The skew dependence of volatility is generally thought of as a monotonic dependence on the option strike, viz a@). The strike dependence is needed in Envy? Maybe these names (charm, color) just prove that some finance guys really wanted to be high-energy physicists.
order to reproduce European option values using the standard Black-Scholes formula. A more complicated non-monotonic dependence is often seen in
FX,
called a
mile".^
Here is a freehand picture illustrating the idea:Volatility smile: o( E ) increases at both ends
Physical Motivation for the FX Volatility Smile
The smile behavior for
FX
can be qualitatively understood using a “fear” idea.For illustration, say spot is
to
= 1.50. USD IGBP
or qo = 0.67. GBP
I USD.
Consider an OTM GBP put for a low strike Ethw = I .40. USD that pays off at expiration if
t*
< Ethw. The fear that the GBP might depreciate this much can induce more investors to buy these puts. The extra demand raises the put price.Thus we get a premium on this volatility a ( E t h w ) relative to the ATM vol, resulting in volatility skew.
Now, consider an OTM call on the GBP for a high strike (say EtHiRh = I .60. USD I
G B P )
that pays off if{*
> EtHiRh. This option, on the other hand, is an OTM put on USD with a low strike in the inverse variable’
More About Skew: A lot more information about skew for equity options is in the next chapter.FX Options 43 EVhw = 0.71. GBPIUSD that pays off if q*
<
EVL””’. The fear that the USD might depreciate this much can also place a premium on this volatility o(EVh”).
Since these two options are the same thing,o(
E V h W ) = o( E g H i r h ) .Thus, vol premia for both low and high strikes can exist; this is the smile.
Because the fear intensities are generally not equal, the currencies generally being of different strengths, the smile will not generally be symmetric.
General Skew, Smile Behavior for FX
A more general phenomenological parameterization of the complexity in the strike dependence of the volatility includes both monotonic skew and non- monotonic smile terms, and the mix is time-dependent. A study along these lines was reported for dollar-yen iii.
Fixed Delta Strangles, Risk Reversals, and Hedging
A strangle is the sum of a call and put, and a risk reversal is the difference of a call and put. The
FX
option vol skewhmile is often quoted using “25-delta’’strangles and risk reversals’. Since 50-delta would be ATM and 0 delta would be completely OTM, the 25-delta is “halfway” OTM. A call Ccoll
(
ocull, Ecull)
and put Cpu,(
opur, Epur )are used with AcU,, = -Apur = 0.25 for a given expiration time t * . We need to find the strikes to produce this 25 delta condition, remembering that o,,,, (E,,,,)
, opur (E,,,)
are functions of the strike. Other delta values are also used, e.g. 10-delta.Let us examine the fixed delta conditions a little more. The relation between the strikes of the call and the put for a given value of delta is related to the forward.
When the deltas are equal and opposite as for the 25-delta condition, we have (5.10) To get an idea of what this implies, consider the situation without skew where this relation simplifies. Up to a volatility term, the forward is related to the product of the call strike and the put strike. From equations (5.4), (5.5) we get
Risk Reversal Convention: The price of a risk reversal is the difference of the call and put prices on the USD. The market is also quoted in terms of the difference in the implied volatilities of the USD call and put. Again, remember that a USD call is an X Y Z put so a positive risk reversal means a bearish market on the XYZ currency. Also, a risk reversal has other names: combo, collar, cylinder,
. . .
(5.1 I ) This means that if we own a call on GBP with 25-delta, ignoring skew, we can hedge it locally with a put on GBP having the same delta (and opposite sign) provided we choose the put strike E,,,, according to the above relation.
Implied Probabilities of FX Rates and Option “Predictions”
Formally, the second derivative of the standard option formula with respect to the strike is the Green function, including the discount factor’. The implied Green function or probability distribution function (pdf), Glmplied p d f , is defined by inserting the market prices for options CMarket Data ( E ) expiring at some time t*
with different strikes. The second derivative is approximated numerically. With x either
I n 6
orI n q ,
we get the relation(5.1 2) Looking at the height of the implied pdf at some level, the option market’s
“prediction” today about the probability that the FX rate will find itself at a that level at time r* in the future can be numerically ascertained.
As mentioned, the physical basis of this “prediction” is just that the fear factor leading purchasers to buy XYZ puts at elevated prices or vols produces an elevated implied pdf value for future XYZ currency depreciation. If these fears are realized, the prediction will come “true”. While the statement that “I am afraid XYZ will drop and will pay extra for put insurance, therefore my prediction is that XYZ will drop” may seem like a tautology, the implied pdf does produce a quantitative evaluation of the effect.
Here is a picture of the idea for a lognormal pdf and a modified implied pdf including skew effects lo. The fat tails coming from the increased volatility for low values of the underlying increase the implied pdf values at low values, with respect to the unskewed lognormal pdf.
Green Functions: We will look at the math later; for right now just try to get the idea.
l o Plot: Actually this plot comes from the S+P 500 index data in Ch. 4, and it serves to
illustrate the idea generally. The implied pdf illustrated is just a lognormal pdf with the skewed vol put in at each value of the underlying and the total renormalized to 1.
FX Options 45
9%
8%
7%
6%
5%
4%
3%
2%
1 % 0%
I
--c Implied pdf -e Lognormal pdf1
If there is substantial real market pressure on the XYZ currency due to selling XYZ spot accompanied by feedback of bearish purchases of XYZ puts, the option market “prediction” will indeed come “true”. However the amount of such depreciation really relies on complex factors including transaction volumes, macroeconomic information, and local political conditions instead of theoretical peaks in the implied option probability distribution. To say the least, the impact of these real factors on the FX market is not easy to evaluate successfully and consistently over time ‘ I .
Still, option pricing including skew information is desirable since information from the option market is included, including the fear-factor distortion.