Formula for the discounted payoff of a digital option

Question

In "Heard on the Street" it states that the expected discounted payoff of a digital option is $$H\exp^{-r(T-t)}N(d_2)$$

where $H$ is the payoff of the option, the exponential is the discounting.

Why do we have the $N(d_2)$, what does it represent and why is it there?

Answer 1

The digital option pays $H$ at time $T$ if $S_T \geq K$ , so its option time at time $t$ is given by

$$V_t=E_t\left[e^{-r(T-t)}H 1_{\{S_T \geq K\}}\right]=e^{-r(T-t)}H* P_t(S_T \geq K)$$

The model used is Black-model, that $$dS_t=rS_tdt+\sigma dW_t$$

or $$S_T=S_te^{\left(r-\frac12 \sigma^2\right)(T-t)+\sigma (W_T-W_t)}{}$$

Calculate $ P_t(S_T \geq K)$

$$ P_t(S_T \geq K)=P_t(S_te^{\left(r-\frac12 \sigma^2\right)(T-t)+\sigma (W_T-W_t)}{} \geq K)=P_t(W_T-W_t \geq\frac{log\frac{K}{S_0}-\left(r-\frac12 \sigma^2\right)(T-t)}{\sigma})$$

$W_T-W_t |W_t$ is centered and normally distributed with variance $T-t$
$$P_t(W_T-W_t \geq\frac{log\frac{K}{S_0}-\left(r-\frac12 \sigma^2\right)(T-t)}{\sigma})=P(Y \geq\frac{log\frac{K}{S_t}-\left(r-\frac12 \sigma^2\right)(T-t)}{\sigma \sqrt{T-t}})$$

where $Y \sim \mathcal{N}(0,1)$

Using the symmetry of the normal distribution,

$$P(Y \geq\frac{log\frac{K}{S_t}-\left(r-\frac12 \sigma^2\right)(T-t)}{\sigma \sqrt{T-t}})=P(Y \leq -\frac{log\frac{K}{S_t}-\left(r-\frac12 \sigma^2\right)(T-t)}{\sigma \sqrt{T-t}})$$

Define $$d_2=-\frac{log\frac{K}{S_t}-\left(r-\frac12 \sigma^2\right)(T-t)}{\sigma \sqrt{T-t}}=\frac{log\frac{S_t}{K}+\left(r-\frac12 \sigma^2\right)(T-t)}{\sigma \sqrt{T-t}}$$

$$P(Y \leq -\frac{log\frac{K}{S_t}-\left(r-\frac12 \sigma^2\right)(T-t)}{\sigma \sqrt{T-t}})=P(Y \leq d_2)=N(d_2)$$

where $N$ is the cdf of a standard normal variable.

Finally,

$$V_t=E_t\left[e^{-r(T-t)}H 1_{\{S_T \geq K\}}\right]=e^{-r(T-t)}H*N(d_2)$$

Answer 2

$N\left(d_2\right)$ is the risk-neutral probability that the spot is greater than the strike at maturity, therefore the RN probability that you get your payoff.

Answer 3

Recall that the price of your contract is \begin{align*} V_t = e^{-r(T-t)} \mathbb{E}^\mathbb{Q} [H1_{\{S_T>K\}}|\mathcal{F}_t] \end{align*} because your option always pays $H$ if $S_T>K$. Next, \begin{align*} V_t &=He^{-r(T-t)} \mathbb{E}^\mathbb{Q} [1_{\{S_T>K\}}|\mathcal{F}_t] \\ &= He^{-r(T-t)} \mathbb{Q} [{\{S_T>K\}}|\mathcal{F}_t] \\ &= He^{-r(T-t)} N(d_2) \end{align*} The fact that the option price equals the discounted (conditional) expectation of the payoff is linked with no-arbitrage via the fundamental theorem of asset pricing.

You can compute the probability of $\{S_T>K\}$ under the Black-Scholes model to obtain $N(d_2)$.