Valuation of Digital Options via European Option Replication

Abstract

Digital options, characterized by their binary payoff structure, present unique challenges in pricing and risk management due to their discontinuous nature. This white paper explores the replication of digital options using European vanilla options, a practical alternative to the analytical Black-Scholes (BS) model. We detail the underlying principles, derive formulas for price and key Greeks (delta, gamma), and discuss the significance of this method in financial modeling and risk management.

1. Introduction

Digital options, also known as binary options, deliver a fixed payoff if the underlying asset price meets a specified condition at expiration. For a cash-or-nothing call, the payoff is $Q$ if $S_T > K$, and zero otherwise; for a put, it’s $Q$ if $S_T < K$. This discontinuity complicates direct pricing and hedging. While the Black-Scholes framework provides an analytical solution, replicating a digital option with European calls offers an intuitive, market-based approach. This method leverages liquid vanilla options to approximate the digital’s behavior, enhancing flexibility and practical applicability.

2. Principle of Replication

The replication method approximates a digital option’s payoff using a portfolio of European call options with strikes offset around the digital’s strike $K$. The core idea is to mimic the digital’s step-function payoff with a steep but continuous transition.

2.1 Payoff Approximation

For a cash-or-nothing call with payoff $Q \cdot 1_{S_T > K}$:

• Construct a portfolio: Buy a call with strike $K - h$ and sell a call with strike $K + h$.
• Payoff at expiration:
  • If $S_T < K - h$: Both calls expire worthless, payoff = 0.
  • If $K - h < S_T < K + h$: The lower-strike call pays $S_T - (K - h)$, the higher-strike call is worthless.
  • If $S_T > K + h$: Payoff is $(S_T - (K - h)) - (S_T - (K + h)) = 2h$.
• Scale the position by $\frac{Q}{2h}$: The portfolio payoff approximates $Q$ over the interval $[K - h, K + h]$, transitioning from 0 to $Q$.

As $h \to 0$, this converges to the digital’s exact payoff, resembling a finite difference approximation of the Heaviside step function.

3. Price Calculation

Under the risk-neutral measure in the BS model, the price of a European call is $C(K) = e^{-rT} [F N(d_1) - K N(d_2)]$, where $F = S_0 e^{(r - q)T}$, $d_1 = \frac{\ln(S_0/K) + (r - q + 0.5\sigma^2)T}{\sigma \sqrt{T}}$, and $d_2 = d_1 - \sigma \sqrt{T}$.

The digital call price via replication is:

$$V = \frac{C(K - h) - C(K + h)}{2h}$$

• As $h$ decreases, this approaches the BS digital price: $V = e^{-rT} N(d_2)$ (for $Q = 1$).
• For a put: $V = \frac{C(K + h) - C(K - h)}{2h}$, approximating $e^{-rT} N(-d_2)$.

Numerical Stability

Choosing an appropriate $h$ balances accuracy and numerical stability. Too small an $h$ amplifies rounding errors; too large distorts the approximation. Typically, $h = 0.01 S_0$ is a practical choice.

4. Greeks Calculation

Greeks measure sensitivity to underlying parameters, crucial for hedging. We derive delta and gamma using the replication approach.

4.1 Delta

Delta ($\Delta = \frac{\partial V}{\partial S_0}$) is the rate of change of price with respect to the underlying price.

• For a call: $\Delta = \frac{\Delta_C(K - h) - \Delta_C(K + h)}{2h}$, where $\Delta_C(K) = e^{-qT} N(d_1)$.
• As $h \to 0$, this approximates the BS delta: $\Delta = \frac{e^{-rT} N'(d_2)}{S_0 \sigma \sqrt{T}}$ (positive for calls, negative for puts after sign adjustment).

4.2 Gamma

Gamma ($\Gamma = \frac{\partial^2 V}{\partial S_0^2}$) captures convexity.

• For a call: $\Gamma = \frac{\Gamma_C(K - h) - \Gamma_C(K + h)}{2h}$, where $\Gamma_C(K) = \frac{e^{-qT} N'(d_1)}{S_0 \sigma \sqrt{T}}$.
• Corrected for sign (as identified earlier): $\Gamma = \frac{\Gamma_C(K + h) - \Gamma_C(K - h)}{2h}$ (negative for calls).
• For a put: $\Gamma = \frac{\Gamma_C(K - h) - \Gamma_C(K + h)}{2h}$ (positive).
• BS gamma: $\Gamma = -\frac{e^{-rT} N'(d_2)}{S_0^2 \sigma \sqrt{T}} \left( \frac{d_2}{\sigma \sqrt{T}} + 1 \right)$ for calls, positive for puts.

The replication gamma aligns with BS as $h$ shrinks, though it’s an approximation sensitive to $h$.

5. Significance of the Replication Approach

Replication uses traded European options, enabling direct hedging with market instruments.
The method offers a tangible link between digital and vanilla options, enhancing understanding of payoff synthesis and finite difference techniques.

References

• Black, F., & Scholes, M. (1973). "The Pricing of Options and Corporate Liabilities."
• Hull, J. C. (2017). Options, Futures, and Other Derivatives.