# Questions tagged [numerical-methods]

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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 ...
298 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 ...
87 views

### Measure of the behavior of Swaption surface

I'm looking to find a different measure than average shift move to explain the behavior of the IR VOL products say Swaption. I know it's a very open question not only touching upon IR VOL scope. Let ...
571 views

### Simulation scheme for SABR beside the standard Euler discretization

QUESTION: Beside Euler Scheme, is there another more robust (and preferably easy to implement) way to simulate asset path with SABR dynamics? Simulation that will withstand even for high volatilities....
410 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 ...
241 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 ...
98 views

### 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 ...
106 views

### 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 ...
87 views

### 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 ...
515 views

### Fastest way to calculate YTM from bond price

I would like to calculate YTM for every top of the book update on the 10-year note traded on Brokertec. There is no closed form solution so have to use a root finding method like Newton-Rhapson. It ...
1 vote
511 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 ...
592 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 ...
295 views

### Numerical Optimizer Matlab Calibration LMM

I am trying to mimimize the following function in order to calibrate the Libor Market Model $$\sum_{i=1}^{n} \left(\sigma_i^{market}-\sigma_i^{Reb}\left(a,b,c,d,\beta\right)/\sqrt{T_i}\right)^2,$$ ...
206 views

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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 ...
1 vote
122 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 ...
1 vote
186 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: ...
1 vote
72 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 ...
145 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 ...
1k 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 ...
401 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, ...
305 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 ...
1 vote
162 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 ...
131 views

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 $$... 5 votes 1 answer 4k views ### Least Squares Monte Carlo Could you explain to me in words (no formulas) the concept of the Least Squares Monte Carlo method to price an American style option? 1 vote 1 answer 149 views ### How to simulate asset prices/returns that display market regimes? Are there any techniques that can make a multivariate random number generating process for stock prices/returns, like geometric Brownian motion via Cholesky, also include the simulation of a finite ... 1 vote 1 answer 170 views ### Overview of frequentist, likelihood and Bayesian approaches to finance problems In quantitative finance tasks (asset pricing, portfolio optimization, option pricing, volatility forecasting, etc), there are frequentist, likelihoodist and Bayesian approaches or interpretations to ... -1 votes 1 answer 59 views ### Good ways to select best decision among N decisions, each with a profit/loss distribution? [closed] I'm working on a problem where an asset owner (e.g., owner of a factory, power plant, etc.) can take a number of possible decisions (say 10). Each of those 10 decisions entails certain actions, but ... 1 vote 0 answers 672 views ### CIR model. Is there a closed-form solution or even a good proxy of analytical solution? Is there a closed-form (analytical) solution for the Cox-Ingersoll-Ross SDE $$dr_t=k_r(\theta_r-r_t)dt+\sigma_r\sqrt{r_t}dW_t\tag{1}$$ ? Notice that \{r_t\} is our ... 2 votes 1 answer 894 views ### How to compute returns from cumulative returns in Python? [closed] If X is a T\times N pandas DataFrame of multivariate asset returns, the cumulative returns can be computed in python as (1 + X).cumprod() - 1 How can I reverse this operation so that I go ... 8 votes 2 answers 582 views ### Improve Finite Difference Scheme I understand how to derive and implement standard finite difference schemes. I wonder how to improve such a standard FD scheme? For example, when solving the standard Black-Scholes equation, the ... 1 vote 1 answer 527 views ### How to find characteristic function in Fourier Cosine method (COS method) by Fang and Oosterlee Fang and Oosterlee (2009) introduced Fourier-Cosine method (COS method) in their paper. The formula to price an option is approximately$$e^{-r\Delta t} \sum_{k=0}^{N-1}' Re\left\{ \phi\left( \frac{k\...
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. ...