Option Delta Units, Conventions, and Portfolio Exposure

# How is delta defined as a unit?


# How is delta defined as a unit?












This is going to be a embarrassingly basic question. But the answer seems to be hard to find.

What does, say, selling, $d$ delta of calls mean? How is the "delta" defined? I am not asking about Greek as in $\frac{\partial V}{\partial S}$ where $V$ is the price of the option and $S$ that of the underlying stock.

## Answer by will (score 4, accepted)

https://quant.stackexchange.com/a/66690

You'll find the same issue with all of the greeks.

I would say that the standard rule for delta is the following:

If you're talking about a single option/strategy, then people will talk about delta percentage*.

If you're talking about a single option/strategy, and you don't plan to hedge the delta when it tades (i.e. you want to trade it live), then you'll talk about the delta in an absolute term, or some other term that you're familiar with - i.e. $35m delta, or 500 lots).

If you're talking about delta of something else - say a portfolio, where the total notional of positions is not obvious/known well by everyone you're talking to, then i would say most people again talk in absolute terms as above.

Some people also prefer to talk about delta in terms of spot, and others forward delta. This is again going to be desk specific.

The delta is normally spoken about scaled to be for a 100% move in the underlying**. Where you have non linear risk, this will be the instantaneous greek scaled to be as if it's the TV change for a 100% move (i.e. $\frac{TV(S+\mathrm{d}S) - TV(S)}{\mathrm{d}S}$)

*There are some cases where the delta will be used to describe the strikes - i.e. a 25d Risk Reversal is a trade where you buy(sell) a put with 25d, and sell(buy) a call with 25d also.

**unless you're talking about dv01, in which case it's for a 1bp move in rates, and is your rates delta.

In answer to Hans' comment -

You can just use finite difference to calculate the difference in the value. Again there are conventions though - do you care about the change in volatility caused by the change in the spot price? Are you valuing your derivatives in a mean reverting model such that moving the spot price moves the forward in a non linear way? Do you have non linear dividends? Are there other implications in your model resulting from changing the spoot price? It is not a clear cut answer, and conventions need to be decided on. For me, there are three kinds of delta i care about:

- Partial delta - i.e. $\frac{\partial \mathrm{TV}}{\partial S}$, that is, the change in price from moving only the spot price, and nothing else.

- Simple Delta Partial delta + vol delta: $\frac{\partial \mathrm{TV}}{\partial S} + \frac{\partial \sigma}{\partial S}\cdot\frac{\partial \mathrm{TV}}{\partial \sigma}$, i.e. the partial delta + the impact on value from the move in vol expected to result from a move in spot price.



## Answer by nbbo2 (score 8)

https://quant.stackexchange.com/a/66694

If you want to split hairs, as I like to do, there are 2 ways to express Delta.

"Pure Delta" is a fraction, i.e. a number between 0 and 1 (for a Call). In terms of units, it is a pure number.

"Position Delta" is equal to Delta times the size of your Call position. If you have Calls on 100 shares and Delta is 0.5 then Position Delta is 100 x 0.5 = 50 shares. As you can see the units for Position Delta is "shares". This is what you actually buy or sell when you hedge or replicate the option by trading in the underlying. (In practice the size of the Call position is often expressed in terms of Contracts, where 1 Contract is equal to 100 shares. So you have to convert from contracts to shares before multiplying by pure delta).

When a textbook says "buy Delta shares" they are assuming you have calls on 1 share, which is fine as an example, but lacks generality and applicability to the real world.

HTH

## Answer by Dorian B. (score 0)

https://quant.stackexchange.com/a/85319

I would say there is five main categories:

- delta (immediate output from Black Scholes model); formula $\frac{\partial V}{\partial S}$; for a call it's $e^{-qt}N(d_1)$ where q is the convenience yield (dividends for stocks for example); change in option price compared to change in spot; delta from a call less the delta from the same put add up to $1$ for non-dividend paying stocks only

- forward delta (immediate output from Black model); formula $\frac{\partial V}{\partial F}$; for a call it's $e^{-rt}N(d_1)$; usually for options on futures products; change in option price compared to change in forward; the call delta less the put delta add up to the discount factor $e^{-rt}$ and not to $1$

- percent delta; $\frac{1}{S}F\frac{\partial V}{\partial F}\times 100\%$ or $\frac{1}{S}S\frac{\partial V}{\partial S}\times 100\%$; it's the same value in both cases (stocks and futures); for a call it's $N(d_1)\times 100\%$; percentage change in option for a percentage change in the spot or in the forward; the call percent delta less the put percent delta is always $100\%$

- delta dollars; $Q\times \delta \times S \times k$ or $Q \times \delta_{fwd} \times F \times k$; the unit is dollars as the name suggests (or whatever currency it's quoted in) and it shows the exposure; $Q$ is the amount of contracts and $k$ is the contract multiplier ($100$ or one lot for stocks, but otherwise different for futures); used in risk management when dealing with a portfolio that has different underlying so you want to see the total exposure as this is additive to the same dollar unit.

- portfolio delta; $Q\times \delta \times k$ or $Q \times \delta_{fwd} \times k$; similar to delta dollars but unitless; counts the number of equivalent underlying quantity you are exposed to; also used in risk management when dealing with the same underlying as it's simpler however it's not additive cross-instruments with different underlyings

Given above, the usage should be obvious from the context.