A stock is at $250 with 28% implied volatility. Price the three month at the money straddle in your head, and say where the constant in the approximation comes from.

A stock is at $250 with 28% implied volatility. Price the three month at the money straddle in your head, and say where the constant in the approximation comes from.

Approach: Use the at the money approximation for the straddle, taking the constant from the normal density at zero, then compare against the exact Black-Scholes value.

$28. The at the money straddle is approximately 0.8 * S * sigma * sqrt(T). The constant comes from 2 * S * phi(0) * sigma * sqrt(T) with phi(0) = 1/sqrt(2*pi), which is 0.3989, so the exact coefficient is 0.7979 and 0.8 is the rounded desk version. Here sqrt(0.25) is 0.5, so the price is 0.8 * 250 * 0.28 * 0.5, which is $28. The exact Black-Scholes value at zero rates is $27.90, so the approximation is 0.35% high. The same formula inverts to read implied volatility off a quoted price, since sigma = straddle/(0.8 * S * sqrt(T)), and a $28 straddle on a $250 stock with three months left implies 28%.

Follow-up: Rates are 5% and the forward sits 1.25% above spot. How does the straddle price change, and which leg carries the difference?

Key concepts: straddle, at the money approximation, implied volatility.