Implied non-recombining trees and calibration for the volatility smile

Abstract

In this paper we capture the implied distribution from option market data using a non-recombining (binary) tree, allowing the local volatility to be a function of the underlying asset and of time. The problem under consideration is a non-convex optimization problem with linear constraints. We elaborate on the initial guess for the volatility term structure and use nonlinear constrained optimization to minimize the least squares error function on market prices. The proposed model can accommodate European options with single maturities and, with minor modifications, options with multiple maturities. It can provide a market-consistent tree for option replication with transaction costs (often this requires a non-recombining tree) and can help pricing of exotic and Over The Counter (OTC) options. We test our model using options data for the FTSE 100 index obtained from LIFFE. The results strongly support our modelling approach.

Keywords:

• Volatility smile
• Implied non-recombining trees
• Calibration
• Non-convex constrained optimization

Acknowledgements

The authors are grateful for financial support from a research grant on Contingent Claims from the University of Cyprus.

Notes

†Other work we are aware of that uses a non-recombining tree is of Talias (2005) where for the calibration he uses genetic algorithms.

†Weights can be related for example to the trading volume of the options.

†For simplicity, we make the assumption that the risk free rate, the dividend yield and the step size do not change across time. Formulas adjusted for time dependence can be found in Appendices B and C.

‡Other non-monotonic functions could also be used for σ (i) but what we have tried proved adequate for our purposes.

§Optimal tree is the one that gives the lowest-value objective function subject to the initial constraints.

¶Probability equation (7) is effectively a martingale restriction (see equation (6) and relevant discussion in Longstaff (1995)). Thus the numerical implementation of the model with this probability equation is restricted to a Markovian stochastic process.

†The BFGS formula was discovered in 1970 independently by Broyden, Fletcher, Goldfarb and Shanno.

‡From now on we will use C(1,1) instead of C _Mod.

†We do not calculate since S(1, 1) is a known, fixed parameter, and thus does not take part in the optimization.

‡FTSE 100 options are traded with expiries in March, June, September, and December. Additional serial contracts are introduced so that options trade with expiries in each of the nearest 3 months.FTSE 100 options expire on the third Friday of the expiry month. FTSE 100 option positions are marked-to-market daily based on the daily settlement price, which is determined by LIFFE and confirmed by the Clearing House.

†In table 1 we note that for the same contract (same underlying asset, same expiration) the number of contracts used in the model changes across months. That is because some contracts were removed because of the filtering rules.

‡This sub-sample has a total of 13 696 observations for the year 2003.

§Also, we compare our model (with respect to over-fitting) with the Black–Scholes model using the Whaley (1982) approach. According to this approach we find the volatility that minimizes the sum of square differences of the Black–Scholes option prices with their corresponding market prices using nonlinear minimization. Results show that the mean (median) absolute error using this approach is 7.36 (5.94) for the full sample and 6.61 (5.60) for the restricted sample which are much higher than the errors obtained using our model for n = 6, 7, 8.

†In the Pearson system there is a family of distributions that includes a unique distribution corresponding to every valid combination of mean, standard deviation, skewness and kurtosis.

‡Copyright 2005 The MathWorks, Inc.

§In rare exceptions only we have implied distributions close to normal or even leptokurtic.