Why Binary Option Vega Can Become Negative
Summary
The document explains why a binary call or put can have negative vega even though plain vanilla calls and puts generally have positive vega. A cash-or-nothing binary payoff can be represented as the limiting value of a narrow call spread divided by its strike width. Because a call spread combines a long option with a short option, its net sensitivity to volatility can be positive or negative depending on the underlying price relative to the strike.
The discussion also connects this representation to trading practice: derivatives traders may price binaries using call spreads with a finite width. The appropriate width can depend on factors such as trade size, liquidity, and other risk considerations, and the spread’s center may vary with whether the trader is buying or selling. The explanation is conceptual and does not provide a numerical example, pricing model, or empirical evidence; the sign of vega depends on the specific structure and market setup.
Key ideas
- A binary call can be approximated by a narrow call spread divided by its strike width.
- A call spread combines long and short option exposures, so its vega may be positive or negative.
- The underlying price relative to the strike influences the spread’s volatility sensitivity.
- Traders may use finite-width call spreads to price binaries, with width shaped by liquidity and risk factors.
Tags
Full text
# Vega of binary option
# Vega of binary option
I'm calculating the greeks for a hypothetical binary option, and I'm getting a symmetrical parabola for the vega's of both put and call options that are OTM, ATM, and ITM. Both of them dip into negative territory however. The vega for the call becomes negative when the binary option moves more into-the-money, while the inverse happens for my put.
I read that calls and puts always have positive vegas, which is why I'm confused about my graphed results (see picture; x-axis represents different spot prices, all else equal). Can anyone shed light into this?
## Answer by Ivan (score 2)
https://quant.stackexchange.com/a/53666
A binary call (a similar argument goes for the put) paying $\mathrm{1}_{S_T>K}$ can be seen as the limit of a call spread divided by the difference in strikes as this difference goes to 0:
$\mathrm{1}_{S_T>K} = lim_{dK\rightarrow0}\frac{Max(S_T-(K+\frac{dK}{2}),0)-Max(S_T-(K-\frac{dK}{2}),0)}{dK}$.
Hence it behaves like a call spread. A call spread being a combination of a long option and a short option, it can have a positive or negative Vega, depending mostly (to simplify) on where $S_t$ is relative to $K$.
In fact binaries are typically priced by derivatives traders as call spreads with a certain width depending on various risk factors such as size, liquidity of the underlying etc. In my example the call spread is centered on the strike but obviously this changes in reality depending on whether you're a buyer or seller of the binary (priced as a call spread).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.