After our focus on Risk Reversal, let’s have a look at another important options strategy very well-known in the market, especially FX, the butterfly. The Risk Reversal lecture gave you an idea of the most important variable in the market, the implied volatility.

In the Black Scholes model, an option pricing (European options) model developed in the early 70s, we assume that the volatility σ of the underlying is constant. However, we saw that implied volatility varies over time (constantly in fact) among the different strike prices, and this discrepancy is know as the volatility skew (or smiles sometimes). In *Options, futures and other financial derivatives* (a must read for traders/sales/brokers), John Hull considers that currencies tend to provide the so–called volatility smile in general, which would mean that the price of a 25 OTM call (maturity 1 week for instance) would be equal to the price 25 OTM put. Which in fact is not true as we saw last time, because it would mean that the RR(25-Delta) would be equalled to zero in that case, therefore we also observe a skew in the FX market.

The Butterfly is a neutral option strategy that uses four call options contracts with the same expiration but three different strikes. It is a limited risk, non-directional options strategy that is designed to earn big (but limited) profits but with a low probability. As you can see it below, there are two types of strategies, the long and short Butterfly spread.

**1. Long Butterfly Spread (Short two calls at middle strike, and long one call each at the lower and upper strike)**

*(Source: The Option Guide)*

Trader is looking for underlying stock to achieve a specific price target at expiration of the options. In this case, he targets a stock at $40 per share at maturity (April this year), therefore the strategy used was:

– Buy one April 30 Call

– Buy one April 50 Call

– Sell two April 40 Calls

At maturity, the trader will generate a profit if the underlying stock trades within the Downside and Upside Breakeven (BE), which is to say between 35 and 45. It means that you expected volatility to remain low in the coming weeks/months.

**2. Short Butterfly Spread (Long two calls at middle strike, and Short one call each at the lower and upper strike)**

*(Source: The Option Guide)*

Trader is looking for a volatility spike which would either increase or decrease the price of the stock sharply. Therefore, the strategy used was:

– Short one April 30 Call

– Short one April 50 Call

– Buy two April 40 Calls

At maturity, the trader will generate a profit if the underlying stock trades outside the Downside and Upside Breakeven (BE) range, which is to say either below 35 or above 45.

The potential profit and loss are both very limited. In essence, a butterfly at expiration has a minimum value of zero and a maximum value equal to the distance between either wing and the body. Even if there is a limited risk exposure, both strategies usually offer small returns (compared to straddles for instance).

**Bloomberg application:** If you pick up one currency pair, let’s say EURUSD for instance. Then if you type OVDV (for option volatility surface), you get the page below which shows ATM implied volatilities (with all the major maturities), the 25 RR and the 25 BF (butterfly). According to the table below, the RR (25-Delta 1 Month) is trading at -0.985 (bid) with a BF (25-Delta 1 Month) at 0.125 (bid).

*(Source: Bloomberg)*

Butterfly is the difference between the average volatility of the call price and put price with the same moneyness level (25-Delta) and the ATM volatility level. For instance a BF 25 could be expressed by the following formula:

BF_{25} = (σ_{25C} + σ_{25P}) /2 – σ_{ATM}

As Risk Reversal measure the slope (skewness), butterfly spreads measure the curvature (kurtosis). As a reminder, in statistics, the kurtosis is a measure of whether the data are peaked or flat relative to a normal distribution. Data sets with high kurtosis tend to have a distinct peak near the mean, decline rather rapidly and have heavy tails. Data sets with low kurtosis tend to have a flat top near the mean rather than a sharp peak. In short, the higher the Butterfly spreads, the more ‘peaked’ is your implied volatility curve.