Digital Options

Digital options come up in work such as pricing a digital embedded in a structured note, or selling a digital close to expiry and hedging it with the underlying.

A fixed amount at expiry, or nothing

A digital option is an option whose payoff depends only on whether the price of the underlying asset at expiry is above the strike price. It is also called a binary option. If the price is above the strike price at expiry, it pays a fixed amount, and otherwise nothing. The amount stays the same however far the price ends above the strike price¹.

An option that pays a fixed amount of cash is called cash-or-nothing, and one that pays the value of the underlying asset itself at expiry is called asset-or-nothing. Both come as a call, which pays when the price at expiry is above the strike price, and a put, which pays when it is below². Unless stated otherwise, this page means cash-or-nothing and looks at a European option, whose condition is checked only at expiry.

Consider a hypothetical example in which a cash-or-nothing option has a current underlying price of 100 yen, a strike price of 100 yen, and a payout of 10 yen at expiry. The table shows the payoff for each price at expiry. For comparison, it also shows the payoff of a standard call option with the same strike price (a vanilla call below).

The payoff of the vanilla call grows with the price at expiry, but the digital call jumps from 0 to 10 yen once the price passes the strike price and grows no further. If you hold both a digital call and a digital put with the same payout and strike price, you receive 10 yen whatever the price at expiry³. Sources differ on what happens when the price at expiry exactly equals the strike price. The simulator pays when the price is "above" the strike⁴.

Asset-or-nothing options and vanilla options

An asset-or-nothing call pays an amount equal to the price of the underlying at expiry if that price is above the strike price. Let $S_T$ be the price at expiry and $K$ the strike price, and write the payoff of a cash-or-nothing call with a payout of 1 as $\mathbf{1}_{\{S_T > K\}}$ (1 if the price is above the strike price, 0 otherwise). The payoff of an asset-or-nothing call is then $S_T \mathbf{1}_{\{S_T > K\}}$².

The payoff of a vanilla call at expiry, $\max(S_T - K, 0)$, splits into these two as follows⁵.

$$\max(S_T - K, 0) = S_T \mathbf{1}_{\{S_T > K\}} - K \mathbf{1}_{\{S_T > K\}}$$

The first term on the right is one unit of an asset-or-nothing call, and the second is $K$ units of a cash-or-nothing call with a payout of 1. If the price at expiry is 110 yen, the asset-or-nothing call pays 110 yen, the cash-or-nothing part takes away 100 yen, and the 10 yen left matches the payoff of the vanilla call. If the price at expiry is 100 yen or less, both are 0 yen.

The price and what moves it

Under the assumptions of the Black-Scholes model (the price of the underlying follows geometric Brownian motion, and interest rates and volatility are constant), the current price of a cash-or-nothing call with a payout of $Q$ is given by the following formula⁶. Here $r$ is the interest rate (continuously compounded), $T$ is the time to expiry in years, and $N$ is the cumulative distribution function of the standard normal distribution. The model's assumptions and the derivation of the formula are covered in The Black-Scholes Model.

$$C_{\text{digital}} = Q e^{-rT} N(d_2)$$

With $S$ as the current price of the underlying and $\sigma$ as the volatility, $d_2$ is as follows (with no dividends).

$$d_2 = \frac{\ln(S/K) + (r - \sigma^2/2)T}{\sigma\sqrt{T}}$$

The put formula replaces $N(d_2)$ with $N(-d_2)$, and the call and put prices add up to $Q e^{-rT}$, the payout discounted from expiry⁷. The price of an asset-or-nothing call with no dividends is $S N(d_1)$, where $d_1 = d_2 + \sigma\sqrt{T}$⁸.

Take the first example (current price 100 yen, strike price 100 yen, payout 10 yen), with 1 year to expiry, an interest rate of 1%, and volatility of 20%. The digital call is priced at 4.75 yen and the digital put at 5.15 yen, and their sum is 9.90 yen, which is 10 yen discounted for 1 year. Under the same conditions, the asset-or-nothing call is worth 55.96 yen. A position of 100 units of a cash-or-nothing call with a payout of 1 yen is worth 47.53 yen. The difference, 8.43 yen, equals the Black-Scholes price of the vanilla call⁹.

The price of a digital call rises as the price of the underlying rises and approaches the discounted payout. Volatility, however, can push the price either way, depending on the price of the underlying and the level of volatility. A vanilla option becomes more expensive when volatility rises, but a digital call behaves as in the following table (strike price 100 yen, 1 year to expiry, interest rate 1%, payout 10 yen)⁹.

Within the table, when the current price is 80 or 90 yen, well below the strike price, higher volatility makes it more likely that the price ends above the strike price at expiry, and the price of the digital call rises. When the current price is 100 to 120 yen, higher volatility makes it more likely that the price ends below the strike price, and the price of the digital call falls. So the buyer of a digital call gains from a rise in volatility in the 80 and 90 yen rows and loses in the 100 to 120 yen rows. The seller's profit and loss is the reverse.

The point where the direction flips is not exactly at the strike price, and it also moves with the level of volatility. Under the same conditions, at a current price of 99 yen, raising volatility from 10% to 20% lowers the price from 4.75 to 4.55 yen. At 98 yen, raising it from 20% to 30% lowers the price from 4.36 to 4.23 yen. Even at 90 yen, raising it from 30% to 80% lowers the price from 3.17 to 2.99 yen⁹. Whether the buyer of a digital call gains or loses from a rise in volatility therefore depends on both the current price and the level of volatility.

Deep diveThe volatility smile and the price of a digital option

The prices above use the same volatility for every strike price. In actual markets, implied volatility differs by strike price (the smile or skew). A digital call is the limit of the difference between two vanilla calls with nearby strike prices, so its price is the slope of the vanilla call price with respect to the strike price, with the sign reversed¹⁰.

$$D(K) = -\frac{\partial C(K)}{\partial K}$$

From this formula, when volatility differs by strike price, the price of a digital call equals the Black-Scholes price with constant volatility minus "vega × skew". Vega is the sensitivity of the vanilla call price to volatility, and skew is the slope of volatility with respect to the strike price¹¹.

The Columbia University lecture note's example has a current price and strike price of 100, an interest rate of 0, 1 year to expiry, and a skew in which volatility falls 2.5 points for every 10% rise in the strike price. The price using only the 25% volatility at the strike price is 45% of the payout, but it becomes 55% once the skew is included¹¹.

With the simulator, you can price a cash-or-nothing option with the Monte Carlo method, which generates many paths for the price of the underlying and averages the payoffs, and compare the result with the formula above (the analytic price). The initial values match the pricing example above (current price 100, strike price 100, 1 year to expiry, interest rate 1%, volatility 20%, payout 10), and the analytic price is 4.75.

Approximating with a call spread

A digital call can be approximated by buying one vanilla call and selling another, with strike prices on either side of the digital's strike price (a call spread). For a digital call with a strike price of $K$ and a payout of $Q$, you buy a call with a lower strike price $K_-$, sell a call with a higher strike price $K_+$, and hold $Q/(K_+ - K_-)$ units of each¹². If the price at expiry is $K_-$ or less, both pay 0, and if it is $K_+$ or more, both pay $Q$. They differ only between $K_-$ and $K_+$. Buying the call with the lower strike price and selling the one with the higher strike price has the same shape as a bull call spread.

In the first example (strike price 100 yen, payout 10 yen), you buy one unit of a call with a strike price of 95 yen and sell one unit of a call with a strike price of 105 yen.

Figure description

Payoff at expiry of buying one unit of a call with a strike price of 95 yen and selling one unit of a call with a strike price of 105 yen. It does not include the premiums. The payoff is 0 yen when the price of the underlying at expiry is 95 yen or less, rises with the price between 95 and 105 yen, and stays at 10 yen at 105 yen or more. A digital call with a strike price of 100 yen and a payout of 10 yen pays 0 yen at 100 yen or less and 10 yen above 100 yen, so the two payoffs differ only between 95 and 105 yen.

The prices before expiry are close too. Under the same conditions as the pricing example (1 year to expiry, interest rate 1%, volatility 20%), the call spread with strikes of 95 and 105 yen is priced at 4.76 yen, 0.01 yen away from the digital call's 4.75 yen. Narrowing the gap to calls with strikes of 99 and 101 yen, 5 units each, gives 4.75 yen⁹. In the limit as the gap goes to 0, the price matches the digital call's¹⁰.

Where you place the two strike prices decides which way the approximation misses the digital. With the lower strike at 90 yen and the higher strike at 100 yen, the same as the digital's, the payoff at expiry is at least the digital's at every price (an approximation from above), and the price is 5.76 yen. With the lower strike at 100 yen, the same as the digital's, and the higher strike at 110 yen, the payoff is at most the digital's (an approximation from below), and the price is 3.82 yen¹³⁹. If the seller of a digital call holds a call spread that approximates it from above, the call spread's payoff covers the digital's payment whatever the price at expiry. In return, the seller buys a call spread that costs more than the digital.

Why hedging near the strike price is hard

The difficulty of hedging a digital option by trading the underlying asset lies in the size of its delta (the change in price when the price of the underlying moves 1 yen) when expiry is near and the price of the underlying is close to the strike price. The delta of a cash-or-nothing call is given by the following formula¹⁴, where $\phi$ is the density function of the standard normal distribution.

$$\Delta_{\text{digital}} = \frac{Q e^{-rT} \phi(d_2)}{S \sigma \sqrt{T}}$$

Because $\sqrt{T}$ is in the denominator, the delta grows without limit as expiry approaches while the price of the underlying stays at the strike price. When the price is away from the strike price, the delta goes toward 0 as expiry approaches¹⁵. Under the example's conditions (strike price 100 yen, payout 10 yen, interest rate 1%, volatility 20%), the delta at a current price of 100 yen is as follows⁹.

One day before expiry, the delta of the digital call is 3.15, so a 1 yen move of the underlying across the strike price shifts the value by about 30% of the 10 yen payout. At the same point, the value is 0.55 yen if the price of the underlying is 98 yen and 9.41 yen if it is 102 yen. Once the price moves as far as 95 or 105 yen, the delta is almost 0⁹. A hedge that keeps matching the delta by trading the underlying therefore has to buy or sell large quantities each time the price crosses back and forth over the strike price. The idea of keeping the delta matched by trading the underlying is covered in Delta Hedging.

Replacing the digital with a call spread dampens these swings. The call spread with strikes of 95 and 105 yen holds one unit of each call, so its delta is the difference between the deltas of the two calls (each between 0 and 1) and never exceeds 1. The cap on the delta is set by the quantity $Q/(K_+ - K_-)$, and it rises as the gap narrows to bring the approximation closer. Approximating with a call spread smooths the shape of the payoff and avoids huge (potentially infinite) delta numbers. On the other hand, near the strike price the approximation and the digital do not pay the same. If required, the strikes can be chosen to approximate from above¹³.

What digital options are used for

Digital options are used as building blocks that shape the payoff of structured notes and similar products. One example is the range note (accrual note), which accrues interest at a high rate while an index such as an interest rate or exchange rate stays within a set range, and at a low rate (sometimes zero) when it moves outside. An investor who buys such a note has in effect sold a series of digital calls struck at the top of the range and digital puts struck at the bottom¹⁷.

From the investor's side, the note can work out well or badly. If the index stays within the range, the investor receives interest at a higher rate than a bond with similar terms, and the narrower the range, the larger this extra amount. When the Federal Reserve Board's manual was written in 1998, most notes checked the index daily, but some checked it monthly, quarterly, or only once before maturity. For notes that check the index daily, interest accrues at the low rate for each day the index is outside the range. The interest drops because the index is outside the range, regardless of how far outside it is¹⁷.

The buyer of a digital option pays the premium up front whether or not the payout is received, and if the condition is not met at expiry, the premium becomes a loss.

OTC trading and the difference from barrier options

Many exotic options, including digital options, trade as OTC derivatives and are less liquid than standard options. A position cannot always be closed out at the quoted price¹⁸.

A digital option looks only at whether the price at expiry is above the strike price. In contrast, a barrier option is an option that disappears or comes into existence depending on whether the price of the underlying touches a preset level (the barrier) before expiry¹⁸.

Notes

1. Board of Governors of the Federal Reserve System, "Trading and Capital-Markets Activities Manual," Section 4330.1 "Options," "Exotic Options" (February 1998 edition, page 2). ↩

2. Durham University, "Chapter 5 Pricing options from around the world (Mathematical Finance Lecture Notes 2024-25)," Section 5.2 "Popular options." The notes define the payoffs of cash-or-nothing and asset-or-nothing options using indicator functions. ↩ ↩²

3. OpenGamma Quantitative Research, "Digital Forex Options" (May 2012, Version 1.1), Section 1 "Introduction." It says that holding a call and a put with the same strike, dates, and payout gives a fixed amount in every case, the same as a zero-coupon bond. ↩

4. The Durham University lecture notes (Section 5.2) pay when the price is "above" the strike, and OpenGamma, "Digital Forex Options," Section 1, pays at the strike "or above." Higham, "Chapter Seventeen. Cash-or-nothing options," 17.1, has the call and the put each pay half the payout when the price exactly equals the strike. ↩ ↩²

5. A rewriting of the vanilla call payoff using the payoff formulas for cash-or-nothing and asset-or-nothing calls in the Durham University lecture notes (Section 5.2). When the price at expiry is above the strike price, both sides equal $S_T - K$, and otherwise both are 0. ↩

6. OpenGamma, "Digital Forex Options," Section 2 "Pricing in Garman-Kohlhagen model." The paper uses the FX formula and writes $d_-$ with the forward price $F$. For a stock with no dividends, $F = S e^{rT}$, and the paper's $d_-$ corresponds to $d_2$ on this page. The paper's discount factor $P_D(0, t_p)$ corresponds to $e^{-rT}$ on this page (when the payment date is the same as expiry). ↩

7. Desmond J. Higham, "Chapter Seventeen. Cash-or-nothing options," 17.1. It uses a no-arbitrage argument to show that the call and put prices add up to the discounted payout. ↩

8. Desmond J. Higham, "Chapter Seventeen. Cash-or-nothing options," 17.7. It derives $S N(d_1)$ as the price of an asset-or-nothing call with no dividends. ↩

9. Prices and deltas are computed with the same calculations as this page's simulator (the analytic price in src/lib/simulators/monte-carlo.ts and the vanilla option price in src/lib/simulators/black-scholes.ts) and rounded to two decimal places. Deltas are computed from the change in price when the price of the underlying is moved slightly up and down. ↩ ↩² ↩³ ↩⁴ ↩⁵ ↩⁶ ↩⁷ ↩⁸

10. Martin Haugh, Columbia University, "Black-Scholes and the Volatility Surface" (2013), Section 6 "Pricing Exotics," "Pricing a Digital Option." It shows the digital call price as the limit of the call spread price divided by the strike gap. ↩ ↩²

11. Haugh, same note, Section 6 "Pricing a Digital Option" and Example 3 "Pricing a digital." The skew is a slope in which volatility changes by 2.5 points for every 10% change in the strike. ↩ ↩²

12. OpenGamma, "Digital Forex Options," Section 3 "Call spread approximation." The paper gives the example $K_\pm = K(1 \pm \epsilon)$ and sets the quantity at $N/(K_+ - K_-)$. ↩

13. OpenGamma, "Digital Forex Options," Section 3 "Call spread approximation," from the last paragraph of page 2 to the first paragraph of page 3. It describes placing the strikes to approximate from above (which the paper calls super-replication) and from below. ↩ ↩²

14. OpenGamma, "Digital Forex Options," the delta formula in Section 2 "Pricing in Garman-Kohlhagen model." The paper's formula is rewritten in the notation for a stock with no dividends ($\phi$ stands for the paper's $\frac{1}{\sqrt{2\pi}}\exp(-\frac{1}{2}d_-^2)$). ↩

15. Desmond J. Higham, "Chapter Seventeen. Cash-or-nothing options," 17.5. It shows that as expiry approaches, the delta grows without limit if the price of the underlying equals the strike price, and goes to 0 if the price is above or below it. ↩

16. OpenGamma, "Digital Forex Options," Section 4 "Risk figures." ↩

17. The same manual, Section 4040.1 "Structured Notes," "Range Notes" (page dated February 1998, pages 4–5). ↩ ↩²

18. The same manual, Section 4330.1 "Options," "Exotic Options" (page dated February 1998, page 2). ↩ ↩²

19. OpenGamma, "Digital Forex Options," Section 1 "Introduction" and Section 3 "Call spread approximation." ↩