The aim of this article is to build an easy model that explains the implied volatility structure observed in the market. In the equity market we have a very pronounced skew whereas in the foreign exchange market the smile is U-shaped. This paper presents a simple model that is easy to use for Simulations (Monte Carlo) as well as for lattices (Partial Differential Equations or Trees) and we derive closed form approximations for the implied volatilities that are very accurate. This allows a better calibration to the market data. This model is then applied to an up-and-in American-style digital option, which is very sensitive to the smile. This option becomes expensive because of the accumulation of two factors: the distributions of the stock at different maturities and the correlation between these maturities.
Model
The distribution at certain maturity T is given as a mixture of different lognormals that have different forward rates and different volatilities:
This model shows that with a probability pi , the distribution of the stock is a lognormal with a forward rate equal to Fi and volatility . This model can be simulated easily by first simulating a discrete variable with probabilities {p i} giving a j and then simulating a lognormal distribution with forward Fj , and volatility j . Also it is quite easy to construct a tree to represent this distribution. It is just a Cox Ross Tree with as many layers as the number of states n. For Partial Differential Equations, the technique is also the same.
This model is a generalisation of the model described in [1]. In their paper, the authors consider a mixture with the same forward leading to a U-Shaped smile.
Implied Volatilities
This model explains a slope for the implied volatilities coming mainly from the different forwards. It also shows curvature coming from the different volatilities (see graph).
This graph was obtained by using the exact formulae of the pricing of the call options. Then the implied volatilities were inverted using a Newton Raphson method.
Pricing & Approximation
Pricing within this framework is easy enough. The price of a call option with maturity and strike is given by:
where BS is the Black Scholes formula.
This formula can be well approximated by the following formula:
where
This model gives a parametrisation of the smile for each maturity that is a quadratic function on the log moneyness. The result is not surprising if you consider for example the Fast Mean Reverting Model developed by Papanicolaou [3].
What about the process?
This model allows matching any distribution and this can be performed efficiently using the analytical formulae. Now the real problem is to create a process that prices correctly the options at maturity T1 and T2 . To do so one has to do some statistics to understand the correlation between ST 1 and ST 2. With few historical data that we have it is really hard to have a good estimate of the correlation. There is however a technique that allows us to do so: the Bootstrap technique for the correlation estimation. It allows us to determine a pretty good estimate [2]. The estimation of the correlation gives an astonishing result. Indeed, it shows that for an upward sloping curve for the implied volatilities the correlation is higher than a simple Black Scholes Model.
Impact on Exotic Options
Consider a one-touch up-and-in option. In this option the buyer gets 100% of the notional if the stock touches a barrier level before maturity. This option can be seen as a mixture of the different up-and-in digitals for all the maturities. In the presence of the smile, an up-and-in European digital is more expensive as it is the limit of the call spread. It corresponds to the reality of hedging. Plus the fact that in our model the different maturities have more correlation, this increases the probability of the barriers getting touched. Finally, the American-style digital is more expensive than it is for a simple Black Scholes model. For the FTSE this difference can easily reach 10-15% of the notional.
References
[1] D. Brigo, F. Mercuro, Fitting volatility smiles with
analytically tractable asset price models.
[2] B. Stine, Bootstrap Methods, Department of Statistics,
Wharton School, University of Pennsylvania, Philadelphia,
Pa 19104.
[3] J.P. Fouque, G. Papanicolaou and R. Sircar: International
Journal of Theoretical and Applied Finance Vol.3, No 1
(2000). 101-142.
This week's Learning Curve was written by Celine Guillemaut, a student of Ecole Polytechnique of Paris, and Adil Reghai, a quantitative analyst at Dresdner Kleinwort Wasserstein in London. Nadhem Meziou helped with the development with this model.
*Formulae can be obtained at: adil.reghai@dresdnerkb.com
