Pricing a Two-Date Asset-or-Nothing Binary Option

# Expectation of N(d2)?


# Expectation of N(d2)?












I am trying to find out the Pricing Equation for certain type of Options under Risk-Neutral pricing. This is the equation I am getting, but I am not sure if this can be solved or not. Any help is appreciated.

$$V = E[I\{S(T_0) \geq B\}N(d_2)]$$

$t_0 < T_0 < T_1$ This is a time line

$$I\{S(T_0) \geq B\} = \begin{cases} 1, & \text{if } S(T_0) \geq B \\ 0, & \text{otherwise} \end{cases}$$

where $S(t_0), S(T_0)$ is the stock price at different times.

$N(d_2)$ is the Black Scholes $N(d_2)$ but Stock Price used in $N(d_2)$ is $S(T_0)$, and time period is $T_1-T_0$. So $N(d_2)$ in itself is a Random Variable in this case.

I am trying to find the Expectation at time $= t_0$

$ d_2 = [ln(S(T_0)/K)+(r-0.5vol^2)(T_1-T_0)] / (vol* sqrt (T_1-T_0)) $

$ S(T_0) = S(t_0) exp((r-0.5vol^2)(T_0-t_0) + vol * sqrt(T_0-t_0)Z) $

Z~N(0,1)

## Answer by LocalVolatility (score 2, accepted)

https://quant.stackexchange.com/a/33730

You are essentially interested in pricing a second order bond-binary option. In its most general form, this contract has a time $T_2$ payoff of

\begin{equation} \mathcal{B}_{\xi_1, \xi_2}^{s_1, s_2} \left( S_{T_1}, S_{T_2}, T_2 \right) = \mathrm{1} \left\{ s_1 S_{T_1} > s_1 \xi_1 \right\} \mathrm{1} \left\{ s_2 S_{T_2} > s_2 \xi_2 \right\}. \end{equation}

In your case, we have $\xi_1 = B$, $s_1 = +1$, $\xi_2 = K$ and $s_2 = +1$. The time $T_1$ value of this option is equal to

\begin{equation} \mathcal{B}_{\xi_1, \xi_2}^{s_1, s_2} \left( S_{T_1}, T_1 \right) = \mathrm{1} \left\{ s_1 S_{T_1} > s_1 \xi_1 \right\} e^{-r \left( T_2 - T_1 \right)} \mathbb{E} \left[ \left. \mathrm{1} \left\{ s_2 S_{T_2} > s_2 \xi_2 \right\} \right| S_{T_1} \right]. \end{equation}

Apart from the additional discounting, this is the same expression as in your question.

This contract is a special case of the generalize multi-period and multi-asset $\mathbb{M}$-binary options analyzed by Skipper and Buchen (2003). Its time $0 \leq t < T_1$ value is given by

\begin{equation} \mathcal{B}_{\xi_1, \xi_2}^{s_1, s_2} \left( S_t, t \right) = e^{-r \tau_2} \mathcal{N}_2 \left( \alpha_{0, 1}, \alpha_{0, 2}; \rho \right), \end{equation}

where $\tau_i = T_i - t$,

\begin{equation} \alpha_{0, i} = \frac{s_i}{\sigma \sqrt{\tau_i}} \left( \ln \left( \frac{S}{\xi_i} \right) + \left( r - \frac{1}{2} \sigma^2 \right) \tau_i \right) \end{equation}

and

\begin{equation} \rho = s_1 s_2 \sqrt{\frac{\tau_1}{\tau_2}}. \end{equation}

Here, $\mathcal{N}_2$ is the bivariate standard normal distribution function with the given correlation. See the original paper for a derivation of this result. You find a similar result in Chapter 2 of the Ph.D. thesis Veiga (2010). See also this related question and the corresponding answers.

References

Skipper, Max and Peter W. Buchen (2003) "The Quintessiential Option Pricing Formula", Working Paper, School of Mathematics and Statistics, University of Sydney, available online

Veiga, Carlos Manuel "Closed Formulas and Rating Schemes for Derivatives", Ph.D. Thesis, Frankfurt School of Finance & Management, available online