From the three main structures dealt on the FX options market, ATM STDL, RR and the VWB it is possible, from the relationships in equations 7.1, 7.2 and 7.3 to immediately retrieve the implied volatilities. As for the at-the-money implied vol. it is synonymous with the quoted ATM STDL (from now on instead of writing
σAT M DN S we write simply σAT M), whereas the implied vol. on a 25 Delta call and
put (actually -25 Delta for the put) can be calculated as in equations 7.5 and 7.6, respectively.
σAT M =AT M ST DL (7.4)
σ25C(t, T) = σAT M(t, T) +vwb(t, T; 25) + 0.5rr(t, T; 25) (7.5)
σ25P(t, T) = σAT M(t, T) +vwb(t, T; 25)−0.5rr(t, T; 25) (7.6)
These examples show the calculations of the implied volatility on a 25D call and put, but applies to any Delta level assuming the corresponding VWB and RR are available.
7.2.1
The Delta-sticky convention
FX derivatives markets quote the strike prices in terms of Delta of the option. This is referred to as the so-called Delta-sticky convention. It’s implications concerns the following:
• The sticky Delta: Once the deal is closed, given the level of the FX spot rate and the implied vol. agreed upon (where the interest rate levels will be taken from the money market), the strike will be set at a level yielding the BS Delta that the two counterparties were dealing.
Practically this means that, if the FX spot rate moves, all other things being equal, the curve of implied vol. vs. Delta will remain unchanged, while the curve of the implied vol. vs. strike will shift.
When the option is quoted with reference to a strike level expressed in terms of Delta, once the option is traded and the FX spot reference rate and the implied vol. are fixed, then the absolute level of the strike can be retrieved by setting
wPf(t, T)Φ(wd1) = ¯∆ (7.7)
which can be expressed as
K =F(t, T)exp
−wσp(T −t)Φ−1(|∆¯|/Pf(t, T)) + 0.5σ2(T −t)
(7.8)
where the at-the-money strike reduces to
KAT M(t, T) =F(t, T)exp
0.5σAT M2 (T −t)
(7.9)
where w = 1 (respectively,w = -1) if a call (put) option. Φ−1 is the inverse of the
cumulative normal distribution function, and the values σ and ∆¯ are the required inputs. So if one wants to find the corresponding strike level to a 25 Delta call, the inputs into equation 7.8 must be the σ at the 25 Delta call level and ∆¯ would be 0.25. The latter enters into the formula as its absolute value, which is relevant when considering put options.
7.2.1.1 Premium included Delta
The Delta convention used in the market for EURUSD and USDJPY has impli- cations on which method to use when converting Deltas into strikes. In the case of the EURUSD a regular Delta is quoted whereas in the case of the USDJPY a premium included/adjusted Delta is the market convention on how to quote Delta (Bloomberg), (Reiswich and Wystrup, 2010).
In order to explain the difference between a regular premium excluded Delta and a premium included Delta we use numbers from the example in (Reiswich and Wystrup, 2010, p. 4) to create Tabel 7.1.
The relationship between the premium included Delta and Delta is
∆P I = ∆− V
S0
Table 7.1: Premium included Delta EURUSD Call Notional 1,000,000 EUR S0 1.3900 K 1.3500 σ - T -
Premium 102,400 USD 73,669 EUR
∆ 60% 600,000 EUR
∆PI 52.63% 526,331 EUR
σand T is not provided, but is also irrelevant in the example.
where the amount of foreign currency units to buy in a hedge of a short position is
F OR =N(∆− V
S0
) (7.11)
If we go short in a EURUSD call option with notional EUR 1 mio. we receive a premium of USD 102,400, which corresponds to EUR 73,669. Lets say the Delta on the option is 60%. Then we have to buy EUR 0.6 mio. in order to keep a local Delta hedge. But considering that we receive something from the trade of the option, the EUR amount to hedge is only EUR 526,331, which corresponds to a premium included Delta of 52.63%.
As just mentioned in the EURUSD case, the market convention is actually to quote the regular premium excluded Delta. So in this case we can rely on equations 7.8 - 7.9 and directly retrieve the strike. This is not the case for the USDJPY where we have to resort to a numerical procedure (based on the Newton-Raphson scheme), when we want to retrieve the strike, since the option premium entering into Delta is a function of the strike itself (Castagna, 2010, p. 35-36). The procedure used in this study follows this scheme and is outlined in the Appendix.
An example of the conversion of the premium included delta to strike is presented in Tabel 9.1 where we calculate both the strike retrieved directly from equations 7.8 - 7.9 and the strike adjusted for the premium included (PI) Delta, calculated numerically, which in this case for the USDJPY is the correct way to retrieve the strike. This is done both for a call and a put.
The premium included Delta will always be less than the regular premium ex- cluded Delta. This is true both when we consider calls and puts (where the Delta on a call is measured along the scale from 0% - 100% and the Delta on a put is measured along the scale from 0% - (-100%) going from OTM - ITM on both scales). This fact would imply that, in the case of a call, the strike retrieved from a PI Delta, without making any adjustments (calculated from 7.8 - 7.9), will always be higher than the
Table 7.2: Conversion of a Premium Included Delta to Strike USDJPY Call USDJPY Put
S0 81.54 S0 81.54 σ25c 11.97% σ25p 13.13% T-t 0.5 T-t 0.5 Pd 0.99828 Pd 0.99828 Pf 0.99786 Pf 0.99786 K 86.596 K 76.905
K adjusted for PI D 86.302 K adjusted for PI D 76.597 4/28/2011
strike adjusted for the PI Delta, calculated numerically. This implies a strike that is more ITM. In case of a put, the strike retrieved from a PI Delta, without making any adjustments, will always be higher than the strike adjusted for the PI Delta, which implies a strike that is less ITM as opposite to the case of the call. This relationship is apparent in Tabel 9.1 where K > ’K adjusted for PI D’ in both the call and put cases.
8
Data description
I. In Chapter 5, Empirical facts:
• Daily spot FX rates from 1/6/2006 - 5/3/2011 on 1.388 weekdays on the EURUSD and the USDJPY.
2. In Chapter 9, Calibration of the models and Chapter 10, Empirical study on the hedging performance:
• Bid and Ask prices on D10 RR, D10 VWB, D25 RR, D25 VWB and the ATM DNS with maturites 1M, 2M, 3M, 6M and 1Y from 1/4/2010 - 06/22/2011
consisting of 18.550 quotes on 371 trading days for both the EURUSD and the USDJPY. The conventions used in the quoting is for the ATM setting: the ATM DNS, for the Delta Premium: Excluded in the case of the EURUSD and Included in the case of the USDJPY, for the Delta style: Spot Delta (up to < 1Y then forward Delta), for the RR: Call−P ut and for the VWB: (Call+P ut)/2−AT M DN S. Preference: Bloomberg BGN and Cutoff: New York 10:00.
• Daily spot FX rates from 1/4/2010 - 6/22/2011 on 371 trading days.
Source: Bloomberg
• Domestic and Foreign interest rates on 1M, 2M, 3M, 6M and 1Y from 1/4/2010 - 6/22/2011 on 371 trading days. More specifically, we use the deposit interest rates Euribor for the EUR, Libor USD for USD and Libor JPY for JPY.
9
Calibration of the models
9.1
Building the market implied volatility surface
We want to go from the data as described in Chapter 8, consisting of 5 structures on each of 5 maturities, to a set of market call option prices with corresponding strike prices. We calibrate to prices and not imp. vols. Furthermore we recognise that we have only the OTM part of the IVS for call options after retrieving the imp. vol. on a 25D call and 10D call by Equation (7.5). In order to get the ITM part of the IVS for call options we have to make an assumption. The following steps are carried out in the procedure to convert our market data into market prices on call options covering a wide range of the IVS:
1. We apply Equations 7.5 - 7.6 in order to get from D10 RR, D10 VWB, D25 RR, D25 VWB and the ATM DNS to σ10C, σ10P, σ25C, σ25P and σAT M losing
the time subscript.
2. From here we retrieve the strikes from Equations (7.8) - (7.9) in case of the EURUSD and use the numerical method applied in Table 9.1 in case of the USDJPY in order to get from a Delta moneyness to strike prices. We use Equation (7.9) in both the EURUSD and USDJPY case to retrieve σAT M as
prescribed in ??.
3. We then assume the put-call parity in 6.8 to hold all though research shows that this is rarely the case (Chalamandaris and Tsekrekos, 2008). Under this assumption we calculate the BS call prices on the 5 pairs of σ10P /Kσ10P,σ25P
/ Kσ25P, σAT M / KσAT M, σ25C / Kσ25C, σ10C / Kσ10C. It is the convention on
every option market that the imp. vol. quoted is the BS imp. vol. which allows us to use the BS model to calculate the prices.
We then end up with market call option prices with the Delta moneyness 90D, 75D, (50)D, 25D and 10D with corresponding strike prices.