SABR Path Simulation, Quadratic Exponential Schemes, and Approximation Error
Summary
The document considers how to simulate asset paths under SABR when volatility and vol of volatility are high. It describes a Monte Carlo comparison between a call value obtained from Hagan's implied volatility approximation and an estimate from a discretized SABR simulation. The reported estimates differ substantially in the example, and increasing the number of paths or reducing the time step did not close the gap.
One response suggests an Andersen style quadratic exponential scheme, which uses different functional approximations across volatility regimes to preserve positive volatility, and notes that a scheme designed for another stochastic volatility model may be adapted. A second response cautions that the simulation result could be closer to the model's value because Hagan's formula is approximate and may diverge when vol of volatility is large. The exchange does not settle which explanation is correct or provide a full implementation or convergence study.
Key ideas
- The document highlights potential disagreement between discretized SABR Monte Carlo prices and Hagan implied volatility prices at high vol of volatility.
- A quadratic exponential scheme is suggested as an alternative to standard Euler discretization.
- The proposed scheme aims to keep simulated volatility positive across different volatility regimes.
- Hagan's implied volatility formula is an approximation and may deviate from the SABR model value.
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Full text
# Simulation scheme for SABR beside the standard Euler discretization
# 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.
## Answer by Vitomir (score 1)
https://quant.stackexchange.com/a/46384
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
## Answer by jaehyukchoi49 (score 0)
https://quant.stackexchange.com/a/77282
It is likely that the price from simulation (5.5637) is correct value of the SABR model. Hagan's formula is just an approximation that deviates from the true value when vol-of-vol is big.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.