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- Adapted Equity Rate Modeling
Adapted Equity Rate Modeling
Compared to fixed maturity note, the modeling challenge with autocallable notes is the uncertain maturity of the product.
With fixed-maturity notes, the natural way of hedging the guaranteed capital is to trade a LIBOR-linked IRS from T0 to T, where the seller receives the fixed leg (the value of the LIBOR leg is 1 - B(T0,T) and offset the IR risk on the capital).
With an autocallable note, we can trade the same swap but we need to be able to cancel the remaining part of it at any callable date Ti.
In a deterministic IR framework, the hedge would be to wait for the callable date to trade the corresponding remaining swap. Obviously, because the evolution of the IRs is also a risk factor, this is not acceptable. A joint model on the equity and on the IR is required.
Regarding the specific IR volatility risk, it is a first-order risk in a sense that an autocallable product is affected by the global level of the rate curve much more than by the the shape of the distribution of forward start swaps or by the autocorrelation of the rate curve.
In other words, a one-factor model on the rate curve is sufficient and a natural choice is an extended Vasicek model whose term structure of volatility has to be calibrated on the set of swaptions of maturity Ti written on the swap from Ti to T for all i, 1 ≤ i < N, to be consistent with the autocallable note definition. This calibration set is a very classical one and is usually referred to as the coterminal calibration set. Technically, this calibration can be done prior to the equity calibration and it is just associated to the dates of the contract.
Regarding the equity, we propose a simple representation using only a term structure of volatility to show the impact of the rate factor on the equity volatility. Specifically, we use a 2-factor Brownian model, one factor for the IR ad the other for the Equity. Moreover, we suppose that the equity is not paying dividends and the repo rate is null.
Under the risk-neutral measure Q, this can be written as:
The last equation does not depend on Q as change of measure affects neither the correlation between BMs nor their volatilities.
For the first equation, we suppose that the volatility has been calibrated according to the coterminal calibration set of the product.
Since the equity is not paying fixed, discrete dividends, its forward F(t,T) is a martingale under the forward neutral measure QT, which is defined as:
The expression of the forward in that case is: 
.
Hence, we have: 
And by combining the 2 first equations, we obtain: 
Hence, 
Moreover, we have:
which leads to the generalization of the BS formula in the sense that, in a setting with stochastic IRs, the IV of the BS framework is given by:
Thus, since the IR model has been calibrated independently to the equity model and supposing that the term structure of IV is observed, it is possible to calibrate the term structure of the volatility of the stock, 
.
From the relation on the volatilities of this simple framework, it is important to note that, depending on the sign of the correlation of the rate factor with the equity factor and of the value of the volatility of the bond, the stock volatility can be either higher or lower than the IV.
This does not affect the probability of recalling the note at the first recalling date (which is not conditional) but the probability of recalling the note at a further recalling date conditionally of not having recalled the note at a prior recalling date.
Furthermore, the addition of a rate factor, by extending the dimension of the model, leads to a vega hedging of larger dimension since the swaptions used in the interest rate calibration set should now be considered as hedging instruments, and to a more complex gamma hedging.
In particular, the "cross-gamma", the second derivative with respect to the spot of the equity and to the spot of the rate, cannot be hedged by any liquid instrument in the market.
Therefore, it is necessary to adopt an approach by robustness in order to control the evolution of the hedged ptf including the autocallable note.
Practically, this can be done by replacing the instantaneous correlation between the 2 factors by an estimation of its upper bound as the cross Gamma, in the case of the autocallable note, is positive.