Asset-or-Nothing Calls: Payoff, Pricing, and Hedging Challenges
Summary
The document identifies a derivative that pays the asset price at maturity when that price exceeds a strike, and pays nothing otherwise. This is an asset-or-nothing call, a binary-style option. In the Black–Scholes framework, one answer gives its value using spot price, dividend yield, time to maturity, and the standard normal cumulative distribution evaluated at d1. The responses characterize this contract as an exotic product that is generally less liquid than a plain vanilla option.
The payoff can also be decomposed into a standard call payoff plus the strike multiplied by a cash-or-nothing binary payoff. Because the binary component changes abruptly at the strike, its delta changes rapidly, making hedging difficult. A narrow call spread can approximate the binary payoff with a less abrupt transition. These comments provide a conceptual description rather than market-specific liquidity data or a full pricing derivation; actual liquidity and hedging conditions depend on the contract and trading venue.
Key ideas
- A payoff equal to the asset price above a strike and zero otherwise is an asset-or-nothing call.
- The Black–Scholes value cited uses spot, dividend yield, time to maturity, and the normal cumulative distribution at d1.
- The payoff decomposes into a vanilla call and a strike-scaled cash-or-nothing binary call.
- The binary payoff changes sharply at the strike, complicating delta hedging.
- A narrow call spread can approximate the binary payoff, while such options are described as less liquid than vanilla options.
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# What is the name of this product?
# What is the name of this product?
Consider the payoff =$S_T1_{S_T>K}$ where $S_T$ is the asset price at maturity.
What is this type derivative called?
and is it a liquid option?
## Answer by Richi Wa (score 4)
https://quant.stackexchange.com/a/15126
This looks like a binary option. Following this wikipedia article it is called an "asset or nothing call". The pricing formula in the Black-Scholes world is
$$ S e^{-q T} \Phi(d_1), $$ where $S$ is the current spot price, $q$ is the dividend yield, $\Phi$ the cdf of a standard normal and $d_1$ is as usual in BS.
To my knowledge such options are much less liquid than there plain vanilla counterparts.
## Answer by sambaixo (score 1)
https://quant.stackexchange.com/a/15145
I don't think that it has a name on its own, but you can write $$ (S_T - K + K)\,1_{S_T>K} = (S_T-K)_+ + K\,1_{S_T>K} $$ so it's a 1 call plus K binary calls.
Binary are hard to hedge, the payoff looks like _|‾, going sharply from out-of-the-money to in-the-money. The delta changes fast and it's difficult to hedge the position.
In practice, you give yourself some cushion by approximating the binary by a call spread with small spread (buy an amount of calls struck at a lower strike, sell the same amount of calls struck at a higher strike). The payoff looks like _/‾.
## Answer by Dreyfus (score 0)
https://quant.stackexchange.com/a/15410
Its a payoff for asset or nothing call, this option is an OTC derivative (more specifically an exotic option) thus relatively illiquid compared to plain vanilla option. Its payoff, with exercise price ($K$), is $S_t$ if $S{t}>K$ and 0 otherwise and $1_{S_t>K}$ is an indicator function which takes value 1 if $S_t>K$ and 0 otherwise. And that's why it is known as binary option.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.