Introducing the Butterfly

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:

BF25 = (σ25C + σ25P) /2 – σATM

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.

Ahead of the FOMC meeting…

The FOMC meeting is starting today and policymakers will reveal tomorrow if they will continue to reduce the quantity of bonds and agency MBS in the asset purchase program that the Fed has been running for a bit more than a year now. Analysts/Strategists are calling for another $10bn cut, which would reduce the quantitative easing program down to $65bn and should benefit to the US Dollar (against most of the currencies).
However, we saw sharp moves in the market lately, with widespread risk aversion and position unwinds. For instance, one of the charts that I like to watch quite a bit in the morning is the US equity market (S&P500) versus AUDJPY. As you can see it, weak macroeconomic indicators starting with a Chinese HSBC PMI (Flash) falling below the 50-threshold to 49.6 in January (vs 50.3 consensus), less dovishness from the Bank of Japan and idiosyncratic drivers – Turkey and Argentina – rose risk aversion and traders started to unwind their carry trade positions and futures positions in the equity market and moved them to ‘safe-haven’ US T-Notes, Gold or even Japanese bonds and Yen (despite that Japan had at that time just posted the largest annual trade deficit at JPY 11.47tr). You can notice than AUDJPY (orange sticks) went down 4.6% between Wednesday and Friday, and the equity market dropped 56 points down to 1,790 (-3%) during the same period.


At the same time, US 10-year yield (in orange) eased from 2.87% to 2.70% and Gold (purple) surged 2.8% to 1,270 as you can see it below.


Moreover, if we quickly review the fundamentals in the US, we saw:
– A disappointing December non-farm payrolls report that printed at 74K, far below analysts’ expectations of 196K,
– A housing market losing momentum, with Existing Home sales that came out at 4.87M (lowest level since October 2012) and New home sales dropping to a seasonally adjusted and annualized 414K (vs 455K expected, biggest miss since July 2013).

In addition, we saw a March T-Bills ‘Panic-Selling’ on upcoming debt-ceiling negotiations in February, with March 16th yields surging from 1bps to 12.75bps in the last few days. Since Congress suspended the debate last October, the US Treasury has been using its assortment of emergency measures in order to delay running out of funds. However, as February is traditionally a big deficit month as tax refunds are paid out, this situation won’t last forever.

With an annual inflation rate still below the Fed’s 2/2.5-percent target range, US policymakers have no rush to decrease the amount of the asset-purchasing program and could potentially decide to stop tapering until the market stabilizes. However, I don’t think the figures and markets reactions we have seen lately are going to be enough for the Fed’s decision; therefore we are going to stick with a $10bn cut as well.

UK update and GBPUSD

In the middle of this Bull US Dollar environment since the Fed announced its Taper last month, the British pound still gets support from traders and investors and remains a good hedge against their USD long positions. As you can see it on the graph below, after it hit its support at 1.6320 (mid-Dec 2013 levels) last week, Friday strong retail sales (+2.6% MoM in December vs +0.4% cons.) and this morning good employment figures (jobless rate went down 0.3% to 7.1% in December, vs analysts’ expectations of a 7.3% print) pushed the pair higher and Cable is trading at 1.6576 at the time I am writing this article.


(Source: Bloomberg)

I added the US-UK 2-year yield spread (white line) in the chart, an indicator that I have been watching since the new BoE Governor introduced the forward guidance last August. We saw this morning that the unemployment rate decreased faster than expected and is approaching the 7-percent threshold presented by the Bank of England (a threshold for considering an interest rate rise). Traders are starting to speculate that the BoE will increase its Official Bank rate (currently at a historical low of 0.5%) in the middle of the year 2015 (vs. August 2016 target the Bank said last August). You can see that the 2-year spread has been increasing again these last few days (from 5bps to 9bps), sending Cable higher by 250 pips.
The publication of the latest BoE minutes this morning didn’t bring any comment on the possible interest rate rise next year, however traders are now watching closely the central bank’s updates on forward guidance as any revision could bring pressure back on the British pound. A potential action from UK policymakers would be to decrease the threshold to 6.5% (like in the US) if the employment keeps improving at a fast pace.

The next resistance on GBPUSD stands at 1.6600 (level reached the last day of last year) and the pair can find some difficulties to break that level. Cable is up 16.5% since its low of 1.4812 (hit on July 9, 2013) and has been trading within the 1.6300 – 1.6600 range for the past month. I think that we will have to see strong fundamentals in the coming weeks to see GBPUSD at higher levels (1.6700 – 1.6800), otherwise the pair will start to lose a bit of strength against the greenback after the sharp increase and go back to lower levels (1.6400 at first).


(Source: Reuters)

Help to Buy Scheme and UK Housing Market

Help To Buy scheme
Help to Buy is a government scheme designed to help borrowers with deposits of just 5 per cent get on to the property ladder.
The first part of the scheme (Equity Loan) was launched in April last year and offers loans up to 20% of the cost of a new-build home (up to the value of £600K). The loan will be interest-free for the first five years, then in the 6th year you will be charged an annual 1.75% fee (of the loan), which will increase every year (using the retail price index plus 1%). The second part of the scheme, called the Mortgage Guarantee, applies to all property purchases (also with a house worth £600K maximum) with a minimum deposit of 5% from the buyers. For instance, as you can see it on the picture below, with a £10K deposit, you can buy a house in the UK worth £200K (it won’t be a house in London though).


(Source: Government website)

UK Housing Market
With an 8% increase in average in 2013 according to indexes such as Nationwide (8.4%) or Halifax (7.5%), the housing market has helped overall UK growth along at the fastest pace in three years. Britain’s Q3 GDP printed eventually at 0.8% QoQ (1.9% YoY); we heard lately the IMF raising its growth forecast from 1.9% to 2.4% for the year 2014.The average asking price in the UK reached new highs of £243,000 in January (Rightmove), with London at the first place (12% YoY increase) and its expensive areas (Average sale price in September 2013 was £4.4 million in Knightsbridge and Belgravia).
If you read analysts expectations for the year 2014, most of them expect an 8% increase also, with Greater London being the best opportunity. Thanks to the Help to Buy scheme, house prices in Greater London will tend to catch up with the prime central London ones. Moreover, as more lenders come on board (Latest: Barclays is offering a fee-free three-year fixed rate at 5.35% and a five year fix at 5.49% since yesterday, Tuesday 21 January), competition is driving rates down little by little and could attract potential buyers.

Start your registration here!

Introducing the Risk Reversal

Let’s focus on the Risk Reversals (RR) in this post, a term that is generally more used in an IB or HF but I believe a term important to know. Generally speaking, a risk reversal is an option strategy that combines the purchase of OTM calls (resp. puts) with the sale of OTM puts (res. calls), similar deltas and same tenors. Let’s have a look at the two different RR strategies you can create:

1. Bullish Risk Reversal (Short OTM put and Long OTM call)
For those who expect an appreciation of the underlying asset. For instance, in this case, an American company will need to raise 1 Mio EUR in 3 months in order to pay the product delivery. EURUSD is now trading at 1.3550, and the company wants to secure a maximum rate of 1.3900.


(Source: Bloomberg)

Strategy: Bullish Risk Reversal will allow the company to buy EURUSD within the range 1.3200 – 1.3900.

At Maturity:
– If the price is below 1.3200, the American company will start to lose money as they will have to buy the pair at 1.3200.
– If Spot rate trades within the 1.3200 and 1.3900 range, the company will buy EURUSD at market.
– If the price is above 1.3900, they will buy the pair at 1.3900 and generate a positive PnL.

2. Bearish Risk Reversal (Short OTM call and Long OTM put)
For those who expect a depreciation of the underlying asset. For instance, in this case, a French company will need to sell 1 Mio EURUSD in 3 months in order to pay the product delivery. The pair is now trading at 1.3550, and the company wants to secure a rate level of 1.3200.


(Source: Bloomberg)

Strategy: Bearish Risk Reversal that will allow the company to sell EURUSD within the range 1.3200 – 1.3900.

At Maturity:
– If the price is below 1.3200, the French company will start to generate some money as they will sell the pair at 1.3200 (above market).
– If Spot rate trades within the 1.3200 and 1.3900 range, the company will sell EURUSD at market.
– If the price is above 1.3900, they will start to lose money as they will have to sell at 1.3900(below market).

This option strategy is generally used for hedging, however it can also be used for leveraged speculation.

Bloomberg application: If you type Risk Reversal on Bloomberg (Ticker: WCRS RR), you get the page below which shows this different RR figures for the G10 currencies (vs the USD) for a maturity of 1 Month.


(Source: Bloomberg)

For example, we have a RR = -0.67 for EUR. In short, it means that the implied volatility (IV) of a 25 Delta Call is less than the IV of a 25 Delta Put.

Let’s start with the definition of a 25 Delta option. A 25-Delta Call refers to a call option OTM (Strike above the current spot rate); if the underlying asset, in that case the pair EURUSD, increases by 1, the call option value will rise by 0.25 (by unit, not percentage). I am sure you’ve already heard of the ‘Greeks’ (Hull study) and the famous Delta (Delta hedge strategy). Delta, in fact, is the first the derivative of the Value of the option with respect to the underlying asset price.

Implied volatility, or IV, is the estimated future volatility. It is usually computed using a Black-Scholes model (or equivalent) for options pricing. You know that a vanilla option depends on 5 parameters: Underlying, Strike, Vols, Interest Rate and Time. As you have the price of the option (by the market), you can ‘reverse’ the formula and get the implied vol of the option (dichotomy method for instance).
The IV (25-Delta call 1M) is at 6.50 and the IV (25-Delta put 1M) is at 7.17. Therefore, from the following relation:

RR (25-Delta 1M) = IV (25-Delta call 1M) – IV (25-Delta pit 1M)

We effectually have a RR 25D 1M of -0.67.

A negative RR 25 on EURUSD 1 month means that prices of puts are more expensive than calls, telling investors the market is more bearish on EURUSD for the month to come.

Usually, the ones that investors look at are the RR 25 Delta 1 Month. However, the RR 1M 10-Delta are also popular with the Butterfly (we will see the definition in another post).

Canada update – USDCAD view

In the middle of December, when I was asked to give a quick update on USDCAD, I said that I was happy to hold a long position as I thought at that time there was further room for depreciation on commo currencies such as the Loonie (CAD) and the Aussie (AUD).
As you can see it on the graph below, the fact that the pair first broke its strong support at 1.0600 in the beginning of December last year, combined with the Fed’s decision to start tapering a couple of weeks later ($10bn cut in QE program down to $75bn), both acted in favour of the greenback. USDCAD is now trading at a three-and-a-half year high of 1.0950 (Sept 2009), up by almost 17% from a low of 0.9403 reached on July 2011 (27th).


(Source: Reuters)

LT Term view:
If we have a look at the main fundamentals, we can see that the economic situation in Canada has worsened since the Great Financial Crisis. First of all, as many analysts/strategists mentioned it lately, the country’s current account (as a share of GDP) has plummeted from a 1% surplus in Q3 2008 to a 3.0/3.5% deficit in early 2010 and has remained steady since then. We know that when a country runs large current account deficits for several years, it tends to put downward pressure on the exchange rate.
Moreover, with an annual inflation that printed at 0.9% in November, slightly below the BoC range of 1.0% – 3.0%, investors and traders are speculating that the central bank will either introduce an easing bias or decrease its overnight lending rate (currently at 1.0%) in order to stimulate the economy.

The graph below represents the 2-year CAD-US yield spread (reversed legend, white line) and the USDCAD spot rate (yellow line). If the spread continues to fall due to a drop in Canadian ST rates (30bps drop since mid-September last year), carry traders will lose their interests in the Loonie, acting in favour of USDCAD (1.1200 looks like a good level to me in the medium term).
As you can see it, since mid-September 2012, the spread (CAD-US) has narrowed by almost 40 bps (trading at 0.6464% at the moment) while USDCAD strengthened from 1.0280 to 1.0950.

(Source: Bloomberg)

Short term:
The pair has been fluctuating within a narrow range (around 1.0950) for the past few days, as traders are waiting for the Bank of Canada’s decision on the overnight rate with the publication of the quarterly Monetary Policy Report (MPR) on Wednesday. We don’t expect sharp moves on USDCAD until the monetary policy meeting; however the pair can continue its bullish momentum in the short term in case of a dovish statement. The next resistance on the topside stands at 1.10.

The J Curve and Japan’s Current Account

Japan Situation
Since Abe’s election in December 2012, the Japanese Yen has dramatically depreciated against the major currencies (USDJPY is up 25%, now trading at 104.30). However, the Ministry of Finance reported on Tuesday that Japan’s current account posted its largest deficit on record in November, printing at 592.8Bn Yen (well above analysts’ expectations of 368.9Bn Yen). The country’s balance of trade goods and services (one of the major components of the current account) widened to ¥1.29tr in November, posting its 17th consecutive deficit as a weaker Yen is pushing import costs.

According to analysts, Abenomics is definitely working as they say that Japan trade’s and current account balances are still in the first stage of a J-Curve effect…

The J-Curve effect
In economics, the J Curve effect is the term used to describe the impact of currency devaluation on a country’s Current Account (Balance of Trade). A weaker currency should definitely boost exports (as they become cheaper) and therefore the country should show a positive Current Account (BoT). However, due to the low price elasticity of demand for the imports and exports in the immediate aftermath, currency depreciation doesn’t have an instantaneous positive impact on the current account (BoT).
If we apply the theory of the J Curve, observations show that in the short term period following currency devaluation, the current account and BoT of that country will decline as the higher exchange rate will firstly correspond to more costly imports and less valuable exports, but as soon as elasticity grows, it will start to improve eventually to better levels.

Theory: Marshall-Lerner
There are many studies which attempt to estimate the J-curves, but we are going to stick with the most famous one, the Marshall-Lerner Condition.
Assume the trade balance in foreign currency terms is expressed as the following equation:

Bf = pfx X – pfm M

Therefore, a change in th trade balance after a sharp depreciation can be denoted as:

ΔBf= (pfx ΔX + X Δpfx) – (pfm ΔM + M Δpfm)                       (0)

With      Vfx = pfx X, the Foreign value of exports                                        (1)
and        Vfm = p fm M, the foreign value of imports                                     (2)
Hence, if we replace (1) and (2) in (0), we have the following equation:

ΔBf = Vfx (ΔX/X + Δpfx/pfx) – Vfm (-ΔM/M – Δpfm/ pfm)           (3)

We can define the elasticities of demand/supply of exports/imports in the following equations:

ex =(ΔX/X) / (Δphx/phx), Home export supply elasticity
ηx =(ΔX/X) / (Δpfx/pfx), Foreign export demand elasticity
em =(ΔM/M) / (Δpfm/pfm), Foreign import supply elasticity
ηm = – (ΔM/M) / (Δphm/phm), Home import demand elasticity

* * *

In mathematics, the formula for the elasticity of Y with respect to X is

e Y X = (dY/dX) * (X/Y),

Where (dY/dX) is the derivative of Y with respect to X

* * *

We know domestic and foreign are related as the following:

pfm = phm r,           with r the exchange rate

With equation (3) and all elasticities expressions, I have now the following expression:

 Vfx * (nx-1) / (1+nx/ex)  +  Vfm * { nm(1+1/em) } / { (nm/em) + 1 }       (4)

ML Condition (MLC):
As it was well defined in Professor Brooks’ dissertation (Currency Depreciation and the Trade Balance, Economics), if prices are fixed in seller’s currencies, then the supply elasticities are infinite:

ex = em = ∞

Definition (Elasticity): As a reminder, the two extremes cases of elasticity are:
– Perfectly inelastic supply, where elasticity = zero
– Perfectly elastic supply, elasticity = ∞ (type of good such that small changes in price cause large changes in quantity supplied)


If we compute equation (4) limit, we have the following equation:

ΔBf = Vfxx – 1) + Vfmm)            (5)

Furthermore, if we assume that Vfx = Vfm (Foreign currency value of exports = Foreign currency value of imports,

ΔBf > 0   eq.   Vfxx – 1) + Vfmm) > 0
ΔBf > 0   eq.   Vfmx – 1) + Vfmm) > 0
ΔBf > 0   eq.   ηx + ηm > 1

In other words, the foreign currency value of the BoT will improve if the sum of the import and export demand price elasticities is greater than unity.
This is known as the Marshall-Lerner condition (see graph below)


(Source: Marshall Lerner’s Blog)