<t>Hi,If I remember correctly, in general the set of convergent sequences does not uniquely determine a topology (this can be proven). In the case of convergence of probability though, it is "well-known" that the functional d(X,Y)=E(f(X,Y)) where f(x,y)=|y-x| / (1 + |y-x|), defines a metric on the s...

<t>Yes, well not only do you need a continuum af strikes but also the ability of trading infinitesimal amounts of each of these options :-)Seriously, it is easy to show that for any epsilon>0 you may find a finite combination af calls/puts whose payoff is epsilon-close to Jiyuan Bao's payoff, and yo...

What exactly do you call an "exponential martingale" ? is it a positive martingale ?I have heard of "exponential martingale associated with a (semi-)martingale" but I've never encountered "exponential martingale" by itself.

<t>Hi everyone,For 2) it's easy to write (or even better, draw) the payoff as a function af X : if C<F it's always 1/C, and in the case of interest C>F, it's 1/F for X<F, then 1/X for F<X<C and finally 1/C for X>C. So it looks kinda like bear spread, but the central segment is hyperbolic instead of ...

<t>I very much doubt that BM is uniformly integrable ! For instance, E|B_t| -> infinity as t -> infinity, so it's not even bounded in L^1. And no, B_t does not converge to +- infinity, or to any other limit : it keeps oscillating between high and low values (it is a recurrent Markov process !).About...

<t>Hi mahfuz,In fact BS is a model ; binomial, trinomial and MC are ways of implementing. "BS" alone doesn't tell you how to price your options (except for vanilla options with closed formulas)."binomial" or 'trinomial" alone doesn't tell you which model you're using (you can use basic CRR which is ...

<t>"Different" BMs doesn't mean anything if you don't specify the joint distribution. On the other hand, using only one BM for the two SDEs would mean that there is a single risk factor in the market, which is too simplistic, so multiple BMs is a better idea.In fact you could either :- write two SDE...

Well, from S_t = S_0 exp( (m - sigma^2 / 2) t + sigma W_t ) you derive: log ST / St = (m - sigma^2 / 2) (T - t) + sigma (W_T- W_t)Divide this by (T - t) and take the expectation ; you get (m - sigma^2 / 2) ; so it's only natural to get a negative value if m = 0. Right ?

Hello,Your simulation seems fine allright, now how do you compute the mean return ?

<r>Now that's a stupid question !Just kidding <E>:-)</E> If f(t) is a deterministic function, the derivative of log f(t) is f'(t) / f(t), which in terms of differentials reads : d(log f(t)) = df(t) / f(t). This is why f'(t)/f(t) is sometimes termed "logarithmic derivative" and df(t) / f(t) "logarith...

<t>Glad I could help. The formula I wrote is just the quotient of the Gamma density by the gaussian density, no reference for this particular case but for the general fact that dP1/dP2 = f1/f2 when P1 has pdf f1, P2 has cdf f2, you can find probably find it in your favorite measure-theoretic probabi...

<t>What do you mean by "characterised by their mean and variance" ? I assume you're talking about probability distributions over the real line R1, that are absolutely continuous i.e. have a density (like a Gamma distribution, unlike a Poisson distribution). If f is pdf (probability density function)...

<t>I agree with TheBridge, this ratio doesn't seem to have any immediate meaning (at least not to me). However, if you think of dW1 and dW2 as dW1 = N1 sqrt(dt) and dW2 = N2 sqrt(dt) where N1,N2 are independent standard Gaussian random variables, then dW1/dW2 = N1/N2 = C and it is well-known that C ...

Not that much better TheBridge ; you were right to mention that a larger class of jump processes than just Levy processes are Feller processes and are applicable to, say, financial modelling (among other things).

<t>As you know from your topic on Markov processes, if (X_t) is such a process then for each function f there is a function g such that E(f(X_(t+s))|F_s)=g(X_s) ; the function g depends linearly upon f and we denote by P_t the corresponding linear operator : g=P_t f. There's a different P_t for each...