Interface Description

Dependencies/Packages

- python 3.8 or higher
- numpy
- scipy
- PyQt5

Usage

python main.py

Graphical User Interface

We are committed to providing users with a brief, efficient, and user-friendly graphical user interface (GUI). Screenshots are provided in the Appendix.

HomePage

Upon launching the program, users will first enter the main interface. In this interface, users can select the desired Option or Implied Volatility calculator by clicking on the corresponding buttons, which will lead them to the respective subpages.

SubPages

Each subpage corresponds to a specific calculator. Users are required to input relevant parameters as indicated by the prompt labels. Once the input is complete, clicking the green “Calculate Price/IV” button will yield the calculation result. To enhance the user experience, we have also designed two auxiliary functions: the orange “Clear Inputs” button allows users to clear all inputs with a single click for easier re-entry, while the “Back” button at the bottom of each subpage allows users to return to the main interface.

Note

When calculating the price of the Geometric Basket Option, the number of spot prices and volatilities entered must be at least two. Therefore, users must input an equal number of spot prices and volatilities, separated by commas (e.g. 100,105) in the respective input fields. If the input format is incorrect, the program will display an error message.

Functionality Description

Description of Directories and Files:

• gui/: This directory contains the code for the graphical user interface.
  • __init__.py: Initializes the gui package.
  • gui.py: Contains the main implementation of the GUI for the option pricer.
• options/: This directory holds the classes that define different types of options.
  • __init__.py: Initializes the options package.
  • american_option.py: Defines the AmericanOption class.
  • asian_option.py: Defines the base AsianOption class and potentially subclasses like GeometricAsianOption and ArithmeticAsianOption.
  • basket_option.py: Defines the base BasketOption class and potentially subclasses like GeometricBasketOption and ArithmeticBasketOption.
  • european_option.py: Defines the EuropeanOption class.
  • kiko_option.py: Defines the KIKOOption class.
  • option.py: Defines the base Option class with common attributes.
• pricer/: This directory contains the classes responsible for the pricing logic of different option types.
  • __init__.py: Initializes the pricer package.
  • binomial_tree_pricer.py: Implements the binomial tree method for pricing American options. (This file is not yet implemented.)
  • implied_volatility_calculator.py: Implements the logic for calculating implied volatility.
  • monte_carlo_pricer.py: Implements the Monte Carlo simulation for pricing various options. (This file is not yet implemented.)
• utils/: This directory can contain utility modules, such as for statistical calculations.(Not yet implemented)
• main.py: This is the main entry point of the application, likely responsible for initializing and running the GUI or providing a command-line interface.

This structure employs OOP principles to create a modular and maintainable option pricer, aiming to separate concerns, making the codebase more organized, maintainable, and easier to understand. Each module focuses on a specific aspect of the option pricer.

Class Diagram and Description for 'options' Module

The class diagram below illustrates the structure and relationships of the classes within the options module. The diagram is represented in a simplified text format for clarity, ignoring class attributes and methods for brevity.

Option <|-- EuropeanOption
Option <|-- AsianOption
Option <|-- BasketOption
Option <|-- AmericanOption
Option <|-- KIKOOption
AsianOption <|-- GeometricAsianOption
AsianOption <|-- ArithmeticAsianOption
BasketOption <|-- GeometricBasketOption
BasketOption <|-- ArithmeticBasketOption

The diagram above illustrates the class hierarchy within the options module.

• The Option class serves as a base class, defining common attributes for various option types, such as spot price, volatility, risk-free rate, maturity, and strike price. It also declares an abstract price() method.

• Specific option types like EuropeanOption, AsianOption, BasketOption, AmericanOption, and KIKOOption inherit from the Option class. This demonstrates inheritance, a key OOP principle.

• AsianOption and BasketOption are further specialized into GeometricAsianOption, ArithmeticAsianOption, GeometricBasketOption, and ArithmeticBasketOption to handle different calculation methods (geometric vs. arithmetic means).

• Each class encapsulates the data and behavior relevant to a particular option type. For example, KIKOOption includes attributes for barriers and rebate, and methods for price calculation and delta calculation, specific to KIKO options.

Test Results & Analysis

Assuming r(risk free interest rate) = 0.05, T(maturity) = 3, S0(spot price) = 100. Below are some test results for different options.

European Option

Tests

σ (volatility)	K (strike price)	q (repo rate)	Type	Price
0.3	100	0.20	Put	34.9281
0.3	110	0.20	Put	42.5734
0.4	100	0.20	Put	38.3535
0.3	100	0.10	Put	23.0717
0.3	100	0.20	Call	3.7385
0.3	110	0.20	Call	2.7767
0.4	100	0.20	Call	7.1639
0.3	100	0.10	Call	11.0827

Analysis Volatility (σ): Higher volatility increases option prices for both calls and puts, as it raises the likelihood of extreme price movements, enhancing the option's value. Strike Price (K): For calls, a higher strike price decreases the option price, while for puts, it increases the price, as it affects the intrinsic value. Repo Rate (q): A higher repo rate reduces call option prices and increases put option prices, as it lowers the expected future price of the underlying asset. Time to Maturity (T): Longer maturity generally increases option prices due to higher time value, allowing more time for favorable price movements. Risk-Free Rate (r): A higher risk-free rate increases call option prices and decreases put option prices, as it impacts the present value of the strike price. Spot Price (S0): Higher spot prices increase call option prices and decrease put option prices, as it directly affects the intrinsic value.

Implied Volatility

Tests

K (strike price)	q (repo rate)	Type	Premium	IV
100	0.20	Put	5	X
100	0.20	Call	5	0.3385
110	0.20	Put	5	X
110	0.20	Call	5	0.3725
100	0.10	Put	5	X
100	0.10	Call	5	0.1792
100	0.20	Put	10	X
100	0.20	Call	10	0.4766

Analysis Strike Price (K): Implied volatility often exhibits a "smile" or "skew" pattern, where options with strike prices far from the current spot price (deep in-the-money or out-of-the-money) tend to have higher implied volatilities.

Option Premium: Higher observed premiums generally lead to higher implied volatilities, as the model adjusts to match the market price.

Time to Maturity (T): Implied volatility can vary with time to maturity, often showing higher values for shorter-term options due to increased sensitivity to market movements.

Repo Rate (q) and Risk-Free Rate (r): Changes in these rates indirectly affect implied volatility by altering the theoretical option price, which the model uses to match the observed premium.

Asian Option

Geometric Asian Option (closed-form formula)

Tests

σ (volatility)	K (strike price)	n (# observations)	Type	Price
0.3	100	50	Put	8.4827
0.3	100	100	Put	8.4311
0.4	100	50	Put	12.5588
0.3	100	50	Call	13.2591
0.3	100	100	Call	13.1388
0.4	100	50	Call	15.7598

Analysis Volatility (σ): Higher volatility increases the option price for both calls and puts, as it raises the likelihood of extreme price movements, enhancing the option's value.

Strike Price (K):

• For calls, a higher strike price decreases the option price, as it reduces the intrinsic value.
• For puts, a higher strike price increases the option price, as it raises the potential payoff. Number of Observations (n): Increasing the number of observations slightly reduces the option price, as averaging over more points smooths out extreme price movements, reducing the option's value.

Option Type: Call options are generally more expensive than put options for the same parameters when the spot price is higher than the strike price, due to the intrinsic value difference.

Arithmetic Asian Option (with MC method with control variate)

Tests

σ (volatility)	K (strike price)	n (# observations)	Type	Use_CV	# Paths	Price	CI
0.3	100	50	Put	False	100000	7.7910	(7.722078287068324, 7.859964997614279)
0.3	100	50	Put	True	100000	7.8023	(7.797849911560479, 7.8067432517246464)
0.3	100	100	Put	False	100000	7.7626	(7.693792516239495, 7.831478742218251)
0.3	100	100	Put	True	100000	7.7545	(7.7501155527589125, 7.7587987868185175)
0.4	100	50	Put	False	100000	11.2777	(11.187985634651469, 11.367440381224927)
0.4	100	50	Put	True	100000	11.2854	(11.277607015719083, 11.293212640382963)
0.3	100	50	Call	False	100000	14.6913	(14.547971128378347, 14.834592305826526)
0.3	100	50	Call	True	100000	14.7328	(14.72207928162502, 14.743591157848412)
0.3	100	100	Call	False	100000	14.6231	(14.481474878060652, 14.764627047724414)
0.3	100	100	Call	True	100000	14.6089	(14.598244584709803, 14.619542606038408)
0.4	100	50	Call	False	100000	18.1572	(17.956540777657548, 18.357873729230892)
0.4	100	50	Call	True	100000	18.2142	(18.193897295219447, 18.234411652013694)

Analysis Volatility (σ): Higher volatility increases the option price for both calls and puts, as it raises the likelihood of extreme price movements, enhancing the option's value.

Strike Price (K):

• For calls, a higher strike price decreases the option price, as it reduces the intrinsic value.
• For puts, a higher strike price increases the option price, as it raises the potential payoff. Number of Observations (n): Increasing the number of observations slightly reduces the option price, as averaging over more points smooths out extreme price movements, reducing the option's value.

Use of Control Variate (Use_CV): Using control variates improves the accuracy of Monte Carlo simulations, leading to more stable and slightly adjusted option prices.

Number of Paths (# Paths): A higher number of simulation paths reduces the confidence interval width, improving the precision of the estimated price.

Basket Option

Geometric Basket Option (closed-form formula)

Tests

S1	S2	σ1	σ2	K	ρ(correlation)	Type	Price
100	100	0.3	0.3	100	0.5	Put	11.4916
100	100	0.3	0.3	100	0.9	Put	12.6224
100	100	0.1	0.3	100	0.5	Put	6.5864
100	100	0.3	0.3	80	0.5	Put	4.7116
100	100	0.3	0.3	120	0.5	Put	21.2891
100	100	0.5	0.5	100	0.5	Put	23.4691
100	100	0.3	0.3	100	0.5	Call	22.1021
100	100	0.3	0.3	100	0.9	Call	25.8788
100	100	0.1	0.3	100	0.5	Call	17.9247
100	100	0.3	0.3	80	0.5	Call	32.5363
100	100	0.3	0.3	120	0.5	Call	14.6855
100	100	0.5	0.5	100	0.5	Call	28.4494

Analysis Spot Prices (S1, S2): Higher spot prices generally increase the price of call options and decrease the price of put options, as they directly affect the intrinsic value of the option.

Volatilities (σ1, σ2): Higher volatilities increase the option price for both calls and puts, as they raise the likelihood of extreme price movements, enhancing the option's value.

Strike Price (K):

• For calls, a higher strike price decreases the option price, as it reduces the intrinsic value.
• For puts, a higher strike price increases the option price, as it raises the potential payoff. Correlation (ρ): Higher correlation between assets increases the price of call options and decreases the price of put options, as it reduces diversification effects and increases the overall basket volatility.

Option Type: Call options are generally more expensive than put options for the same parameters when the spot prices are higher than the strike price, due to the intrinsic value difference.

Arithmetic Basket Option (Monte Carlo simulation with/without control variate)

Tests

S1	S2	σ1	σ2	K	ρ(correlation)	Type	Use_CV	# Paths	Price	CI
100	100	0.3	0.3	100	0.5	Put	False	100000	10.4947	(10.400587253254375, 10.58891027105442)
100	100	0.3	0.3	100	0.5	Put	True	100000	10.5778	(10.565708926266245, 10.589920463327639)
100	100	0.3	0.3	100	0.9	Put	False	100000	12.3403	(12.235396799206924, 12.44526458138919)
100	100	0.3	0.3	100	0.9	Put	True	100000	12.4273	(12.424579436314026, 12.43005646081458)
100	100	0.1	0.3	100	0.5	Put	False	100000	5.4840	(5.426963969348094, 5.540936349189497)
100	100	0.1	0.3	100	0.5	Put	True	100000	5.5218	(5.513275437068496, 5.530254635575429)
100	100	0.3	0.3	80	0.5	Put	False	100000	4.2185	(4.163473175181289, 4.273509275891397)
100	100	0.3	0.3	80	0.5	Put	True	100000	4.2498	(4.242131184170442, 4.257538026355852)
100	100	0.3	0.3	120	0.5	Put	False	100000	19.7546	(19.621174565629453, 19.88804658649657)
100	100	0.3	0.3	120	0.5	Put	True	100000	19.8794	(19.863121946768093, 19.895739867820826)
100	100	0.5	0.5	100	0.5	Put	False	100000	20.9408	(20.79479633274701, 21.086882338303507)
100	100	0.5	0.5	100	0.5	Put	True	100000	21.0787	(21.050526809250954, 21.106827227374183)
100	100	0.3	0.3	100	0.5	Call	False	100000	24.5399	(24.298262378433883, 24.781493598896297)
100	100	0.3	0.3	100	0.5	Call	True	100000	24.4989	(24.467841044184553, 24.529861991034622)
100	100	0.3	0.3	100	0.9	Call	False	100000	26.3310	(26.055477767340346, 26.606513782017462)
100	100	0.3	0.3	100	0.9	Call	True	100000	26.3546	(26.348236277024604, 26.360913745824455)
100	100	0.1	0.3	100	0.5	Call	False	100000	19.5388	(19.365515649235302, 19.7120119465996)
100	100	0.1	0.3	100	0.5	Call	True	100000	19.4450	(19.42586074736659, 19.464182209297366)
100	100	0.3	0.3	80	0.5	Call	False	100000	35.4778	(35.20851058547995, 35.74704937561642)
100	100	0.3	0.3	80	0.5	Call	True	100000	35.3814	(35.3493343854587, 35.41341795150587)
100	100	0.3	0.3	120	0.5	Call	False	100000	16.5856	(16.37569171631144, 16.795468831833663)
100	100	0.3	0.3	120	0.5	Call	True	100000	16.5885	(16.55922127974048, 16.617835476973926)
100	100	0.5	0.5	100	0.5	Call	False	100000	34.9923	(34.520485466349086, 35.46420839737982)
100	100	0.5	0.5	100	0.5	Call	True	100000	34.9886	(34.88320310212911, 35.09394896352364)
100	100	0.3	0.3	100	0.5	Put	False	1000000	24.5295	(24.453128632570863, 24.605970896525665)
100	100	0.3	0.3	100	0.5	Put	True	1000000	24.4954	(24.48556162703866, 24.50523515961164)

Analysis Spot Prices (S1, S2): Higher spot prices generally increase the price of call options and decrease the price of put options, as they directly affect the intrinsic value of the option.

Volatilities (σ1, σ2): Higher volatilities increase the option price for both calls and puts, as they raise the likelihood of extreme price movements, enhancing the option's value.

Strike Price (K):

• For calls, a higher strike price decreases the option price, as it reduces the intrinsic value.
• For puts, a higher strike price increases the option price, as it raises the potential payoff. Correlation (ρ): Higher correlation between assets increases the price of call options and decreases the price of put options, as it reduces diversification effects and increases the overall basket volatility.

Use of Control Variate (Use_CV): Using control variates improves the accuracy of Monte Carlo simulations, leading to more stable and slightly adjusted option prices.

Number of Paths (# Paths): A higher number of simulation paths reduces the confidence interval width, improving the precision of the estimated price.

KIKO Option

Tests

σ	T	L(Lower Bound)	U(Upper Bound)	N (# Observations)	R (rebate)	Price	Delta	CI
0.20	2.0	80	125	24	1.5	6.0092	-0.0931	5.9454, 6.0731
0.20	2.0	80	125	48	1.5	6.0990	-0.1793	6.0353, 6.1628
0.20	2.0	80	125	24	2.0	6.2190	-0.0677	6.1557, 6.2823

Analysis Volatility (σ): Higher volatility increases the option price as it raises the likelihood of the underlying asset hitting the barriers, enhancing the option's value.

Time to Maturity (T): Longer maturity generally increases the option price, as it provides more time for the underlying asset to hit the barriers.

Lower and Upper Barriers (L, U):

• A lower lower barrier (L) increases the likelihood of a knock-in event, raising the option price.
• A higher upper barrier (U) decreases the likelihood of a knock-out event, also raising the option price. Number of Observations (N): More frequent observations slightly increase the option price, as the barriers are checked more often, increasing the chance of hitting them.

Rebate (R): A higher rebate increases the option price, as it provides additional value in the event of a knock-out.

American Option

Tests

S	σ (volatility)	rate	T	K	Option	Steps	Price
50	0.4	0.1	2	40	Put	200	2.9462
50	0.4	0.1	2	50	Put	200	6.1882
50	0.4	0.1	2	70	Put	200	15.9395

Analysis Spot Price (S): Higher spot prices generally decrease the price of put options, as they reduce the intrinsic value of the option. Volatility (σ): Higher volatility increases the option price for both calls and puts, as it raises the likelihood of extreme price movements, enhancing the option's value. Risk-Free Rate (rate): A higher risk-free rate generally increases the price of call options and decreases the price of put options, as it affects the present value of the strike price. Time to Maturity (T): Longer maturity generally increases option prices due to higher time value, allowing more time for favorable price movements. Strike Price (K):

• For puts, a higher strike price increases the option price, as it raises the potential payoff. Steps: More steps in the binomial tree generally lead to more accurate option prices, as they provide a finer resolution of the underlying asset's price movements.

Extensions

Basket Option with more than 2 assets For Basket Option, Geometric version can handle more than 2 assets, but Arithmetic version can only handle 2 assets here.

S1	S2	S3	σ1	σ2	σ3	K	ρ(correlation)	Type	Price
100	100	100	0.3	0.3	0.3	100	0.5	Put	12.3920
100	100	100	0.3	0.3	0.3	100	0.9	Put	16.2390
100	100	100	0.1	0.3	0.3	100	0.5	Put	8.3141
100	100	100	0.3	0.3	0.3	80	0.5	Put	5.0169
100	100	100	0.3	0.3	0.3	120	0.5	Put	22.9624
100	100	100	0.5	0.5	0.5	100	0.5	Put	26.9640
....	....	....	....	....	....	...	...	...	...

Appendix (Screenshots)

HomePage European Option Implied Volatility Geometric Asian Option Arithmetic Asian Option Geometric Basket Option Arithmetic Basket Option KIKO Option American Option with Binomial Tree