Fair value of a binary cash-or-nothing option with a barrier

Question

I want to find the fair value of a European cash-or-nothing option that pays \$1 if $S_t>K$ and $S$ breached the level $M<0<K$, where $S$ is the risk-neutral process $dS_t=\sigma dW_t$.

My idea is to define a first passage time $\tau$ to level $M$ (since the other condition must be met anyway to get \$1 at time $T$) $(\tau=\min\{t;S_t=M\})$ and use the reflection principle of the Brownian motion to determine the probability density of $\tau$.

Integrating both sides of the SDE we find its solution $S_t=S_0+\sigma W_t$. Then, we applying the reflection principle and change of variable in integration $\nu=w/\sqrt{t} \Rightarrow d\nu=dw/ \sqrt{t}$:

\begin{align*} \mathbb{P}(\tau\leq t)&=\mathbb{P}(\tau\leq t,S_t\geq M)+\mathbb{P}(\tau\leq t,S_t\leq M) \\ & = 2\mathbb{P}(\tau\leq t,S_t\geq M) \\ &=2\mathbb{P}(S_t\geq M) \\ & = 2\int_{M}^{\infty}\frac{1}{\sqrt{2\pi t}}e^{-w^2/2t}dw \\ & = 2\int_{M/\sqrt{t}}^{\infty}\frac{1}{\sqrt{2\pi t}}e^{-\nu^2/2}d\nu \\ & = 2-2\Phi\left(\frac{M}{\sqrt{t}}\right) \end{align*}

The fair value of a standard cash-or-nothing option is $\mathbb{E}^\mathbb{Q}[\mathbb{I}_{\{S_t>K\}}]$. In this case, I think that we need to multiply that by $\mathbb{P}(\tau\leq t)$, i.e. the price of the cash-or-nothing option with barrier is:

$$\mathbb{E}^\mathbb{Q}[\mathbb{I}_{\{S_t>K\}}]\times\mathbb{P}(\tau\leq t)$$ Do you think this is correct?

Answer 1

As Daneel mentioned in his comment, you can't simply split your expectation of product into a product of two expecations as the two quantities are far from being independent...

Now, to answer your question w.r.t. how you could compute the expectation of the joint event of being in the money while having hit the barrier, you were right in using the reflexion principle. But I'd say you weren't ambitious enough :p

From the question, it appears your barrier is a down and in barrier.

Try using the reflexion principle to determine the joint law of $(S_t, \min_{s \leq t} S_s)$. Looking at the derivation you did in your question, this should be easy for you.

After you have done that, you can simply express the quantity that you want, which is: $$\mathbb{Q}\left( S_t>K, \min_{s \leq t} S_s < M \right)$$