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- Risk Analysis (without DIP)

# Risk Analysis (without DIP)

In order to price the autocallable digitals effectively, one can compute the undiscounted **conditional probabilities** of receiving the coupons. These values can enable one to quickly check whether the pricing makes sense. After these probabilities are computed, they should be discounted and multiplied by the coupons to be received. This gives us the price of the autocallable digitals.

The first digital option is a classical European digital. The undiscounted probability of striking the second year is conditional on not autocalling at the end of the first year. The undiscounted probability of striking the third year is conditional on not autocalling at the end of the first and second years. As time goes by, the probabilities of coupons being paid decrease and the value of the last path-dependent digitals can be very small because the conditional probabilities of striking would be low. In this case, the seller must be careful when offering a very large digital with a low probability.

The risks associated with a single asset autocallable structure are similar to those associated with single asset digitals.

The trader selling the autocallable is:

- short the skew

- short the forward: short IR, long dividends, long borrowing costs

- his vega position depends on the autocall level and the forward price of the underlying

- short vega if forward price < trigger level
- long vega if forward price > trigger level

The Vega hedge will consist of a set of European options with strikes matching as closely as possible the autocall trigger dates. The **overall volatility sensitivity is split over these Vega buckets**, and each of these sensitivities will change as the market moves. **If an autocall event is about to happen, the short-term Vega will increase and the Vega in the other buckets will decrease**, in line with the higher probability of autocalling. A Vega hedge set at inception will need to be readjusted if the market moves significantly in relation to the autocall triggers.

**Interest Rate / Equity Correlation **

The autocall is an excellent example of a structure where the correlation between equity and IR has an effect on the price.

The autocallable is redeemed at a time in the future that is a function of the path the underlying equity takes. Assume the investor is paying the bank LIBOR in exchange for this equity exposure, then the duration of the swap is dependent on the equity and thus the structure is sensitive to the correlation between interest rates and the equity underlying. In the case where an autocallable is structured to provide equity exposure as part of a note, then the investor is in this case implicitly short the floating leg of an interest rate swap and the same thing holds.

In the case of the autocallable, the pricing of this correlation effect is typically done by taking a small margin. This can be specified by first deciding on the level of correlation and also the length of the trade. To get an idea about this effect we use a hedging argument. Setting aside the equity component that will be Delta hedged using the underlying equity, we look at the IR hedge of the seller.

The seller of a 2y annual autocallable will go long ZC bonds with respective maturities of 1 and 2 years.

*First case: Assume that the equity/interest rate correlation is positive. *

**If the underlying increases**, then the probability of early redemption at the first autocall date increases, and to adjust the IR hedge accordingly the seller will increase the amount of 1-year bonds held and sell some of the 2-year bonds. Because of the positive correlation in this case between the underlying equity and IRs, we expect that IRs will also increase on average, and thus the price of the ZC bonds will decrease. Since the bond with the longer maturity decreases more in value than the bond with the shorter maturity, the seller nets a loss on the rebalancing of this hedge because the seller is buying one bond but selling the one that decreased more in value.

**If the underlying decreases**, then the probability of the structure autocalling early decreases. In this case the seller must adjust the IR Delta hedge by selling some of the 1-year ZC bonds held and buying more of the 2-year bonds. On average we expect that IRs will also decline because of the positive correlation. This implies that the 2-year bond will increase in value more than the 1-year bond, and again the seller thus nets a loss on the rebalancing of this hedge.

*Second case: Assume that the equity/interest rate correlation is negative. *

**If the underlying increases**, the opposite happens. We expect the IR in this case to decline, on average, and the same rebalancing as the case of increased possibility of early redemption as above will in this case net the seller a profit.

**If the underlying decreases**, then again the opposite happens: negative correlation means that we expect rates to go up and thus reduce the price of the 2-year bond more than the 1-year bond. The decreased probability of early redemption means the seller will need to buy more of the 2-year bond and sell some of the 1-year bond, thus netting a profit.

The upshot of this analysis is that the autocallable’s price should be higher when a positive correlation is assumed between IRs and the underlying equity, and lower if the correlation is negative.

The question thus arises as to whether we should employ a model that includes stochastic rates, and thus be able to enter a value for this correlation and include its impact in the price?

Arguments in favour of the use of such models are discussed by Giese (2006) for example, and the impact on pricing is discussed and concluded to be important. However, although using such models allows one to see this impact, they do not give us additional information regarding the hedging of the equity interest rate correlation. Specifically, since the sign of this correlation governs whether there is a cost or a benefit, deciding on which correlation to use and adding a cost accordingly can be done without having to employ a stochastic rates model.

The magnitude of this cost will be a function of the maturity of the structure. If we had assumed a 3y annual autocallable then the same hedging argument holds, only the Delta hedge for IRs will include the 1- and 3-year bonds. The impact of a move in IRs is greater on a 3-year bond than on a 2-year bond and thus the impact of the correlation is greater the longer the maturity of the autocall structure.

The use of stochastic IR models and the importance of the correlation between rates and equities becomes more significant when pricing hybrid products. Generally we can say here that since this correlation cannot be hedged in a straightforward manner, and perhaps not hedged at all, the best thing to do is decide on the level for this correlation and add a cost accordingly. To trade this correlation, and thus hedge this correlation risk, one would need a liquid structure involving equity and IRs, from which one could extract this correlation by hedging away the other parameters.

Note that, from the investor’s point of view, a positive correlation would imply that he is likely to get his above market autocall coupon and his money back in a high interest rate environment. This is a good scenario for the investor.

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