FX Options
Overview
FINCAD has implemented 12 new foreign exchange specific
option models to price FX options with American and European exercise,
including functions for Asian, barrier, double barrier, binary, and
combinations thereof. The functions listed
below are accessible through the Function Finder, Foreign Exchange, FX Option
Models.
The ease of use of these new FX functions arises from the
fact that your inputs and outputs are currency based instead of FX rate
based. That is, one can replicate the
results of the new functions utilizing standard models like aaBSG (Black-Scholes Generalized),
however the standard model outputs / inputs require further manipulation to get
currency based results.
Foreign exchange options are an alternative to forward
contracts when hedging an FX exposure because options allow the company to
benefit from favorable FX rate movements, while a forward contract locks in the
FX rate for a future transaction. Of
course this “insurance” from the option is not free, while it costs nothing to
enter into a forward transaction.
When pricing foreign exchange options, the underlying is
the spot or forward foreign exchange rate.
If we assume that exchange rates follow geometric Brownian motion (same
type of stochastic process as a stock), using USD / JPY as the FX rate, the
foreign currency (JPY) is analogous to a stock providing a known dividend
yield, where the owner of the foreign currency receives a “dividend yield”
equal to the risk-free rate in the foreign currency. These and other assumptions allow us to
utilize generic option models, such as Black-Scholes, in the valuation of FX
options.
The FX functions listed below value call or put options on
the principle currency. This can be chosen to be the either the
primary or secondary currency. If the
principle is the primary (secondary) currency, it is the risk free rate of the
primary (secondary) currency which is analogous to the dividend yield on a
stock. The fair value in the secondary (primary) currency will then match the
fair value of the corresponding option on the stock providing that dividend
yield, provided the FX rate – the underlying – is quoted as secondary per one
primary (primary per one secondary).
For FX barrier options on a principle currency which provide
rebates, the rebate is always in the other currency. If the principle is the primary (secondary)
currency, the rebate is thus in the secondary (primary) currency. This can be checked explicitly through the
“rebate currency” output of these functions.
For model details, please refer to the specific math
document for the related FINCAD functions listed below.
Formulas & Technical Details
FX options can be confusing and sometimes require a little
extra thought because one customer will consider the option a CALL and another
will consider the same FX option a PUT.
It is always important to understand what the expected payoff is because
once the payoff is known the inputs to the option functions will be clear.
As an example, if the strike rate is 1.45 CAD/USD ($1.45
CAD buys $1.00 USD) then the payoff for a long call on USD would look like the
illustration below. Remember that a call
on USD is the same as a put on CAD, so if the CAD/USD rate rises to 1.50, a
1.45 PUT on CAD is in the money.
Figure 1 Payoff
for a Long Call on USD
To explain further, we have the right to buy $1
USD for $1.45 CAD (which is equivalent to the right to sell $1.45 CAD for $1
USD). When the CAD/USD rate ends at 1.50
we would exercise the option and sell $1.45 CAD in exchange for $1 USD, and
could immediately exchange that $1 USD for $1.50 CAD, for a profit of $0.05
CAD.
FINCAD Functions
European and American exercise
Calculates fair value
and risk statistics for a European exercise foreign exchange option
aaFX_BIN
Calculates fair value
and risk statistics for an American exercise foreign exchange option
Asian (average price)
Calculates fair value
and risk statistics for an Asian foreign exchange option
Barrier (single and double): European and
American exercise
Calculates fair value
and risk statistics for a European exercise single barrier foreign exchange
option. Both call and put options can be
valued, and all types of barrier (up-and-in, up-and-out, down-and-in,
down-and-out) are supported.
Calculates fair value
and risk statistics for an American exercise single barrier foreign exchange
option. As in the European case, both
call and put options can be valued, and all types of barrier (up-and-in,
up-and-out, down-and-in, down-and-out) are supported.
Calculates fair value
and risk statistics for a double barrier (American/European exercise) foreign
exchange option (either a call or put).
This is a double knock-out barrier option: initially the holder owns a
call or put binary option, but if at any time either barrier is breached, the
option is lost or knocked-out.
Calculates fair value
and risk statistics for a double barrier (American/European exercise) with
binary payoff foreign exchange option (either a call or put). This is a double knock-out barrier option:
initially the holder owns a call or put binary option, but if at any time
either barrier is breached, the option is lost or knocked-out.
Binary (digital)
Calculates fair value
and risk statistics for a binary foreign exchange option
Calculates fair value
and risk statistics for a binary barrier foreign exchange option (with a payoff
of a fixed amount of cash, if the barrier is breached, or nothing, if the
barrier is never breached). Both up-and-in
and down-and-in options can be valued.
Calculates fair value
and risk statistics for a binary barrier foreign exchange option (with a payoff
of a fixed amount of cash, if the barrier is not touched, or nothing, if the
barrier is touched). Both up-and-out and
down-and-out options can be valued.
Calculates fair value
and risk statistics for a binary barrier foreign exchange option (with a payoff
of a fixed amount of cash, if the barrier is touched and the option is in the
money at expiry, or nothing, otherwise).
Both up-and-in and down-and-in options can be valued.
Calculates fair value
and risk statistics for a binary barrier foreign exchange option (with a payoff
of a fixed amount of cash if the barrier is not touched and the option is in
the money at expiry, or nothing, otherwise).
Both up-and-out and down-and-out options can be valued.
Root
Finding Functions
Some
of these FINCAD functions have their inverse (root finding) versions:
These
“_ix” (implied x, where x is any input
parameter) functions find the value of any input parameter for a given value of
an output statistic. More details can be
found in the General Root Finding (ix) Functions
FINCAD Math Reference document.
Description of Outputs
|
fair value in Primary currency |
|
fair value in Secondary currency |
|
delta of principal currency |
|
delta of Primary currency |
|
delta of Secondary currency |
|
gamma of principal currency |
|
gamma of Primary currency |
|
gamma of Secondary currency |
|
vega of principal currency |
|
vega of Primary currency |
|
vega of Secondary currency |
|
theta (per day) of principal currency |
|
theta (per day) of Primary currency |
|
theta (per day) of Secondary currency |
|
rho of Primary currency rate |
|
rho of Primary rate for Primary currency |
|
rho of Primary rate for Secondary currency |
|
rho of Secondary currency rate |
|
rho of Secondary rate for Primary currency |
|
rho of Secondary rate for Secondary currency |
|
|
|
intrinsic value in Primary currency |
|
intrinsic value in Secondary currency |
|
hedge amount of Primary currency |
|
hedge amount of Secondary currency |
|
notional of Primary currency |
|
notional of Secondary currency |
|
premium ratio for option on the principal currency |
|
premium ratio for calling/putting Primary currency |
|
premium ratio for putting/calling Secondary currency |
|
|
|
rebate currency |
|
rebate value in Primary currency |
|
rebate value in Secondary currency |
|
|
|
probability of breaching barrier |
|
probability of early exercise |
|
probability of hitting the lower barrier |
|
probability of hitting the lower barrier (early exercise
considered) |
|
probability of hitting the upper barrier |
|
probability of hitting the upper barrier (early exercise
considered) |
|
probability that the underlying hits a barrier |
|
probability that the underlying hits a barrier (early
exercise considered) |
|
|
|
value of lower rebate in Primary currency |
|
value of lower rebate in Secondary currency |
|
value of upper rebate in Primary currency |
|
value of upper rebate in Secondary currency |
For details about the calculation of Greeks, see
the Greeks of Options on non-Interest Rate
Instruments FINCAD Math Reference document.
Related Functions
European and American exercise
aa_BSG, aa_BIN2
FINCAD Math Reference documents: Black-Scholes Generalized; Standard American Style
Options (Cox Rubinstein Binomial)
Asian (average price)
FINCAD Math Reference document: Asian Options
Barrier (single and double): European and American exercise
aaBarrier_eu; aaBarrier_am; aaBarrier_dbl; aaBarrier_dbl_bin
FINCAD Math Reference document: Double Barrier Options
Binary (digital)
aaDigital_AON; aaBinary_bar_hit_cash; aaBinary_bar_nohit_cash; aaBinary_bar_in_cash; aaBinary_bar_out_cash
FINCAD Math Reference document: Binary Options
Disclaimer
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information contained in it. This document should not be relied on as a
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financial, accounting or other advisors.
This information is subject
to change without notice. FINCAD assumes no responsibility for any errors in
this document or their consequences and reserves the right to make changes to
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