Exotic Options and Complex Derivatives › Non-Path-Dependent Exotics

Not all exotic options depend on the path of prices. Some have modified payoff structures at expiration or involve options on options.

Digital options (also called binary options) have discontinuous payoffs. They pay a fixed amount if the underlying finishes above (for a call) or below (for a put) the strike price, and zero otherwise.

The defining characteristic of digital options is their all-or-nothing nature. Unlike vanilla options where the payoff scales with how far in-the-money the option finishes, digital options pay the full amount for any positive intrinsic value, no matter how small. This creates a discontinuity at the strike. One cent in-the-money, you receive the full payout. One cent out-of-the-money, you receive nothing.

For a cash-or-nothing digital call paying amount $Q$:

$$\text{Payoff} = \begin{cases} Q & \text{if } S_T > K \\ 0 & \text{otherwise} \end{cases}$$

where $Q$ is the fixed cash payout amount, $S_T$ is the underlying asset price at expiration, and $K$ is the strike price.

Under risk-neutral pricing, the value is:

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

where:

• $S_0$: current stock price
• $K$: strike price
• $Q$: cash payout amount
• $\sigma$: volatility
• $r$: risk-free rate
• $T$: time to expiration
• $d_2 = \frac{\ln(S_0/K) + (r - \sigma^2/2)T}{\sigma\sqrt{T}}$: distance to strike in standard deviations (risk-adjusted)
• $N(d_2)$: risk-neutral probability that the option finishes in-the-money

This is simply the present value of the cash amount times the risk-neutral probability of finishing in-the-money.

This formula has a direct probabilistic interpretation. The term $N(d_2)$ represents the probability, under the risk-neutral measure, that the stock price will exceed the strike at expiration. This is the same $N(d_2)$ that appears in the Black-Scholes formula, but here it stands alone rather than being multiplied by other terms. The option value is the payout amount ($Q$), discounted to present value ($e^{-rT}$), times the probability of receiving it ($N(d_2)$). No other option pricing formula offers such a transparent connection between probability and price.

For an asset-or-nothing digital call that pays the asset itself if $S_T > K$:

$$C_{\text{asset-digital}} = S_0 N(d_1)$$

where:

• $S_0$: current stock price
• $K$: strike price
• $r$: risk-free rate
• $T$: time to expiration
• $\sigma$: volatility
• $d_1 = \frac{\ln(S_0/K) + (r + \sigma^2/2)T}{\sigma\sqrt{T}}$: standardized distance measure
• $N(d_1)$: factor adjusting for the conditional expectation of the asset price

The asset-or-nothing formula uses $N(d_1)$ rather than $N(d_2)$, and this difference is mathematically significant. While $N(d_2)$ gives the probability of finishing in-the-money, $N(d_1)$ incorporates an additional adjustment that accounts for the correlation between the probability of exercise and the value received upon exercise. When the stock price is high (making in-the-money more likely), the asset payment is also worth more. This positive correlation means we cannot simply multiply the expected asset value by the exercise probability. Instead, $N(d_1)$ represents the exercise probability under the "stock measure," which naturally accounts for this correlation.

This delta explosion near expiration creates practical hedging problems. A market maker selling digital options near the strike at short maturities faces the prospect of needing to trade enormous amounts of the underlying for small price movements, incurring substantial transaction costs.

Compound options are options on options, where the underlying asset is itself an option. This structure creates a two-stage exercise decision.

Compound options present interesting valuation challenges. At the first expiration date, the holder decides whether to pay the compound strike to acquire the underlying option. This decision depends on the underlying option's value at that time, which depends on the stock price. However, the underlying option retains time value and depends on how the stock price will evolve until the second expiration. This creates a chain of contingent decisions that must be valued simultaneously.

The four basic types are:

• Call on call (CoC): Right to buy a call option at a specified price
• Call on put (CoP): Right to buy a put option
• Put on call (PoC): Right to sell a call option
• Put on put (PoP): Right to sell a put option

Consider a call on call with compound strike $K_1$ expiring at $T_1$, written on an underlying call with strike $K_2$ expiring at $T_2$, where $T_2 > T_1$. At $T_1$, the holder of the compound option will exercise if the value of the underlying call exceeds $K_1$.

$$\text{Payoff at } T_1 = \max(C(S_{T_1}, K_2, T_2 - T_1) - K_1, 0)$$

where:

• $S_{T_1}$: asset price at compound option expiration $T_1$
• $C(\cdot)$: value of the underlying European call option
• $K_1$: strike price of the compound option (price to buy the underlying)
• $K_2$: strike price of the underlying option
• $T_1$: expiration of the compound option
• $T_2$: expiration of the underlying option

The critical stock price $S^*$ plays a central role in compound option pricing. It divides the state space at time $T_1$ into two regions: when $S_{T_1} > S^*$, the underlying call is worth more than $K_1$ and it makes sense to exercise the compound option; when $S_{T_1} < S^*$, the underlying call is worth less than $K_1$ and the compound option expires worthless. Finding $S^*$ requires solving a nonlinear equation, as the Black-Scholes formula gives option value as a complex function of the stock price. Fortunately, since call prices are monotonically increasing in the stock price, a unique solution always exists and can be found efficiently using root-finding algorithms.

The closed-form pricing formula for a call on call involves the bivariate normal distribution:

$$C_{\text{CoC}} = S_0 M(a_1, b_1; \rho) - K_2 e^{-rT_2} M(a_2, b_2; \rho) - K_1 e^{-rT_1} N(a_2)$$

where:

• $S_0$: current stock price
• $K_1$: strike price of the compound option
• $K_2$: strike price of the underlying option
• $T_1$: expiration of the compound option
• $T_2$: expiration of the underlying option
• $S^*$: critical stock price at $T_1$ where underlying option value equals $K_1$
• $r$: risk-free rate
• $\sigma$: volatility
• $M(a, b; \rho)$: bivariate standard normal cumulative distribution function
• $N(\cdot)$: cumulative standard normal distribution
• $\rho = \sqrt{T_1/T_2}$: correlation coefficient
• $a_1 = \frac{\ln(S_0/S^*) + (r + \sigma^2/2)T_1}{\sigma\sqrt{T_1}}$
• $a_2 = a_1 - \sigma\sqrt{T_1}$
• $b_1 = \frac{\ln(S_0/K_2) + (r + \sigma^2/2)T_2}{\sigma\sqrt{T_2}}$
• $b_2 = b_1 - \sigma\sqrt{T_2}$

The formula consists of three value components.

1. $S_0 M(a_1, b_1; \rho)$: The current asset price weighted by the joint probability of exercising both options (under the stock measure).
2. $- K_2 e^{-rT_2} M(a_2, b_2; \rho)$: The present value of the underlying strike $K_2$, contingent on both exercises.
3. $- K_1 e^{-rT_1} N(a_2)$: The present value of the compound strike $K_1$, contingent on exercising the compound option.

The appearance of the bivariate normal distribution $M(a, b; \rho)$ reflects the joint nature of the exercise decisions. We need to compute the probability that two conditions are both satisfied: the stock price exceeds $S^*$ at time $T_1$ (triggering exercise of the compound option), and the stock price exceeds $K_2$ at time $T_2$ (making the underlying call finish in the money). These two events are correlated because they both depend on the evolution of the same underlying stock price. The correlation coefficient $\rho = \sqrt{T_1/T_2}$ captures this dependence, representing how much of the total uncertainty about the terminal price at $T_2$ is already resolved by time $T_1$.

The parameters $a_1$ and $a_2$ relate to the first exercise decision at $T_1$, measuring the standardized distance from the current price to the critical price $S^*$. The parameters $b_1$ and $b_2$ relate to the second exercise decision at $T_2$, measuring the standardized distance from the current price to the underlying strike $K_2$. The subscript 1 versions incorporate the extra drift adjustment associated with the stock measure, while the subscript 2 versions use the risk-neutral measure directly.