Questions tagged [monte-carlo]
Monte Carlo simulation methods are a broad class of computational algorithms that rely on repeated random sampling to obtain numerical results.
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How can we simulate daily return based on multi-factor model?
This is an interesting question for simulation. The question is a bit lengthy but I'm trying my best to make it super clear here.
Now I have some multi-factor model, say some US barra risk model from ...
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Monte Carlo simulation with SABR model
I have to price European Options using only the classical Monte Carlo method.
The models I have to select are Lévy models and SABR. Consider for instance the simplest Lévy model: a Geometric Brownian ...
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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 ...
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Satisfying put-call parity in Monte Carlo option valuation
I am trying to price European call and put options on a stock using the Monte Carlo method, given some dynamics for the underlying that may or may not have a closed-form solution (e.g. Black-Scholes, ...
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Clustering of Maximum Drawdown Values in Monte Carlo Simulations (Jaekle & Tomasini example)
Hope this question isn't too naive. I've been trying to replicate the Monte Carlo method using sampling without replacement as described in the Jaekle & Tomasini book (Trading Systems: A New ...
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cap/floor valuation in a hypothetical scenario using monte carlo simulation
How can I do a cap/floor valuation in a hypothetical scenario using monte carlo simulations of some interest rate model?
My conditions:
(For example) I want to do valuation of a cap option under a ...
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Monte Carlo option pricing
Can someone please confirm if I understood this correctly.
The Monte Carlo method for pricing path-dependent options essentially gives you a multitude of price processes, which you use to determine ...
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Why do we use a simple average for pricing options in MonteCarlo?
I was recently reviewing my notes on the Binomial Trees and MonteCarlo (MC) methods for option pricing. I've taken this for granted and just used the method....but I started questioning why we take a ...
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Why do we adjust the drift in the geometric brownian motion
I am building a monte carlo based on the GMB, and I am having a hard time understanding why we subtract 1/2 variance from the drift. If I have a drift of 12% and a volatility of 50%, that would give ...
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Understanding the application of Asset-Correlation to credit risk models
Suppose we have a portfolio of $n$ credits. In order the estimate the Portfolio Value at Risk (99,9) we use a standard vasicek model with the Ability to pay variable $A_i=\sqrt{\rho}x+\sqrt{1-\rho}z_i$...
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Monte Carlo methods: Choosing the best measure
When pricing derivatives using Monte Carlo methods, we take outset in the risk neutral pricing formula which states that we need to calculate the expected value of the discounted cashflows. To do this,...
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Method for using Historical Simulation method on an Instrument priced using Monte Carlo
I was speaking to a very esteemed professional in Financial Risk and he mentioned that he always prefers to use Historical Simulation as the method for his VaR even if he prices his Exotic derivatives ...
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Commodity forward curve Monte-Carlo
I need to value an Asian commodity option using Monte Carlo and a log-normal model. The inputs are the commodity forward curve and the volatility surface for futures/options expiry. Unfortunately, all ...
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Forward Black Implied Volatility For Within Risk Neutral European Option Pricing
Going to preface this question with an acknowledgement with how silly the ask is, but alas that is the working world; if anyone can share any ideas I'm all ears.
We're pricing an exotic option in risk ...
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asian geometric option valuation-- unable to get monte carlo simulation to converge to analytic value
I'm trying to price asian put options in which the averaging window begins immediately (T=0). currently, I'm trying to match up geometric averaging between my Monte Carlo simulations and my attempt at ...
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Closed-form equation for geometric asian call option
I'm looking to use the geometric asian option as a control variable for a monte carlo simulation. However, I have an issue with the closed-form equation to get the geometric price.
I'm using the ...
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Pricing European Call Closed Form Spread Options in Python
I am currently trying to correctly price European Call Closed Form Spread Options using Python. The main problem I am currently running into is that I have nothing to compare the option price so that ...
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Generating normally distributed random numbers using Sobol generator in QuantLib
I am trying use low discrepancy Sobol RNG to generate normally distributed random numbers and fill an Eigen matrix with those random numbers. The matrix represents a basket of 5 assets (rows) each ...
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How can I use Monte Carlo to price a Zero-coupon bond in the Cox-Ingersoll-Ross model?
Let me prefix this by saying that, yes, Cox-Ingersoll-Ross (C.I.R.) is deprecated when used to model interest rates. Yet integrals of the form
$$P(0,T) = E\left(\exp\left(-\int_0^Tr_s ds\right)\right) ...
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Black-Scholes implied volatility using a GARCH model
Why I'm not getting the same Black-Scholes implied volatility values as the ones given in the paper "Asset pricing with second-order Esscher transforms" (2012) by Monfort and Pegoraro?
The ...
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Is it possible to calibrate Mertons Jump Diffusion Model such that it matches mean and vola from a normal process without jumps? [closed]
I'm currently playing around with Mertons version of jump diffusion processes where i'm testing the predicitions of a trading model given a time series with and without jumps to isolate the effects of ...
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Backward induction: equation including expected values of stochastic process
Given the following SDE: $$ d\psi_t = \rho dt + \mu \psi_t dX_t$$, where $G(t) = \rho t$, $\rho = \frac{1}{T}$ $\psi_0 =0$, $T=1$, $\mu > 0$ and $X_t$ is a standard Brownian Motion (assume we know ...
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Derivatives without analytic expressions? [closed]
I was wondering if there exist options or other derivatives that do not have a known closed-form analytic expression (i.e., some sort of Black-Scholes PDE) and are usually priced using Monte Carlo ...
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Benchmark Model for Path-Dependant Monte Carlo Simulations?
As part of my research for my masters thesis, I'm testing out the effectiveness of some different models in Monte Carlo simulations for path dependant options.
I will be attempting a model-free ...
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LSMC for Out of The Money paths
In the Longstaff & Schawartz article they condition on using In-The-Money (ITM) paths only for the regression. The reason for this is to obtain more accurate results and also reduce the ...
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Pricing VIX derivatives using Monte Carlo
I am looking at pricing VIX options and futures using Monte Carlo simulation. Most papers recognise that VIX can be written in terms of the S&P500 index itself, namely as:
$$
VIX_t = \sqrt{-\frac{...
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Continuation value in Longstaff-Schwartz: Why the expected value?
In the paper by Longstaff and Schwartz on American option pricing, the continuation value at time $t_k$ is given by:
\begin{align}
F(\omega;t_k) = \mathbb{E}_Q\Big[\sum_{j=k+1}^Kexp\Big(-\int_{t_k}^{...
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Problem matching prices of Black-Scholes vs. GARCH(1,1) in Duan (1995)
In the paper of Duan (1995) the author compare European call option prices using Black-Scholes model vs. GARCH(1,1)-M model (GARCH-in-mean). To be brief, the author fits the following GARCH(1,1)-M ...
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Piecewise constant Heston model Monte Carlo simulation
I am studying this time dependent Heston model
\begin{equation}
dS_t=(r-q) dt +\sqrt{V_t} dW_t^1 \\
dV_t=\kappa_t(\theta_t-V_t) dt + \sigma_t dW_t^2 \\
S_0=s_0\\
V_0=v_0\\
...
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Monte Carlo Derivative Pricing
In order to try and price some derivatives with payoff $H(S_T).$
I am going to calibrate a few models (BS, Heston and CEV) to some real world data. Then I will calculate option prices as follows:
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Is it possible to estimate 'future zero curves' using Hull-White 1 factor model?
With regards to the following:
https://www.mathworks.com/help/fininst/price-bermudan-swaptions-with-different-interest-rate-models.html
It seems to me that the HW1 model is used to generate the ...
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Correlating multiple two-factor hull white processes in the context of CVA
Let's say we have two rates processes (EUR and USD) both following a 2-factor hull white.
Since it is in the context of CVA calculation, we calibrate both processes using Cap/Floors and swaptions (or ...
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Use `LocalVolTermStructureHandle` in Python QuantLib
I would like to simulate a local volatility underlying
$$ dS_t = S_t\sigma(t, S_t)dW_t $$
and have looked at QuantLib's LocalVolTermStructureHandle to do so.
So far:...
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How to simulate a delta hedged option strategy
I'd like to do a montecarlo simulation of a $\Delta$ hedged strategy (long OTM call) to see how the PnL distributes on cases like:
$\sigma_{bought} < \sigma_{realized}$
$\sigma_{bought} > \...
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Simulating of short rate model
I'm trying to simulate the risk factor of PFE from the interest rate model.
For example, under Vasicek model :
$$dr_t = k(\theta-r_t)dt + {\sigma}dW_t$$
with the analytic solution, we can simulate N ...
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Understanding Monte Carlo Simulation with Stratified Sampling
I am studying Monte Carlo Simulation with Stratified Sampling following Martin Haugh's lectures notes. As is it explained the variance of the estimator is
$$
Var(\hat{\theta}_{st,n}) = \sum_{j=1}^{m} \...
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Convergence rate of Bermudan to American option
When trying to value an American option we often use grid-based methods (e.g. Monte Carlo in combination with Longstaff Schwartz; or Finite Difference Methods). As such, we are in fact estimating the ...
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Quantlib Piece-wise Heston for Monte Carlo Path Generation
I am trying to use a piece-wise heston to generate paths for a Monte Carlo Simulation. I create and calibrate a ql.PiecewiseTimeDependentHestonModel as in the example on the ql doc python site:
https:/...
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Quantlib Monte Carlo using regular Volatility Surface, not Local Volatility surface
I am trying to run a Quantlib Python Monte Carlo simulation using either the ql.BlackScholesMertonProcess or the ql.GeneralizedBlackScholesProcess. I have a vol surface that I have generated using ql....
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Quantlib vol surface issue 'the black vol surface is not smooth enough'
I create a vol surface from the market and do smoothing(interpolation and extrapolation), and explicitly correct for any total variance decreasing on a given strike as we increase maturity. I create a ...
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Question regarding PRIIPs Category 3 product (life-cycle) calculation
I need your help regarding PRIIPs calculations. I’m working on life-cycle product, which I believe should be treated as Category 3 product regarding PRIIPS regulation. Let’s assume there are two funds:...
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Adjoint Automatic Differentiation
I'm currently benchmarking several Monte Carlo computing methods for sensitivities computation purposes.
I've already done some implementation in Python using Numpy and MyGrad libraries and it's ...
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Price of a simple autocall - Sebastien Bossu Advanced Equity derivatives
I am reading Advanced Equity Derivates by Sebastien Bossu and trying to do the exercises.
In chapter 1 we have the following question :
Consider an exotic option expiring in one, two, or three years ...
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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 ...
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MonteCarlo Value At Risk for futures portfolio
I wanted to ask, suppose I have a portfolio of futures of gasoline and other oil products eg ULSD (Ultra Low Sulphur Diesel), WTI (West Texas Intermediate) for different months. I want to compute the ...
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Speeding up Cutting Edge Quantitative Models on GPUs? [closed]
I have a very strong interest in the use of GPUs in quantitative finance, and am in search of algorithms/simulations/models that can have their runtime heavily reduced by GPU computation.
What models, ...
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Pricing options in underlying problem
Let us look at options, which are cash settled, but instead of receiving cash, you receive the proportion
from underlying asset with the same value as cash. Moreover, you can pay for these options in ...
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Optimize interest rate swap calculations in Monte Carlo Simulation
I’m running a simulation in which I want to calculate the NPV of 100 swaps over 1000 (or even much more) different interest rate curves.
It looks like Quantlib is not really fast in performing these ...
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How do I estimate volatility for MPR historical data
How can I estimate volatility with historical data for Monetary Policy Rate (MPR) to use in a short rate model?
I could use regular techniques like simple standard deviation or max likelihood, but the ...
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Problem with pricing a call option using the Monte Carlo Vasicek model
I am trying to price a call option on a zero coupon under the Vasicek Model using Monte Carlo method:
$$C_0 = B(0,\theta) \ \mathbb{E}^{\mathbb{Q}_T}[(B(\theta,T)-K)^{+}]$$
The problem is that the ...
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