Deriving the At-the-Money Straddle Approximation from the Lognormal Integral
Summary
The document examines the at-the-money-forward straddle approximation under a Black–Scholes geometric Brownian motion model. It starts from the straddle payoff’s risk-neutral expectation, expresses the terminal price using a lognormal density, and changes variables to log returns. A first-order expansion of the exponential terms then reduces the payoff approximately to the absolute value of a shifted normal variable. The familiar leading term is proportional to spot, volatility, and the square root of time, with the normal absolute-deviation constant.
The writer’s derivation leaves a drift-related shift of order volatility squared times time inside the absolute value, suggesting that the simple approximation is best when this quantity is small. The document asks whether that reasoning is sound or whether integration offers a better route. It supplies no resolution, numerical comparison, or error bound. Its argument is therefore an exploratory approximation under a specific model, and does not establish accuracy for large volatility or longer maturities, or outside the stated assumptions.
Key ideas
- An at-the-money-forward straddle can be written as a risk-neutral expectation of the absolute difference between terminal price and strike.
- Under geometric Brownian motion, transforming to log returns gives a normal variable inside the payoff integral.
- A first-order exponential expansion produces the leading volatility and square-root-of-time dependence.
- The expansion leaves a drift shift proportional to volatility squared times time.
- The document raises, but does not resolve, the approximation’s accuracy or provide an error bound.
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# Straddle Approximation - Directly from Integral
# Straddle Approximation - Directly from Integral
The ATMF straddle approximation formula, given by
$V_\text{Str}(S, T) \approx \sqrt{\frac{2}{\pi}} S_0 \sigma \sqrt{T}$
where $S_0$ is the current underlying spot price, $T$ is the time remaining until expiration, and $\sigma$ is the volatility of log returns. The derivation using the Black-Scholes formula and Taylor expanding the normal cdf is easily found, for instance here: https://brilliant.org/wiki/straddle-approximation-formula/.
However, we have that for $Z \sim N(0, 1)$, $\mathbb{E}[\sigma|Z|] = \sigma \sqrt{2/\pi}$ (the MAD). The ATMF straddle value is, under risk-neutral measure, $\mathbb{E}[|S - K|]$ (assuming $S$ follows GBM of course). My question is, how to argue this approximation directly from integration?
My attempt so far: since $K = S_0 e^{rT}$
$\int_0^\infty |S - S_0 e^{rT}| p_T(S) dS = S_0 \int_0^\infty |S/S_0 - e^{rT}| p_T(S) dS$
where $p_T$ is the risk neutral pdf of prices (lognormal as defined by GBM). Next, let $y = \log(S/S_0)$ so $dy = dS/S$. Then, after altering the limits of integration based on the substitution,
$\int_{-\infty}^\infty |e^y - e^{rT}| \varphi\left(\frac{y - (r - \sigma^2/2)T}{\sigma \sqrt{T}} \right) dy$
where $\varphi$ is the standard normal pdf. In other words, $Y = (r-\sigma^2/2)T + \sigma \sqrt{T} Z$. Now, we can Taylor expand $e^{rT} \approx 1 + rT$ and $e^y \approx 1 + y$ so, performing another variable change to convert to standard normal notation
$\int_{-\infty}^\infty |1 + (r-\sigma^2/2)T + \sigma\sqrt{T}Z - (1+rT)| \varphi(z) dz = \int_{-\infty}^\infty |\sigma\sqrt{T}Z - \sigma^2T/2| \varphi(z) dz$
which seems to suggest that the result holds when $\sigma^2T/2 \approx 0$, i.e. $\sigma$ and $T$ are not too large. This relates to the more common derivation, since we see that the linear approximation of the normal cdf gets worse at higher variances here: https://www.desmos.com/calculator/rvmmhlxxx9.
See any issues? Or a different way to accomplish it?Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.