/ 
Math-Business-BlackScholesMerton-1.25
River stage one • 2 direct dependents • 2 total dependents
/ Math::Business::BlackScholesMerton::Binaries

NAME

Math::Business::BlackScholesMerton::Binaries

SYNOPSIS

use Math::Business::BlackScholesMerton::Binaries;

# price of a Call option
my $price_call_option = Math::Business::BlackScholesMerton::Binaries::call(
    1.35,       # stock price
    1.36,       # barrier
    (7/365),    # time
    0.002,      # payout currency interest rate (0.05 = 5%)
    0.001,      # quanto drift adjustment (0.05 = 5%)
    0.11,       # volatility (0.3 = 30%)
);

DESCRIPTION

Prices options using the GBM model, all closed formulas.

Important(a): Basically, onetouch, upordown and doubletouch have two cases of payoff either at end or at hit. We treat them differently. We use parameter $w to differ them.

$w = 0: payoff at hit time. $w = 1: payoff at end.

Our current contracts pay rebate at hit time, so we set $w = 0 by default.

Important(b) :Furthermore, for all contracts, we allow a different payout currency (Quantos).

Paying domestic currency (JPY if for USDJPY) = correlation coefficient is ZERO. Paying foreign currency (USD if for USDJPY) = correlation coefficient is ONE. Paying another currency = correlation is between negative ONE and positive ONE.

See [3] for Quanto formulas and examples

SUBROUTINES

call

USAGE
my $price = call($S, $K, $t, $r_q, $mu, $sigma)

PARAMS
$S => stock price
$K => barrier
$t => time (1 = 1 year)
$r_q => payout currency interest rate (0.05 = 5%)
$mu => quanto drift adjustment (0.05 = 5%)
$sigma => volatility (0.3 = 30%)

DESCRIPTION
Price a Call and remove the N(d2) part if the time is too small

EXPLANATION
The definition of the contract is that if S > K, it gives
full payout (1).  However the formula DC(T,K) = e**(-rT) N(d2) will not be
correct when T->0 and K=S.  The value of DC(T,K) for this case will be 0.5.

The formula is actually "correct" because when T->0 and S=K, the probability
should just be 0.5 that the contract wins, moving up or down is equally
likely in that very small amount of time left. Thus the only problem is
that the math cannot evaluate at T=0, where divide by 0 error occurs. Thus,
we need this check that throws away the N(d2) part (N(d2) will evaluate
"wrongly" to 0.5 if S=K).

NOTE
Note that we have call = - dCall/dStrike
pair Foreign/Domestic

see [3] for $r_q and $mu for quantos

put

USAGE
my $price = put($S, $K, $t, $r_q, $mu, $sigma)

PARAMS
$S => stock price
$K => barrier
$t => time (1 = 1 year)
$r_q => payout currency interest rate (0.05 = 5%)
$mu => quanto drift adjustment (0.05 = 5%)
$sigma => volatility (0.3 = 30%)

DESCRIPTION
Price a standard Digital Put

d2

returns the DS term common to many BlackScholesMerton formulae.

expirymiss

USAGE
my $price = expirymiss($S, $U, $D, $t, $r_q, $mu, $sigma)

PARAMS
$S => stock price
$t => time (1 = 1 year)
$U => barrier
$D => barrier
$r_q => payout currency interest rate (0.05 = 5%)
$mu => quanto drift adjustment (0.05 = 5%)
$sigma => volatility (0.3 = 30%)

DESCRIPTION
Price an expiry miss contract (1 Call + 1 Put)

[3] for $r_q and $mu for quantos

expiryrange

USAGE
my $price = expiryrange($S, $U, $D, $t, $r_q, $mu, $sigma)

PARAMS
$S => stock price
$U => barrier
$D => barrier
$t => time (1 = 1 year)
$r_q => payout currency interest rate (0.05 = 5%)
$mu => quanto drift adjustment (0.05 = 5%)
$sigma => volatility (0.3 = 30%)

DESCRIPTION
Price an Expiry Range contract as Foreign/Domestic.

[3] for $r_q and $mu for quantos

onetouch

PARAMS
$S => stock price
$U => barrier
$t => time (1 = 1 year)
$r_q => payout currency interest rate (0.05 = 5%)
$mu => quanto drift adjustment (0.05 = 5%)
$sigma => volatility (0.3 = 30%)

[3] for $r_q and $mu for quantos

notouch

USAGE
my $price = notouch($S, $U, $t, $r_q, $mu, $sigma, $w)

PARAMS
$S => stock price
$U => barrier
$t => time (1 = 1 year)
$r_q => payout currency interest rate (0.05 = 5%)
$mu => quanto drift adjustment (0.05 = 5%)
$sigma => volatility (0.3 = 30%)

DESCRIPTION
Price a No touch contract.

Payoff with domestic currency
Identity:
price of notouch = exp(- r t) - price of onetouch(rebate paid at end)

[3] for $r_q and $mu for quantos

upordown

USAGE
my $price = upordown(($S, $U, $D, $t, $r_q, $mu, $sigma, $w))

PARAMS
$S stock price
$U barrier
$D barrier
$t time (1 = 1 year)
$r_q payout currency interest rate (0.05 = 5%)
$mu quanto drift adjustment (0.05 = 5%)
$sigma volatility (0.3 = 30%)

see [3] for $r_q and $mu for quantos

DESCRIPTION
Price an Up or Down contract

common_function_pelsser_1997

USAGE
my $c = common_function_pelsser_1997($S, $U, $D, $t, $r_q, $mu, $sigma, $w, $eta)

DESCRIPTION
Return the common function from Pelsser's Paper (1997)

get_stability_constant_pelsser_1997

USAGE
my $constant = get_stability_constant_pelsser_1997($S, $U, $D, $t, $r_q, $mu, $sigma, $w, $eta, $p)

DESCRIPTION
Get the stability constant (Pelsser 1997)

ot_up_ko_down_pelsser_1997

USAGE
my $price = ot_up_ko_down_pelsser_1997($S, $U, $D, $t, $r_q, $mu, $sigma, $w)

DESCRIPTION
This is V_{RAHU} in paper [5], or ONETOUCH-UP-KNOCKOUT-DOWN,
a contract that wins if it touches upper barrier, but expires
worthless if it touches the lower barrier first.

ot_down_ko_up_pelsser_1997

USAGE
my $price = ot_down_ko_up_pelsser_1997($S, $U, $D, $t, $r_q, $mu, $sigma, $w)

DESCRIPTION
This is V_{RAHL} in paper [5], or ONETOUCH-DOWN-KNOCKOUT-UP,
a contract that wins if it touches lower barrier, but expires
worthless if it touches the upper barrier first.

get_min_iterations_pelsser_1997

USAGE
my $min = get_min_iterations_pelsser_1997($S, $U, $D, $t, $r_q, $mu, $sigma, $w, $accuracy)

DESCRIPTION
An estimate of the number of iterations required to achieve a certain
level of accuracy in the price.

_get_min_iterations_ot_up_ko_down_pelsser_1997

USAGE
my $k_min = _get_min_iterations_ot_up_ko_down_pelsser_1997($S, $U, $D, $t, $r_q, $mu, $sigma, $w, $accuracy)

DESCRIPTION
An estimate of the number of iterations required to achieve a certain
level of accuracy in the price for ONETOUCH-UP-KNOCKOUT-DOWN.

_get_min_iterations_ot_down_ko_up_pelsser_1997

USAGE

DESCRIPTION
An estimate of the number of iterations required to achieve a certain
level of accuracy in the price for ONETOUCH-UP-KNOCKOUT-UP.

range

USAGE
my $price = range($S, $U, $D, $t, $r_q, $mu, $sigma, $w)

PARAMS
$S stock price
$t time (1 = 1 year)
$U barrier
$D barrier
$r_q payout currency interest rate (0.05 = 5%)
$mu quanto drift adjustment (0.05 = 5%)
$sigma volatility (0.3 = 30%)

see [3] for $r_q and $mu for quantos

DESCRIPTION
Price a range contract.

americanknockout

American Binary Knockout

Description of parameters:

$S - spot $H1 - lower barrier $H2 - upper barrier $K - payout strike $tiy - time in years $sigma - volatility $mu - mean $r - interest rate $type - 'c' for buy or 'p' for sell

Reference: http://www.phy.cuhk.edu.hk/~cflo/Finance/chohoi/afe.pdf