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Questions tagged [euler]

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Euler scheme to simulate trajectories + Python Code

Question: Euler scheme to simulate trajectories of S + Python code to get independent trajectories of S? For solving a problem I have the following assumptions: stochastic basis (Ω,FT,(Ft)t≥0,P) ...
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0answers
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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 ...
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1answer
141 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 ...
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309 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 ...
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120 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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114 views

Weak and strong convergence Euler, Milstein, QE scheme for Heston model

I wonder what the weak and strong convergence for the Euler, Milstein, QE discretization schemes is for the Heston model. I do not know how to compute them, nor where to find them.
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Comparison of GBM paths with exact formulation and euler Discretization

I wrote the following program to compare the simulation path of a GBM using Euler discretization of the stock price, Euler discretization of the log-stock price and the exact formulation with the ...
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153 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 ...
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1answer
149 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: ...
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243 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 $...
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200 views

Euler discretization bias, heston model

I am performing option pricing using Heston model and Euler discretization. I'm getting the following result: ...
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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 ...
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1answer
82 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 $\delta x_t = x_t-x_{t-1}$
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2answers
341 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 ...
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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 ...
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209 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 ...