Questions tagged [numerical-methods]
The numerical-methods tag has no usage guidance.
141
questions
1
vote
1
answer
178
views
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 ...
- 219
0
votes
1
answer
80
views
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 ...
- 636
6
votes
0
answers
173
views
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
votes
1
answer
206
views
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
vote
0
answers
114
views
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
vote
2
answers
255
views
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
votes
0
answers
116
views
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
7
votes
0
answers
319
views
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
1
vote
0
answers
80
views
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 ...
- 2,386
2
votes
0
answers
120
views
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) ...
- 485
-4
votes
1
answer
113
views
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 ...
- 93
1
vote
1
answer
113
views
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
6
votes
1
answer
246
views
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
1
vote
0
answers
105
views
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
1
vote
1
answer
59
views
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 ...
- 35
5
votes
0
answers
131
views
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 ...
- 14.9k
7
votes
2
answers
223
views
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
votes
2
answers
560
views
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 ...
- 609
3
votes
0
answers
298
views
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, ...
- 129
7
votes
1
answer
680
views
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 ...
- 73
5
votes
4
answers
234
views
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 ...
- 678
1
vote
0
answers
103
views
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
4
votes
1
answer
123
views
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
0
votes
1
answer
1k
views
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 ...
- 89
-1
votes
1
answer
82
views
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:
...
- 17
3
votes
0
answers
58
views
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
3
votes
1
answer
1k
views
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 ...
- 2,386
1
vote
0
answers
41
views
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 ...
- 89
1
vote
0
answers
70
views
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 ...
- 89
0
votes
1
answer
102
views
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{ ...
- 127
2
votes
0
answers
242
views
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
$$...
- 127
-1
votes
1
answer
58
views
Good ways to select best decision among N decisions, each with a profit/loss distribution? [closed]
I'm working on a problem where an asset owner (e.g., owner of a factory, power plant, etc.) can take a number of possible decisions (say 10). Each of those 10 decisions entails certain actions, but ...
- 101
1
vote
0
answers
526
views
CIR model. Is there a closed-form solution or even a good proxy of analytical solution?
Is there a closed-form (analytical) solution for the Cox-Ingersoll-Ross SDE
\begin{equation}
dr_t=k_r(\theta_r-r_t)dt+\sigma_r\sqrt{r_t}dW_t\tag{1}
\end{equation}
?
Notice that $\{r_t\}$ is our ...
1
vote
1
answer
686
views
How to compute returns from cumulative returns in Python? [closed]
If X is a $T\times N$ pandas DataFrame of multivariate asset returns, the cumulative returns can be computed in python as
(1 + X).cumprod() - 1
How can I reverse this operation so that I go ...
- 2,885
1
vote
1
answer
135
views
How to simulate asset prices/returns that display market regimes?
Are there any techniques that can make a multivariate random number generating process for stock prices/returns, like geometric Brownian motion via Cholesky, also include the simulation of a finite ...
- 2,885
8
votes
2
answers
446
views
Improve Finite Difference Scheme
I understand how to derive and implement standard finite difference schemes. I wonder how to improve such a standard FD scheme? For example, when solving the standard Black-Scholes equation, the ...
- 678
1
vote
1
answer
425
views
How to find characteristic function in Fourier Cosine method (COS method) by Fang and Oosterlee
Fang and Oosterlee (2009) introduced Fourier-Cosine method (COS method) in their paper.
The formula to price an option is approximately
$$e^{-r\Delta t} \sum_{k=0}^{N-1}' Re\left\{ \phi\left( \frac{k\...
- 830
1
vote
1
answer
149
views
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 ...
- 2,885
1
vote
2
answers
181
views
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.
...
- 2,885
1
vote
0
answers
75
views
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 ...
1
vote
0
answers
131
views
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?...
- 11
1
vote
0
answers
130
views
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
vote
1
answer
173
views
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 ...
- 691
4
votes
0
answers
131
views
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). ...
- 10.8k
1
vote
1
answer
436
views
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 ...
- 113
5
votes
1
answer
484
views
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
2
votes
1
answer
4k
views
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{...
- 111
3
votes
2
answers
266
views
Produce the random variable for an asset from a uniformly distributed random varible
I'm working on a quant interview question from the book called Quant Job Interview Questions And Answers (by Mark Joshi and other authors). I cannot understand the following question(not the answer, ...
- 587
1
vote
1
answer
92
views
Calibrate a model parameter with an error function
Suppose I want to find the implied volatility using an option model from market prices. Surely I can find the implied volatility for each strike price ($k$ different strike prices) for a given ...
- 1,346
3
votes
1
answer
90
views
When is a numerical solution the only way to obtain a solution to BS?
I am only now reading into Mathematical Finance, I understand the derivation of the BS equation with vanilla European options. On the next page of my book it starts to delve into obtaining exact ...
user42442