At-the-Money Black–Scholes Approximation and Delta Hedging

Summary

The document addresses two practical questions about at-the-money options: how to approximate a Black–Scholes price quickly and how much underlying to trade for a delta hedge. One answer gives a rough call-price estimate based on the underlying price, volatility, and the square root of time to expiry, using a constant near 0.4. That shortcut assumes the strike equals the underlying price and uses a normal-price approximation; it is not the standard Black–Scholes model, which assumes geometric Brownian motion and lognormal prices.

For hedging, the answers use an at-the-money delta of about one half, so the hedge is roughly half the shares represented by the option contract, with direction depending on the option position. These are approximations, not universal exact values: actual delta and price depend on model assumptions and inputs such as rates, dividends, and contract details. The exchange excerpt provides no worked example or empirical validation.

Key ideas

• An at-the-money option price can be approximated from the underlying price, volatility, and time to expiry under simplifying assumptions.
• The quick price approximation relies on normally distributed prices and is not the standard Black–Scholes assumption.
• An at-the-money option’s delta is approximately one half in the examples discussed.
• A delta hedge therefore uses about half the contract’s share exposure, with trade direction set by the position.

Tags

# BS and delta hedging questions


# BS and delta hedging questions












I have two related questions concerning Black Scholes and delta hedging. I thought about this two questions, but I could not come up with an answer, so maybe you guys & girls can help me:

- If an option is at the money, how can the Black Scholes price be calculated in a very fast way (possibly without any big calculations)?

- If an option is at the money, how many shares do you have to buy in order to delta-hedge?

## Answer by Dmitri Nesteruk (score 3)

https://quant.stackexchange.com/a/4893

- See this question

- You have to buy/sell $\Delta$ shares. $\Delta_{ATM} \approx 0.5$.

## Answer by Matt Wolf (score 3)

https://quant.stackexchange.com/a/4894

- stock price * volatility * 0.4 * sqt(T), where T denotes time to expiration in years and 0.4 is coming from sqt(1/(2*pi)). The simplifying assumption here is (and that is very important and you will most likely be asked to state the assumptions should such question be asked in the interview): strike price equals underlying asset price AND asset prices are NORMALLY DISTRIBUTED (unlike the assumption in B-S) which assumes the asset price to follow an ARITHMETIC Brownian motion.

- As the delta is approximately (stress, not equal) 0.5, you need to hedge with about 1/2 the amount of the underlying asset that the options contract stipulates.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.