Questions tagged [numerical-methods]
The numerical-methods tag has no usage guidance.
144
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Antoine Savine's store
In his book "Modern Computational Finance, AAD and Parallel Simulation", Antoine Savine writes page 263 in the footnote : "We could have more properly implemented the store with GOF’s ...
- 93
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103
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State-of-the-art grid construction techniques
I am wondering what the state-of-the-art regarding grid definition and construction, for solving PDEs using finite differences. I know some techniques are described in Duffy's Finite difference ...
- 588
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2
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82
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Does discretizing a diffusion model make it look like a jump diffusion model?
Can we distinguish a sample generated from a diffusion model with large time steps from a sample generated from a jump diffusion model. Not mathematically but numerically (if we ask a computer to tell ...
- 181
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1
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396
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Basket option value calculation
I am reading the article, where different approximations for the pricing of basket options are presented. I have tried to reproduce the result obtained by the Gentle's method in Python.
We define the ...
- 241
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1
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178
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Choosing a time step in Monte Carlo simulation of forward rates in LIBOR Market Model
Lets talk about the Monte Carlo simulation of forward rates in Euler discretization scheme under the $T_N$-forward measure, a so called terminal measure. Suppose that we have a number of time steps ...
- 764
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196
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SABR-LMM: best way to perform a MC simulation
I am working on a SABR-LMM model with the following system of SDEs under a numeraire $N$:
$$
\begin{align}
&\mathrm{d} F_i(t) = \sigma_i (t) (F_i(t) + s)^{\beta} \Big( \mu^f_i (t) \mathrm{d}t ...
- 428
2
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1
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343
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Black Scholes PDE discretization
We can solve the Black Scholes PDE by numerical methods like Euler
\begin{equation}
\frac{\partial V}{\partial t} + rS\frac{\partial V}{\partial S}+\frac{1}{2} \sigma^2 S^2\frac{\partial^2 V}{\partial ...
- 47
1
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1
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201
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Issues with calculating IV with options bar data
I am currently working with some options OHLC data (30 minute bars) from IBKR for a range of strike prices, maturities and for both calls/puts. For each bar, I am trying to back out the IV (crudely ...
- 93
1
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2
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290
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Estimating conditional expectation using monte carlo and least squares regression
I'm looking to understand the problem of least squares monte carlo that is used in valuation of bermudan options, but from a simpler context.
Say I have random variables $X$ and $Y$ which are uniform [...
- 123
2
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0
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119
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Numerical scheme for this HJB equation
Without dwelling on details on how to obtain the HJB equation for this problem, I would like to know if the scheme I wrote for solving it numerically is viable or did I miss something.
I need to solve ...
- 98
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344
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Solving option market making problem
I am currently working on a paper for quoting option as a market maker from Bastien Baldacci , Philippe Bergault & Olivier Guéant
Without dwelling on details on how to obtain the HJB equation for ...
- 98
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0
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91
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Choice of grid for numerical integration
I have to compute an integral involving the characteristic function for pricing options in a model and it so happens that accurate approximation seems to be mostly about putting lots of points in ...
- 2,436
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0
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145
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Bermudan pricing in Black-Scholes
Is there an "analytical" method to price American options (approximated as daily Bermudans) in the Black-Scholes model using backward induction?
$$V_T(S) = \max(K-S, 0)$$
$$V_{T-\Delta t}(S) ...
- 495
-4
votes
1
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153
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How to identify between Analytical, Numerical and ML Model based option pricing? [closed]
I am new to Quantitiative Finance. Coming from Computer Science domain, I wanted to clear the key distinguishing factor between analytical, numerical and ML based models for option pricing.
As far as ...
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1
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120
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Method to retrieve implied density for a mixture of local volatility model
Given a mixture model of two local volatility models, the price for an option is given by:
$$V(K,T) = p V_{loc1}(K,T) + (1-p) V_{loc2}(K,T)$$
where $V_{loc}(K,T)$ is the price of the option given a ...
- 125
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Importance sampling for Monte Carlo with local volatility in practice
I am given a diffusion with a local volatility to price barrier options:
$$dX(t)=X(t)\mu dt+X(t)\sigma(t,X)dW_t$$
I want to use Importance Sampling to price barrier options "far" out of the ...
- 125
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173
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Programming the Milstein method and computing the increments
In the wikipedia article on the Milstein method, the following python code to simulate a geometric Brownian motion is presented:
...
- 201
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1
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68
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Fitting parameters given an inverse function. (Orosi, 2015)
In trying to replicate Orosi's (2015) 5-parameter implied volatility model, but I can't wrap my head around the parameter fitting procedure Orosi proposes. My main goal is to calibrate the model to my ...
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144
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Integrated Delta does not seem to be smooth (ATM, Heston)
I am interested in an integrated call option that removes the dependence on time, $$I(S)=\int_0^\infty C(S,t)\text{d}t.$$ Because the value of a call option is a smooth function, I expect this ...
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276
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Likelihood ratio and pathwise sensitivity method for coupled SDEs
I have two coupled SDEs
\begin{align*}
dS_t=rS_tdt+V_tdW_t^{(1)},\\
dV_t=aV_tdt+b(V_t)dW_t^{(2)},\\
\end{align*}
where $W_t^{(1)}$ and $W_t^{(2)}$ are independent Brownian motions, initial input data ...
- 609
2
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2
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874
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Quasi Monte Carlo and Brownian bridge (how to combine them)
I am trying to understand how quasi Monte Carlo (QMC) and the Brownian bridge (BB) can be combined to price an asset, but I am having a hard time understanding how. I am just considering a European ...
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369
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Implementation of solvers for curve construction
I'd be really interested to hear people's experiences of implementing global solvers for curve construction, especially with regard to how robust the approach is in practice, numerical performance, ...
- 139
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1
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951
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Hyperbolic and Elliptic PDEs in Quant Finance
Parabolic PDEs (e.g. heat equation) are closely linked to finance via the Feynman Kac Theorem.
Do other types of PDEs appear in quant finance? Elliptic PDEs don't contain a time dimension (so perhaps ...
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4
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286
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Asymptotics of Call Option as $S\to0$
Let $C(S)$ denote the (initial) value of a call option with underlying spot price $S$. I assume that the underlying has continuous sample paths (not necessarily a geometric Brownian motion though).
As ...
- 688
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0
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139
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Implicit Scheme for Cox-Ingersoll-Ross Model PDE
I am considering the PDE for the price of a bond $V(r,t)$ with maturity $T$ under the Cox-Ingersoll-Ross model,
$$V_t+\frac12\sigma^2rV_{rr}+\nu(\theta-r)V_r-rV=0\quad r>0, t\in(0,1)$$
with ...
- 609
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votes
1
answer
126
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Maximum norm stability for implicit Black-Scholes equation
I am trying to prove maximum norm stability for the following implicit approximation to the Black-Scholes equation
$$\frac1{\Delta t}\left(U_j^{(n+1)}-U_j^{(n)}\right)+\frac{rS_j}{\Delta S}\left(U_{j+...
- 609
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1
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1k
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C++ code Thomas algorithm for solving a pentadiagonal Ax=b [closed]
I am looking to solve $Ax=b$ for $x$ where $A$ is pentadiagonal square matrix (elements on the upper and lower diagonals can however equal to zero) and $x$, $b$ two vectors of the same size.
The ...
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Improving control variate for variance reduction
I have tried stock price as control variate for my monte carlo simulation, and I am trying to reduce the variance of my estimated price for European Put option. And the code look like this:
...
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0
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64
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Verify mean-square convergence of the Euler-maruyama scheme numerically
I have a question about the order of convergence of the Euler-Maruyama scheme and how one verifes this numerically.
I have read that the Euler-Maruyama scheme is mean-squared convergent of order 1/2 ...
- 153
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1
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1k
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Anyone has detailed explanation on how to use epstein-zin preferences in asset pricing models
I'd be interested to know how Epstein-Zin preferences are used in, say, consumption-based asset pricing models. I'm looking for specific derivations (how you get the SDF) and possible numerical ...
- 2,436
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0
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Quick Discretization question for finite difference and finite element methods
Assume we have the discretization in space $x_1, x_2, ... , x_M$ and time $t_1, t_2, ... , t_N$ for a finite difference or finite element method for option pricing and we want to solve for the option ...
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boundary conditions in finite element method
In the appendix A of this paper, https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.227.5073&rep=rep1&type=pdf, a finite element method is demonstrated to price a straddle. The same ...
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Accuracy of Explicit Euler method (finite difference) decreases as Δx decreases, shouldn't it increase?
The price of a commodity can be described by the Schwartz mean reverting SDE
$$dS = \alpha(\mu-\log S)Sdt + \sigma S dW, \qquad \begin{array}.W = \text{ Standard Brownian motion} \\ \alpha = \text{ ...
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0
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Solve the Schwartz mean reverting PDE for option pricing using Euler explicit method (matlab)
Objective: Implement the Euler Explict Method for solving the PDE for option
prices under the Schwartz mean reverting model.
The price evolution of a commodity can be described by the Schwartz SDE
$$...
- 147
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1
answer
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Good ways to select best decision among N decisions, each with a profit/loss distribution? [closed]
I'm working on a problem where an asset owner (e.g., owner of a factory, power plant, etc.) can take a number of possible decisions (say 10). Each of those 10 decisions entails certain actions, but ...
- 101
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0
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637
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CIR model. Is there a closed-form solution or even a good proxy of analytical solution?
Is there a closed-form (analytical) solution for the Cox-Ingersoll-Ross SDE
\begin{equation}
dr_t=k_r(\theta_r-r_t)dt+\sigma_r\sqrt{r_t}dW_t\tag{1}
\end{equation}
?
Notice that $\{r_t\}$ is our ...
2
votes
1
answer
847
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How to compute returns from cumulative returns in Python? [closed]
If X is a $T\times N$ pandas DataFrame of multivariate asset returns, the cumulative returns can be computed in python as
(1 + X).cumprod() - 1
How can I reverse this operation so that I go ...
- 2,980
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1
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144
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How to simulate asset prices/returns that display market regimes?
Are there any techniques that can make a multivariate random number generating process for stock prices/returns, like geometric Brownian motion via Cholesky, also include the simulation of a finite ...
- 2,980
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2
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547
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Improve Finite Difference Scheme
I understand how to derive and implement standard finite difference schemes. I wonder how to improve such a standard FD scheme? For example, when solving the standard Black-Scholes equation, the ...
- 688
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1
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497
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How to find characteristic function in Fourier Cosine method (COS method) by Fang and Oosterlee
Fang and Oosterlee (2009) introduced Fourier-Cosine method (COS method) in their paper.
The formula to price an option is approximately
$$e^{-r\Delta t} \sum_{k=0}^{N-1}' Re\left\{ \phi\left( \frac{k\...
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Overview of frequentist, likelihood and Bayesian approaches to finance problems
In quantitative finance tasks (asset pricing, portfolio optimization, option pricing, volatility forecasting, etc), there are frequentist, likelihoodist and Bayesian approaches or interpretations to ...
- 2,980
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2
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215
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Are asset return means difficult to predict because they have no lower bound?
In finance, it is widely known that the volatility of asset returns ($\sigma$) are easier to predict than the expected value of asset returns ($\mu$) , otherwise known as the average return or mean.
...
- 2,980
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Risk-Neutral covariance matrix of arbitrage-free Nelson Siegel
For my thesis on a Bayesian sampling routine for a modification on arbitrage-free Nelson-Siegel I came across an equation that involves a matrix exponential within an integral, i.e.
$\int_{0}^{\Delta ...
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0
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175
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Longstaff Schwartz method (LSM): how to increase accuracy?
In the LSM method, I am currently (as they do in the paper) using weighted Laguerre polynomials as basis functions, about 3-5 of them.
If I wish to increase the accuracy of my model, what should I do?...
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143
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How is it possible that the measurement uncertainty in Kalman Filter is less than 0?
In Euan Sinclair's Option Trading, Pricing and Volatility Strategies and Techniques, it mentions that the true value of the price can be estimated via Kalman Filter:
$$S_\mathrm{new} = S + k (S_b − S)...
- 11
1
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1
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199
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L2 Assumptions of the Longstaff Schwartz method
In page 121 of the original LS Paper they use the fact that the space of functions they are dealing with (payoffs of American options), belong to the $\mathcal L^2$ space.
They use this assumption ...
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Optimized search for yield-to-worst of a callable bond
Suppose that I need to find the yield-to-worst of a callable bond, and that the option is American (call any time). The bond may have step-up coupons and/or non-constant call price (oprion strike). ...
- 11.9k
1
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1
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533
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QuantLib returns slightly different bondYield when backtested
I am just starting to get familiar with QuantLib (in particular, fixed rate bond pricing functions). I read a number of examples, from which I am able to calculate bond price and bond yield.
The ...
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564
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In Carr-Madans option pricing method, why do they use FFT?
In the famous fourier option pricing method by Carr-Madan, (http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.348.4044&rep=rep1&type=pdf), the crucial formula is
They evaluate this by ...
- 53
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votes
1
answer
5k
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Simulation of Geometric Brownian Motion in R
Using R, I would like to simulate a sample path of a geometric Brownian motion using
\begin{equation*}
S(t) = S(0) \exp\left(\left(\mu - \frac{\sigma^{2}}{2}\right)t + \sigma B_{t}\right),
\end{...
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