Digital options

I recently came across the digital option—while reading Volatility: Practical Options Theory—A.Iqbal—and it really peaked my curiosity! So, I thought I would write about it in an attempt to grasp my understanding of it.

Delta as the probability of an ITM expiry

To begin, we need to build from the concept of delta—the rate of change of an option-price ($V_t$) relative to it's underlying ($S_t$)—a greek (options) symbol that represents the partial-derivative of $S_t$ in the function for pricing an option:

$$ \Delta \equiv \Delta(S_t, t, \sigma_i) = \frac{\partial V(S_t, t, \sigma_i)}{\partial S} $$

As a first-order derivative of the underlying, delta is fundamental in understanding how an option is priced so let's quickly outline some of it's most important qualities:

• It not only represents the change in an option price but also the probability in which that option price will expire in-the-money (ITM) where $\Delta = \text{Prob}(S_T > K)$. (todo: write about how/why this is the case)
• It's PDF follows a normal-distribution $~N(\sigma, \mu)$ where $\sigma$ represents volatility and $\mu$ is the current price of the underlying.
• It's CDF is the indicator for the probability of the price given at expiry
• Delta is greatest at $K$ which is at-the-money (ATM) where $\Delta \approx 0.5$
  • In other words, the highest point of the PDF curve
  • This point is also where the gradient of the delta partial-derivative function is steepest
  • The deeper OTM or ITM an option falls within, the smaller the gradient of this function
• It can be positive or negative, ranging from $[-1, 0]$ for puts and $[0, 1]$ for calls
  • Closer to $|1|$ indicates a very high probability that an option will expire ITM
  • Closer to $0$ indicates a very high probability that an option will expire OTM

Most of these points can be nicely summarised in the figure below, where the the orange line illustrates the PDF of a call ATM and the yellow line represents the rate of change that delta takes-on as the price of the underlying changes.

An useful point-of-view for practical options trading is to think of $\Delta_t$ as the time $t$ conditional probability that an option expires ITM, a fast and intuitive way to (approximately) identify relative value opportunities in options.

The Digital option

This brings us to the digital (or binary) option, which is a beautiful illustration of how delta represents this probability ($S_T > K$) and gives the trader an opportunity to directly trade that probability in a binary fashion using our option value function at time $t$:

$$ V_t = \mathbb{E}_t[V_T] = \mathbb{E}_t[\max(S_T - K, 0)] $$

i.e. the price of an option today is equal to the market's expectation of the payoff at time $T$. By applying the above equation to the price of strike $K$ by our digital ($D_t$) we can see that:

$$ D_t = \mathbb{E}_t[1_{S_T > K}] = \text{Prob}_t(S_T > K) $$

In words, the price of the digital is exactly the probability that a vanilla call option with strike K expires ITM. Leaving us with the resulting binary pay-off diagram.

The Unit call-spread

Now that we understand what a digital option is, how can we trade such a product? The simplest way is to synthesise a unit call-spread, simply put, a unit call-spread can be manufactured by taking our strike $(K)$ and both purchasing / selling a call-option around that strike to given delta $(\delta)$.

The payoff profile generated by this synthetic (purchasing a call at $K - \delta$ and selling a call at $K + \delta$) is demonstrated below. Notice, that as our $\delta$ becomes smaller, it converges to the payoff profile of the digital presented earlier.

Suppose instead that the notional is set to ${1}/{2\delta}$. In this case the payoff would be shown in our diagram above using the largest delta (in red). We can see that the payoff of the call spread is not too dissimilar to that of the digital in that, above 1.39 both the digital and the call spread pay 1 USD and below 1.35 they both pay nothing, the payoff is capped at 1 USD. Pricing a unit call-spread can be summarised as follows:

$$ \frac{1}{2\delta}(V(S_t, K - \delta, t, \sigma_i) - V(S_t, K + \delta, t, \sigma_i)) $$

Hedging digital exposures

We'll get there eventually...