Questions tagged [euler]

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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 ...
-1
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1answer
75 views

Capital Allocation, VaR, Expected Shortfall

Are there any serious drawbacks / weaknesses in the Euler allocation method, when used to allocate VaR capital (and potentially Expected Shortfall) to risk factors in a portfolio? I notice that ...
1
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0answers
53 views

Vasicek Short rate simulation - analytical formula vs discretization

I've been using two approaches to simulate Vasicek short rate paths and I'm wondering if one of them is more correct than the other. The first approach is based on the analytical formula (see code ...
2
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0answers
60 views

Average individual consumption growth vs average aggregate consumption growth

Consumption growth is an essential thing in most asset pricing models and usually the Euler equation defines the return of an asset as a covariance between consumption frowth and the cash-flows of ...
3
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0answers
69 views

Discretisation of OU (mean reverting) process with a jump process

I have a question about how to apply the Euler approximation on OU process with a jump process. The stochastic process $X_t$ has dynamic $$dX_t=\alpha(\beta-X_t)dt+\sigma dW_t+dY_t$$ where $dY_t=...
2
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0answers
40 views

How does this transformation for Euler Scheme in mean reverting SDEs alleviate instability?

I saw this text in the book - Interest Rate Modelling by Andersen volume 1 on Page 112: I am unable to understand: How does instability arise when we use the Euler scheme on X(t)? What change does ...
3
votes
1answer
305 views

How to determine the order of convergence of the Euler-Maruyama method?

To make this simple let us consider the Geometric Brownian Motions. My questions: 1. How can I show that the Euler-Maruyama Method is convergent using GBM? 2. How can I determine the order of ...
3
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0answers
69 views

Euler discretization with jumps

There is a process $B_t = B_0\prod_{i=1}^{N_t}(1-Z_n)$, where $Z_n=e^{-ξ_n}$ for i.i.d exponentially distributed random variables $(ξn)_{n≥1}$ with rate $ρ=20$. ${N_t}$ is a counting process ...
3
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1answer
1k views

Calculate drift of Brownian Motion using Euler method

I am working on a project to approximate numerically the solution $X_t$ of a stochastic differential equation (SDE) using the Euler method. I have do to this for the Brownian motion with drift. I am ...
6
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0answers
613 views

(C++) Monte Carlo pricer for SABR model to test Hagan / Paulot formulas

I'm trying to test the so-called Hagan formula (p.6 of this paper) and the Paulot formula, order 1 only (eq. (43) p.19 of this paper. For this, i'm trying to use both Euler and Milstein scheme ...
2
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0answers
202 views

Problem of negative local volatility:

Consider the displaced log-normal process: $$dS(t) = \lambda(t)(a(t)+b(t)S(t))dW(t), S(0) = S_0>0, $$ where $W(t)$ is a one-dimensional Brownian motion. We suppose that $(\forall t \ge 0) : \...
0
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0answers
301 views

Euler discretization of SDE, combined with antithetic sampling

let's say we have a GBM $dS_t = r S_t dt + \sigma S_t dW_t$, where $W_t$ is standard Brownian motion, and we have an European option $C$ with payoff $f(S_T)$. I want to use an Euler discretization ...
0
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1answer
375 views

Euler discretisation error for stochastic volatility model

Given the following model$$dS_t=S_t(\mu dt+\sigma(t,S_t)dW_t)$$ Using Monte Carlo Pricing method, I want to determine the price of the option. However I have been encountered the following problems: ...
7
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0answers
307 views

simulating from the CIR++

I am looking at the CIR++ model which is described in interest rate models by Brigo et al, and was wondering on how to actually simulate from this model. The model reads $$r_t=x_t+\phi(t),$$ where $...
2
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0answers
370 views

Euler discretization bias, heston model

I am performing option pricing using Heston model and Euler discretization. I'm getting the following result: ...
3
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0answers
49 views

Regularity requirement for convergence of Euler scheme for stochastic integral?

Let $S_t$ be follow Black Scholes, then I am interesting in simulating the process $\int ^t _0 e^{-rt}1_{\{S_t\leq K\}}dS_t$ which is like a naive hedge of a European put, which does not work in ...
2
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1answer
128 views

How to use Euler discretization for this interest rate model?

How can I perform Euler discretization on this model where $\delta t=1$ and $\delta x_t = x_t-x_{t-1}$
1
vote
2answers
475 views

European call down and out option (geometric Brownian motion, Monte Carlo, Euler)

I need to estimate the expected value and the Greeks of an European call down and out option, assuming geometrical Brownian motion of the asset, with Monte Carlo simulation employing Euler ...
3
votes
2answers
2k views

How to discretize a GBM under P- and Q-measures?

Under the P-measure, a geometric Brownian motion can be specified using the following SDE: $$dS_t=\mu S_tdt+\sigma S_tdW_t^P$$ and its Euler discretization is $$S_{t+\Delta t}=S_t + \mu S_t \Delta ...
2
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2answers
281 views

What are some examples of non-solvable SDE where Monte Carlo discretization is necessary

Reading Glasserman - "Monte Carlo Methods in Finance" it says in the introduction to Chapter 6 - Discretization Methods, that moste models arising in derivatives pricing can be simulated only ...