Digital Option（European）

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1. Forex Trading Center - Forex Digital Options

The exercise methods of digital options (Digital, Binary) are divided into European and American styles. The following are the application scenarios and explanations of European digital options

1.1 Application Scenarios

Foreign exchange European digital options have the following main application scenarios:

1. Hedging purpose: Enterprises can buy foreign exchange digital call options to hedge their foreign exchange exposure risk. If the foreign exchange rate rises at maturity and causes losses to the enterprise, the options can compensate for some of the losses. Enterprises can also purchase foreign exchange digital put options to hedge against foreign trade collection risks.
2. Diversification of overall investment portfolio: Investors will allocate a certain proportion of their investment to foreign exchange options in order to diversify their investment portfolio. If the movement of foreign exchange rates is beneficial to investors, they can earn investment returns higher than deposit interest.
3. Arbitrage trading: When there is a significant difference between the spot exchange rate and the implied exchange rate of options, traders can engage in arbitrage trading to profit. For example, if the spot exchange rate of the US dollar to the Chinese yuan is 6.5, but the implied exchange rate of a call option is only 6.2, the trader can sell the spot and buy the option, and then close the position to profit when the option expires.
4. Profit purpose: Some speculative traders will actively buy foreign exchange digital options in an attempt to predict the direction and magnitude of foreign exchange rates and make profits at maturity. Although the risk is high, the potential returns are also high.
5. Enhance leverage: Due to the option amount being much smaller than the underlying asset, a higher leverage effect can be obtained. Speculators can control large foreign exchange positions by paying only a small amount of option fees, thereby amplifying investment returns. But the risk will also increase.
   So foreign exchange digital options are mainly used for hedging, portfolio diversification, arbitrage trading, and speculative purposes, which can meet the needs of different investors. However, there are also certain risks that investors need to evaluate carefully.

1.2 Explanation of Digital Option Varieties Traded by the China Foreign Exchange Trading Center

Refers to a fixed amount agreed upon by both parties. When the observed value on the expiration date is higher or lower than the agreed exercise price, if it is an in the money option, the option buyer can exercise and receive the fixed amount.
The reference exchange rate can be based on the benchmark price published by the trading center. At present, the foreign currency pair (G10) market supports selecting benchmark prices at 10:00, 11:00, 14:00, 15:00, and 16:00 as reference exchange rates.
『Note』
The trading rules for digital options are the same as those for regular European options, with the only difference being the treatment of option expiration dates: European digital options track the reference exchange rate compared to the exercise price. If it is within the price, the option can be exercised, and the option buyer receives a fixed amount of cash agreed upon at the time of the transaction. The specific judgment relationship is shown in the table below.

Option type	Reference exchange rate and strike price relationship	Option status	Corresponding relationship
Call	Reference rate>Strike price	ITM and effective	Cash Or Nothing Call
Call	Reference rate<=Strike price	OTM and invalid	Cash Or Nothing Call
Put	Reference rate<Strike price	ITM and effective	Cash Or Nothing Put
Put	Reference rate>=Strike price	OTM and invalid	Cash Or Nothing Put

At present, European barrier options and European digital options only support the foreign currency pair (G10) market.

Extract：P53，Product Guidelines of China Foreign Exchange Trading Center V4.2.pdf

So, essentially, digital options in trading centers are a type of Cash or Notions Options. Reference Pricing Principle

2. Introduction to more types of digital options

The following is an explanation of the types of digital options and their Payoff expressions. The characteristic of digital options is that the payment result is a fixed amount or the value of the underlying asset, and is usually based on whether the underlying asset has reached a certain price level or touched a certain price barrier.

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2.1. Classification of Digital Options

2.1.1 Vanilla Digital Options

• CASH-OR-NOTING CALL and PUT
• At maturity, if the underlying asset price meets the conditions, the buyer will receive a fixed cash payment, otherwise no amount will be paid.
• Asset Or Dating CALL and PUT
• At maturity, if the price of the underlying asset meets the conditions, the buyer will receive the value of the underlying asset. If the conditions are not met, no payment will be made.

2.1.2 Barrier Digital Options

• At-Touch Options
• If during the term of the option, the underlying asset price touches the barrier price, cash or asset payment shall be made immediately.
• Knock-In Options
• If the price of the underlying asset touches the barrier price during the term of the option, the option will be activated and payment will be made according to the conditions upon expiration.
• Knock-Out Options
• If the price of the underlying asset touches the barrier price during the term of the option, the option becomes invalid and the payment is zero.

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3. Payoff function for each digital option

3.1 Basic Digital Options

CASH_OR_NOTHING_CALL

• At the expiration date，if $S_T > K$，pay a fixed amount of $C$；Otherwise,pay 0.

• Payoff:

  $$\text{Payoff} = \begin{cases} C, & \text{if } S_T > K \\ 0, & \text{if } S_T \leq K \end{cases}$$

ASSET_OR_NOTHING_CALL

• At the expiration date，if $S_T > K$，Pay asset value $S_T$；Otherwise,pay 0.
• Payoff:

  $$\text{Payoff} = \begin{cases} S_T, & \text{if } S_T > K \\ 0, & \text{if } S_T \leq K \end{cases}$$

CASH_OR_NOTHING_PUT

• At the expiration date，if $S_T < K$，Pay a fixed amount of $C$；Otherwise,pay 0.
• Payoff:

  $$\text{Payoff} = \begin{cases} C, & \text{if } S_T < K \\ 0, & \text{if } S_T \geq K \end{cases}$$

ASSET_OR_NOTHING_PUT

• At the expiration date，if $S_T < K$，Payment asset value $S_T$；Otherwise,pay 0.
• Payoff:

  $$\text{Payoff} = \begin{cases} S_T, & \text{if } S_T < K \\ 0, & \text{if } S_T \geq K \end{cases}$$

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2.2 Barrier Digital Option

DOWN_CASH_AT_TOUCH

• If the price of the underlying asset first touches the barrier price $H$（and $H < S_0$）during its existence，a fixed amount $C$shall be paid.
• Payoff:

  $$\text{Payoff} = \begin{cases} C, & \text{if } \min(S_t) \leq H \\ 0, & \text{otherwise} \end{cases}$$

DOWN_ASSET_AT_TOUCH

• If the price of the underlying asset first touches the barrier price of $H$(and $H < S_0$) during its existence, pay the asset value of $S_t$.

• Payoff:

  $$\text{Payoff} = \begin{cases} S_t, & \text{if } \min(S_t) \leq H \\ 0, & \text{otherwise} \end{cases}$$

UP_CASH_AT_TOUCH

• If the price of the underlying asset first touches the barrier price $H$(and $H > S_0$) during its existence, a fixed amount $C$ shall be paid.

• Payoff:

  $$\text{Payoff} = \begin{cases} C, & \text{if } \max(S_t) \geq H \\ 0, & \text{otherwise} \end{cases}$$

UP_ASSET_AT_TOUCH

• If the price of the underlying asset first touches the barrier price of $H$(and $H > S_0$) during its existence, pay the asset value of $S_t$.

• Payoff:

  $$\text{Payoff} = \begin{cases} S_t, & \text{if } \max(S_t) \geq H \\ 0, & \text{otherwise} \end{cases}$$

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2.3 Knock-In Digital Option

DOWN_IN_CASH_AT_EXPIRY

• If the price of the underlying asset touches the obstacle price $H$(and $H < S_0$) during its existence, a fixed amount $C$ will be paid upon maturity based on whether the conditions are met:

• Payoff:

  $$\text{Payoff} = \begin{cases} C, & \text{if } \min(S_t) \leq H \text{ and } S_T > K \\ 0, & \text{otherwise} \end{cases}$$

UP_IN_ASSET_AT_EXPIRY

• If the price of the underlying asset touches the barrier price of $H$(and $H > S_0$) during its existence, pay the asset value of $S_T$ at maturity:

• Payoff:

  $$\text{Payoff} = \begin{cases} S_T, & \text{if } \max(S_t) \geq H \text{ and } S_T > K \\ 0, & \text{otherwise} \end{cases}$$

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2.4 Knock-Out Digital Option

DOWN_OUT_CASH_AT_EXPIRY

• If the price of the underlying asset does not touch the barrier price $H$(and $H < S_0$) during its existence, a fixed amount $C$ will be paid upon maturity:

• Payoff:

  $$\text{Payoff} = \begin{cases} C, & \text{if } \min(S_t) > H \text{ and } S_T > K \\ 0, & \text{otherwise} \end{cases}$$

UP_OUT_ASSET_AT_EXPIRY

• If the price of the underlying asset does not touch the barrier price of $H$(and $H > S_0$) during its existence, the asset value of $S_T$ shall be paid upon maturity:

• Payoff:

  $$\text{Payoff} = \begin{cases} S_T, & \text{if } \max(S_t) < H \text{ and } S_T > K \\ 0, & \text{otherwise} \end{cases}$$

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2.5 其他组合（Knock-In or Knock-Out + Call/Put）

Similar to the Knock In and Knock Out options mentioned above, payment rules can be combined based on whether the obstacle price has been touched and the option type (CALL or PUT). For example:

DOWN_IN_CASH_CALL

• If the price of the underlying asset touches the barrier price $H$（and $H < S_0 \））during its existence, cash $C$ will be paid when $S_T > K$ at maturity:

• Payoff:

  $$\text{Payoff} = \begin{cases} C, & \text{if } \min(S_t) \leq H \text{ and } S_T > K \\ 0, & \text{otherwise} \end{cases}$$

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Summary

• Basic digital options (Cash or Nothing and Asset or Dating) have simple payments that depend on the relationship between the underlying price and the strike price.
• Barrier based digital options (At Touch, Knock In, Knock Out) have added barrier conditions, making payment rules more complex and flexible.
• The payment functions of various digital options can be flexibly combined, suitable for different perspectives and risk preferences on the volatility of underlying asset prices.