12 votes
Accepted

Least Squares Monte Carlo

To compute the price of an American option or a callable instrument in general, at each potential exercise date, one is required to compare its continuation value (discounted risk-neutral expectation ...
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10 votes
Accepted

Anyone has detailed explanation on how to use epstein-zin preferences in asset pricing models

Recursive Utility The traditional approach to consumption-based asset pricing includes time separable (additive) expected utility functions, $$U(C_t,C_{t+1})=u(C_t)+\beta \mathbb{E}_t[u(C_{t+1})],$$ ...
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10 votes
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Hyperbolic and Elliptic PDEs in Quant Finance

PDE Classification (Background) Linear second-order PDEs can be classified as either elliptic, parabolic or hyperbolic. A general PDE in two dimensions for $u=u(x,y)$ would look like $$Au_{xx}+2Bu_{xy}...
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8 votes
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Speeding up computations: when to use Quasi and standard Monte-Carlo in pricing

By definition the fair value of an option is given by an expectation value of the payoff, $\mathbf{E}\left[\textrm{payoff}(\textit{paths})\right]$. The probability distribution of the ...
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  • 1,839
8 votes
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Is there a good closed-form approximation for Black-Scholes implied volatility?

The method described in Hallerbach (2004) always worked well for me. We derive an estimator for Black-Scholes-Merton implied volatility that, when compared to the familiar Corrado & Miller [...
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7 votes

Is there a good closed-form approximation for Black-Scholes implied volatility?

Let's Be Rational uses exactly two iterations to give full machine accuracy for all inputs. It can be viewed as a three-stage analytical formula if you like. The code is free to download at www....
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7 votes

Heston Model Integration Oscillations

There has been a huge amount of work on this. Generally a Fourier transform approach is used. First, be careful to use the form of the characteristic function that does not wind about zero in order ...
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  • 6,743
6 votes
Accepted

Covariance matrix and Cholesky decomposition

I am not sure if I understood your question correctly but I will try to answer it anyway. If you have a standard normal random vector $z \sim N(\mathbb{0},I_n)$ (where $z,0 \in \mathbb{R}^{n\times1}$ ...
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  • 2,874
6 votes
Accepted

What does "convergence" in Monte Carlo simulation mean?

To keep things simple let's assume you have a perfect random number generator (i.e. I will discuss only the statistics not the numerics of the problem). I will also focus on the practical matter and ...
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  • 1,933
6 votes
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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 ...
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  • 1,223
6 votes
Accepted

Produce the random variable for an asset from a uniformly distributed random varible

The question requires you to provide a method which uses uniform random variables and transforms them to generate realizations of the described asset values. To give a bit more general answer: this ...
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  • 76
6 votes
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Simulation of Geometric Brownian Motion in R

The issue is that you do not plot one sample path but for each time point $t$, you simply plot one possible realisation of the random variable $S_t(\omega)$. Thus, you don't get a connected path. (...
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  • 13.8k
6 votes
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In Carr-Madans option pricing method, why do they use FFT?

Indeed, the FFT was a notable improvement in computational option pricing in 1999, but further investigation has shown that it can be easily optimized both in terms of speed and accuracy. For instance,...
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  • 1,401
6 votes
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Are asset return means difficult to predict because they have no lower bound?

To answer, the assertion that volatility is easier to predict than expected return requires clarification. The phrase "easier to predict" is particularly ambiguous. To me this means that the ...
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  • 3,260
6 votes
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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 ...
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  • 1,359
6 votes
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Asymptotics of Call Option as $S\to0$

This is more of a math question than a quant question. Under Black Scholes dynamics (assuming $r=0$ for simplicity), as everyone knows we have $$C=SN(d_1)-KN(d_2)$$. In this case, we are interested ...
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  • 13.9k
5 votes

Is there a good closed-form approximation for Black-Scholes implied volatility?

There are some other references: Li and Lee (2009) [download] An adaptive successive over-relaxation method for computing the Black–Scholes implied volatility Stefanica and Radoicic (2017) An ...
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  • 1,060
5 votes
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What is an efficient method to find implied volatility?

Peter Jaeckel wrote a paper just on how to solve this problem: By Implication (July 2006; Wilmott, pages 60-66, November 2006). Probably the most complicated trivial issue in financial mathematics: ...
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  • 6,743
5 votes
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QuantLib returns slightly different bondYield when backtested

I would start by saying that yes, this is an acceptable precision. However, the reason you are not getting the same result is because, by default, QuantLib has ...
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  • 5,295
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 ...
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  • 456
4 votes

Usage of Brownian Bridge?

You can find a brief but useful explanation of Brownian bridge techniques in Andersen and Piterbarg (page 125), which includes references for further reading. It's probably the best place to start. ...
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4 votes

What does "convergence" in Monte Carlo simulation mean?

the output of an MC simulation depends on the random numbers used and if the distribution used is not too weird, after 10,000 runs you will get an answer that is distributed $$ \mu + \frac{\sigma}{\...
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  • 6,743
4 votes
Accepted

Quadratic exponential method (by Andersen) in Heston model

There is a qualitative shift in the shape of the density. When V is small it is monotone decaying. When V is large it looks more like a Gaussian. Another reason he uses two schemes is that he wants ...
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  • 6,743
4 votes

Heston Model Integration Oscillations

I'd use FFT or similar rather than direct integration. Here is an old paper with Heston example: Option pricing using fractional FFT
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  • 4,217
4 votes

How should I develop my coding ability in order to set myself up for a quant role?

To test your programming skills, try QuantLib. Can you do interest-rate modelling with QuantLib? Can you debug the 10-level C++ template? Do you know how to use day count? Do you know how to use ...
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  • 2,255
4 votes

Architecture of a global pricing library with immutable payoffs

That's the best question that nearly no one asks. I'm with you on Quantlib and Strata, haven't really seen a very good design around but I've seen quite a few bad ones. It is definitely doable and has ...
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4 votes

What are the industry standards and rules of thumb when it comes to numerical methods?

1> Analytical - Black Scholes formula for Vanilla European options, Digitals. These valuations are just an "interpolation" of traded options. We interpolate the implied volatility from the traded ...
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  • 948
4 votes

C++ code Thomas algorithm for solving a pentadiagonal Ax=b

You are looking to solve a linear system with a band diagonal system matrix $A$ of dimensions $N \times N$ and having the property $m_1 = 2, \, m_2 = 2$, respectively the number of sub-diagonals under ...
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  • 356
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{...
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