# Simulation scheme for SABR beside the standard Euler discretization

QUESTION: Beside Euler Scheme, is there another more robust (and preferably easy to implement) way to simulate asset path with SABR dynamics? Simulation that will withstand even for high volatilities.

The method I am talking about is presented here:

http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.296.732&rep=rep1&type=pdf (slide 12: Monte Carlo simulation of SABR)

PROBLEM:(You can skip this)

$$\alpha=0.15$$, $$\beta=0.5$$, $$\rho = -0.6$$, $$v=2.5$$. $$v$$ is the vol-of-vol parameter.

I have implemented this method to simulate from SABR: http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.296.732&rep=rep1&type=pdf (slide 12: Monte Carlo simulation of SABR)

By Hagans orignial formula I can compute the implied volatility for $$T=2,f_o=spot=100, K=strike=95$$ $$IV_{SABR}=0.07786$$. Putting this into a Black Scholes call price forum (with interest rate zero) we get $$C_{BS}(f_0,K,T,r=0, 𝐼𝑉_{𝑆𝐴𝐵𝑅})=7.2364$$ Let us use that as our benchmark.

$$C_{BS}(f_0,K,T,r=0, 𝐼𝑉_{𝑆𝐴𝐵𝑅})=E^{Q^*}_0[(f_T-K)^+]$$where under $$Q^*$$, $$f$$ has the SABR dynamics $$\left(df_t=\sigma_tf_T^\beta dW_t,d\sigma_t= v\sigma_t dV_t.... \text{ etc} \right)$$.

When I run the simulation with timestep $$dt=0.001$$ and $$n=10000$$ simulations, take the mean value to estimate $$E^{Q^*}_0[(f_T-K)^+]$$ I get $$5.5637$$ which is far from the benchmark at 7.2. Hence this method is not ideal with these parameters. I have tried to increase the number of simulations and decrease timestep.

In general, with high volatility $$\alpha,v$$ then this method is not a good simulation scheme. When $$\alpha,v$$ is low then there is no problem.

## 1 Answer

You can use quadratic exponential method. For larger values of vol uses exponential function while for lower uses a quadratic ones, thus keeping the vol always positive.

This paper by Andersen as a reference to QE and other schemes too to apply to stochatic vol models (he refers to Heston, but you can adapt to SABR).

https://pdfs.semanticscholar.org/5730/bbf4f6437ab1775e222b5fef1ccdac1ddfc0.pdf