---
# System prepended metadata

title: Heston Discretisation Schemes

---

# Heston Discretisation Schemes

## Single asset Schemes 

### Euler
The standard one.

### Andersen-Brotherton-Ratcliffe Scheme 1 (Log-normal)
- Article:
    - Extended Libor Market Models with Stochastic Volatility
    - Andersen, Brotherton-Ratcliffe
    - 2001
- Description:
    - Log-normal approximation
    - Revied in Kahl-Jackel
    - Variance is always positive
    - Unstable for suitable time steps

### Andersen-Brotherton-Ratcliffe Scheme 2 (Moment matched Log-normal)
- Article:
    - Extended Libor Market Models with Stochastic Volatility
    - Andersen, Brotherton-Ratcliffe
    - 2001
- Description:
    - Log-normal approximation + Moment matching
    - Revied in Kahl-Jackel
    - Variance is always positive
    - Simulation stable

### Kahl-Jackel Scheme
- Article:
    - Fast strong approximation Monte-Carlo schemes for stochastic volatility models
    - Kahl, Jackel
    - 2005
- Description:
    - *"pathwise approximations"* is used
    - If the Feller's condition is satisfied, the variance is always positive.
    - Reived in Andersen

### Broadie-Kaya Scheme
- Article:
    - Exact simulation of stochastic volatility and other affine jump diffusion processes
    - Broadie, Kaya
    - 2006
- Description:
    - The exact simulation of the non-central chi square distribution
    - Computationaly costy

### Andersen Scheme
- Article:
    - Efficient Simulation of the Heston Stochastic Volatility Model
    - Andersen
    - 2005
- Description:
    - The famous one
    - Mimicing the non-central chi square distribution by using moment matching
    - Variance is always guaranteed to be positive.

### Zhu Scheme
- Article:
    - A Simple and Exact Simulation Approach to Heston Model
    - Zho
    - 2008
- Description:
    - Converting the Heston process to the Ornstein-Uhlenbeck process
    - But the mean reviersion is stochastic and unstable with big time steps.


## Multi-Asset


