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I have a formula that uses Black-Scholes to compute the implied pricing of a "Cash or Nothing" binary option on the price of a currency.

The option is priced/traded in the same currency as S, K and the payout is 1 unit of same currency if S(T) > K and 0 if S(T) < K. The option's price ranges from 0 - 1. I'm trying to figure out how to change the formula from what I believe is the cash_or_noting formula to what I think should be the asset_or_nothing formula.

S = 110 #current_price
K = 100 #ATM strike
v = 1.20 #annualized volatility
r = 0.00 #interest rate
T =  0.44 #days remaining (annualized)

from scipy.stats import norm
from math import exp, log, sqrt

d2 = (log(S/K) + (r - 0.5 * v**2) * T) / (v*sqrt(T))
print (exp(-r * T) * norm.cdf(d2))
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  • $\begingroup$ You need to be more specific about your payoff: what is cash or nothing and what is asset or nothing. $\endgroup$ – Gordon Jul 30 '18 at 19:18
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    $\begingroup$ A cash-or-nothing option is a digital option with (unit) payout $ 1\{ S_T \geq K\}$. An asset-or-nothing option on the other hand corresponds to a payout $ S_T 1\{ S_T \geq K\} $. These are the 2 terms involved in the price of a European call option. More specifically the binary option price evaluates as $DF(0,T) N(d_2)$ the other one is $DF(0,T) F(0,T) N(d_1)$ where $DF(0,T)$ is the discount factor and $F(0,T)$ the forward price. $\endgroup$ – Quantuple Jul 30 '18 at 19:47
  • $\begingroup$ @Gordon I added some additional details that I hope makes things more clear. I'm struggling as its an option on the price of a currency, the option is traded in the same currency and the option price is bound between 0 and 1 with a payout in the same currency. $\endgroup$ – Snapula Jul 30 '18 at 21:32
  • $\begingroup$ @Snapula: See comments by Quantuple above, which I fully agree. $\endgroup$ – Gordon Jul 31 '18 at 13:32
  • $\begingroup$ Thanks to you both. @Quantuple it doesn't matter that the currency that the option is trading in is also the asset that is received in the event of a payoff? I thought that could be a complicating factor. eg - an option on the price of EUR/USD, traded in Euros with the payout being Euros. $\endgroup$ – Snapula Jul 31 '18 at 20:09

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