Euan Sinclair, the author of Volatility Trading (Wiley, 2013) and the newer Positional Option Trading (Wiley, 2020) has said:
Some people say the Black-Scholes option pricing model breaks near expiration. These people are wrong.
Sinclair is right
The notion that the Black-Scholes option pricing model “breaks” near expiration is a misunderstanding. The model remains valid but must be interpreted correctly, especially as time to maturity approaches zero. Let’s elaborate with some algebra and a few numerical examples.
Recap
The Black-Scholes formula for a European call option is given by:
\[ C(S, t) = S \Phi(d_1) - K e^{-r(T-t)} \Phi(d_2) \]
where:
- \( S \) is the current stock price.
- \( K \) is the strike price.
- \( T \) is the time to expiration.
- \( r \) is the risk-free interest rate.
- \( \sigma \) is the volatility of the stock.
- \( \Phi \) is the cumulative distribution function of the standard normal distribution.
As time to expiration \( T-t \) approaches zero, the behavior of the Black-Scholes model can be analyzed through the terms \( d_1 \) and \( d_2 \).
\[d_1 = \frac{\ln(S/K) + (r + \sigma^2/2)(T-t)}{\sigma \sqrt{T-t}}\]And
\[d_2 = d_1 - \sigma \sqrt{T-t}\]If \( t \rightarrow T\)
\[\lim_{ (t \to T} d_1 \approx \frac{\ln(S/K) + (r + \sigma^2/2) \cdot 0}{\sigma \sqrt{0}} \rightarrow \infty \text{ or } -\infty\]And
\[\lim_{ t \to T} d_2 \approx d_1 - \sigma \sqrt{0} = d_1\]Three Cases: In Out At the Money
In-the-Money ( \( S > K \) )
In this case, \( \ln(S/K) \) is positive, \( d_1 \) and \( d_2 \) both tend to \( \infty \), \( \Phi(d_1) \rightarrow 1 \) and \( \Phi(d_2) \rightarrow 1 \).
Thus
\[C(S, t) \approx S - K e^{-r(T-t)} \approx S - K\]Out-of-the-Money ( \( S < K \) )
In the OTM case, \( \ln(S/K) \) is negative, \( d_1 \) and \( d_2 \) both tend to \( -\infty \), \( \Phi(d_1) \rightarrow 0 \) and \( \Phi(d_2) \rightarrow 0 \).
Thus
\[C(S, t) \approx 0\]At-the-Money ( \( S \approx K \) )
Right at the money, \( \ln(S/K) \approx 0 \), \( d_1 \approx \frac{(r + \sigma^2/2) (T-t)}{\sigma \sqrt{T-t}} = \frac{(r + \sigma^2/2) \sqrt{T-t}}{\sigma} \), \( d_2 \approx d_1 - \sigma \sqrt{T-t} = \frac{(r - \sigma^2/2) \sqrt{T-t}}{\sigma} \). As \( T-t \rightarrow 0 \), both \( d_1 \) and \( d_2 \) tend to zero, \( \Phi(d_1) \rightarrow 0.5 \) and \( \Phi(d_2) \rightarrow 0.5 \).
Hence
\[C(S, t) \approx S \cdot 0.5 - K e^{-r(T-t)} \cdot 0.5 \approx 0.5(S - K)\]The following numerical calculations show that Black-Scholes model’s behavior near expiration aligns very well with the intrinsic value of the option and the probability of the option finishing in-the-money.
Intrinsic Value: For a call option, the intrinsic value is \( \max(S - K, 0) \). As time to expiration \( (T - t) \) approaches zero, the option’s price should approach its intrinsic value.
Probability of Finishing In-the-Money: The probability of an option finishing in-the-money (ITM) is represented by \( \Phi(d_2) \). The parameters \( \Phi(d_1) \) and \( \Phi(d_2) \) indicate the likelihood that the stock price will be above the strike price at expiration.
Illustrative examples
In-the-Money ( \( S = 105 \), \( K = 100 \) )
Intrinsic Value is \( \max(105 - 100, 0) = 5 \)
\[d_1 = 2.45 \\), \\( d_2 = 2.25\] \[\Phi(d_1) = 0.993 \\), \\( \Phi(d_2) = 0.988\]ITM Option Price
\[C = 105 \cdot 0.993 - 100 \cdot e^{-0.05 \cdot 0.01} \cdot 0.988 \approx 5.02\]As the option is ITM, the price \( 5.02 \) is close to its intrinsic value \( 5 \). Here, \( \Phi(d_2) = 0.988 \) indicates a very high probability (98.8%) of finishing ITM.
Out-of-the-Money ( \( S = 95 \), \( K = 100 \) )
Intrinsic Value: \( \max(95 - 100, 0) = 0 \)
\[d_1 = -2.35 \\), \\( d_2 = -2.55\] \[\Phi(d_1) = 0.009 \\), \\( \Phi(d_2) = 0.005\]OTM Option Price
\[C = 95 \cdot 0.009 - 100 \cdot e^{-0.05 \cdot 0.01} \cdot 0.005 \approx 0.40\]As the option is OTM, its price \( 0.40 \) is near zero, consistent with its intrinsic value of zero. The value of \( \Phi(d_2) = 0.005 \) indicates a very low probability (0.5%) of finishing ITM.
At-the-Money ( \( S = 100 \), \( K = 100 \) )
Intrinsic Value: - \( \max(100 - 100, 0) = 0 \)
\[d_1 = 0.025 \\), \\( d_2 = 0.005\] \[\Phi(d_1) = 0.510 \\), \\( \Phi(d_2) = 0.502\]ATM Option Price
\[C = 100 \cdot 0.510 - 100 \cdot e^{-0.05 \cdot 0.01} \cdot 0.502 \approx 0.80\]This option price \( 0.80 \) is slightly above zero due to the time value and volatility. Yet \( \Phi(d_2) = 0.502 \) indicates about a 50.2% probability of finishing ITM.
Key Points
- Intrinsic Value Convergence: As \( T-t \rightarrow 0 \), the option price approaches its intrinsic value. For ITM options, the price approaches \( S - K \). For OTM options, the price approaches zero.
- Probability Reflection: The values \( \Phi(d_1) \) and \( \Phi(d_2) \) represent probabilities of the option finishing ITM. Near expiration, these probabilities are in line with the actual likelihood of the option’s payoff.
The Black-Scholes model correctly reflects the intrinsic value and the probability of finishing ITM as expiration nears.
The Black-Scholes model does not “break” near expiration.
Instead, the behavior of the model is consistent with the intrinsic value of the option and the probability of the option finishing in-the-money. The correct interpretation and application of the model’s output near expiration are crucial.