dummy variables, p, i , ,fin is the row vector of expectations, and C is the positive definite matrix of covariances. In the case of 2,2.17 , X is normal with expectation and variance aj, Y is normal with expectation...

## J A

Having proved 2.3,23 for Q-measurable indicator random variables X, we may use the standard machine developed in the proof of Theorem 1.5.1 of Chapter 1 to obtain this equation for all -measurable random variables X for which XY is integrable. This shows that XE Y i satisfies the partial-averaging condition that characterizes XY Q , and hence XE Y Q is the conditional expectation E XY . iii If we estimate X based on the information in Q and then estimate the estimate based on the smaller amount...

## Mm o rnimi2[fwtj2tjl

This will cause us to obtain 4.10.22 when we take the limit in 4.10.23 . Prove 4,10.24 in the following steps. iii Under the assumption that E fJ f W t 2dt is finite, show that Warning The random variables Zi,Z , , ,Zn i are not independent. Prom iii , we conclude that Zj converges to its mean, which by ii is Exercise 4.15 Creating correlated Brownian motions from independent ones . Let Wj t , ,Wd t J be a d-dimensional Brownian motion. In particular, these Brownian motions are independent of...

## E[Wti1 Wti 0 Var[iiI iTfe tilU

Associated with Brownian motion there is a filtration T t , t gt 0, such that for each t gt 0 and u gt t, W t is T t -measurable and W u - tf t is independent of T t . Brownian motion is both a martingale and a Markov process. Its transition density is 1 lt y- gt 2 p r,x,y e . v2nT This is the density in the variable y for the random variable W s t given that Wis x. A profound property of Brownian motion is that it accumulates quadratic variation at rate one per unit...

## U

Whenever Sj, B-i,.,,, are disjoint Borel subsets of R. Now let f x be a real-valued function defined on R. For the following construction, we need to assume that for every Borel subset B of R, the set z x B is also a Borel subset of R. A function with this property is said to be Borel measurable Every continuous and piecewise continuous function is Borel measurable. Indeed, it is extremely difficult to find a function that is not Borel measurable. We wish to define the Lebesgue integral JRf x...

## F [ I[QxMxdxd Fuj Jq JO

Is equal to both EX and 0 1 - F x dx. Exercise 1.6. Let u be a fixed number in R, and define the convex function tp x eux for all x E R. Let X be a normal random variable with mean fi EX and standard deviation a E X t 2 2, i.e., with density ii Verify that Jensen's inequality holds as it must Exercise 1.7. For each positive integer n, define fn to be the normal density with mean zero and variance n, i.e., i What is the function f x lim oc fn x 7 ii What is lim f fn x dxl in Note that Explain...

## Filiale der staatlichen Bauerschaft K0LX03 a

Mathematics Subject Classification 2000 60-01, 60H10, 60165. 91B28 Library of Congress Cataloging-in-Publication Data Shreve, Steven E. Stochastic calculus for finance Steven E. Shreve. p. cm. Springer finance series Includes bibliographical references and index. Contents v. 2. Continuous-time models. ISBN 0-387-40101-6 alk. paper 1. Finance Mathematical models Textbooks. 2. Stochastic analysis Textbooks, I, Title. H. Springer finance. HG106.S57 2003 332' .01 '51922 dc22 2003063342 ISBN...