Foreign Exchange Asian Options – Pricing Principles

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For an introduction to foreign exchange Asian options, refer to the article Foreign Exchange Asian Options.

Asian options are a type of option contract where the payoff depends on the average value of the underlying asset over a specified period. These options are particularly popular in over-the-counter energy markets and other illiquid commodity markets. The averaging feature reduces price volatility, making the options cheaper and less susceptible to market manipulation.

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Classification of Asian Options

Asian options can be categorized along multiple dimensions:

1. Average Price Type:

   • Average Strike (Floating Strike): The strike price is determined by the average price over the observation period.
   • Average Rate (Fixed Strike): The payoff depends on the difference between the average price and a fixed strike price.

2. Average Price Calculation:

   • Arithmetic Average: The arithmetic mean of observed prices.
   • Geometric Average: The geometric mean of observed prices.

3. Observation Period:

   • Discrete: Observation dates are specified (e.g., daily, weekly, monthly).
   • Continuous: Every day from the start date to the expiration date is an observation date.

4. Exercise Style:

   • American: Can be exercised at any time before expiration.
   • European: Can only be exercised at expiration.

Additionally, Asian options can be divided based on their life cycle:

• Pre-Averaging Period: The option has not yet entered the averaging period.
• In-Averaging Period: The option is within the averaging period.

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Pricing and Valuation Parameters

The following table outlines the key parameters for pricing and valuing Asian options:

Parameter	Description	Example	Notes
Call/Put	Option Type	Call	Call/Put
Strike Type	Fixed or Floating Strike	Fixed	Fixed/Floating
Average Type	Arithmetic or Geometric Average	Arithmetic	Arithmetic/Geometric
Exercise Type	European or American Exercise	European	European/American
Reference Date	Calculation Date	2022/10/8
Premium Date	Date for Premium Payment	2022/10/10
Expiry Date	Option Expiration Date	2023/10/8
Settlement Date	Settlement Date	2023/10/10
Spot Price	Current Spot Price	6.8
Strike Price	Option Strike Price	6.9
First Average Date	Start of Averaging Period	2022/05/24	May differ from the Reference Date
Fixing Frequency	Frequency of Observations	Monthly	Daily, Weekly, Monthly
Cumulative Average Rate	Average Rate for In-Averaging Options	7.0	For options in the averaging period
Number of Fixings	Total Number of Observations	12	Calculated based on dates
Number of Fixings Done	Completed Observations	2	Calculated based on dates

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Payoff Calculation

The payoff for Asian options depends on the average price type:

Payoff	Call	Put
Average Strike	max(S - A, 0)	max(A - S, 0)
Average Rate	max(A - X, 0)	max(X - A, 0)

Where:

• S: Underlying Price
• X: Strike Price
• A: Average Price (Strike or Rate)

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Average Price Calculation

Arithmetic Average

$$\text{Average}_{\text{Arithmetic}} = \frac{\sum_{i=1}^n S_i}{n}$$

Geometric Average

$$\text{Average}_{\text{Geometric}} = \exp\left(\frac{\sum_{i=1}^n \log S_i}{n}\right)$$

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Pricing Methods

Asian options can be priced using several methods:

1. Analytical Models (Closed-Form Solutions): Suitable for geometric average options.
2. Binomial/Trinomial Trees: Useful for discrete averaging.
3. Monte Carlo Simulation: Flexible and accurate for complex scenarios.
4. PDE Models: Numerical solutions for partial differential equations.

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Closed-Form Solutions

For geometric average options, closed-form solutions exist due to the log-normal distribution of geometric averages. The pricing formula for geometric average-rate options is derived from the Black-Scholes framework.

For arithmetic average options, closed-form solutions are not available due to the non-log-normal distribution of arithmetic averages. However, approximations can be used.

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Monte Carlo Simulation

Monte Carlo simulation is widely used for pricing Asian options, especially for arithmetic averages and complex scenarios. The steps include:

1. Define Payoff: Calculate the payoff based on the average price.
2. Generate Simulated Paths: Use geometric Brownian motion to simulate underlying price paths.
3. Calculate NPV: Discount the average payoffs to obtain the option price.

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Comparison of Methods

Scenario	Closed-Form	Monte Carlo
Geometric Average Options	Accurate and Fast	Accurate but Slower
Arithmetic Average Options	Approximations Available	Highly Accurate
In-Averaging Period Options	Limited Accuracy	Highly Accurate

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Performance Comparison (GPU vs. CPU)

With advancements in computing power, particularly GPU technology, Monte Carlo simulations can be significantly accelerated. Below is a performance comparison:

Simulations	Closed-Form	Monte Carlo (Single Thread)	Monte Carlo (10 Cores)	Monte Carlo (GPU)
100,000	0.168 sec	1.2 sec	0.112 sec	0.012 sec
1,000,000	0.168 sec	9.2 sec	0.902 sec	0.013 sec
10,000,000	0.168 sec	90 sec	8.30 sec	0.223 sec

Hardware Specifications:

• CPU: Intel(R) Core(TM) i9-10900K CPU @ 3.70GHz
• GPU: NVIDIA GeForce RTX 3080 Ti

For more details on Asian option solutions, refer to:

• Meridian Asian Option Solutions