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How to replicate a single asset auto call through call spreads ?

Single asset auto call: Definition and pay off profile is clear. Just want to know the method to replicate it through vanilla call spreads.

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You can only replicate the digital part with call spreads.

To be clear, let's decompose the Autocall as:

  • the first part made of digital options (paying the coupons in case of early redemption)

  • the second part made of a down and in put (this is the part responsible for the fact that, at maturity, you benefit from a conditional capital protection as long as the underlying's final level is above the DIP barrier; otherwise, capital loss equal to underlying's loss

Replication of the first part:

The digitals can be replicated using call spreads. It can be shown that a digital is equal to a call spread with the width between the strike tending towards 0. The more aggressive you are, the tighter the width between the strikes of the call spread. From the seller's point of view of the Autocall, you are short the digitals. So, when building your replication, you want to price (sell) the digitals at a price slightly higher than the actual "true" price of the digital. You do that with the spread "on the left" of the digital barrier. Example: consider a digital option paying a 10% coupon if underlying > 100% ==> replication by the Autocall seller : 5 * call spread 98%-100% (if spot below 98%, then payoff = 0 just as for the digital; same thing if above 100%; between 98% and 100%, the call spread will pay something positive whereas digital pays 0 hence it is conservative from the seller's point of view).

Replication of the second part:

Via a combination of put spreads + put (I can develop that one if you'd like)

A good reference for the above: Exotic Options and Hybrids: A Guide to Structuring, Pricing and Trading from Mohamed Bouzoubaa

Edit: down and in put replication

From the Autocall seller's point of view (who is buying the down and in put): it is replicated by a certain number of put spreads + one more put.

For example, let's say you want to replicate an at-the-money down and in put with barrier at 70% - so, if the spot at maturity decreased by at least 30%, the DIP behaves like a vanilla ATM put, otherwise it pays 0 at maturity.

You can replicate this with 10 put spreads 67%-70% (i.e long 70% strike puts, and short 67% strike puts) + long 1 put with strike 70%.

  • If at maturity the spot is at 71%, the DIP should pay 0 and the replicating portfolio pays 0 as well
  • if the spot is at 67%, the DIP should pay 33% and the replicating portfolio pays 10*3% (from the put spreads) + 3% (from the additional put) = 33% (so same as DIP, all good)
  • if at maturity the spot is between 67% and 70%, say 68%, the DIP should pay 32% but the replicating portfolio pays 10*2% (put spreads) + 2% (additional put) = 22% < 32%

We see than in all cases, the replicating portfolio either pays the same amount or a smaller amount than the down and in put. Hence the replicating portfolio is a bit cheaper than the real DIP, hence it is conservative from the Autocall seller's (i.e. from the DIP's buyer) point of view. The larger the width between the 2 strikes of the put spreads (and the smaller the number of put spreads) (for example long 6 put spreads 65%-70% + long 1 put with strike 70%), the more conservative the replicating portfolio is.

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  • $\begingroup$ could you develop the put part please ? $\endgroup$ – Cedric_W Jun 11 '18 at 9:29
  • $\begingroup$ sure, I edited the post to add the DIP part $\endgroup$ – Alex Jun 11 '18 at 13:17

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