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 ...
Jan Stuller's user avatar
  • 6,098
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 ...
jherek's user avatar
  • 1,394
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 ...
oliversm's user avatar
  • 1,389
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 \...
Antoine Conze's user avatar
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 ...
Antoine Conze's user avatar
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 ...
Peter A's user avatar
  • 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 ...
Antoine Conze's user avatar
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{...
Gordon's user avatar
  • 21.1k
4 votes
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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....
AKdemy's user avatar
  • 8,739
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 ...
Jesper Tidblom's user avatar
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.
FunnyBuzer's user avatar
  • 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, ...
AKdemy's user avatar
  • 8,739
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 ...
Luigi Ballabio's user avatar
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.
river_rat's user avatar
  • 980
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 ...
Hans's user avatar
  • 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 ...
ir7's user avatar
  • 5,043
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 ...
ir7's user avatar
  • 5,043
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-...
Antoine Conze's user avatar
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 ...
Yoda And Friends's user avatar
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 ...
millovanovic's user avatar
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 ...
LocalVolatility's user avatar
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 ...
Quantuple's user avatar
  • 14.6k
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 ...
LocalVolatility's user avatar
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 ...
Yian Pap's user avatar
  • 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 ...
Antoine Conze's user avatar
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 ...
Sanjay's user avatar
  • 1,657
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 ...
Antoine Conze's user avatar
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 ...
Vincent C.'s user avatar
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 ...
AdB's user avatar
  • 704

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