Barrier digital options and pricing

What do you call options which behave like barrier options but for a digital option?

That is, given $0 < t < T$, then if $S_t > K_t$, the binary option $B(K_T,T)$ comes into play, i.e. which pays out 1 unit if $S_T > K_T$. If however $S_t < K_t$, then we get nothing ,and of coruse if $S_t > K_t$but $S_T < K_T$, then we get nothing as well.

What are these options called, and where can I find their prices? For example, in a Black Scholes setting?

• I guess they are called barrier digital options. They are path-dependent and for this reason there isn't a closed form solution. You can use Monte-Carlo or Binomial trees. Alternatively you can approximate a digital option as a N call spreads, with N big. – NSZ Mar 23 '17 at 10:44
• @NSZ: I disagree with your statement that these options don't have a closed-form solution in the Black-Scholes framework. These options are one of the two elementary building blocks of barrier options with a plain vanilla payoff (which is widely known to have a closed-form solution). The other component is a barrier option on an asset-or-nothing payoff. – LocalVolatility Mar 23 '17 at 10:56
• Do you have some references? If not, then perhaps prices can be obtained in a simpler continuous framework (e.g., arithmetic brownian motion)? – Tony Mar 23 '17 at 11:00
• Sounds to me like an up-and-in binary (or digital) option. – Quantuple Mar 23 '17 at 11:17
• @Quantuple: This wasn't fully clear to me from the question. I read this as a down-and-out call. But maybe Tony can clarify this. In any case - the framework that I referenced can be used for this. – LocalVolatility Mar 23 '17 at 11:18

Disclaimer: This answer derives the prices of two different binary options within the Black/Scholes framework. Note that this is not an appropriate valuation model to use for non-European contracts in most real-world markets.

Up-and-In Binary Call

After reading your question for a second time, I agree with Quantuple's comment that you seem to be looking for the solution to an up-and-in binary call option.

Formally, let

$$\nu = \inf \left\{ t \in \mathbb{R}_+ : S_t \geq K \right\}$$

be the first hitting time of $S$ to the strike $K$. The option has a unit payoff conditional on $\nu \leq T$ and $S_T \geq K$, i.e.

$$V_T = \mathrm{1} \left\{ S_T \geq K \right\} \mathrm{1} \left\{ \nu \leq T \right\}.$$

Note however that $S_T \geq K \; \Rightarrow \; \nu \leq T$ and thus $\left\{ S_T \geq K \right\} \subseteq \left\{ \nu \leq T \right\}$. Consequently, we can skip the second indicator and your payoff is just

$$V_T = \mathrm{1} \left\{ S_T > K \right\}.$$

I.e. the price of an up-and-in binary call option is the same of that of a normal binary call option. You thus have the standard result that

$$V_0 = e^{-r T} \mathcal{N} \left( d_- \right),$$

where

$$d_- = \frac{1}{\sigma \sqrt{T}} \left( \ln \left( \frac{S_0}{K} \right) + \left( r - \frac{1}{2} \sigma^2 \right) T \right).$$

Down-and-Out Binary Call

A more interesting case is the down-and-out binary call. This is how I initially understood your question. Now let

$$\nu = \inf \left\{ t \in \mathbb{R}_+ : S_t \leq K \right\}$$

and

$$V_T = \mathrm{1} \left\{ S_T \geq K \right\} \mathrm{1} \left\{ \nu > T \right\}.$$

This option knocks out, should the spot price breach the barrier before maturity. Otherwise it has a digital payoff of one.

Let $\tau = T - t$ be the time-to-maturity. The valuation function $\tilde{V}(S, \tau)$ of this option satisfies the initial boundary value problem

\begin{eqnarray} \mathcal{L} \left\{ \tilde{V} \right\} (S, \tau) & = & 0 \qquad (S, \tau) \in \mathcal{D},\\ \tilde{V}(K, \tau) & = & 0, \qquad \forall \tau \in \mathbb{R}_+\\ \tilde{V}(S, 0) & = & \mathrm{1} \{ S \geq K \}, \end{eqnarray}

where $\mathcal{L}$ is the Black/Scholes forward operator and $\mathcal{D} = \left\{ (S, \tau): S > K, \tau \in \mathbb{R}_+ \right\}$. Using the method of images, see e.g. Buchen (2001), the solution can be shown to be

$$\tilde{V}(S, \tau) = \mathcal{B}_K^+(S, \tau) - \stackrel{K}{\mathcal{I}} \left\{ \mathcal{B}_K^+(S, \tau) \right\},$$

where

\begin{eqnarray} \mathcal{B}_K^+ (S, \tau) & = & e^{-r \tau} \mathcal{N} \left( d_- \right),\\ d_- & = & \frac{1}{\sigma \sqrt{\tau}} \left( \ln \left( \frac{S}{K} \right) + \left( r - \frac{1}{2} \sigma^2 \right) \tau \right),\\ \stackrel{K}{\mathcal{I}} \left\{ \mathcal{B}_K^+ (S, \tau) \right\} & = & \left( \frac{S}{K} \right)^{2 \alpha} \mathcal{B}_K^+ \left( \frac{K^2}{S}, \tau \right),\\ \alpha & = & \frac{1}{2} - \frac{r}{\sigma^2}. \end{eqnarray}

References

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

• How do you pick the bs vol to use though? – will May 23 '17 at 22:57
• That's a valid though very different question. Tony was asking about the valuation formula assuming that the underlying follows a GBM. Since this model only poorly reflects observable market prices and dynamics, I wouldn't recommend using it in the first place to value any non-European payoff. – LocalVolatility May 25 '17 at 9:47
• Fair, thought they do ask where they can find their pricing - yeah they say for example in a BS setting, but i think you should definitely note in the answer that this is not a valid way to value barrier options, indeed even local vol isn't correct. – will May 25 '17 at 9:58