Pricing a Cash-or-Nothing Call with the Normal CDF

Summary

The document explains how to apply the cash-or-nothing call formula at the valuation date. The option pays a fixed cash amount if the underlying finishes above the strike, and its value is the discounted payout multiplied by the risk-neutral probability of finishing in the money, represented by the standard normal cumulative distribution function evaluated at d.

At time zero, the time to maturity is the full maturity T. The reply notes that calculating d also requires the current underlying price S and strike E, in addition to the payout, interest rate, volatility, and maturity. Once those inputs are available, calculate d and evaluate the normal CDF. The question supplies payout, rate, maturity, and volatility, but omits S and E, so it does not provide enough information for a numerical option price. It offers a formula setup rather than a worked calculation or discussion of assumptions such as dividends or alternative pricing models.

Key ideas

• At valuation time zero, time to maturity equals the option's full maturity.
• The d parameter depends on spot price, strike, interest rate, volatility, and time to maturity.
• The normal cumulative distribution function evaluated at d supplies the probability factor in the pricing formula.
• A numerical price cannot be calculated without the spot price and strike.

Tags

# Cash-or-Nothing Call Option


# Cash-or-Nothing Call Option












I am trying to price a cash or nothing call option and I know know that the Cash or Nothing formula for a call option is $C(t,s)=Xe^{-r(T-t)}*N(d)$

If I have payoff X=100 r=0.03 T=2 $\sigma=0.3$

I would have $C(t,s)=100e^{-0.03(2-t)}N(d)$

but how would I find $N(d)$ and (T-t)as where $$ d=\dfrac{\ln(S/E)+(r-\sigma^2/2)(T-t)}{\sigma\sqrt{T-t}} $$

to calculate the price at time 0

## Answer by Xman (score 0)

https://quant.stackexchange.com/a/51260

In your formula you have the following variables:

- t= 0, T = the maturity of the call option which is known.

- S is the spot value of the underlying asof t = 0 (today) which is known.

- E is the strike of the option which is known

- r and σ are known.

In other word all the variables are known and thus it's a straight forward formula to get N(d)

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.