5

The PDE is defined for $x \in ]-\infty, +\infty[$ but the finite difference scheme requires a truncated domain $[x_{\min}, x_{\max}]$, and the choice of $x_{\min}$ and $x_{\max}$ will affect the quality of the result, regardless of the scheme being explicit, implicit, or mixed. A good rule of thumb is to choose the truncation $[x_{\min}, x_{\max}]$ such ...


5

They have written the equation to be solved as $$ -\frac{\partial C}{\partial t} + r S \frac{\partial C}{\partial S} + ... = 0 $$ instead of the more usual $$ \frac{\partial C}{\partial t} + r S \frac{\partial C}{\partial S} + ... = 0 $$ This means that in their setup $t$ represents the time to maturity, that is $t = T - \text{time}$. So they start from $t ...


4

Ikonen and Toivanen don't say that the LCP is solved exactly, they simply say that the modified back-substitution is a valid algorithm to solve the LCP. A numerical error may arise around the location of optimal exercise, since it does not fall directly on the finite difference grid. I think that however, the error is of the same order as the discretization ...


3

You can consult Seydel pages 99-106 for explicit FD or for a short summary this link. The idea is that you cannot choose $k$ and $\frac{h^2}{2}$ independently for stability.


2

Actually recombining binomial trees are only a particular case of an explicit FDM scheme. But they have obvious limitations, the foremost being that they cannot accomodate local volatilities. Also 1/2 explicit 1/2 implicit FDM schemes (Crank-Nicolson) have faster convergence with respect to the size of the time step. And FDM schemes can accomodate all sorts ...


2

If you are asking whether it is possible to price path-dependent American options in tree based models, the short answer is yes. You simply construct your tree/grid and evaluate the rules in each node (analogous to what you would do in your MC simulations). These rules can be arbitrarily complex. Note, however, that you can only evaluate them at a discrete ...


1

Unfortunately not written in Python, but in R. If you have experience with R this real life example posted on an underground quant blog has step by step what you may be looking for: (Scroll down to conclusion) https://quantipy.wordpress.com/2017/08/21/implementation-of-dupires-formula-for-local-volatilities/ (I do not take credit for this persons work), ...


1

You need to add an auxiliary state variable that represents the current strike $K_t$, with dynamics $K_{t} = K_{t^-}$ if $S_t > 0.8 K_{t^-}$, $K_{t} = S_t$ if $S_t \leq 0.8 K_{t^-}$. You will get a jump/PDE with 2 state variables which you can then solve. Some people call that "1.5" PDE because the second state variable updates only depend on the first ...


1

The usual approach to deal with path dependency in finite differences/lattices solvers is to capture the path dependency trough one or more auxiliary variable(s) that make the problem non path dependent in the augmented space, and to discretize along these auxiliary variable(s). For instance that's easily done for asian options where the path dependency is ...


1

This depends on quite a few other inputs. If you're comparing deltas, what's the volatility of the underlying and the moneyness of the derivative (whether call, put, or something more exotic). I've done valuations that need 10 million simulations (5x out of the money options with 10 years until expiration) and others where 10,000 is perfectly sufficient (at ...


1

For the maturity, choose a grid $\{t_0=0,t_1,\dots,t_n=T\}$ such that $T$ is the option's maturity. For the underlying, if it is positive, you might choose an upper boundary by selecting a grid $\{S_0=0,S_1,\dots,S_{\max}\}$ such that the derivative's delta at $t_{n-1}$ is above a threshold $D$ in order to specify a boundary condition such as: $$ \frac{\...


1

Answer is very straightforward: always center your grid on the current spot value and make sure that it covers sufficiently many standard deviations on each side to cover enough of the terminal distribution at maturity. In particular it should cover the forward at maturity. NB: By definition when you use a FDM you will get back the grid of all values of ...


1

Another approach would have been to use some projection as in Pooley and Vetzal Convergence remedies for non-smooth payoffs in option pricing. In your case, it may be a projection of the initial condition to the RBF space (I have read your paper, and it looks interesting). I wonder a bit how the two approaches compare.


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