By Damien Lamberton
Creation DISCRETE-TIME types Discrete-time formalismMartingales and arbitrage possibilities whole markets and alternative pricing challenge: Cox, Ross and Rubinstein version optimum preventing challenge AND AMERICAN concepts preventing time The Snell envelope Decomposition of supermartingales Snell envelope and Markov chains program to American concepts BROWNIAN movement AND STOCHASTIC DIFFERENTIAL EQUATIONS General reviews on continuous-time processesBrownian movement Continuous-time martingales Stochastic critical and Itô calculus Stochastic differential equations THE BLACK-SCHOLES version Description. Read more...
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Additional info for Introduction to Stochastic Calculus Applied to Finance, Second Edition
Then we shall state the concept of martingales in a continuous-time set up, and, nally, we shall construct the stochastic integral and introduce the dierential calculus associated with it, namely the Itô calculus. 51 52 CHAPTER 3, BROWNIAN MOTION AND SDE It is advisable that, upon rst reading, the reader passes over the proofs in small print, as they are rather technical. 1 General comments on continuous-time processes What do we exactly mean by continuous-time processes? 1. Let (E, E ) be a measurable space.
Prove that the super-replication price V0 is greater than V¯0 = sup E p1 ,p2 ,p3 p1 >0,p2 >0,p3 >0 p1 +p2 +p3 =1 p1 a+p2 b+p3 c=r 4. Prove that ∗ ∗ V¯0 ≥ VCRR = E p ,0,1−p f (SN ) (1 + r)N f (SN ) (1 + r)N . , where p∗ is such that p∗ a + (1 − p∗ )c = r (note that, using equations p1 + p2 + p3 = 1 and p1 a + p2 b + p3 c = r, we can express p1 as α(p2 ) and p3 as β(p2 ) and that E α(p2 ),p2 ,β(p2 ) (f (SN ))) is a continuous function of p2 ). Give an interpretation for p∗ and for VCRR in a Cox-Ross-Rubinstein model with d = 1 + a and u = 1 + c.
The following denition is an extension of the one in discrete-time. 1. Let us consider a probability space (Ω, A , P) and a ltration (Ft )t≥0 on this space. e. E(|Mt |) < +∞, for any t is • a martingale if, for any s ≤ t, E(Mt |Fs ) = Ms ; • a supermartingale if, for any s ≤ t, E(Mt |Fs ) ≤ Ms ; • a submartingale if, for any s ≤ t, E(Mt |Fs ) ≥ Ms . 2. It follows from this denition that, if (Mt )t≥0 is a martingale, then E(Mt ) = E(M0 ) for any t. Here are some examples of martingales. 3. If (Xt )t≥0 is a standard Ft -Brownian motion, then: 1.