Approximating ATM Short-Term Straddle Value with Implied Volatility
Summary
The note explains why an at-the-money short-dated straddle can be approximated using implied volatility. In a Black–Scholes setting, put-call parity reduces the value of an ATM-forward straddle to twice the call value. The ATM call value is approximated as a constant times the underlying level, implied volatility, and the square root of time to expiry; for a straddle on returns, the underlying level drops out.
Applying this approximation to a ratio of basket and constituent straddle values makes the common constant and time factor cancel. The resulting ratio is approximately the basket’s ATM implied volatility divided by the weighted sum of constituent ATM implied volatilities. This is a simplifying approximation, not an exact pricing identity: it relies on Black–Scholes assumptions, ATM-forward positioning, and the stated treatment of dividends and yields. Deviations can matter when those conditions do not hold or when the approximation is less accurate.
Key ideas
- Put-call parity relates the value of an ATM-forward straddle to twice the call value.
- An ATM Black–Scholes call value is approximated using implied volatility and the square root of time to expiry.
- For straddles written on returns, the underlying price level cancels from the approximation.
- The basket-to-constituent straddle ratio can therefore be approximated by a ratio of implied volatilities.
Tags
Full text
# at-the-money short term straddle and the implied vol
# at-the-money short term straddle and the implied vol
Here is a passage from "Advanced Equity Derivatives: Volatility and Correlation" by Sebastien Bossu, Wiley (2014).
We see the prox $\beta_0,$ it seems to use the approximation that `the at-the-money short term straddle is same as the implied vol?` But I can not obtain this result. Here we may use approximate formula of at-money call/put with risk free rate $r=0:$ $$c= p = 0.4S\sigma\sqrt{T-t}?$$
## Answer by Daneel Olivaw (score 2, accepted)
https://quant.stackexchange.com/a/36769
Throughout the answer we assume a Black-Scholes framework, noting $C_{BS}(t,T)$ and $P_{BS}(t,T)$ the prices at $t$ of a call and a put option written on an underlying $X_t$ and with maturity $T$. In general, the subscript $BS$ will designate the Black-Scholes price of a derivative.
A straddle of strike $K$ corresponds to a simultaneous long position in a call option and in a put option both with strike $K$. Letting $V(t,T)$ be the value at $t$ of a straddle with maturity $T$, by call-put parity:
$$ V(t,T) = C(t,T)+P(t,T)=2C(t,T)+e^{-r(T-t)}K-X_t$$
Assuming no revenue (i.e. dividends) or cost yield, if the straddle is at-the-money (ATM) forward:
$$ V(t,T) = 2C(t,T)$$
Now, letting $\sigma_X^{\star}$ be the ATM volatility of underlying $X_t$, a useful approximation of the Black-Scholes formula for ATM calls is:
$$ C_{BS}(t,T) \approx 0.4X_t\sigma_X^{\star}\sqrt{T-t} $$
Thus:
$$ V_{BS}(t,T) \approx 0.8X_t\sigma_X^{\star}\sqrt{T-t}$$
Moreover, note that your straddle is written on returns instead of prices/levels, hence when the straddle is ATM the approximation simplifies to:
$$ V_{BS}(t,T) \approx 0.8\sigma_X^{\star}\sqrt{T-t}$$
In your formula, the scaling factor $0.8$ and the square root will cancel, leaving:
$$ \beta_0 = \frac{V_{BS}(t,T,B_t)}{\sum_{i=0}^n{w_iV_{BS}(t,T,S^{(i)}_t)}} \approx \frac{\sigma_B^{\star}}{\sum_{i=0}^n{w_i\sigma_i^{\star}}} $$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.