# Tag Info

1

I would suggest that you use a more 'modern' method to recover option prices from characteristic functions. The approach of this papers (for practical calculations of option prices) is somewhat outdated. The backbone of affine models (such as SVJJ) is the characteristic function $\psi(u)$ of the log-price distribution, which is known in closed form. The ...

3

About the integration problem: Your integrand is highly oscillatory, and the adaptive quadrature of Matlab doesn't handle such integrands very well. In general, I would recommend Mathematica when Matlab's standard procedures don't perform well. In this case, a Levin-type method would perform much better. The reason that quadv produces NaN values is because ...

4

If by 'solve' you mean how do we know that $\ln S_t$ is the right change of variable, then you can go by the following (not rigorous) line of thought: Ito's fomula suggests that given an SDE $$dX_t = \mu(X_t,t)dt+\sigma(X_t,t)dW_t$$ and a function $f(x,t)$: the SDE for the process $Y_t=f(X_t,t)$ will satisfy $$dY_t = [f_t(X_t,t) + f_x(X_t,t)\mu(X_t,t) + ... 3 These are all examples on Ito Formula in its general form (with quadratic variations): 11 My understanding is because the Ito's integration definition keeps the martingale property. With Brownian motion W(t, \omega) defined, to define stochastic integration in a Riemann–Stieltjes style:$$\int_0^t f(t, \omega) d W(t, \omega) = \lim_{\| \Delta_n\| \to 0 } \sum_{i=1}^{n} f(\tau_i,\omega) \left ( W(t_i, \omega) - W(t_{i-1}, \omega) \right ) $$, ... 9 In fact Ito and Stratonovich calculus are both mathematically equivalent. In the following paper you can e.g. see that both derivations lead to the same result, i.e. the Black-Scholes equation: Black-Scholes option pricing within Ito and Stratonovich conventions by J. Perello, J. M. Porra, M. Montero and J. Masoliver From the abstract: Options ... 1 Let me begin to say that this was one of the most interesting and well written questions I've read in a long time. Even though you have already answered your own question I would like to clear out some terminology and also present my theory as for why many people make the same mistake as Shreve. First I would like to point out that it makes no sense to say ... 1 OK, I think now I got the point, after comparing to Shreve's "Stochastic calculus for finance I, The binomial asset pricing model", the simpler case. The pricing theory in continuous time is: Defi ne the wealth process X(t), by de finition, it is self-fi nancing:$$d X(t) = \Delta(t) dS(t) + r (X(t) - \Delta(t)S(t)) dt$$De fine risk-neutral measure ... -2 Shreve did not in my opinion approach the problem rigorously. The only economically meaningful portfolio is one that is self-financing (in Shreve's words, you cannot decided tomorrow what to invest in today). However he did not explicitly state in his derivation that the portfolio was self-financing, and that is where I feel he lacks rigor. Shreve ... 1 You seem particularly frustrated that these formula derivations are not excluding, e.g., American options. But keep in mind that, up through the derivation of the PDE, there is nothing in them that assumes a particular payoff condition. The PDEs such as$$\begin{equation*} f_t(t, x) + \beta(t, x)f_x(t, x) + \frac{1}{2}\gamma^2(t, x)f_{xx}(t, x) = rf(t, x) ...

3

It is true that the self-financing property of the replicating portfolio seems not explicitly presumed nor shown in Shreve's derivation of the Black-Scholes formula. One may note that a replicating portfolio is by definition a self-financing portfolio which replicates the payoff. The problem as I see is that Shreve is just suggesting some portfolio and ...

Top 50 recent answers are included