Pricing and Hedging Phoenix and Snowball Autocallable Notes

# How to price a phoenix and snowball type autocallable options?


# How to price a phoenix and snowball type autocallable options?












I'm currently studying the pricing of autocallable options, especially snowball (accumalated coupon) and phoenix (accumlated coupon, but the coupon may also be autocalled if the underlying price touches the lower barrier) type with down-and-in put option embedded (The underlying asset could be single asset or a basket of assets), but I cannot find any materials talking about this topic.

I'm confused about the following questions:

- What are we pricing in this product in reality? The option premium given the coupon rate or the coupon rate that set the option premium to be zero?

- How to price the option? If Monte Carlo simulation is the only way for pricing, or we can find PDE and solve it numerically, or there are some other better ways?

- How is the coupon much higher than the other fixed-income products? And how to implement the hedging strategy for this kind of products?

Thanks!

## Answer by will (score 6)

https://quant.stackexchange.com/a/47264

- Typically structures like this are traded as notes. They will be sold at a face value of 100%, where that is normally the combination of a zcb (ie 1y usd, say 97.5%), expected coupon (say +10%), short Knock In put (also knocked out by the autocall feature, say -8%), and some profit for the issuer (in this case, 100%-97.5%-10%+8%=0.5%). Sometimes these are traded is "swap format", where there is no notional exchange, instead libor+spread is paid on the notional (normally). Essentially, the short components are set so that they can fund some coupons.

- Montecarlo is normally the best way, the payoffs are very frequently tweaked, so you need the flexibility a "generic montecarlo" offers you to quickly add in these features as potential clients make up their minds on what they want.

- The coupons are higher when you are giving up something more likely to happen. The reality is that investors are somewhat unbothered by the volatility of the underlying, despite this being what sets the value of the Knock In put you nearly always sell in this family of structures. If it's traded on something like eurusd then you're looking at a single digit vol handle - that KI put is worth very little. If we trade it on natgas on the other hand, it's worth a lot, so you get a better coupon. In these cases, the barrier on the fx phoenix is probably going to have to be around 95% to get the same coupon as a 50% barrier on natgas. A 95% barrier is not terribly palatable to investors though, so it's not going to sell too well, so you lower the coupon and push the barrier farther away, to make it more attractive. To hedge them, you dump them into the rest of your portfolio and hedge the residual. Unfortunately, you'll not be able to hedge some of the risks with you end up with from your portfolio.

## Answer by Haixuan Ye (score 2)

https://quant.stackexchange.com/a/73274

For anyone who is still interested in this. A snowball can be priced through PDE by using autocallable + doubleNoTouch + doubleOutPut - upOutPut. The solver is easily constructed by playing around the BCs and barrier conditions.

## Answer by wxu (score -2)

https://quant.stackexchange.com/a/61343

Recently because of some personal reason, I tried to price Snowball Autocall using MC and PDE, assuming single underlying.

12 months Snowball, Monthly autocall observations, Daily Put Down & In. Payoff:

- if Autocall, then 100% principal + autocall coupon

- if Knock in and No Autocall, then client lose because of short put

- if No autocall and No Knock in, them principal + bonus coupon.

so if you want to price it using PDE, one simple way I used is to price these 3 components and then combine it into one which is the fair price of Snowball. For 1) and 3), it's easy to deal with the boundary; for 2), since it's knock-in, we can price it using normal product - normal product with knock-out; for example, for simple down and in put, you can price a) a put, as well as b) a put with down and out, then a - b is what you want.

let me know if you need more information. I can share my code and simple report if you want.