Conditional Probability - Geometric Brownian Motion

Question

Background

I am trying to find a way to price a variant of a gap option by using closed-end expressions. What makes this option a bit tricky is that it can be exercised at four predetermined dates (t=1, 2, 3, 4) and that the strike/the barrier to exercise, H, increases for each period. On the other hand, the strike determining the size of the payoff, K, is so small compared to the value of the underlying asset that it will always be optimal to exercise early, which somewhat simplifies the pricing.

Question

In this connection, I need to estimate the (risk-neutral) probability of the option being in-the-money at t=2 assuming it is out-of-the-money at t=1, and further estimate the probability of the option being in-the-money at t=3 assuming it is out-of-the-money at both t=1 and t=2 etc. Assuming that the underlying asset follows a geometric Brownian motion where it’s price at t can be written as:

$S_t = S_0e^{(\alpha-\delta-0.5\sigma^2)t+\sigma\sqrt{t}z}$

By using the Black-Scholes formula, the probability of the option being in-the-money can be estimated as N(d2).

How can I estimate the probability of the option being in-the-money at t=3, assuming it is out-of-the-money at t=1 and t=2? And further estimate the probability of the option being in-the-money at t=4 assuming it is out-of-the-money at t=1, t=2 and t=3?

Answer 1

One way to obtain many multi-period risk-neutral probabilities related to geometric Brownian motion processes is to use the valuation function for higher-order binaries. These contracts are special cases of the multi-asset multi-period $\mathbb{M}$-binaries introduced by Skipper and Buchen (2003)

Definition

The time $T_n$ terminal value of a $n$-th order bond binary is given by

\begin{equation} \mathcal{B}_{\xi_1, \xi_2, \ldots, \xi_n}^{s_1, s_2, \ldots, s_n} \left( S_{T_1}, S_{T_2}, \ldots, S_{T_n}, T_n \right) = \prod_{i = 1}^n \mathrm{1} \left\{ s_i S_{T_i} > s_i \xi_i \right\}. \end{equation}

I.e. this contract has a unit payoff conditional on the asset prices at all times $T_i$ being above ($s_i = 1$) or below ($s_i = -1$) the levels $\xi_i$, respectively. Its time $0 \leq t < T_1$ value is given by

\begin{equation} \mathcal{B}_{\xi_1, \xi_2, \ldots, \xi_n}^{s_1, s_2, \ldots, s_n} \left( S_t, t \right) = e^{-r \tau_n} \mathcal{N}_n \left( \mathbf{\alpha}_-; \mathbf{C} \right), \end{equation}

where $\tau_i = T_i - t$ and $N_n(\mathbf{x}; \mathbf{C})$ is the $n$-variate standard normal cumulative distribution function evaluated at $\mathbf{x}$ and with correlation matrix $\mathbf{C}$. The elements of $\mathbf{\alpha}_-$ are given by

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

and

\begin{equation} \mathbf{C}_{i, j} = s_i s_j \sqrt{\frac{\min \{ s_i, s_j \}}{\max \{ s_i, s_j \}}}. \end{equation}

Joint Probability

Using these results, you obtain the joint probability of $S_{T_n}$ being above $K_n$ and each $S_{T_i}$ being below $K_i$ for $i \in \{ 1, 2, \ldots, n - 1 \}$ as

\begin{equation} \mathbb{P} \left\{ S_{T_n} > K_n, S_{T_i} < K_i \; \forall \; i < n \right\} = e^{r \tau_n} \mathcal{B}_{K_1, K_2, \ldots, K_n}^{-, -, \ldots, +} \left( S_t, t \right). \end{equation}

From this, you can then compute the conditional probability.

References

Buchen, Peter W. (2001) "Image Options and the Road to Barriers", Risk Magazine, Vol. 14, No. 9, pp. 127-130

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