ASC570 Financial Economics II UITM  Assignment Sample Malaysia

ASC570 Financial Economics II is a course offered at UITM in Malaysia. This Financial Economics  course focuses on the application of hedging strategies using options. Students will learn about the principles of pricing options, modeling stock prices using lognormal distribution, and the impact of Brownian motion on option prices. Additionally, the course covers the use of Ito’s Lemma to analyze and evaluate option prices. Through this course, students will gain a deeper understanding of financial economics and how options can be utilized for risk management and investment strategies.

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Assignment Activity 1: Analyse the Black-Scholes formula for option pricing based on different underlying assets, varying option Greeks and highlight the risk management practices of modern institution.

The Black-Scholes formula is a mathematical model used to calculate the theoretical price of European-style options. It assumes that the underlying asset’s price follows a geometric Brownian motion with constant volatility and that there are no dividends paid during the option’s life. The formula takes into account various factors, including the underlying asset price, strike price, time to expiration, risk-free interest rate, and volatility.

The formula for calculating the price of a call option is:

C = S * N(d1) – X * e^(-rT) * N(d2)

And the formula for calculating the price of a put option is:

P = X * e^(-rT) * N(-d2) – S * N(-d1)

Where:

• C and P represent the call and put option prices, respectively.
• S is the current price of the underlying asset.
• X is the strike price of the option.
• T is the time to expiration in years.
• r is the risk-free interest rate.
• N(d1) and N(d2) are cumulative standard normal distribution functions.
• d1 = (ln(S/X) + (r + (σ^2)/2) * T) / (σ * sqrt(T))
• d2 = d1 – σ * sqrt(T)

The Black-Scholes formula provides an estimate of the fair value of options based on the inputs mentioned above. It is widely used in financial markets for option pricing and risk management purposes. However, it has certain assumptions and limitations, such as the assumption of constant volatility, efficient markets, and no transaction costs. Deviations from these assumptions can lead to discrepancies between the model’s predictions and real-world prices.

Risk management practices in modern institutions involve monitoring and managing the various risks associated with options. These practices include:

• Sensitivity Analysis: Assessing the impact of changes in underlying asset prices, volatility, interest rates, and time to expiration on the option’s value. This analysis involves measuring the option Greeks, which are indicators of an option’s sensitivity to these factors (delta, gamma, theta, vega, and rho).
• Portfolio Hedging: Constructing portfolios of options and underlying assets to offset the risks associated with fluctuations in market conditions. This involves adjusting the portfolio’s composition to maintain desired risk exposures.
• Value-at-Risk (VaR) Analysis: Quantifying the potential loss in the value of an options portfolio under adverse market conditions, considering the statistical properties of the underlying assets.
• Stress Testing: Simulating extreme market scenarios to assess the impact on options portfolios and identify vulnerabilities.
• Risk Limits and Controls: Establishing risk limits and implementing control mechanisms to manage exposures and prevent excessive risk-taking.
• Volatility Management: Monitoring and managing the volatility of the underlying asset through strategies such as implied volatility trading and volatility arbitrage.
• Risk Monitoring Tools: Utilizing sophisticated risk management systems and technologies to monitor, measure, and report on options-related risks in real-time.

Assignment Activity 2: Construct risk management techniques under delta hedging method.

Delta hedging is a risk management technique used to minimize the risk exposure of an options position to small changes in the price of the underlying asset. Delta measures the sensitivity of an option’s price to changes in the underlying asset price. By maintaining a delta-neutral position, the option’s value becomes less sensitive to small price movements in the underlying asset.

The steps involved in delta hedging are as follows:

• Calculate the delta of the option: Delta represents the rate of change of the option price concerning changes in the underlying asset price. For a call option, delta ranges from 0 to 1, while for a put option, it ranges from -1 to 0.
• Determine the number of options to hedge: The number of options needed to hedge is determined by dividing the delta of the options position by the delta of the underlying asset. For example, if the delta of the options position is 0.5 and the delta of the underlying asset is 0.4, you would need to hedge by selling 1.25 times the number of options.
• Adjust the hedge regularly: As the underlying asset price changes, the delta of the options position will change. To maintain a delta-neutral position, the hedge needs to be adjusted by buying or selling additional options or the underlying asset.
• Monitor and rebalance: Continuously monitor the delta of the options position and the underlying asset to ensure they remain in balance. Rebalance the hedge as needed to maintain a delta-neutral position.

Delta hedging helps reduce the exposure to small price movements in the underlying asset, but it does not eliminate all risks. It is most effective in stable market conditions where the options position can be frequently adjusted. In volatile markets, the effectiveness of delta hedging may be limited, as the delta of the options position can change rapidly.

Assignment Activity 3 :Differentiate some basic kinds of exotic options, including Asian, barrier, compound, gap and exchanges.

Exotic options are derivative contracts that possess complex features beyond the standard call and put options. Here are some basic types of exotic options:

Asian Options:  Asian options derive their payoff from the average price of the underlying asset over a specific period rather than its price at expiration. They are used to manage the risk associated with price fluctuations during the life of the option.

Barrier Options: Barrier options have an additional feature that depends on the underlying asset’s price reaching or breaching a predetermined barrier level before or at expiration. These options can be further classified as:

• Up-and-In: The option becomes active and can be exercised only if the underlying asset price reaches the barrier level.
• Up-and-Out: The option ceases to exist if the underlying asset price reaches the barrier level.
• Down-and-In: The option becomes active if the underlying asset price falls to the barrier level.
• Down-and-Out: The option expires worthless if the underlying asset price falls to the barrier level.

Compound Options: Compound options are options on options. They give the holder the right to buy or sell another option at a later date, at a predetermined price. Compound options can be used to capture specific market movements or to delay the decision to enter another options contract.

Gap Options: Gap options provide a payoff based on the difference between the underlying asset’s price at expiration and its price at the start of the option. They are designed to provide protection against extreme market movements.

Exchange Options: Exchange options are options contracts where the underlying asset is not a traditional security but rather an exchange rate between two currencies or an interest rate. They are used to hedge against foreign exchange rate fluctuations or interest rate movements.

These exotic options offer investors additional flexibility and customization in managing their risk exposure and investment strategies. However, they are often more complex and less liquid than standard options, requiring sophisticated pricing models and risk management techniques.

Assignment Activity 4:Explain the parameters of the Lognormal Distribution on stocks.

The lognormal distribution is commonly used to model the distribution of stock prices because it allows for the possibility of positive returns while restricting them from going below zero. 

The lognormal distribution is characterized by two key parameters:

• Drift (μ): The drift represents the expected or average rate of return of the stock over time. It determines the center of the lognormal distribution curve and influences the long-term trend of the stock price.
• Volatility (σ): Volatility measures the degree of variation or dispersion in stock returns. It represents the standard deviation of the continuously compounded returns and determines the width or spread of the lognormal distribution curve. Higher volatility indicates greater price fluctuations and vice versa.

The lognormal distribution assumes that stock returns are continuously compounded and follow a normal distribution after taking the natural logarithm. The resulting distribution of stock prices is positively skewed, meaning that extreme positive returns are more likely to occur than extreme negative returns.

The parameters of the lognormal distribution are often estimated using historical data. The historical returns are used to calculate the average return (drift) and the standard deviation of returns (volatility). These estimates are then used in pricing models like the Black-Scholes model to determine option prices or in risk management techniques to assess the potential range of future stock prices.

Assignment Activity 5 :Formulate Brownian motion, Ito’s Lemma and Monte Carlo simulation in valuating the price of the option assuming lognormal stock prices.

Brownian Motion: Brownian motion is a mathematical model used to describe the random movements of particles in continuous motion. In the context of finance, it is used to model the stochastic movement of stock prices. Brownian motion assumes that stock price changes occur randomly and are independent of previous changes. It is characterized by two parameters:

• • Drift (μ): The average rate of return or the expected growth rate of the stock price.
  • Volatility (σ): The standard deviation of the stock price returns, representing the degree of randomness or uncertainty.
  • Ito’s Lemma: Ito’s Lemma is a mathematical formula used in stochastic calculus to calculate the differential of a function of a stochastic process. In option pricing, it is employed to derive the differential equation governing the movement of the option price with respect to time and the underlying asset’s price. This equation, known as the Black-Scholes equation, is then used to solve for the option price. 

    Monte Carlo Simulation: Monte Carlo simulation is a computational technique used to estimate the value of an option by simulating numerous possible future stock price paths and calculating the option’s payoff for each path. It involves the following steps:

    • Generate a large number of random paths for the stock price using the parameters of the lognormal distribution (drift and volatility).
    • For each path, calculate the corresponding option payoff based on the option’s contract specifications.
    • Discount the expected payoff back to the present value using the risk-free interest rate.
    • Average the discounted payoffs across all simulated paths to obtain the estimated option price.

Monte Carlo simulation allows for the incorporation of complex option features, such as path-dependent or exotic options, that cannot be easily priced using closed-form formulas. It provides a flexible and robust method for valuing options under various market conditions and can be used to assess the impact of different parameters and market scenarios on option prices.

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