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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 infinity: consequently, the term $N'(d_1)$ goes to zero, whilst the term $N(-d_2)$ goes to 1. Therefore, deep ITM puts can have a positive Theta, with a limit ...

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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 ...

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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 is both mathematically nicer to handle, analyse, and computationally has much better solvers than other generic PDE solvers. Don't solve the Black-Scholes PDE, ...

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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 $t ... 5 The PDE is defined for$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 ... 4 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 ... 4 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 shows how to set up the finite difference scheme so that all vanillas with strikes and expiries on the grid are matched exactly. 4 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{align*} Moreover, there existsj_0such that \begin{align*} &\ \frac{\max_j|U_j^{(n+1)}| - a_{j_0}\max_j|U_j^{(n)}| - c_{j_0}\max_j|U_j^{(n)}|}{b_{j_0}} \\ ... 3 You can consult Seydel pages 99-106 for explicit FD or for a short summary this link. The idea is that you cannot choosek$and$\frac{h^2}{2}$independently for stability. 3 From a cursory look, the FdBlackScholesBarrierEngine seems to do what you want; when the localVol parameter is set to true, it will use the local volatility contained in the passed process. I'd suggest you to check the code, though. As a further note: the GeneralizedBlackScholesProcess class converts the Black volatility to the local one internally (see the ... 3 The problem with your formula is the equation sign$=$. The second order finite difference is only an approximation to the true gamma: $$f^{\prime \prime}(x) \approx \frac{f(x+h)-2f(x)+f(x-h)}{h^2}.$$$h$can not be a result. Ideally, it should be small (whatever that means), so your original choice of$1\text{bp}$seems appropriate for this ... 3 Who gave you that idea? You absolutely can use Finite Differences for other PDEs. They are routinely used to solve hyperbolic PDEs (wave equation, both first and second order) and elliptic PDEs (steady state diffusion/heat equation). You can even mix and match the equation types and create PDEs that have characteristic of both hyperbolic and parabolic ... 3 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 operator as for example stated in Hung-Ju Kuo and N. S. Trudinger, On the discrete maximum principle for parabolic difference operators which can be applied to ... 3 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 normal rv, and $$S_T (Z) = S_0\eta (Z),$$ $$S_T^{up} (Z) = (S_0+dS)\eta(Z) = (S_0+dS)S_0^{-1} S_T (Z)$$ $$S_T^{dn} (Z) = (S_0-dS)\eta(Z) = (S_0-dS)S_0^{-1} S_T (... 3 Note that:$$ 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 $$as S\rightarrow \infty, for all t, and one still expects the put-call parity to hold:$$ C(t,S) - P(t,S) = S-K{\rm e}^{-r(T-t)}$$for all$S$. 3 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-Planck you may want to start from$t=\delta t$using a gaussian approximation for the density on the first step, so as to start from a smooth density. An alternative ... 3 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 0.2 vol and 1 year tenor. Shifts smaller than ~ 0.00008 result in an error for Gamma. Delta seems to be less sensitive for this, and it is fine for at least 1e-7 ... 2 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 integration to solve the smoothing integral. 2 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 value. This is possible as any value$V_{i + 1, j}$at time$\tau_{i + 1}$only depends on the values$V_{i, j}$at time$\tau_i$. When you use a scheme that is ... 2 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 claim that may but need not be contingent on the asset$X$, with$t$-value$N_t$; Define a probability measure$\mathbb{Q}^N$associated to the asset$N$such ... 2 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 dynamics. Consider for example European plain vanilla options. These have a (quasi) closed-form solution when the underlying follows an exponential Levy processes ... 2 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 equation. This means that the algebraic system you will solve will not involve any$V_{N,j,k+1}$values directly (so no$C_{k+1}$contribution from the F-... 2 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 with all terms roughly of the same size. For instance in the "log normal" SABR case ($\beta=1$) it would be appropriate to work in$(x, y) \in ]-\infty, +\infty[...

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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 formula directly to the implied vol surface rather than to the option price surface. The Dupire formula when applied to the implied vol surface $\Sigma(K,T)$ ...

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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 ...

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I got a lot of mileage out of Daniel Duffy's Finite Difference Methods in Financial Engineering: A Partial Differential Equation Approach.

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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 FD solvers, which is linear (zero second spatial derivative on the boundary). This means that on the boundary the PDE \$a\frac{\partial U}{\partial x} + b \frac{...

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Our company chose to use FDM for calculating American Options. According to colleagues I talked with, binomial trees are efficient and accurate When there are a small number of option values. But it has a couple of weaknesses: (1) Binomial tree models are generally inefficient when cash dividends should be taken into consideration; (2) Compared with FDM, ...

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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 ...

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