Quantitative Finance

Question 1

Geometric Brownian Motion (GBM) SDE

Answer 1

dSₜ = rSₜdt + σSₜdWₜ
where dWₜ ≈ Z√Δt

Question 2

Solution to the Geometric Brownian Motion (GBM) SDE

Answer 2

Question 3

Binomial Option Pricing Model

Answer 3

Question 4

Black-Scholes-Merton European Option Pricing Formulas

Answer 4

Question 5

Black–Scholes–Merton (BSM) PDE

Answer 5

Question 6

Ito’s Lemma

Answer 6

Ito’s Lemma is a fundamental result in stochastic calculus that provides a way to compute the differential of a function of a stochastic process. It is especially important in finance for deriving the dynamics of derivative prices in models where the underlying asset follows a stochastic process, such as the Black-Scholes model

dSₜ = μdt + σdWₜ
where dWₜ ≈ Z√Δt

df = (∂𝑓/∂t)dt + (∂𝑓/∂S)dSₜ + 1/2(∂²𝑓/∂S²)(dSₜ)²

Question 7

Black-Scholes-Merton Model

Assumptions, derivation, limitations

Answer 7

1. Efficient Markets: The model assumes markets are frictionless, meaning there are no transaction costs, taxes, or other frictions. Securities can be traded without impacting their price.
2. No Dividends: The model assumes that the underlying stock does not pay dividends during the life of the option.
3. Log-Normal Distribution of Stock Prices: Stock prices are assumed to follow a geometric Brownian motion with constant volatility and a continuous path. This implies that the returns of the stock are normally distributed.
4. Constant Risk-Free Interest Rate: The model assumes that the risk-free rate (e.g., rate on Treasury bills) is constant over the life of the option.
5. Constant Volatility: The model assumes that volatility, which measures the variability of stock returns, is constant over the option’s life.
6. European-Style Options: The model applies only to European options, which can only be exercised at expiration, unlike American options that can be exercised at any time.
7. No Arbitrage: The model assumes no arbitrage opportunities, meaning there are no opportunities for riskless profit.
8. Continuous Trading: It assumes that the underlying asset can be traded continuously in time, allowing for constant rebalancing of portfolios.

Question 8

Risk-Neutral Measure

Answer 8

The risk-neutral measure is a probability framework used in derivative pricing, where all assets are assumed to grow at the risk-free rate, regardless of their actual risk. This simplifies pricing by allowing us to compute the expected payoff of a derivative and discount it back to the present using the risk-free rate. Under this measure, the discounted price of any asset becomes a martingale, meaning its expected future price is equal to its current price when adjusted for time. It’s a key tool in models like Black-Scholes, enabling arbitrage-free pricing without needing to account for investors’ risk preferences.

Question 9

Vanna

Answer 9

Vanna is a second-order Greek that measures how an option’s delta changes with volatility and how its vega changes with the underlying asset’s price.

Question 10

Change of Measure

Answer 10

The change of measure is a technique in financial mathematics that involves switching from one probability measure to another, such as from the real-world measure P to the risk-neutral measure Q. This transformation simplifies derivative pricing by ensuring that all risky assets, under the new measure, have an expected return equal to the risk-free rate.

Question 12

Vomma/Volga

Answer 12

Volga is a second-order Greek that measures the sensitivity of an option’s vega to changes in volatility.

Question 13

Martingale Processes

Answer 13

A martingale process is a stochastic model in finance where, under the assumption of no arbitrage, the future value of an asset’s price equals its current value on average, reflecting a “fair game” dynamic used for pricing derivatives and assessing market behavior.

Question 14

Variance Reduction Techniques

Answer 14

• Antithetic Variates: This technique uses pairs of dependent random variables to reduce variance. When simulating a random variable, its complement or negative (the “antithetic” variable) is also simulated. Averaging both results often leads to variance reduction, as they tend to balance each other.
• Control Variates: In this method, a known variable (the control) with a known expected value is introduced into the simulation. The control variable is correlated with the variable of interest, and its known expected value is used to adjust the estimate, thereby reducing the variance.

Question 15

Newton-Raphson Method

Answer 15

The Newton-Raphson method is an iterative numerical technique used to approximate the roots of a real-valued function by successively refining an initial guess using the formula

Question 16

Lagrange multipliers

Answer 16

Lagrange multipliers are a technique used in constrained optimization to find the local maxima or minima of a function while considering equality constraints. By introducing a new variable (the multiplier) for each constraint, the method transforms the constrained problem into an unconstrained one, allowing for easier analysis and solution.

Question 17

Gradient Descent

Answer 17

Gradient descent is an optimization algorithm used to minimize a function by iteratively moving in the direction of the steepest descent, as defined by the negative of the gradient. It adjusts parameters to find the lowest point on a cost function, making it essential for training machine learning models.

Question 18

Autocorrelation

Answer 18

Autocorrelation in finance refers to the correlation of a time series with its own past values. It occurs when past price movements or returns influence future movements, which can distort statistical analysis and model performance. Autocorrelation can indicate inefficiencies in the market, violate assumptions of classical models like ordinary least squares (OLS), and lead to erroneous conclusions in predicting asset returns or volatility. Detecting and adjusting for autocorrelation is crucial in financial modeling to ensure accurate forecasting, risk management, and portfolio optimization.

Question 19

Value at Risk (VaR)

Answer 19

Value at Risk (VaR) is a widely used risk management metric that quantifies the potential loss in value of an asset, portfolio, or firm over a defined period for a given confidence level. VaR helps financial institutions, portfolio managers, and traders estimate the maximum expected loss with a certain level of confidence, making it an essential tool for assessing market risk.

Limitations:
* VaR does not capture extreme events beyond the specified confidence level.
* It assumes normal market conditions and might underestimate risk in times of financial crises.
* Does not consider risks beyond the selected time horizon.

Question 20

Parametric (Variance-Covariance) VaR

Answer 20

• Assumes a normal distribution of returns.
• Uses the mean and standard deviation of returns to calculate VaR.
• Advantage: Simple and fast to compute.
• Limitation: May not capture non-linear risks (e.g., options) or fat-tailed distributions.

Question 21

Historical VaR

Answer 21

• Based on historical price data to estimate potential losses.
• Advantage: Does not assume a normal distribution and directly reflects past market conditions.
• Limitation: Heavily dependent on historical data, which may not accurately reflect future risk.

Question 22

Monte Carlo VaR

Answer 22

• Uses simulations to generate a range of possible future price scenarios.
• Advantage: Can model complex portfolios and non-linear risks.
• Limitation: Computationally intensive.

Question 23

Modern Portfolio Theory (MPT)

Answer 23

• Modern Portfolio Theory (MPT) aims to optimize portfolios by maximizing return for a given level of risk.
• It emphasizes diversification, reducing overall risk by combining assets with low correlation.
• Risk is measured by the standard deviation of returns, and MPT focuses on minimizing this risk.
• The efficient frontier represents the set of portfolios that offer the best risk-return balance.
• MPT introduces the concept of the Sharpe Ratio, measuring risk-adjusted returns.

The optimization problem is typically formulated as:

Question 24

Sharpe Ratio

Answer 24

The Sharpe Ratio evaluates an investment’s risk-adjusted return by comparing its excess return to its volatility. A higher ratio indicates better performance relative to the risk taken, with values above 1.0 typically considered good.

Question 25

Supervised Learning

Answer 25

Supervised learning is a machine learning paradigm where a model is trained on a labeled dataset. Each training example is paired with an output label, and the goal of the model is to learn a mapping from inputs (features) to the desired output (target or label). After training, the model can be used to predict labels for new, unseen data.

Supervised Learning Models:
* Linear Regression
* Logistic Regression
* Support Vector Machines (SVM)
* Decision Trees
* Random Forest
* K-Nearest Neighbors (KNN)
* Naive Bayes
* Gradient Boosting Machines (GBM)
* Artificial Neural Networks (ANN)
* Linear Discriminant Analysis (LDA)

Question 26

Unsupervised Learning

Answer 26

Unsupervised learning is a type of machine learning where a model learns patterns from data without any explicit labels or outcomes. The goal is to identify hidden structures or relationships within the data. Unlike supervised learning, which works with labeled data (where each input comes with a corresponding target or label), unsupervised learning works with data that has no labels. Common techniques in unsupervised learning include clustering, dimensionality reduction, and association rules.

Question 27

Mean-Variance Optimization (MVO)

Answer 27

Mean-variance optimization (MVO) is a mathematical framework used to construct portfolios that maximize expected return for a given level of risk, or minimize risk for a given return. It calculates optimal portfolio weights based on asset returns, risks, and correlations, generating the efficient frontier of optimal portfolios. MVO is widely used in finance but is sensitive to estimation errors in inputs, leading to potential instability in the results.

Question 28

The Capital Asset Pricing Model (CAPM)

Answer 28

The Capital Asset Pricing Model (CAPM) is a foundational concept in finance that describes the relationship between expected return and risk for an individual security or portfolio. It is used to estimate the expected return on an asset, given its risk relative to the overall market. The core idea behind CAPM is that investors need to be compensated in two ways: time value of money and risk.

Key Assumptions:
* Investors are rational and risk-averse.
* Markets are perfectly competitive, and there are no transaction costs or taxes.
* All investors have the same expectations regarding the future performance of assets.
* All assets are infinitely divisible and tradeable.
* The risk-free rate is available for borrowing and lending.
* Investors are only concerned with mean and variance of returns (i.e., they prefer higher expected return for a given level of risk).

The CAPM equation is:

Question 29

Charm

Answer 29

The charm is one of the lesser-known Greeks used in options trading. It measures the rate of change in delta with respect to the passage of time (also known as delta decay). In other words, charm indicates how the delta of an option is expected to change as time passes, holding other factors like the price of the underlying asset constant.

Question 30

Vasicek Model

Answer 30

The Vasicek model is a short-rate model used in finance to describe the evolution of interest rates. It was introduced by Oldřich Vasicek in 1977 and is one of the first one-factor models used for modeling the term structure of interest rates.

The Vasicek model describes the short-term interest rate 𝑟(𝑡) as a mean-reverting process governed by the following stochastic differential equation (SDE):
drₜ = a(b - rₜ)dt + σdWₜ

Where:
rₜ = Short-term interest rate
a = Speed of reversion
b = Long-term mean rate
σ = Volatility
dWₜ = Wiener process

Question 31

Risk-Neutral Valuation

Answer 31

Risk-neutral valuation is a method of pricing derivatives assuming that investors are indifferent to risk. In this framework, the expected returns of all securities are equal to the risk-free rate. The value of an asset is calculated as the discounted expected payoff under the risk-neutral probability measure.

Question 32

Efficient Frontier

Answer 32

The Efficient Frontier, a concept from Modern Portfolio Theory, is a set of optimal portfolios that offer the highest expected return for a given level of risk or the lowest risk for a given level of return. Portfolios on the Efficient Frontier dominate those that are below it, which means that they offer better returns for the same level of risk or less risk for the same return. The shape of the Efficient Frontier is usually concave, and it is derived from the mean-variance optimization.

Question 33

Monte Carlo Simulation

Answer 33

Monte Carlo simulation in finance is used to model the probability of different outcomes in complex systems, such as asset prices, by running a large number of random scenarios. It helps in estimating risk, pricing options, and portfolio optimization by incorporating volatility and uncertainty into the model. This method is widely used to simulate paths for stock prices or interest rates, helping traders and risk managers make informed decisions.

Question 34

Binomial Trees

Answer 34

In finance, binomial trees are used to model the potential future movements of asset prices over time by breaking them into discrete intervals. Each node in the tree represents a possible price of the underlying asset, where it can either move up or down by a specified factor, simulating market volatility. Binomial trees are often employed in option pricing, allowing for the calculation of the option’s value at each node, working backward from expiration to the present.

Question 35

Futures Contract

Answer 35

A standardized agreement traded on an exchange to buy or sell an asset at a specified future date at a predetermined price. Futures are marked-to-market daily, and margin requirements must be maintained.

Question 36

Forward Contract

Answer 36

A private agreement between two parties to buy or sell an asset at a specified future date at a predetermined price. Unlike futures, forwards are not standardized, traded on exchanges, or marked-to-market, leading to higher counterparty risk.

Question 37

Duration

Answer 37

In fixed income markets, Duration is a measure of the sensitivity of a bond’s price to changes in interest rates. It helps investors understand how much the price of a bond is likely to fluctuate with changes in interest rates. Essentially, it’s an indicator of the bond’s interest rate risk.

Macaulay Duration:
This is the weighted average time until cash flows (interest payments and the principal repayment) from a bond are received. It is expressed in years. Macaulay duration reflects the time it takes for an investor to be repaid the bond’s price by the bond’s cash flows. However, Macaulay Duration is less commonly used in practice for estimating bond price sensitivity to interest rates.

Question 38

Convexity

Answer 38

Convexity measures the curvature of the price-yield relationship of a bond, or how the duration of a bond changes as interest rates change. It improves the accuracy of bond pricing, particularly when large changes in interest rates occur.

Question 39

Risk-Neutral Probability

Answer 39

Risk-neutral probability is a probability measure under which the present value of all assets, discounted at the risk-free rate, equals their expected future payoffs. In a risk-neutral world, all investors are indifferent to risk, so they price assets based solely on their expected return at the risk-free rate. This concept is central to derivatives pricing, where the expected payoff of the derivative is discounted at the risk-free rate to find its current price under the risk-neutral measure.

Question 40

Vega

Answer 40

Vega is one of the “Greeks,” a set of metrics used to measure different sensitivities of options prices. Specifically, vega measures the sensitivity of an option’s price to changes in the volatility of the underlying asset.

Question 41

Cointegration

Answer 41

Cointegration is a statistical property of two or more time series whereby a linear combination of them has a constant mean and variance over time, even if the individual series themselves are non-stationary. In finance, cointegration is often used in pairs trading, where two assets with a cointegrated relationship are traded with the expectation that their prices will revert to a mean value.

Question 42

Principal Component Analysis (PCA)

Answer 42

Steps involved in a principal component analysis (PCA) for financial time series:
1. Collect Data: Gather the time series data of interest (e.g., returns of multiple assets).
2. Normalize Data: Standardize the data so that each variable has a mean of zero and variance of one.
3. Covariance Matrix: Compute the covariance matrix of the standardized data.
4. Eigenvalue Decomposition: Perform eigenvalue decomposition of the covariance matrix to extract the principal components (eigenvectors) and their associated eigenvalues.
5. Interpret Components: The first few principal components explain the majority of the variance in the data. Use these components to reduce dimensionality or as input into a model.

In finance, PCA is used for risk management (e.g., analyzing yield curves), portfolio construction, and factor analysis.

Question 43

Fundamental Theorem of Asset Pricing (FTAP)

Answer 43

The Fundamental Theorem of Asset Pricing (FTAP) states that a financial market is arbitrage-free if and only if there exists an equivalent martingale measure (risk-neutral measure) under which the discounted price process of every tradable asset is a martingale. Furthermore, if the market is complete (i.e., all contingent claims can be replicated by trading), then this measure is unique.

This theorem underpins modern derivative pricing theory, as it justifies pricing options and other derivatives by taking expectations under the risk-neutral measure and discounting them at the risk-free rate.

Question 44

Arbitrage Pricing Theory (APT)

Answer 44

1. Multi-Factor Model: APT posits that asset returns are influenced by multiple macroeconomic factors, such as inflation, interest rates, and GDP growth, rather than just a single market factor (as in CAPM).
2. No Arbitrage Assumption: The theory is based on the idea that mispriced assets create arbitrage opportunities, and market forces will correct these mispricings, ensuring that assets are fairly priced based on their exposure to risk factors.
3. Flexible Factor Selection: Unlike the Fama-French model, APT does not specify the factors to be used; they can be chosen based on empirical analysis or market context, providing flexibility in application.
4. Linear Relationship: The expected return of an asset is modeled as a linear function of its sensitivities (factor loadings) to the various risk factors, making the pricing relationship straightforward to calculate.
5. Diversifiable Idiosyncratic Risk: APT assumes that the idiosyncratic risk (asset-specific risk) can be diversified away in a well-constructed portfolio, leaving only systematic (factor-based) risks that determine returns.

Question 45

Fama-French Three-Factor Model

Answer 45

The Fama-French three-factor model expands on the Capital Asset Pricing Model (CAPM) by including additional factors to explain stock returns. Here are five key bullet points summarizing the model:

1. Market Risk Premium: The model retains the market risk premium from CAPM, representing the excess return of the market portfolio over the risk-free rate.
2. Size Factor (SMB): Small minus Big (SMB) captures the excess return of small-cap stocks over large-cap stocks, based on the idea that smaller companies tend to outperform larger ones.
3. Value Factor (HML): High minus Low (HML) captures the excess return of high book-to-market value stocks (value stocks) over low book-to-market value stocks (growth stocks).
4. Equation: The model expresses a stock’s return as:
   Rᵢ = R𝑓 + βₘ (Rₘ - R𝑓) + βₛ * SMB + βₕ * HML + ϵ
   where Rᵢ is the expected return, R𝑓 is the risk-free rate, Rₘ is the market return, and βₘ, βₛ, βₕ are factor loadings for the market, SMB, and HML, respectively.
5. Empirical Success: The Fama-French model has been empirically validated as improving the explanation of stock returns compared to the CAPM, particularly by addressing anomalies like the small-cap and value premium.

Question 46

Linear Regression

Answer 46

• Linear Equation: Linear regression models the relationship between dependent and independent variables with a linear equation, typically y = β₀ + β₁x + ϵ.
• Simple vs. Multiple: Simple linear regression uses one predictor variable, while multiple linear regression involves two or more predictors.
• Assumptions: Assumes linearity, independence of errors, homoscedasticity (constant variance of errors), and normally distributed errors.
• Goal: Minimizes the sum of squared residuals (differences between observed and predicted values) to find the best-fitting line.
• Applications: Commonly used in fields like finance, economics, and biology to predict outcomes, identify trends, or quantify relationships.

Question 47

Logistic Regression

Answer 47

Logistic Regression is a supervised learning algorithm used for binary classification problems. It models the relationship between one or more independent variables (features) and a dependent variable (target) by estimating the probabilities using a logistic function (sigmoid function). Unlike linear regression, which predicts continuous values, logistic regression predicts probabilities that can be mapped to one of two classes (typically 0 or 1).

Question 48

Theta

Answer 48

Theta is one of the options Greeks, which measures the sensitivity of the price of an option to the passage of time. Specifically, Theta quantifies how much the price of an option is expected to decrease with each passing day, assuming all other factors remain constant.

Question 49

Treynor Ratio

Answer 49

Treynor Ratio is a risk-adjusted return metric that measures how much excess return (return above the risk-free rate) is earned per unit of market risk (systematic risk). It is similar to the Sharpe ratio, but instead of using total risk (standard deviation), it uses beta, which represents the portfolio’s sensitivity to market movements.

The formula for the Treynor Ratio is:

Treynor Ratio = (Rₚ - R𝑓) / βₚ

Where:
- Rₚ is the portfolio’s return.
- R𝑓 is the risk-free rate.
- βₚ is the portfolio’s beta, or its volatility relative to the market.

Key Points:
- The Treynor Ratio focuses on systematic risk, making it more useful for evaluating well-diversified portfolios where unsystematic risk has been minimized.
- A higher Treynor Ratio indicates that the portfolio has provided a higher return per unit of systematic risk.

This ratio is often used by portfolio managers and investors to compare the performance of various portfolios, especially when looking at their risk-adjusted returns relative to the market.

Question 50

VWAP (Volume Weighted Average Price)

Answer 50

VWAP (Volume Weighted Average Price) is a trading benchmark used by traders that gives the average price a security has traded at throughout the day, based on both volume and price. VWAP is calculated by taking the total dollar amount traded for every transaction (price multiplied by the number of shares traded) and then dividing it by the total shares traded.

The formula for VWAP is:
VWAP = Σ(Price x Volume) / Σ(Volume)

Key Points:
1. Intraday Indicator: VWAP is typically used during a single trading day and resets at the start of each new day.
2. Usage: Traders use VWAP to assess whether a security was bought or sold at a good price relative to the market average. Institutional traders, in particular, use it as a benchmark to gauge their performance against market liquidity.
3. Price Relative to VWAP:
- Buy Signal: Price below VWAP indicates that an asset might be underpriced, and traders may look for buying opportunities.
- Sell Signal: Price above VWAP suggests that an asset is overpriced, making it a potential sell opportunity.

It’s widely used in algorithms for trading large orders to avoid market impact by not deviating too much from the average price at which most participants are trading.

Question 51

Copula

Answer 51

A copula in finance is a mathematical function used to describe the dependency between different financial assets or risk factors. It helps model the joint distribution of asset returns, capturing how they co-move, especially during extreme market conditions. Copulas are particularly valuable in risk management, portfolio optimization, and the pricing of multi-asset derivatives.

Question 52

Gamma Hedging

Answer 52

Gamma hedging is a risk management strategy used in options trading to reduce the risk associated with changes in the underlying asset’s price. It involves adjusting a portfolio’s delta hedge to neutralize the effects of gamma, the rate of change of delta relative to the underlying asset’s price. By maintaining a gamma-neutral position, traders can limit the portfolio’s exposure to large price swings and minimize the need for frequent rebalancing.

Question 53

Delta Hedging

Answer 53

Delta hedging is a risk management strategy used in options trading to minimize the directional risk of an options position. It involves adjusting the quantity of the underlying asset held to offset changes in the option’s value, keeping the portfolio delta-neutral. As the option’s delta changes with the movement of the underlying asset, the hedge must be frequently rebalanced to maintain neutrality.

Question 54

Risk Parity

Answer 54

Risk Parity is an investment strategy that allocates risk equally across different asset classes, aiming for balanced portfolio risk and improved diversification. By adjusting asset weights based on their volatility, it ensures each contributes equally to overall portfolio risk, often using leverage for less volatile assets. While this can lead to lower volatility and more consistent returns, it may underperform in strong equity markets and introduces leverage risk.

Question 55

Sortino Ratio

Answer 55

The Sortino Ratio is a variation of the Sharpe Ratio, which measures the risk-adjusted return of an investment or portfolio. Unlike the Sharpe Ratio, which considers total volatility (both upside and downside), the Sortino Ratio focuses only on downside risk, as investors typically view upside volatility (gains) as favorable and are more concerned with downside volatility (losses).

Formula:
Sortino Ratio = (Rₚ - R𝑓) / Downside Deviation

Where:
* Rₚ = Portfolio return or investment return
* R𝑓 = Risk-free rate of return (often using Treasury bills)
* Downside Deviation = Standard deviation of returns below a defined threshold, typically the risk-free rate or a target return

Question 56

Statistical Arbitrage

Answer 56

Statistical arbitrage (StatArb) is a quantitative trading strategy that exploits price inefficiencies between correlated assets. It typically involves buying one asset and selling another with the expectation that their prices will converge over time. StatArb strategies rely on advanced statistical models, such as mean reversion and cointegration, to identify trading opportunities. These strategies are usually market-neutral, minimizing exposure to overall market movements and focusing on relative price changes. Robust risk management is essential, as sudden market shifts or model failures can lead to significant losses.

Question 57

Inerest Rate Caps and Floors

Answer 57

Interest Rate Cap:
An interest rate cap is a derivative that limits the maximum interest rate on a floating-rate loan or investment. If the market rate exceeds the cap, the cap provider compensates the buyer for the difference. It is used by borrowers to protect against rising interest rates, providing a financial ceiling on potential rate increases.

Interest Rate Floor:
An interest rate floor is a derivative that sets a minimum interest rate on a floating-rate loan or investment. If the market rate falls below the floor, the floor provider compensates the buyer for the shortfall. It is commonly used by investors to ensure a minimum return, offering protection against falling interest rates.