# Digital Options

> ***`The Game is a win-or-lose battle,`*** \
>     ***`where bullas and beras combat with`*** \
>     ***`Spears and Shields,`*** \
>     ***`for prizes claimable at expiry.`***  \
>                                      ***`-- anon`***:diving\_mask:

Divergence v1 offers European-style digital options, each backed by an ERC-20 token collateral. They are exercised at expiry, paying a fixed amount or nothing at all. 

## What are Digital Options?

<table data-card-size="large" data-view="cards"><thead><tr><th align="center"></th><th></th><th data-hidden></th><th data-hidden data-card-cover data-type="files"></th></tr></thead><tbody><tr><td align="center"><strong>Digital Call</strong></td><td><ul><li>Premium costs less than 1 collateral</li><li>Buyer has the right to claim 1 collateral, if the underlying price <em><strong>settles at or above</strong></em> the strike. </li><li>Otherwise, it expires worthless.</li></ul></td><td></td><td><a href="https://205922869-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2F-MXDQ_f-eqqSVmsimay0%2Fuploads%2FxJKXxBejU6Scj0ORl7Qy%2Fbulla_gitbook.png?alt=media&#x26;token=68e7a414-1d04-4739-b2d3-6b5754f344d5">bulla_gitbook.png</a></td></tr><tr><td align="center"><strong>Digital Put</strong></td><td><ul><li>Premium costs less than 1 collateral</li><li>Buyer has the right to claim 1 collateral, if the underlying price <strong>settle</strong><em><strong>s below</strong></em> the strike. </li><li>Otherwise, it expires worthless. </li></ul></td><td></td><td><a href="https://205922869-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2F-MXDQ_f-eqqSVmsimay0%2Fuploads%2FHdrk8zZvXj4SOE6zcRhy%2Fbera_gitbook.png?alt=media&#x26;token=47ccee7d-bed2-4c79-b74e-39180abd1c79">bera_gitbook.png</a></td></tr></tbody></table>

## How Much Does it Cost to Long or Short a Digital Option?

<table><thead><tr><th width="194">Position</th><th width="130">Call</th><th>Put</th></tr></thead><tbody><tr><td>Long</td><td>Premium</td><td>Premium</td></tr><tr><td>Short</td><td>1 - Premium</td><td>1 - Premium</td></tr></tbody></table>

## Risk & Reward for a Digital Option

<table><thead><tr><th width="194">Settlement</th><th width="130">Long Call</th><th width="133">Short Call</th><th width="124">Long Put</th><th>Short Put</th></tr></thead><tbody><tr><td>Price ≥ Strike</td><td>Gain 1 - Premium</td><td>Lose 1 - Premium</td><td>Lose Premium</td><td><p>Gain</p><p>Premium</p></td></tr><tr><td>Price &#x3C; Strike</td><td>Lose Premium</td><td>Gain Premium</td><td>Gain 1 - Premium</td><td><p>Lose 1 - </p><p>Premium</p></td></tr></tbody></table>

## How to trade Digital Options?

When using Divergence v1 smart contracts, one can come across:

* &#x20;:crossed\_swords: ***`Battle`*** an AMM pool for digital options
* &#x20;:stadium: ***`Arena`*** deployer and admin of *`Battle`*&#x20;
* <img src="https://205922869-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2F-MXDQ_f-eqqSVmsimay0%2Fuploads%2Ftejg5ghEh1qfogsQmMsH%2FSpear_icon2.png?alt=media&#x26;token=00efed32-5bf7-40d0-b538-54c0ca8e7f9b" alt="" data-size="line">***`Spear`*** a European digital call option token &#x20;
* <img src="https://205922869-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2F-MXDQ_f-eqqSVmsimay0%2Fuploads%2FWs9b8z36xI2FH2LGAIkN%2FShield_icon2.png?alt=media&#x26;token=a067e97e-30d2-4571-8b12-685771546945" alt="" data-size="line">***`Shield`*** a European digital put option token

Each Battle mints Spear and Shield,  ERC-20-compliant options tokens of a specific underlying, strike price, collateral, and expiry. A Battle's contract address is not re-used after expiry. A new Battle can be created for the following expiry. &#x20;

{% hint style="info" %}
*Suppose a digital call costs 0.01 DAI. Its underlying is ETHUSD, and its strike is $2,000. If ETH settles above or equal $2,000, the option pays 1 DAI. Otherwise, it expires worthless. The buyer of this call earns 1 - 0.01 = 0.99 DAI less fees, i.e., a 99x return.*
{% endhint %}

The v1 interface is one of the many ways one may interact with the protocol. The following pages provide concise, step-by-step guides on how to trade digital options:

<table data-card-size="large" data-view="cards"><thead><tr><th></th><th data-hidden></th><th data-hidden></th><th data-hidden data-card-target data-type="content-ref"></th><th data-hidden data-card-cover data-type="files"></th></tr></thead><tbody><tr><td><strong>Long Options</strong></td><td></td><td></td><td><a href="../../user-guide/long-options">long-options</a></td><td><a href="https://205922869-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2F-MXDQ_f-eqqSVmsimay0%2Fuploads%2FZyuhLg6GrukMZZdHY409%2Flong.png?alt=media&#x26;token=c196bd29-0156-48e3-933a-1f1e91b286c2">long.png</a></td></tr><tr><td><strong>Short Options</strong></td><td></td><td></td><td><a href="../../user-guide/short-options">short-options</a></td><td><a href="https://205922869-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2F-MXDQ_f-eqqSVmsimay0%2Fuploads%2FKp3q0cEWeB2NeJST7vi3%2Fshort.png?alt=media&#x26;token=1f09ac92-f8cd-4a76-af49-7e49f003e8a6">short.png</a></td></tr></tbody></table>

## How to Price Digital Options?

Simply put, each and every individual user of the protocol decides the price, as they buy and sell tokenized digital options. The protocol does not use a theoretical pricing model to arbitrarily set option prices. **The price for an option is simply the amount of collateral paid divided by the number of options received**. [This model-free approach](https://docs.divergence-protocol.com/overview/protocol-basics/triangular-swaps) enables real-time price discovery:

* There's not a one-size-fits-all theoretical price model that works at all times.&#x20;
* Commonly used pricing models typically require discretionary inputs and real-time adjustments. The costs of these adjustments need not be imposed on all participants. Instead, the calculations can be done off-chain by individuals, before a transaction.&#x20;

Spear and Shield tokens are priced between 0.01 and 0.99 collateral units. Their prices suggest market expectations about the direction of the underlying price.&#x20;

{% hint style="info" %}
*Suppose you pay 40 USDC to long 100 Spear. You spend 40/100 = 0.4 USDC for a spear. You are estimating a 40% chance that the underlying will settle above the strike price.*
{% endhint %}

For more theoretical and practical guides to digital options pricing, see the [reference](https://docs.divergence-protocol.com/overview/protocol-basics/references) section.&#x20;

## What is Put-Call Parity?

A crucial underpinning of Divergence v1's design is **the price parity of digital options:**

$$
Digital Call Price + Digital Put Price = 1
$$

In other words, at maturity, the probability of a call being profitable and the probability of a put being profitable add up to 100%.  One can expect the payoff for a long or short digital option to be:

![A Long Spear is a Short Shield. | A Long Shield is a Short Spear. ](https://205922869-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2F-MXDQ_f-eqqSVmsimay0%2Fuploads%2FK99B1Tle6hm3EZDEZGPq%2F12.png?alt=media\&token=65eb2536-0aa8-403e-956d-add622d90eb2)

* **Long Digital Call = Short Digital Put**&#x20;

{% hint style="info" %}
*Suppose Spear/Shield trades at .40/.60.*

* *A trader who longs a Spear at .40 expects to receive a 1.0 payout, which includes the initial investment of .40 plus .60 profit* *if the underlying settles at or above the strike.*&#x20;

* *A trader who shorts a Shield at .60 expects to keep the collected premium as profit under the same condition.*&#x20;
  {% endhint %}

* **Long Digital Put = Short Digital call**

{% hint style="info" %}
*Suppose Spear/Shield trades at .40/.60.*&#x20;

* *A trader who longs a Shield at .60 expects to receive a 1.0 payout, which includes the initial investment of .60 plus .40 profit if the underlying settles below the strike.*&#x20;
* *A trader who shorts a Spear at .40 expects to keep the collected premium as profit under the same condition.*
  {% endhint %}

{% embed url="<https://youtu.be/ngLl54mY8D0>" %}
Put-Call Parity Explained
{% endembed %}

## Why Digital Options?

Digital options can be used to replicate the payoff structure of any financial asset. They are useful as a hedge against sudden, large price movements in illiquid underlying assets.&#x20;

{% embed url="<https://www.youtube.com/watch?t=1s&v=B0N35uDgKMo>" %}
**Binary Calls Explained**
{% endembed %}

{% embed url="<https://www.youtube.com/watch?v=XCrwqzAC4XQ>" %}
**Binary Puts Explained**
{% endembed %}

## Cash-or-Nothing, Asset-or-Nothing or ???

<img src="https://205922869-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2F-MXDQ_f-eqqSVmsimay0%2Fuploads%2Ftejg5ghEh1qfogsQmMsH%2FSpear_icon2.png?alt=media&#x26;token=00efed32-5bf7-40d0-b538-54c0ca8e7f9b" alt="" data-size="line">Spears and <img src="https://205922869-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2F-MXDQ_f-eqqSVmsimay0%2Fuploads%2FWs9b8z36xI2FH2LGAIkN%2FShield_icon2.png?alt=media&#x26;token=a067e97e-30d2-4571-8b12-685771546945" alt="" data-size="line">Shields can be considered as: &#x20;

* **Cash-or-Nothing** when pool collateral is a **stablecoin;**
* **Asset-or-nothing** when the **underlying asset** is used as collateral;&#x20;

In some cases, a pool may be funded with a collateral asset that is different from the underlying asset, and in this case, the payout is a **correlated or uncorrelated asset.**

{% embed url="<https://www.youtube.com/watch?v=W-CxXh1JQyE>" %}
**Asset-or-Nothing & Cash-or-Nothing Calls Explained**
{% endembed %}

{% embed url="<https://www.youtube.com/watch?t=2s&v=y4IhQmt-fR0>" %}
**Asset-or-Nothing & Cash-or-Nothing Puts Explained**
{% endembed %}

## Composability of Digital Options

Digital options are the basic components of many structured products. One can use a basket of digital options, or combine digital options with other financial products to compose defined risk strategies. They can be a versatile, powerful tool for [hedging a DeFi portfolio](https://medium.com/divergence-protocol/how-to-hedge-your-defi-portfolio-using-options-on-divergence-1dafe6afba9f).

Digital Options are also [the building blocks of standard options](https://en.wikipedia.org/wiki/Black%E2%80%93Scholes_model#Cash-or-nothing_call).  As noted in above video explainers, Asset-or-Nothing and Cash-or-Nothing options can replicate vanilla options:&#x20;

* **Standard European call = Asset-or-Nothing call - Cash-or-Nothing call** \
  \&#xNAN;*where the cash-or-nothing call payoff equals the strike value*
* **Standard European put = Cash-or-Nothing put - Asset-or Nothing put**&#x20;

  *where the cash-or-nothing put payoff equals the strike value*

Therefore they also present opportunities to arbitrage or hedge against standard options available in DeFi and beyond.:diving\_mask: