Option Delta Quotes as Hedge Ratios and Volatility Curve Inputs

# What is the "delta" option quoting convention about?


# What is the "delta" option quoting convention about?












At my work I often see option prices or vols quoted against deltas rather than against strikes. For example for March 2013 Zinc options I might see 5 quotes available for deltas as follows:

```
ZINC MARCH 2013 OPTION DELTA -10 POINTS
ZINC MARCH 2013 OPTION DELTA -25 POINTS
ZINC MARCH 2013 OPTION DELTA +10 POINTS
ZINC MARCH 2013 OPTION DELTA +25 POINTS
ZINC MARCH 2013 OPTION DELTA +50 POINTS
```

The same 5 values -10, -25, +10, +25, + 50 are always used. What are these deltas? What are the units? Why are those 5 values chosen?

## Answer by Richard Herron (score 14, accepted)

https://quant.stackexchange.com/a/34

Delta is the partial derivative of the value of the option with respect to the value of the underlying asset. An option with a delta of 0.5 (here listed as +50 points) goes up \$0.50 if the underlying asset goes up \$1.00. Or goes down \$0.50 if the underlying asset goes down \$1.00. Keep in mind that delta is an instantaneous derivative, so the value will change both in time (charm is the change in delta with time) and with changes in value of the underlying asset (gamma is the change in delta with the underlying asset, which is also the second partial derivative of the option value with respect to the underlying asset value).

The actual delta is a little different from +50, +25, and so on, but they're close enough. I am sure that you can find the real delta values. I guess they're listed like this because if you're hedging a portfolio you really care about the delta, not the strike. E.G., if I only wanted to delta hedge, and owned one share, I could buy two -50 delta puts, which sum to a delta of zero.

The wikipedia page on Greeks is a pretty good starting point.

## Answer by Jesper Harder (score 5)

https://quant.stackexchange.com/a/86

I handle volatility curves where moneyness is quoted in delta by an iterative guess:

- Use an initial guess for delta of 0.5 (call)/-0.5 (put)

- Look up the volatility on the curve using the guess for delta

- Calculate delta for the option using the vol found in 2.

- Repeat using Newton-Raphson, until the difference in delta is small enough.

## Answer by glyphard (score 4)

https://quant.stackexchange.com/a/616

Quoting prices in delta makes it easier for a trader to delta hedge their portfolio. (The trader knows teh delta they are trying to add or cut, so the price quoted in delta gives them the contract qty that they need to trade quickly, without needing a calculator, let alone a pricing model)

## Answer by user639 (score 4)

https://quant.stackexchange.com/a/803

Deltas represent hedge ratio; i.e. 5%, 10%, 25%...i.e. buy two 50 delta puts, buy 100 shares of stock for perfect hedge at price, done. Delta volatility "smile" should be represented with the smallest delta having the highest volatility to the largest delta having the smallest volatility, being the at the money option, struck at the price of the stock. 50 delta put on \$100 IBM is 100 strike. Buy 2 50 delta puts, sell 100 shares IBM at \$100. Higher volatility options have less chance of ending up in the money at expiration. easy way to think of this is 5 delta has 5% chance of ending up in the money at expiration. Instead of hedging with the underlying contract, lower deltas can be offset by selling other options against...buy two 5 deltas and sell one 10 delta against it.

## Answer by user59 (score 3)

https://quant.stackexchange.com/a/376

I'm going to disagree with richardh here, since it seems unusual to quote options with 10% delta.

Instead, I think "+25 POINTS" means an option whose strike price is 25 points above zinc's current March 2013 futures price (not zinc's current spot price). In other words, an option that's 25 points out of the money (I'm assuming these are calls).

Quoting option prices this way is much more stable. The price of an option on X with a fixed strike price varies as X's price varies. However, the price of an option with strike price X+25 is relatively stable (since X+25 itself varies with X).

Similarly, option volatility follows a "V" pattern when plotted against strike price (the "volatility smile"), with the lowest volatility when the strike price is equal to the current price. Therefore, you only need 4 volatilitys (2 on each side of the current price) to obtain the volatility-vs-strike-price graph. I'm not sure why they give you 5 (and why the 5th one isn't zero delta).

To be absolutely pedantic, Black-Scholes looks at Log[strikeprice/currentprice], not strikeprice minus currentprice. However, if the strike is close to the current price, these two numbers are nearly equal.

## Answer by David-Michael Lincke (score 3)

https://quant.stackexchange.com/a/394

The question may be motivated by the way closing implied volatilities are reported for LME traded third wednesday prompt metals contracts. Function LMIV on Bloomberg provides vol quotes for 50 delta options and corresponding premiums/discounts for 10, 25 delta puts and calls.

LME metals futures are really forward contracts with constant maturity forwards being electronically quoted for various terms (e.g. LMCADS03 Comdty for 3 month copper). Most of the liquidity is clustered in the third wednesday prompt contracts for which there are no continuous electronic quotes (at least not on Bloomberg) and which thus are frequently quoted by premium/discount from the nearest forward contract.

## Answer by Rustam (score 0)

https://quant.stackexchange.com/a/9074

For future use: Here's a good link with exact formula - http://www.quantessential.com/messageview.cfm?catid=3&threadid=24601

And here's and article with same formulas - http://www.wilmott.com/pdfs/050527_haug.pdf (see paragraph 2.3)