Are you ready to dive into the exciting world of quantitative finance and tackle those tricky math problems that often pop up in PSEi Quant interviews? Well, buckle up, because we're about to break down some key concepts and strategies to help you nail those questions. We're going to focus on making these problems easier to understand and solve. This guide is crafted to equip you with the knowledge and confidence to shine. Let's get started, guys!

Understanding the Core Concepts

Before we jump into specific problems, let's make sure we're all on the same page with some fundamental concepts. Quantitative finance relies heavily on a few key areas of mathematics, including calculus, probability, statistics, and linear algebra. Let’s explore each of these to ensure we have a solid base.

Calculus: At the heart of many financial models, calculus provides the tools to understand continuous change. Concepts like derivatives and integrals are crucial for pricing derivatives and understanding risk management. Derivatives allow us to determine rates of change, essential for optimizing portfolios and analyzing market trends. Integrals help us accumulate values over time, useful for calculating present and future values of cash flows. To be effective in quantitative finance, a solid understanding of limits, continuity, differentiation, and integration is indispensable. These concepts enable you to model complex financial phenomena and make informed decisions based on mathematical insights. For instance, understanding how interest rates change over time or how stock prices evolve requires a firm grasp of calculus principles.

Probability: Probability is fundamental in finance because it allows us to quantify uncertainty. In financial markets, very little is certain, so understanding the likelihood of different outcomes is vital. Key concepts include random variables, probability distributions, and expectation. A random variable is a variable whose value is a numerical outcome of a random phenomenon. Probability distributions describe the likelihood of each possible value of a random variable. Expectation is the average value we expect a random variable to take. These concepts allow us to model various financial scenarios, from stock price movements to the likelihood of default on a loan. For example, the Black-Scholes model, a cornerstone of options pricing, relies heavily on probability distributions to estimate the probability of an option expiring in the money. Furthermore, understanding probability allows us to make informed decisions when faced with uncertain market conditions, helping to minimize risk and maximize returns. The use of probability extends to risk management, where it is used to assess the likelihood of adverse events and develop strategies to mitigate their impact.

Statistics: Closely related to probability, statistics allows us to analyze data and make inferences about populations. In finance, we often use statistical methods to analyze historical data and make predictions about future market behavior. Key concepts include hypothesis testing, regression analysis, and time series analysis. Hypothesis testing allows us to test claims about a population based on sample data. Regression analysis helps us understand the relationship between two or more variables. Time series analysis allows us to analyze data that is collected over time, such as stock prices or interest rates. For example, we might use regression analysis to understand the relationship between interest rates and stock prices, or time series analysis to forecast future stock prices based on historical data. Statistics also plays a crucial role in portfolio management, where it is used to estimate the risk and return of different assets and construct portfolios that meet specific investment objectives. By applying statistical methods, financial professionals can make data-driven decisions, improve their understanding of market dynamics, and enhance their ability to manage risk.

Linear Algebra: Linear algebra provides the mathematical tools for dealing with systems of linear equations and transformations. In finance, it is used in portfolio optimization, risk management, and the pricing of complex financial instruments. Key concepts include vectors, matrices, and linear transformations. Vectors are used to represent portfolios of assets, matrices are used to represent relationships between different variables, and linear transformations are used to transform data from one form to another. For example, linear algebra is used to solve systems of equations that arise in portfolio optimization problems, where we seek to find the portfolio that maximizes return for a given level of risk. It is also used in risk management to calculate the covariance matrix of asset returns, which is a measure of how the returns of different assets move together. Moreover, linear algebra is essential in pricing complex derivatives, where it is used to solve partial differential equations that describe the evolution of the derivative's price over time.

Practice Problems and Solutions

Okay, theory is great, but let's get our hands dirty with some practice problems. These are designed to mirror the types of questions you might encounter in a PSEi Quant interview. Let's work through them step-by-step so you understand the methodology.

Problem 1: Option Pricing with Black-Scholes

Question: A stock is currently trading at ₱100. The risk-free interest rate is 5% per annum, and the stock's volatility is 20%. Using the Black-Scholes model, what is the price of a European call option with a strike price of ₱105 and an expiry date in six months?

Solution:

The Black-Scholes model is a cornerstone in option pricing, and being comfortable with it is essential. The formula is:

C = S * N(d1) - X * e^(-rT) * N(d2)

Where:

• C = Call option price
• S = Current stock price
• X = Strike price
• r = Risk-free interest rate
• T = Time to expiration
• N(x) = Cumulative standard normal distribution function
• d1 = [ln(S/X) + (r + (σ^2)/2) * T] / (σ * sqrt(T))
• d2 = d1 - σ * sqrt(T)

Let's plug in the values:

• S = ₱100
• X = ₱105
• r = 0.05
• T = 0.5 (6 months)
• σ = 0.20

First, calculate d1 and d2:

d1 = [ln(100/105) + (0.05 + (0.20^2)/2) * 0.5] / (0.20 * sqrt(0.5))

d1 ≈ -0.0476

d2 = -0.0476 - 0.20 * sqrt(0.5)

d2 ≈ -0.1887

Now, find N(d1) and N(d2). You'll need a standard normal distribution table or a calculator with this functionality. For simplicity, let's assume:

N(-0.0476) ≈ 0.4809

N(-0.1887) ≈ 0.4253

Finally, calculate the call option price:

C = 100 * 0.4809 - 105 * e^(-0.05 * 0.5) * 0.4253

C ≈ 48.09 - 105 * 0.9753 * 0.4253

C ≈ 48.09 - 43.44

C ≈ ₱4.65

So, the approximate price of the European call option is ₱4.65. Remember, accuracy is key here, so make sure you're comfortable using a standard normal distribution table or a calculator for precise values.

Problem 2: Portfolio Optimization

Question: An investor wants to create a portfolio using two assets: Asset A and Asset B. Asset A has an expected return of 10% and a standard deviation of 15%. Asset B has an expected return of 15% and a standard deviation of 20%. The correlation between the returns of the two assets is 0.3. What is the portfolio weight of Asset A that minimizes the portfolio's variance?

Solution:

Portfolio optimization is a common theme in quant interviews. The goal here is to find the weights that minimize the portfolio's risk (variance).

The formula for the variance of a two-asset portfolio is:

σp^2 = wA^2 * σA^2 + wB^2 * σB^2 + 2 * wA * wB * ρAB * σA * σB

Where:

• σp^2 = Portfolio variance
• wA = Weight of Asset A
• wB = Weight of Asset B
• σA = Standard deviation of Asset A
• σB = Standard deviation of Asset B
• ρAB = Correlation between Asset A and Asset B

Since wA + wB = 1, we can write wB = 1 - wA. Substitute this into the variance formula:

σp^2 = wA^2 * σA^2 + (1 - wA)^2 * σB^2 + 2 * wA * (1 - wA) * ρAB * σA * σB

To minimize the variance, we need to take the derivative of σp^2 with respect to wA and set it to zero:

d(σp^2)/dwA = 2 * wA * σA^2 - 2 * (1 - wA) * σB^2 + 2 * (1 - 2 * wA) * ρAB * σA * σB = 0

Now, let's plug in the values:

• σA = 0.15
• σB = 0.20
• ρAB = 0.3

2 * wA * (0.15)^2 - 2 * (1 - wA) * (0.20)^2 + 2 * (1 - 2 * wA) * 0.3 * 0.15 * 0.20 = 0

0. 045 * wA - 0.08 + 0.08 * wA + 0.018 - 0.036 * wA = 0

1. 045 * wA + 0.08 * wA - 0.036 * wA = 0.08 - 0.018

2. 089 * wA = 0.062

wA = 0.062 / 0.089

wA ≈ 0.6966

So, the portfolio weight of Asset A that minimizes the portfolio's variance is approximately 69.66%. This means the investor should allocate about 69.66% of their portfolio to Asset A and the remaining 30.34% to Asset B to minimize risk.

Problem 3: Expected Value Calculation

Question: A trader is considering investing in a project with the following possible outcomes: There is a 40% chance of making a profit of ₱50,000, a 30% chance of breaking even (₱0 profit), and a 30% chance of losing ₱20,000. What is the expected value of this investment?

Solution:

Expected value problems are common and test your understanding of probability-weighted outcomes.

The formula for expected value (EV) is:

EV = Σ (Probability of Outcome * Value of Outcome)

In this case:

• Outcome 1: Profit of ₱50,000 with a probability of 40% (0.4)
• Outcome 2: Profit of ₱0 with a probability of 30% (0.3)
• Outcome 3: Loss of ₱20,000 with a probability of 30% (0.3)

Now, let's calculate the expected value:

EV = (0.4 * ₱50,000) + (0.3 * ₱0) + (0.3 * -₱20,000)

EV = ₱20,000 + ₱0 - ₱6,000

EV = ₱14,000

Therefore, the expected value of this investment is ₱14,000. This means that, on average, the trader can expect to make a profit of ₱14,000 if they invest in this project.

Key Strategies for Success

So, you've got the concepts and you've seen some examples. Now, let's talk strategy. How do you walk into that interview room and crush it? Here are a few tips:

1. Practice, Practice, Practice: The more problems you solve, the more comfortable you'll become with the concepts. Seek out practice problems from various sources, and don't be afraid to tackle challenging questions. Consistent practice will help you develop a deeper understanding of the material and improve your problem-solving skills.

2. **Understand the