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Questions tagged [euler]

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0answers
36 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 ...
2
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0answers
57 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 ...
1
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0answers
49 views

Simulation of Stochastic Volatility with Correlated Jumps (SVCJ) price paths

I am trying to simulate price paths for Monte Carlo option pricing of the Stochastic Volatility with correlated jumps model as presented in Dufffie et al.(2000), Eraker et. al. (2003) and Eraker (2004)...
3
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0answers
51 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
votes
1answer
405 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 ...
4
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0answers
389 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 ...
3
votes
0answers
137 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) : \...
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0answers
213 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
votes
1answer
219 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: ...
8
votes
0answers
275 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
241 views

Euler discretization bias, heston model

I am performing option pricing using Heston model and Euler discretization. I'm getting the following result: ...
4
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0answers
44 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
votes
1answer
98 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
368 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
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2answers
1k 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 ...
1
vote
2answers
217 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 ...