The Infinite-Maturity Limit of an At-the-Money Black–Scholes Call

Summary

The document evaluates the Black–Scholes price of an at-the-money European call as maturity grows without bound. Starting from the standard call pricing formula, it considers the limiting behavior of the two normal-distribution terms under nonnegative interest rates and positive volatility.

As maturity increases, the probability term multiplying the spot price approaches one. The discounted strike contribution vanishes when rates are positive; when the rate is zero, the associated normal-distribution term instead approaches zero. In either stated case, the call price tends to the current underlying price. This result relies on the assumptions specified in the derivation, including a fixed at-the-money strike and the Black–Scholes framework. It is a limiting theoretical result, not a practical valuation rule for finite maturities or a discussion of dividends, negative rates, or other market features.

Key ideas

• For a fixed at-the-money strike, the Black–Scholes call price approaches spot as maturity tends to infinity under the stated assumptions.
• The normal term multiplying spot tends to one as maturity grows.
• The strike contribution vanishes for positive rates through discounting.
• At a zero rate, the strike contribution vanishes because its normal-distribution term tends to zero.
• The conclusion is specific to the model assumptions and asymptotic maturity.

Tags

# What is the value of an ATM call under the Black Scholes Framework when $T \rightarrow \infty$?


# What is the value of an ATM call under the Black Scholes Framework when $T \rightarrow \infty$?












In the Black Scholes framework what is the value of an at-the-money vanilla European call option as time to maturity goes to infinit ($T \rightarrow \infty$)?

## Answer by Gordon (score 2, accepted)

https://quant.stackexchange.com/a/30890

The Black-Scholes call option price is given by \begin{align*} C = S_0 N(d_+) - K e^{-rT}N(d_-), \end{align*} where $$d_{\pm}= \frac{\ln \frac{S_0}{K}+(r\pm\frac{1}{2}\sigma^2) T}{\sigma \sqrt{T}}.$$ Here, we assume that the interest rate $r\ge 0$ and $\sigma >0$. For an at-the-money call option, that is, $K=S_0$, we note that $\lim_{T\rightarrow \infty} d_+ = \infty$, that is, $$\lim_{T\rightarrow \infty} N(d_+) = 1.$$ In addition, if $r>0$, then $$\lim_{T\rightarrow \infty}Ke^{-rT} N(d_-) = 0,$$ and if $r =0$, then $\lim_{T\rightarrow \infty} d_- = -\infty$, and $\lim_{T\rightarrow \infty} N(d_-) = 0.$ Therefore, $$\lim_{T\rightarrow \infty} C = S_0.$$

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