Equity and FX modelling papers

Modelling the FX Skew

Presentation by Dherminder Kainth and Nagulan Saravanamuttu

In this presentation we show that the Double-Heston model allows accurate calibration to the liquid instruments in the FX markets and captures the aspects of the market dynamics that are important in the pricing of exotic options. Initially, we outline the empirical features of the FX markets and note that certain types of barrier products are very liquidly traded. In particular, the Double-No-Touches show price visilibility which implies that any pricing model has to recover these prices. We then show how the prices of barrier products probe the market-implied forward smile dynamics and note that one of the crucial features of the FX markets is the stochastic nature of the risk-reversals. We proceed to investigating the applicability of a number of models and show that the Double-Heston process satisfies our criteria and calibrates well to vanillas and DNTs as well as pricing other exotic barrier products to within bid/offer spread of the market-standard model.

Unconstrained Fitting of Non-Central Risk-Neutral Densities Using a Mixture of Normals

Paper by Riccardo Rebonato and Teresa Cardoso
Figures to the paper (in powerpoint).

One of the most important problems in calibrating option models (eg, stochastic volatility, local-volatility, jump-diffusion, etc) is obtaining a reliable smile surface from the (often noisy and non-contemporaneous) market prices of plain vanilla options. For some models it can be debated whether using ’undoctored’ noisy prices might be better than regularizing the input, but, for some approaches, such as local-volatility models, having a smooth input smile surface is a must.

Pricing discretely sampled path-dependent exotic options using replication methods

Paper by Mark Joshi

A semi-static replication method is introduced for pricing discretely sampled path-dependent options. It depends upon buying and selling options at the reset times of the option but does not involve trading at intervening times. The method is model independent in that it only depends upon the existence of a pricing function for vanilla call options which depends purely on current time, time to expiry, spot and strike. For the special case of a discrete barrier, an alternative method is developed which involves trading only at the initial time and the knockout time or expiry of the option.We also show that the future smiles determine the finite-dimensional distributions of the martingale measure.

Log-type models, homogeneity of option prices and convexity

Paper by Mark Joshi (2002)

It is shown that the properties of convexity of call prices with respect to spot price and homogeneity of call prices as a function of spot and strike hold for a large class of models of stock price evolution. It is also shown that these properties hold for wider classes of options. A converse result is also demonstrated.

The Kolmogorov Project (working version Nov 02)

Paper by Riccardo Rebonato and Mark Joshi (2002)
Presentation by Riccardo Rebonato at King's College London (2002)

The overall goal of this work is therefore the process-independent (model-independent) valuation of complex derivatives by means of weak static replica-tion through the specification of future smile surfaces in such a way that it should not be possible to arbitrage the resulting price in a model-independent way. Input to the proposed pricing approach is the model-independent arbitrage-free dynamics of the smile surface.

An efficient and general method to value American-style equity and FX options in the presence of user-defined smiles and time-dependent volatility

Paper by Peter Jäckel and Riccardo Rebonato (2000)

It is possible to extend replication to the pricing of American style options. In this document, the technique is explained with particular view on the effect that different assumed smile dynamics have on the price. Also, the relation to conventional finite difference methods is discussed which helps to explain why the replication method for American style options displays a remarkable convergence behaviour.