8
votes
Accepted
Negative theta for a short put
Theta on a European Put option on a non-dividend paying stock is:
$$\Theta=-\frac{S_t \sigma}{2\sqrt{\tau}}N'(d_1)+rKe^{-r\tau}N(-d_2) $$
For deep in-the-money Puts, $d_1$ and $d_2$ go to negative ...
- 6,308
6
votes
Accepted
Brennan-Schwartz algorithm for pricing American options
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 ...
- 1,444
6
votes
Accepted
Improve Finite Difference Scheme
Don't solve the Black-Scholes PDE, solve the heat equation
One of the major results of mathematical finance is showing that the Black-Scholes PDE can be mapped to the heat equation. The heat equation ...
- 1,399
5
votes
Finite difference: move forwards or backwards?
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 \...
- 5,702
5
votes
Accepted
Errors on Finite Differences + Implicit Scheme + Black & Scholes
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 ...
- 5,702
5
votes
Improve Finite Difference Scheme
Some of the standard tricks are mentioned in this paper, Finite Difference Schemes with Exact Recovery of Vanilla Option Prices
https://papers.ssrn.com/sol3/papers.cfm?abstract_id=3530561
which also ...
- 494
4
votes
Binomial Trees vs FDM
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 ...
- 5,702
4
votes
Accepted
Maximum norm stability for implicit Black-Scholes equation
Note that
\begin{align*}
U_j^{(n)} &= \frac{U_j^{(n+1)} - a_jU_{j-1}^{(n)} - c_jU_{j+1}^{(n)}}{b_j}\\
&\le \frac{\max_j|U_j^{(n+1)}| - a_j\max_j|U_j^{(n)}| - c_j\max_j|U_j^{(n)}|}{b_j}.
\end{...
- 21.2k
4
votes
Accepted
Finite Difference Method in Greeks (Options)
Agree with @Brian B. With BS, you cannot have the issue in (1). Tree, grid, Monte Carlo could all result in errors though.
(2) is a likely reason. I just tried in Julia for ATM, 0 div and rates plus 0....
- 9,394
4
votes
Boundary condition issues for Black-Scholes PDE using finite-differences
Alright, so I solved my issues here. Since it might be useful to others stuck on the same thing, I mention it here.
The problem I described about assuming the second order term was equal to zero is ...
- 731
3
votes
Explicit Euler stability for the Heat Equation (FDM)
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.
- 1,012
3
votes
Black Scholes Theta Finite difference
@Sanjay's answer is correct but there is an important consideration from a practical perspective.
Closed form theta in BS is the change per unit time (the change after one year). In other words, ...
- 9,394
3
votes
Accepted
how to price barrier option under local vol model using QuantLib
From a cursory look, the FdBlackScholesBarrierEngine seems to do what you want; when the localVol parameter is set to ...
- 7,853
3
votes
Canonical text on numerical PDEs in finance
I got a lot of mileage out of Daniel Duffy's Finite Difference Methods in Financial Engineering: A Partial Differential Equation Approach.
- 1,080
3
votes
Accepted
Implicit finite difference method always guarantees positive and stable price of derivative?
For the original PDE, the positivity can be deduced from the maximum principle for a parabolic operator. There is also a discrete version of the maximum principle for the finite difference parabolic ...
- 2,806
3
votes
Accepted
Greeks: Estimate gamma by Monte Carlo finite difference
Pathwise finite difference Gamma formula is indeed:
$$\Gamma(S_0,T, dS; Z) = (dS)^{-2} \left[ (S_T^{up} (Z) - K)^+ -2 (S_T (Z) - K)^+ + (S_T^{dn} (Z) - K)^+ \right], $$
where $Z$ is a standard ...
- 5,123
3
votes
Accepted
Boundary Conditions for Call Option in Black Scholes Model
Note that:
$$ C(t,S) =S-K{\rm e}^{-r(T-t)} $$
as $S\rightarrow \infty$, for all $t$.
Basically because one can easily accept
$$ P(t,S) =0 $$
as $S\rightarrow \infty$, for all $t$,
and one still ...
- 5,123
3
votes
Negative Density in Local Stochastic Volatility (LSV) Model Calibration
Yes it should preserve positivity. However due to numerical noise you may observe very small negative values on the edges of the lattice, that you can truncate to zero.
If you solve using Fokker-...
- 5,702
3
votes
Accepted
Question on boundary conditions when using Finite Difference
1 - The Neumann boundary condition is actually named after Carl Neumann, not John von Neumann.
There is another boundary condition not often mentioned but used very often in practice in Quant finance ...
- 5,702
3
votes
Numerically stable method for estimating $\partial_t \mathbb{E}[f(X_t)]$ where $X_t$ is an n-dim Ito process and $f:\mathbb{R}^n\rightarrow\mathbb{R}$
I would recommend to have a look at: https://www.youtube.com/channel/UC9RbRnYPhO9lpiY-6wWNHWg
It is quite long but worth it. It covers exactly the topic you are looking for.
More precisely, this ...
- 382
2
votes
Accepted
Smoothing of the payoff function as a terminal condition for numerical option pricing
I just wanted to say that I solved the problem using the symbolic/analytical features of Mathematica and Matlab to perform the inverse Fourier transform and then I used high-order numerical ...
- 116
2
votes
Accepted
Pricing an American derivative with finite differences
When you use a fully explicit finite difference scheme, you can simply apply the backward induction step and afterwards ensure that the option price at each node is at least equal to the intrinsic ...
- 6,084
2
votes
Accepted
Other numerraire choices when applying Feynman Kac
Assuming that you
Have an (or a set of) SDE(s) describing the dynamics of an asset $X$, with $t$-value $X_t$;
Define $V$ as a claim contingent on the asset $X$, with $t$-value $V_t$;
Define $N$ as a ...
- 14.7k
2
votes
Which options do not have a closed form pricing formula like BS?
The question about the availability of closed-form solutions can generally not be answered for types of options alone but only for the combination of a payoff function and the underlying asset ...
- 6,084
2
votes
Accepted
SABR PDE spot/forward upper boundary condition implementation
Wherever your discretized PDE references $V_{N,j,k+1}$, you will use your expression for $V_{N,j,k+1}$ in terms of $V_{N−1,j,k+1}$ and $V_{N−2,j,k+1}$ to eliminate $V_{N,j,k+1}$ from the discretized ...
- 521
2
votes
Accepted
Unable to obtain correct Finite Difference Results
a few pointers:
did you use appropriate boundary conditions ?
how did you truncate the space domain ? in particular it is common to first do a change of variable to get a better behaved $A$ matrix ...
- 5,702
2
votes
Black Scholes Theta Finite difference
First and foremost I do not agree with you Closed Form value. I get $\Theta=-8.963$. There are various of BS calculator you can use the check your results and in general you should do that. Here is ...
- 1,667
2
votes
Accepted
Local Volatility implementation
The usual way is to fit a surface (e.g. smoothing splines) to the grid and to compute derivatives off the surface. Note however that the entire process tends to be more stable when applying the Dupire ...
- 5,702
2
votes
Is it possible to model path-dependent clauses using finite difference methods?
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 ...
- 714
2
votes
Local Volatility calculation in Python
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 ...
- 141
Only top scored, non community-wiki answers of a minimum length are eligible