The collocating local volatility framework – a fresh look at efficient pricing with smile

ABSTRACT

It is a market practice to price exotic derivatives, like callable basket options, with the local volatility model [B. Dupire, Pricing with a smile, Risk 7 (1994), pp. 18–20; E. Derman and I. Kani, Stochastic implied trees: Arbitrage pricing with stochastic term and strike structure of volatility, Int. J. Theor. Appl. Finance 1 (1998), pp. 61–110.] which can, contrary to stochastic volatility frameworks, handle multi-dimensionality easily. On the other hand, a well-known limitation of the nonparametric local volatility model is the necessity of a short-stepping simulation, which, in high dimensions, is computationally expensive. In this article, we propose a new local volatility framework called the collocating local volatility (CLV) model which allows for large Monte Carlo steps and therefore it is computationally efficient. The CLV model is by its construction guaranteed to be almost perfectly calibrated to implied volatility smiles/skews at a given set of expiries. Additionally, the framework allows to control forward volatilities without affecting the fit to plain vanillas. The model requires only a fraction of a second for complete calibration to simple vanilla products.

KEYWORDS:

• Parametric local volatility
• stochastic collocation method
• SCMC sampler
• Monte Carlo
• basket options
• efficient pricing

2010 MATHEMATICS SUBJECT CLASSIFICATIONS:

• C63
• G12
• G13
• C15

Acknowledgements

The author would like to thank his former colleagues from the PMV team at Rabobank Nederland for a fruitful cooperation.

Disclosure statement

No potential conflict of interest was reported by the author.

Notes

1. In [7], it is shown that ${ϵ}_N\to 0$ exponentially in N.

2. Standard expiries for EUR/JPY market are: {1D, 2B, 3B, 4B, 1W, 2W, 1M, 2M, 3M, 6M, 9M, 1Y, 2Y, 3Y, 4Y, 5Y} where D,B,W,M,Y represent day, business day, week, month and year respectively.

3. For each time $T_i$ the collocation points $x_{i,j}^{\mathcal{N}(0,1)}\equiv x_j^{\mathcal{N}(0,1)}$ are the same for $j=1,\dots ,N$.

4. A normal process $Z\left(t\right)$ is standarized as $\left(Z\right(t)-\mathbb{E}[Z\left(t\right)\left]\right)/\mathrm{std}\left(Z\right(t\left)\right)$.

5. i5-4670 CPU @ 3.40 GHz with 8.00 GB ram simulated with Matlab.

6. The implied volatilities for the CLV framework were calculated from the call option prices obtained from the Monte Carlo simulation of the CLV framework. Note that under the CLV framework the option prices are obtained in a split-second.