# 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