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Could anyone give me some references where I could find
(a) discrete version(s) of Ito's lemma
(b) a proof how it converges to the continuous form in the limit
(c) its usage within stochastic difference equations
(d) a deduction of a discrete version of the Black Scholes model.

Every little bit of information would help.

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5 Answers

up vote 8 down vote accepted
  1. Stochastic Calculus for Finance II: Continuous-Time Models by Shreve or
  2. Shreve or Øksendal's Stochastic Differential Equations
  3. Øksendal
  4. Williams' Probability with Martingales or Shreve
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Terry Tao wrote a nicely motivated discussion of the discrete Black-Scholes equation.

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Szabados and Sz\'{e}kely. Stochastic integration based on simple, symmetric random walks. 2009. MRNumber 2472013

Szabados. A discrete It\^o's formula. 1990. MRNumber 1116806

Cs\"org\"o and Rev\'esz. On strong invariance for local time of partial sums. 1985. (Note: the authors actually had two papers in that year, and Szabados cites the wrong one.) MRNumber 805116

Kudzma. It\^o's formula for a random walk. 1982. MRNumber 684465.

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If the goal is a discrete version of Black-Scholes, this seems backwards to me. The standard discrete description of dynamic hedging using discrete steps for time and price on a binomial tree (you can easily extend it to a recombining multinomial tree, too) is for me the best discrete version of Black-Scholes. In particular, this description makes it quite clear why you want to use pseudo-probabilities and not real probabilities.

The continuous version of Black-Scholes is easily derived from this by taking a limit.

I guess I'm a Luddite, but I've never understood the need to know Ito's lemma and stochastic calculus when doing mathematical finance. The real world is discrete. Simple continuous limits such as Black-Scholes are extremely useful, but I've never fully understood why anyone would want to use overly sophisticated continuous stochastic models. The real world is far too noisy.

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Williams' discrete version of Black-Scholes is pretty brief. –  Steve Huntsman Feb 23 '10 at 15:02
    
Yes, it looks very nice and concrete. –  Deane Yang Feb 23 '10 at 15:13
    
I think one reason is that calculations in the continuous limit are simpler because you only have to deal with the highest order terms, much like calculating $\int_a^b x^n dx$ is easier than $\sum_a^b j^n$. It's also easier to deal with the Gaussian distribution than various distributions which tend to Gaussian in the continuous limit etc... You pay one time price of learning stochastic calculus, but it pays off in the long run. –  Mio Apr 4 '10 at 2:21
    
@Mio: Any chance you'd be willing to elaborate on a particular example? –  Deane Yang Apr 5 '10 at 1:46
3  
@Deane sure, well, one obvious example is that once I know stochastic calculus it's much easier for me to solve $dS=\mu S dt + \sigma S dW_t$ than to mess with the binomial model in order to get the Bernoulli distribution and take the limit. You have a simple algebra of $(dW_t)^2 = dt$ and all the higher powers of $dt$ and $dW_t$ vanish and with it you can do a lot of things that would be much more complicated to do in a discrete setting, like solve all kinds of SDE's. –  Mio Apr 6 '10 at 0:00
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