Binary Options

Overview

The simplest Binary (also known as Digital) options are cash-or-nothing and asset-or-nothing options. In a cash-or-nothing option, a predetermined amount is paid if the asset is, at option expiry, above (call) or below (put) some strike level, independent of the path taken. An asset-or-nothing option is similar to a cash-or-nothing option except that the holder of the option is entitled to either the full asset value – at option expiry – or nothing. Both of these options are handled by the function aaDigital_AON().

A simple generalization of the asset-or-nothing option is a digital-gap-option. A gap option has a payout profile equal to the asset value, less the gap value, depending on whether the asset finishes above or below the strike price. It is clear that a digital gap option is simply the difference between an asset or nothing and a cash-or-nothing digital option with the cash amount set to the gap value. Gap options are handled by the function aaDigital_gap().

The next types of digital options are Binary Barrier Options for which an option’s payout depends on whether the asset touched a barrier level at some time during the life of the option. There are two classes of binary barrier options. The first are options where a payout of cash (or asset) is made if the barrier is hit (or not hit) during the life of the option. Functions dealing with this class of options are summarized in the following table:

Binary Barrier Options (Barrier hit or not hit)

Function	Barrier	Payment Type
aaBinary_bar_hit_cash()	Hit	cash or nothing
aaBinary_bar_hit_asset()	Hit	asset or nothing
aaBinary_bar_nohit_cash()	Not hit	cash or nothing
aaBinary_bar_nohit_asset()	Not hit	asset or nothing

For example, aaBinary_bar_hit_cash() handles binary barrier options with a payoff of a fixed amount of cash if the barrier is hit during the life of the option, or nothing if the barrier is never hit.

The payout is made either when the barrier is hit, or at option expiry. For cash payouts, this distinction will only affect the period of time over which the payment is discounted. For asset payouts, however, the distinction is more subtle. If the payout is made when the barrier is touched, then the present value of the payout is equal to the discounted barrier value – since this is the asset value when the barrier is touched. On the other hand, if the payout is made at option expiry, then the present value of the payout is equal to whatever the asset value happens to be at the expiry date, discounted back to the valuation date.

The second class is the options where a payout of cash (or asset) is made if the barrier is hit (or not hit) during the life of the option and if the option is in the money at expiry. These are the types of knock-in and knock-out binary barrier options. Functions dealing with this class of options are summarized in the following table:

Binary Barrier Options (Knock-in or Knock-out)

Function	Barrier Type	Digital Option Type	Call/Put
aaBinary_bar_in_cash()	knock-in	cash or nothing	Y
aaBinary_bar_in_asset()	knock-in	asset or nothing	Y
aaBinary_bar_out_cash()	knock-out	cash or nothing	Y
aaBinary_bar_out_asset()	Knock-out	asset or nothing	Y

For example aaBinary_bar_in_asset() deals with binary barrier options which has a payout of an asset-or-nothing binary option if the barrier is hit during the life of the option. In other words, if the barrier is hit during the life of the option, a payment equivalent to the value of the underlying asset is made if the option is in the money at expiry; otherwise (if the barrier is not hit or the option is out of the money at expiry) no payment will be made.

There are (inverse) root finding versions of all FINCAD Binary functions above. They are listed and described in the General Root Finding (_ix) Functions FINCAD Math Reference document.

There are other types of digital options available within the FINCAD options functions. There are various flavors of double barrier binary options. Specific digital option related functions are:

aaBarrier_dbl_bin()

Combines a binary option with knock-outs at the two barriers. Rebates may be specified at the barriers and at expiry

aaBarrier_dbl_bin_dbltouch()

Binary option is knocked-in or out if both barriers are touched

aaBarrier_dbl_bin_hit_cash()

Cash is paid out if barriers are hit or not hit during the life of the option

aaBarrier_dbl_bin_mix()

A binary option is knocked-in or out depending on whether the upper or the lower barrier is hit first

aaBarrier_dbl_bin_onetouch()

A binary option is knocked-in or out if either barrier is touched.

We also note that within most of the single barrier option related functions (aaBarrier*) and the double barrier option related functions (aaBarrier_dbl*), there is often the ability to input a rebate amount at any of the barriers (or at expiry if none of the barriers is hit). Since these functions allow features like discretely sampled barriers, exercise windows, continuously sampled barriers, varying level barriers, etc., it is possible to construct a very wide range of binary options using the rebate feature. For more information, see the barrier and double barrier option documents.

In the family of Binary functions there are two, aaBinary_exp and aaBinary_hit, that are kept as a legacy to older versions. They have full functionality and are embedded into the workbooks, however we encourage users to employ other available FINCAD functions among listed below.

Finally, we note the following convention: the fair value and risk statistics for binary barrier options which have hit the barrier historically, and for which payment is made when the barrier is hit, are zero, where “historically” includes the present day. For example, in the case of a binary option with an up-and-in barrier which pays out a fixed amount of cash when the barrier is hit, and for which the underlying price is greater than the barrier value, all statistics are equal to zero except the probability of hitting the barrier, which is equal to one.

FINCAD Functions

aaDigital_AON(price_u, ex, cash, d_v, d_exp, payoff, vlt, rate_ann, cost_hldg, option_type, stat)

aaBinary_exp(price_u, ex, bar, d_v, d_exp, bar_type, payoff, cash, vlt, rate_ann, cost_hldg, option_type, stat)

aaDigital_Gap(price_u, ex, cash, d_v, d_exp, vlt, rate_ann, cost_hldg, option_type, stat)

aaBinary_hit(price_u, bar, d_v, d_exp, payoff, cash, vlt, rate_ann, cost_hldg, stat)

aaBinary_bar_hit_cash(price_u, bar, d_v, d_exp, barrier_type, cash, paytime_type, vlt, rate_ann, cost_hldg, stat)

aaBinary_bar_hit_asset(price_u, bar, d_v, d_exp, barrier_type, paytime_type, vlt, rate_ann, cost_hldg, stat)

aaBinary_bar_nohit_cash(price_u, bar, d_v, d_exp, barrier_type, cash, vlt, rate_ann, cost_hldg, stat)

aaBinary_bar_nohit_asset(price_u, bar, d_v, d_exp, barrier_type, vlt, rate_ann, cost_hldg, stat)

aaBinary_bar_in_cash(price_u, ex, bar, d_v, d_exp, barrier_type, cash, vlt, rate_ann, cost_hldg, option_type, stat)

aaBinary_bar_in_asset(price_u, ex, bar, d_v, d_exp, barrier_type, vlt, rate_ann, cost_hldg, option_type, stat)

aaBinary_bar_out_cash(price_u, ex, bar, d_v, d_exp, barrier_type, cash, vlt, rate_ann, cost_hldg, option_type, stat)

aaBinary_bar_out_asset(price_u, ex, bar, d_v, d_exp, barrier_type, vlt, rate_ann, cost_hldg, option_type, stat)

Other Types of Binary Options

· Quanto: each of the functions above has a Quanto version.

· FX Specific: There are several FX specific versions of binary option functions

· Double Barrier

· Single Barrier

Root Finding Functions for Binary Options

Many of these FINCAD functions have their inverse (root finding) versions:

aaBinary_bar_hit_cash_ix()

aaBinary_bar_hit_asset_ix()

aaBinary_bar_nohit_cash_ix()

aaBinary_bar_nohit_asset_ix()

aaBinary_bar_in_cash_ix()

aaBinary_bar_in_asset_ix()

aaBinary_bar_out_cash_ix()

aaBinary_bar_in_asset_ix()

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 Inputs

Input Argument	Description
price_u	current value of the underlying asset
ex	exercise price (used only in the functions which have a option type switch)
cash	amount of cash payment (used in the functions which deal with options with a payoff of cash or nothing)
d_v	value date
d_exp	option expiry date
payoff	1 = cash, 2 = asset (used in aaDigital_AON() only).
vlt	The annualized volatility of the underlying asset
rate_ann, cost_hldg	Also denoted rate1 and rate2, respectively. These rates are quoted on an annually compounded, Act / 365 (fixed) basis. · If the underlying is an equity, rate2 is the annualized dividend yield. · If the underlying is a forward price, rate2 should be set equal to the risk-free rate1. · If the underlying is an FX rate, and quoted on a domestic per foreign basis, rate1 should be the risk-free domestic rate and rate2 the risk-free foreign rate. · If the underlying is an FX rate, and quoted on a foreign per domestic basis, rate1 should be the risk-free foreign rate and rate2 the risk-free domestic rate. · If the underlying is a commodity, then rate2 should be set to the annualized holding cost of the commodity, including storage and insurance costs as well as marginal convenience value.
option_type	1 = call: 2 = put Note: Functions dealing with the first class of binary barrier options, i.e., options with payouts depending only on whether a barrier is hit, do not have this switch.
stat	any set of 1…8. (any set of 1,2 for aaDigital_gap()). See the outputs in the examples below.
barrier_type	1 = up and in; 2 = down and in (used in the binary barrier functions only).
paytime_type	1 = pay at barrier; 2 = pay at expiry (used in aaBinary_bar_hit_cash() and aaBinary_bar_hit_asset only).

Description of Outputs

Output Statistics	Description
fair value	The fair value of the option.
delta	The rate of change in the fair value of the option per one unit change in the current value of the underlying asset. This is the derivative of the option price with respect to the underlying current value.
gamma	The rate of change in the value of delta per one unit change in the current value of the underlying asset. This is the second derivative of the option price with respect to the underlying current value.
theta	The rate of change in the fair value of the option per one day decrease of the option time. This is the negative of the derivative of the option price with respect to the option time (in years), divided by 365.
vega	The rate of change in the fair value of the option per 1% change in volatility. This is the derivative of the option price with respect to volatility, divided by 100.
rho of rate	The rate of change in the fair value of the option per 1% change in the risk-free rate, rate_ann. This is the derivative of the option price with respect to the risk-free rate, divided by 100.
rho of dividend yield	The rate of change in the fair value of the option per 1% change in the holding cost, cost_hldg. This is the derivative of the option price with respect to the holding cost, divided by 100. If the underlying is futures, this statistic is not available.
probability of hitting the barrier	The probability (the risk neutral probability) that the underlying price reaches the barrier during the life of the option. This output is available only for the binary barrier functions.

Note: For the function aaDigital_gap() only the delta of its risk statistics is given. Remember that aaDigital_gap() is equivalent to aaDigital_AON() with payoff=asset minus aaDigital_AON() with payoff=cash. One can obtain other risk statistics of aaDigital_gap() from those of aaDigital_AON() easily.

For details about the calculation of Greeks, see the Greeks of Options on non-Interest Rate Instruments FINCAD Math Reference document.

Examples

Context examples are given for binary options on stocks and stock indices.

Example 1: Binary option

Consider a binary call option on an equity with a spot price of 100. The strike price is 100. The payoff of the option is 3 if the option is in the money on the expiry date, and 0 otherwise. Today’s date is Jun. 13, 1998. The option expires at Jan. 24, 1999. Suppose the relevant annual risk free rate is 6%, the stock’s annual volatility is 20% and its annual yield is 2%. Calling FINCAD function aaDigital_AON() we get the following result:

aaDigital_AON

Argument	Description	Example Data	Switch
price_u	underlying price	100
ex	exercise price	100
cash	cash payment	3
d_v	value (settlement) date	13-Jun-1998
d_exp	expiry date	24-Jan-1999
payoff	cash or asset	1	cash
vlt	volatility	0.2
rate_ann	rate - annual - Actual/365	0.06
cost_hldg	holding cost - annual - Actual/365	0.02
option_type	option type	1	call
stat	stat list	1…7

Results

Statistics	Description	Value
1	fair value	1.5307044
2	delta	0.0733304
3	gamma	-0.001072
4	theta	5.91E-05
5	vega	-0.013207
6	rho of rate	0.0337437
7	rho of dividend yield	-0.04432

Example 2: Binary Barrier option with payout of cash if the barrier is hit

Consider a binary barrier option of type down and in on a stock with a spot price of 100. The barrier is 85. The payout of the option is 5 at the time the barrier is hit, and 0 if the barrier is never hit during the option period. Today’s date is Jan. 13, 1998. The option expires on Jan. 24, 1999. Suppose the relevant annual risk free rate is 6%, the stock’s annual volatility is 20% and its annual yield is 2%. Calling FINCAD function aaBinary_bar_hit_cash() we get the following result:

aaBinary_bar_hit_cash

Argument	Description	Example Data	Switch
price_u	underlying price	100
bar	barrier	85
d_v	value (settlement) date	13-Jan-1998
d_exp	expiry date	24-Jan-1999
bar_type	barrier type	2	down and in
cash	cash payment	5
paytime_type	pay time type	1	pay at the time when the barrier is hit
vlt	volatility	0.2
rate_ann	rate - annual - Actual/365	0.06
cost_hldg	holding cost - annual - Actual/365	0.02
stat	stat list	1…8

Results

Statistics	Description	Value
1	fair value	1.907380
2	delta	-0.139647
3	gamma	0.008136
4	theta	-0.002677
5	vega	0.137279
6	rho of rate	-0.085299
7	rho of dividend yield	0.079981
8	probability of hitting the barrier	0.391968

Note: “Down and in” means that the underlying price at the value date is bigger than the barrier. Similarly “up and in” means that the underlying price is smaller than the barrier.

Example 3 Knock-in Binary Barrier option

Consider a binary barrier call option of type down and in on a stock with a spot price of 100. The barrier is 85. The payoff of the option is the value of the stock if the barrier is hit and the underlying exceeds 90, the strike price. Today’s date is Jan. 13, 1998. The option expires on Jan. 24, 1999. Suppose the relevant annual risk free rate is 6%, the stock’s annual volatility is 20% and its annual yield is 2%. Calling FINCAD function aaBinary_bar_in_asset() we get the following result:

aaBinary_bar_in_asset

Argument	Description	Example Data	Switch
price_u	underlying price	100
ex	exercise price	90
bar	barrier	85
d_v	value (settlement) date	13-Jan-1998
d_exp	expiry date	24-Jan-1999
barrier_type	barrier type	2	down and in
vlt	volatility	0.2
rate_ann	rate - annual - Actual/365	0.06
cost_hldg	holding cost - annual - Actual/365	0.02
option_type	option type	1	call
stat	stat list	1…8

Results

Statistics	Description	Value
1	fair value	13.16518
2	delta	-1.13234
3	gamma	0.08407
4	theta	-0.03199
5	vega	1.29355
6	rho of rate	-0.15453
7	rho of dividend yield	0.02754
8	probability of hitting the barrier	0.39197

Example 4: Gap option

Consider a European style call option on a stock with a spot price of 100. The strike is 100. When exercised, the payoff of the option is the stock price minus 3, a fixed amount of cash. Today’s date is Jan. 13, 1998. The option expires on Jan. 24, 1999. Suppose the relevant annual risk free rate is 6%, the stock’s annual volatility is 20% and its annual yield is 2%. Calling FINCAD function aaDigital_gap() we get the following result:

aaDigital_gap

Argument	Description	Example Data	Switch
price_u	underlying price	100
ex	exercise price	100
cash	cash payment	3
d_v	value (settlement) date	13-Jan-1998
d_exp	expiry date	24-Jan-1999
vlt	volatility	0.2
rate_ann	rate - annual - Actual/365	0.06
cost_hldg	holding cost - annual - Actual/365	0.02
option type	option type	1	call
stat	stat list	1…2

Results

Statistics	Description	Value
1	fair value	58.90208
2	delta	2.39163

References

[1] Hudson, Mike, (March 1991), ‘The Value in Going Out’, Risk.

[2] Rubinstein, Mark and Reiner, Eric, (October 1991), ‘Unscrambling the Binary Code’, Risk.

Disclaimer

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