**Payoff **:

– Behaves like a down & out call spread; assuming first fixing is at 0%, there’s a cap on the maximum payout which is 8.65%, (- 8.65% is the **down & out barrier**)

– Before the first Observation short term volatility of the structure is negligible incase its sufficiently ITM since the upper cap hasn’t been fixed till 1st Observation, but incase the spot starts approaching the -8.65% barrier structure becomes short volatility & since the expected maturity of the structure decreases – being Long Napoleon decreases your short term vega. (As a seller when markets fall your vega increases – makes money since volatility also rises)

**Risks**

**– Vega with spot: **

Case 1: When markets fall: **negative Vega in the nearest time bucket increases in magnitude since** the probability of knocking out increases & **Vega magnitude in the farther time buckets falls**. Incase its deep OTM i.e. much below the 8.65% barrier Vega becomes slightly positive in the nearest time bucket.

Case 2: When markets are up: it behaves like an ITM call spread and hence is short volatility. **Magnitude of Vega is more in the farther time buckets then the near month time bucket**.

**Note**: Structure is shorter vega when the spot are higher as compared to when spots are lower since increasing volatility at a higher spot increases the probability of structure Knocking out more than it is at a lower spot ( At a lower spot in any case it’s going to knock out – so increasing volatility doesn’t have much affect there)

**Price & Vega with vol**

Price decreases with volatility as increasing volatility increases the probability of Knocking out

**Note: Structure is Long vol of vol because of the positive convexity of price with volatility as shown in the figure above**

Vega expected to be higher at lower volatility levels since increasing volatility would cause a greater increase in probability of knocking out as compared to a scenario when the volatilities are already high ( At higher vols structure is at is knocking out hence no major affect by increasing volatility)

**Vega hedging Napoleon as a seller**

**Case 1:**

**When markets fall**, spot falls & volatility rises (especially the short term volatility) Expected maturity of the structure decreases

From the discussion above as a seller I get **massively longer vega in the near month time bucket** & slightly shorter vega in the father time buckets.

Step1: Perform extensive study of the volatility regimes of the underlying

Step2: Sell short term vega at a point where you believe the short term vol to be at the highest level, thus making money

Farther maturity buckets can be left un-hedged for low notional, since long term vol movements aren’t significant

**Case 2: When markets rise,** Spot rises, volatility falls

As a seller I get shorter Vega in the all the time buckets quite evenly. Again a similar exercise of spotting the lowest volatility point should be done and Vega rebalanced accordingly. **Since in this case Vega has to be rebalanced for all the time buckets – a single mid term maturity option can be bought incase options of farther maturity aren’t available.**

**Delta hedging Napoleon**

**Seller is long gamma**. Consider a case when the first observation has been fixed at 0%, cap fixed at 0% -> abrupt change in payoff at 0%.

Now on the second observation date in case the spot is around 0%, the seller is massively long gamma as compared to any other spot positions. If the stock is volatile enough, good money can be made on delta hedging proactively on this date. When spot is below the -8.5% barrier, seller becomes short gamma.

**Hedging Vega convexity: **As a seller I am short convexity, & an easy way to hedge it would be going long a strangle & short a straddle in appropriate ratio.

The combination of a short straddle and a long strangle, although short volatility, is long vega convexity, as illustrated above.

**The problem**: This is not a static hedge, with significant spot moves the straddle – strangle portfolio loses its convexity.

(OTM/ITM options have price convexity with respect to volatility, & ATM option price varies linearly with volatility – during significant spot moves ITM options won’t remain ITM & OTM options won’t remain OTM hence affecting the convexity of the straddle – strangle portfolio), During massive spot movements the straddle – strangle portfolio has to be rebalanced and a significant cost can be incurred by the seller during that process.

Initially seller is: long a strangle & short a straddle

Incase spot moves, One of the legs of strangle becomes ATM & straddle becomes ITM.

**Rebalancing**: buy back the straddle (ITM) and sell an ATM straddle (**Cost incurred: X), **short the strangle & buy an OTM strangle **(Money made: Y)**

**Definitely the magnitude of X is more than Y & hence rebalancing it during significant spot moves incurs significant costs.**

**Skew: **Structure is long skew: Can be hedged by trading standard risk reversals or a portfolio of gamma & variance swaps (Easy to construct the hedge portfolio since most of the Napoleons are on standard indices- ITM & OTM options on indices are fairly liquid)

**Conclusion:**

– Napoleons are a good structure to sell since it provides sellers an opportunity to **buy gamma, volatility**

– **Vol of Vol exposure** is significant & hence one should clearly account for cost of hedging the convexity effectively

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