# Multivariate fractional Brownian motion and generalizations of SABR model

@inproceedings{Musiela2019MultivariateFB, title={Multivariate fractional Brownian motion and generalizations of SABR model}, author={Marek Musiela}, year={2019} }

The SABR model is a generalization of the Constant Elasticity of Variance (CEV) model. It was introduced and analyzed by Hagan et al. (2002). Rapidly it has become the market standard for quoting cap and swaption volatilities thanks to the approximate formula for implied volatility which allowed real time risk management of large books of caps and swaptions. Later on it was also used in FX and equity markets. The generalization introduces stochastic volatility to the CEV model. The volatility…

## 2 Citations

### A partial rough path space for rough volatility

- Mathematics, Economics
- 2022

We develop a variant of rough path theory tailor-made for the analysis of a class of ﬁnancial asset price models, the so-called rough volatility models. As an application, we prove a pathwise large…

### On asymptotically arbitrage-free approximations of the implied volatility

- Mathematics
- 2022

Following-up Fukasawa and Gatheral (Frontiers of Mathematical Finance, 2022), we prove that the BBF formula, the SABR formula, and the rough SABR formula provide asymptotically arbitrage-free…

## References

SHOWING 1-10 OF 14 REFERENCES

### Short-time at-the-money skew and rough fractional volatility

- Mathematics, Economics
- 2015

The Black–Scholes implied volatility skew at the money of SPX options is known to obey a power law with respect to the time to maturity. We construct a model of the underlying asset price process…

### Volatility is rough

- Economics
- 2014

Estimating volatility from recent high frequency data, we revisit the question of the smoothness of the volatility process. Our main result is that log-volatility behaves essentially as a fractional…

### Pricing under rough volatility

- Mathematics
- 2015

From an analysis of the time series of realized variance using recent high-frequency data, Gatheral et al. [Volatility is rough, 2014] previously showed that the logarithm of realized variance…

### MANAGING SMILE RISK

- Economics
- 2002

Market smiles and skews are usually managed by using local volatility models a la Dupire. We discover that the dynamics of the market smile predicted by local vol models is opposite of observed…

### Arbitrage with fractional Brownian motion

- Mathematics
- 2007

In recent years fractional Brownian motion has been suggested to replace the classical Brownian motion as driving process in the modelling of many real world phenomena, including stock price…

### Basic properties of the Multivariate Fractional Brownian Motion

- Mathematics
- 2010

This paper reviews and extends some recent results on the multivariate fractional Brownian motion (mfBm) and its increment process. A characterization of the mfBm through its covariance function is…

### On multivariate fractional brownian motion and multivariate fractional Gaussian noise

- Mathematics2010 18th European Signal Processing Conference
- 2010

This work evaluates several parameters of the model that allow to control the correlation structure at lag zero between all the components of the multivariate process and specifies an algorithm that allows the exact simulation of multivariate fractional Gaussian noises and thus fractional Brownian motions.