Advanced Modeling

While the equity/rate issue is truly the specificity of autocallable notes, two other specificities of the equity market need to be taken into consideration for such products: the smile modeling and the dividends. 


1. Smile

Regarding the smile issue, the difference between the price of an up binary option in the BS framework with a fixed volatility and the price of the same up binary option in the presence of a smile is equal to the vega of the call of the same strike time the opposite of the local skew, Local skew, of the IV (this result is a direct application of the fact that an up binary option is bounded by two call spreads). 


Hence, in the equity market, the presence of skew increases the price of the up binary option. 

Applying this result to the autocallable note means that the probability of recalling the note at the closest recalling date is higher if the volatility skew is considered than it is in a setting like the one we proposed earlier and as a consequence, the term structure of the rate sensitivity is affected. 

However, while the risk associated to the skew of volatility is therefore important to model, the calibration of a two-factor model including this risk is more complex and requires the use of numerical tools such as PDE resolutions. In particular, it is not possible to use the local volatility formula as it is specific to a monofactor model. 


2. Dividends

The dividend risk is not so much of an issue in the case of the autocallable note we presented previously. However, it could become more important in structures where the customer chose to risk a loss on his capital by selling a DIP in exchange of higher coupons in case of an early termination. 

Nevertheless, under the forward neutral measure QT, all assets of the form xt/B(t,T), with xt a spot process, are martingales. 

Yet, with the addition of fixed discrete dividends, the forward of the equity is including discount factor terms from each dividend dates to the considered maturity.  Hence, it is no longer a martingale and, in the presence of stochastic IRs, the calibration of its volatility is more complex than what we have proposed earlier and also requires the use of advanced numerical tools because of the convexity adjustments due to fixed dividends. 





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