Give an estimate for an at-the-money call option on a stock without dividends, with low interest rates and near-term expiration

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Give an estimate for an at-the-money call option on a stock without dividends, with low interest rates and near-term expiration.
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In the Black-Scholes model, the call option value c is typically:

c = S * N(d1) - e^(-rT) * K * N(d2)

where:
• S is the current stock price,
• K is the strike price,
• N(.) is the cumulative normal distribution function,
• r is the risk-free interest rate,
• T is the time to maturity,
• d1 and d2 and are variables calculated using the stock price, strike price, risk-free rate, volatility, and time to maturity.

d₁ = (ln(S₀/X) + (r + σ²/2)T) / (σ√T)
d₂ = d₁ - σ√T

Given that in a low-interest environment where the interest rate r is close to zero, the term e^(-rT) in the formula approximates to 1 and for an at-the-money option, S equals K, the formula simplifies to:

c ≈ S * (N(d1) - N(d2))

The difference N(d1) - N(d2), which is the area under the normal distribution curve between d1 and d2 is:

N(d1) - N(d2) ≈ ∫ from d2 to d1 of (1 / √(2 * π)) * e^(-1/2 * x^2) dx

Because d1 and d2 are close, we can use the value of the PDF of the normal distribution at 0, φ(0) = (1 / √(2 * π)), as a constant multiplier. The approximation considering that the area under the curve is a rectangle with an height of φ(0) and a width of (d1 - d2) gives:

N(d1) - N(d2) ≈ φ(0) * (d1 - d2)

N(d1) - N(d2) ≈ (d1 - d2) * (1 / √(2 * π)).

Plugging this back into the formula for c gives us the first approximation for the call option's value:

c ≈ S * (d1 - d2) * (1 / √(2 * π)).

In the context of at-the-money options with short maturity and low interest rates, where the current stock price (S) equals the strike price (K), the terms d1 and d2 in the Black-Scholes formula simplify because the log(S/K) part becomes log(1), which equals 0.

The formulas for d1 and d2 typically are:

d1 = (log(S/K) + (r + (σ^2)/2) * T) / (σ * √T)
d2 = d1 - σ * √T

When S equals K, and considering r is approximately 0 for short maturities, the formulas reduce to:

- d1 ≈ σ/2 * √T
- d2 ≈ -σ/2 * √T

Substituting the simplified d1 and d2 values into the approximation, we get:

N(d1) - N(d2) ≈ (1 / √(2π)) * (σ/2 * √T - (-σ/2 * √T))

This simplifies to:

N(d1) - N(d2) ≈ (1 / √(2π)) * σ * √T

Since φ(0), the value of the standard normal PDF at zero, is approximately 0.4 for small σ, we use this as a rule of thumb:

N(d1) - N(d2) ≈ 0.4 * σ * √T

Finally, given that c ≈ S * (N(d1) - N(d2)) is:

c ≈ 0.4 * σ * S* √T

This gives a rough estimate of the at-the-money call option's price, which is particularly useful for short-dated options where precise calculations are less essential.

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