Question

Q8. SPX index at 3000. Assume normal distribution for index price. Consider a Binary CALL option with strike of 3250.

a. If IV is 30%, what is fair price for the BINARY call (5 points)?

b. If the binary call is price at 0.20, what is the implied volatility (aka implied stdev) (5 points)?

Answer 1

There are two forms of binary options: cash-or-nothing and asset-or-nothing. A cash-or-nothing binary option either pays you a fixed amount of money or nothing at all. The asset-or-nothing option is basically the same, but your payment equals the price of the asset underlying the option. A binary call option pays off the corresponding amount if at maturity the underlying asset price is above the strike price and zero otherwise. The binary put option pays off that amount if the underlying asset price is less than the strike price and zero otherwise.

We are assuming Cash-or-nothing binary call option:

The price of the option can be found by the formulas below, where S the initial stock price, X the strike price, T the time to maturity, q the dividend rate, σ the volatility and r the risk-free interest rate. N denotes the cumulative distribution function of the normal distribution.

Cash-or-nothing call: Ccn = e−rTN(d2)

Assumptions:

Risk-free rate (r) = Long-term average of US 10-year treasury yields = 4.47%

Historical Dividend yield (q) of S&P 500 index = 4.32%

On NADEX, binary options have terms of hours, 1 day, or 1 week. So, Time to maturity (T-t) = 1 day

Given,

Spot Price (S)	3000
Strike Price (X)	3250
Interest Rate ( r )	4.47%
Dividend Yield (q)	4.32%
Time to maturity (T-t)	0.08
Annualized Implied Volatility (σ)	30%
d1	-0.88
d2	-0.97
Binary Call price (Ccn) ($)	0.17

(b)

Spot Price (So)	3000
Strike Price (K)	3250
Interest Rate ( r )	4.47%
Dividend Yield (q)	4.32%
Time to maturity (T)	0.08
Annualized Implied Volatility (σ)	35.11%
d1	-0.74
d2	-0.84
Binary Call price (Ccn) ($)	0.20