Utility Gap Slides The Utility Gap
Risk Aversion & Expected Utility Theory
Graduate Quantitative Finance | Session 01
The St. Petersburg Paradox
Consider a game where a fair coin is flipped until it shows heads for the first time on the \(k\)-th toss. The payoff is \(2^k\).
Mathematical Expectation:
\[ E[X] = \sum_{k=1}^{\infty} \left(\frac{1}{2^k}\right) 2^k = \sum_{k=1}^{\infty} 1 = \infty \]
The Friction:
The expected value is infinite, yet no rational agent would pay more than a few dollars to play.
Discussion: Why does the law of large numbers fail to describe human behavior here?
Expected Utility Hypothesis (vNM)
The Bernoulli Resolution
Agents do not maximize dollar value, but the **utility** of wealth: \(u(w)\).
\[ E[u(W)] = \sum p_i u(w_i) \]
"The utility resulting from any small increase in wealth will be inversely proportionate to the quantity of goods previously possessed." — Daniel Bernoulli
Common Utility Functions:
Logarithmic:
\(u(w) = \ln(w)\) (Bernoulli)
Power (CRRA):
\(u(w) = \frac{w^{1-\gamma}}{1-\gamma}\)
Exponential (CARA):
\(u(w) = -e^{-\alpha w}\)
Quantifying Risk Aversion
Pratt-Arrow Measure
Absolute Risk Aversion (ARA):
\[ A(w) = -\frac{u''(w)}{u'(w)} \]
Concavity is King
A risk-averse agent has a **concave** utility function (\(u'' < 0\)).
The Risk Premium (\(\pi\))
The amount an agent is willing to pay to avoid risk. Defined by the equation:
\[ u(E[W] - \pi) = E[u(W)] \]
Certainty Equivalent:
\(CE = E[W] - \pi\)
The guaranteed amount that yields the same utility as the risky prospect.
Jensen's Inequality:
\(u(E[W]) \geq E[u(W)]\)
Fundamental proof for why risk aversion implies choosing the mean over the gamble.
Executive Scenario
An investor with wealth \(W_0 = \$1,000,000\) and utility \(u(w) = \ln(w)\) is offered a 50/50 gamble: Lose \$100,000 or Gain \$120,000.
1. Expected Value Analysis:
Is the EV positive? Yes. (\(+\$10,000\))
2. Expected Utility Analysis:
Do they take the bet? Calculate \(E[u(W)]\).
Why might a hedge fund manager and a retiree value this expected value differently?
Psychological Anchor Marginal Utility Risk Budgeting
Utility Function Worksheet Utility & Risk
Quantitative Finance: Problem Set 01
NAME:
DATE:
1
Resolving the St. Petersburg Paradox
Recall the St. Petersburg Paradox: A coin is flipped until heads appears at toss \(k\), paying \(2^k\). Assume an investor has initial wealth \(W_0 = \$1,000,000\) and a logarithmic utility function \(u(w) = \ln(w)\).
A) Write the infinite series expression for the Expected Utility \(E[u(W_0 + X)]\).
B) Calculate the maximum price \(P\) this investor would be willing to pay to play the game once. (Hint: Solve \(u(W_0) = E[u(W_0 + X - P)]\), or approximate the series sum).
2
Constant Relative Risk Aversion (CRRA)
An investor has power utility \(u(w) = \frac{w^{1-\gamma}}{1-\gamma}\) where \(\gamma > 0\) is the coefficient of relative risk aversion.
A) Derive the Absolute Risk Aversion \(A(w) = -\frac{u''(w)}{u'(w)}\) for this function.
B) Explain how \(A(w)\) changes as wealth \(w\) increases. What does this imply about the investor's willingness to take a fixed dollar amount of risk as they get wealthier?
3
Calculating the Risk Premium
An investor has wealth \(W_0 = 100\) and utility \(u(w) = \sqrt{w}\). They face a gamble where they win \(11\) with probability \(0.5\) and lose \(11\) with probability \(0.5\).
A) Calculate \(E[W]\).
B) Calculate \(E[u(W)]\).
C) Find the Certainty Equivalent (CE) and the Risk Premium (\(\pi\)). Round to two decimal places.
Efficient Frontier Slides The Efficient Frontier
Portfolio Optimization & Mean-Variance Analysis
Graduate Quantitative Finance | Session 02
The Diversification Effect
Imagine 10 technology stocks, each with a high expected return but even higher volatility (30%+).
The Challenge:
Can we combine these high-risk assets to create a portfolio with lower risk than the least volatile stock in the group?
Mathematically, if correlations are less than 1, the answer is Always Yes.
Asset A Volatility 25%
Asset B Volatility 25%
Portfolio (\(\rho = 0\)) Volatility 17.7%
Expected Value in Vector Form
We treat the returns of \(n\) assets as a random vector \(\mathbf{r}\) with expected returns \(\mathbf{\mu}\).
The Linear Operator
\[ E[R_p] = \mathbf{w}^T \mathbf{\mu} = \sum_{i=1}^n w_i \mu_i \]
Subject to the constraint: \(\mathbf{w}^T \mathbf{1} = 1\) (fully invested).
Key Insight:
Portfolio expectation is a simple linear combination. It does not account for the interaction between assets (correlations).
The Covariance Matrix (\(\Sigma\))
Quadratic Form
\[ \sigma_p^2 = \mathbf{w}^T \Sigma \mathbf{w} \]
While expectation is linear, risk is **quadratic**. This non-linearity is why optimization requires calculus.
Inside the Matrix:
\[ \Sigma = \begin{bmatrix} \sigma_1^2 & \sigma_{12} \\ \sigma_{21} & \sigma_2^2 \end{bmatrix} \]
Diagonal
Individual asset variances (Idiosyncratic Risk).
Off-Diagonal
Covariances between pairs (Systematic Dependency).
Markowitz Optimization
How do we find the weights \(\mathbf{w}\) that yield the minimum variance for a target expected return \(\mu_p\)?
Lagrange Multiplier Setup:
\[ L = \mathbf{w}^T \Sigma \mathbf{w} - \lambda_1(\mathbf{w}^T\mathbf{\mu} - \mu_p) - \lambda_2(\mathbf{w}^T\mathbf{1} - 1) \]
Minimize
Total Portfolio Variance
Constraint 1
Hit Target Return \(\mu_p\)
Constraint 2
Weights sum to 1.0
Portfolio Construction Workshop Portfolio Workshop
Mean-Variance Optimization & Matrix Algebra
ANALYST:
SESSION:
1
Asset Universe Parameters
Consider a two-asset universe with the following expected returns (\(\mu\)) and covariance matrix (\(\Sigma\)):
Expected Returns
\[ \mathbf{\mu} = \begin{bmatrix} 0.12 \\ 0.08 \end{bmatrix} \]
Covariance Matrix (\(\Sigma\))
\[ \Sigma = \begin{bmatrix} 0.04 & 0.01 \\ 0.01 & 0.02 \end{bmatrix} \]
A) Calculate the expected return and variance for an equally weighted portfolio (\(w_1 = 0.5, w_2 = 0.5\)).
2
The Minimum Variance Portfolio (MVP)
We wish to find the weights \(w_1\) and \(w_2\) that minimize portfolio variance subject to \(\sum w_i = 1\).
A) Express the portfolio variance \(\sigma_p^2\) as a function of \(w_1\) only (substituting \(w_2 = 1 - w_1\)).
B) Take the first derivative \(\frac{d\sigma_p^2}{dw_1}\), set to zero, and solve for the optimal weights \(w_1^*\) and \(w_2^*\).
C) What is the expected return of this Minimum Variance Portfolio?
3
Diversification Limits
Assume Asset 1 and Asset 2 are perfectly positively correlated (\(\rho = 1\)).
Does an MVP still exist that has lower risk than both individual assets? Prove mathematically or explain via the diversification effect.
Critical Discussion:
In practice, the covariance matrix \(\Sigma\) is estimated from historical data and is often "noisy" or non-stationary. How might an error in estimating a single correlation coefficient \(\rho_{12}\) impact the resulting "optimal" weights?
Neutral Grounds Slides Neutral Grounds
Risk-Neutral Pricing & The Law of One Price
Graduate Quantitative Finance | Session 03
The Arbitrage Puzzle
How can we price a derivative contract without knowing the **actual** probability of the stock going up or down?
The Secret:
If we can replicate the payoff using a combination of the underlying stock and a risk-free bond, the price **must** be the cost of that replication.
"No Free Lunch"
The existence of a risk-neutral measure \(Q\) is equivalent to the absence of arbitrage opportunities in the market.
The \(Q\) World
In the risk-neutral world, investors are indifferent to risk. Therefore, every asset must grow at the **risk-free rate** \(r\).
Fundamental Pricing Equation
\[ V_0 = e^{-rT} E^Q [V_T] \]
Key Properties:
Discrete Time: The Binomial Model
Stock price \(S\) can move to \(Su\) or \(Sd\). The risk-neutral probability \(q\) is:
\[ q = \frac{e^{r\Delta t} - d}{u - d} \]
where \(d < e^{r\Delta t} < u\) must hold for no-arbitrage.
Pricing Step-by-Step:
Calculate payoffs at expiration (\(T\)).
Compute risk-neutral probability \(q\).
Work backward through the tree to find \(V_0\).
Delta Hedging
The risk-neutral price is valid because we can perfectly replicate the option using \(\Delta\) shares of stock.
The Hedge Ratio (\(\Delta\))
\[ \Delta = \frac{f_u - f_d}{S_u - S_d} \]
This tells us how many shares of stock to hold per option to eliminate risk.
"The option price is just the value of the equivalent portfolio of stock and cash."
Option Architect Activity Option Architect
Binomial Asset Pricing & Risk-Neutral Measures
ANALYST:
DATE:
1
The One-Step Tree
Market Data:
Current Price (\(S_0\)): $100
Strike Price (\(K\)): $105
Up Factor (\(u\)): 1.2
Down Factor (\(d\)): 0.8
Risk-free rate (\(r\)): 5% (annual, continuous)
Time step (\(\Delta t\)): 1 year
A) Calculate the risk-neutral probability \(q\) of an upward move.
B) Calculate the payoff of a European Call option at both nodes at \(T=1\).
Payoff (Up)
Payoff (Down)
C) Determine the no-arbitrage price \(C_0\) of the call option today.
2
Replication Strategy
Suppose you sell the call option from Problem 1. You want to create a synthetic portfolio of \(\Delta\) shares of stock and \(B\) amount in bonds to hedge your position perfectly.
A) Calculate the hedge ratio \(\Delta\). Explain what this value represents for the portfolio manager.
B) Calculate the amount \(B\) that must be invested (or borrowed) at the risk-free rate to complete the replication.
C) Verification: Show that the cost of your replication portfolio (\(\Delta S_0 + B\)) equals the option price \(C_0\) found in Problem 1.
3
Dynamic Hedging (Conceptual)
In a multi-period model, the risk-neutral probability \(q\) remains constant at each node if \(u, d,\) and \(r\) are constant.
Why must we rebalance our \(\Delta\) (the number of shares held) as the stock moves from \(S_0\) to \(S_u\) or \(S_d\)? What does this imply about the "expected return" of the option seller?
Arbitrage Check
If the market price of the call was $10 instead of your calculated price, describe the exact series of trades (buy/sell stock, buy/sell option, borrow/lend) you would execute to capture a risk-free profit.
Tail Risk Slides Beyond the Normal
Value at Risk (VaR) & Expected Shortfall (ES)
Graduate Quantitative Finance | Session 04
The Illusion of Safety
In 2008, risk models estimated a "1-in-10,000 year" event. Then it happened.
The Fatal Flaw:
Assuming a **Normal Distribution** for market returns severely underestimates the probability of extreme negative outcomes ("Fat Tails").
Kurtosis Matters
Standard Deviation measures average wiggle. We need metrics that measure the **Impact of the Abyss**.
Value at Risk (VaR)
VaR is the maximum loss expected over a given time period with a specified confidence level \(\alpha\).
Mathematical Definition:
\[ P(L > \text{VaR}_\alpha) = 1 - \alpha \]
The Problem with VaR:
Expected Shortfall
Conditional Expectation
ES answers: "If things go bad, how bad do they get on average?"
\[ ES_\alpha = E[L | L > \text{VaR}_\alpha] \]
Coherent Risk Measure
Unlike VaR, ES is subadditive and satisfies all axioms of a coherent risk metric.
Normal vs. T-Distribution
For a normal distribution, VaR and ES are closely linked. For fat-tailed distributions (Student-T), the gap between VaR and ES widens significantly.
Normal
Exponential decay of tail probability
Fat Tails
Polynomial decay (Higher Risk)
Crisis Simulation
A portfolio has a 1-day 99% VaR of $5 Million.
During a liquidity freeze, the distribution shifts from Normal to a Cauchy distribution (heavy tails).
VaR Impact
The VaR threshold might stay the same, but the **frequency** of breaches triples.
ES Impact
The Expected Shortfall could jump from $7M to $50M+ (Infinite Variance).
Discussion: Is a risk metric that assumes finite variance useful in a market that exhibits infinite variance characteristics during crashes?
Risk Metrics Worksheet Tail Risk Audit
Metrics: Value at Risk (VaR) & Expected Shortfall (ES)
RISK OFFICER:
QUARTER:
1
Parametric VaR Estimation
A portfolio has a daily expected return of \(0.01\%\) and a daily standard deviation of \(1.2\%\). Assume the returns are normally distributed and the total portfolio value is \(\$100\) Million.
A) Calculate the 1-day 95% VaR (in dollars). Recall \(z_{0.95} \approx 1.645\).
B) Calculate the 1-day 99% VaR (in dollars). Recall \(z_{0.99} \approx 2.326\).
C) Scale your 95% VaR to a 10-day horizon using the "square root of time" rule. Under what assumptions does this scaling hold?
2
Conditional Loss (ES)
For a standard normal distribution \(N(0,1)\), the Expected Shortfall at level \(\alpha\) is given by: \[ ES_\alpha = \frac{\phi(z_\alpha)}{1-\alpha} \] where \(\phi(z)\) is the PDF of the standard normal distribution.
A) Using the data from Problem 1 (\(\mu=0.0001, \sigma=0.012\)), calculate the 1-day 95% Expected Shortfall. Given \(\phi(1.645) \approx 0.103\).
B) Compare the ES value to the VaR value from 1A. Why is ES considered a "conservative" or "coherent" alternative to VaR?
3
The Kurtosis Gap
Historical data shows that a specific hedge fund's returns exhibit a kurtosis of 8.0 (Leptokurtic).
How would the relationship between VaR and ES differ for this fund compared to the normal distribution modeled in Problem 2? If you used a normal-based VaR to set risk limits for this fund, what specific danger would you face during a market downturn?
Subadditivity Check:
One property of a coherent risk measure is subadditivity: \(\rho(X + Y) \leq \rho(X) + \rho(Y)\). Briefly describe a scenario where combining two portfolios with non-normal returns might result in a VaR that is higher than the sum of their individual VaRs.
Stochastic Path Slides Stochastic Paths
Monte Carlo Simulation & Numerical Integration
Graduate Quantitative Finance | Session 05
The Limits of Calculus
The Black-Scholes model provides a closed-form solution for "Vanilla" options. But what about **Asian Options**?
The Path Dependency:
Payoff is based on the **average** price over the life of the option, not just the final price. There is no simple formula for this integral.
Numerical Salvation:
If we can't solve the integral analytically, we simulate thousands of possible futures and calculate the average payoff.
The Path Engine
To simulate price, we use the SDE for GBM in its discrete Euler-Maruyama form:
\[ S_{t+\Delta t} = S_t \exp\left( (r - \frac{1}{2}\sigma^2)\Delta t + \sigma\sqrt{\Delta t} Z \right) \]
\(Z \sim N(0,1)\) is a standard normal random variable.
Simulation Steps:
Generate random normal \(Z\).
Step price forward by \(\Delta t\).
Repeat until maturity \(T\).
Record final payoff.
Convergence to Expectation
Monte Carlo Estimator
\[ \hat{V}_0 = e^{-rT} \frac{1}{M} \sum_{i=1}^M \text{Payoff}_i \]
As \(M \to \infty\), the sample mean \(\hat{V}_0\) converges almost surely to the true expected value \(E^Q[V_T]\).
Standard Error of Simulation
Numerical error decreases at a rate of \(1/\sqrt{M}\). To double the precision, you must quadruple the number of paths.
SE = \(\frac{\text{StdDev}(\text{Payoffs})}{\sqrt{M}}\)
A 95% Confidence Interval is \(\hat{V}_0 \pm 1.96 \times \text{SE}\)
The Path-Dependent Edge
Monte Carlo isn't just for pricing; it's for understanding the entire distribution of potential outcomes.
Advantage 1: Flexibility
Can handle any payoff structure (lookback, barrier, chooser, Asian).
Advantage 2: Dimensionality
Efficient for multi-asset baskets where tree methods become exponentially slow.
NEXT: Implementing an Asian Option pricing engine.
Monte Carlo Engine Project Monte Carlo Engine
Path-Dependent Pricing & Simulation Metrics
Architect:
Iteration:
1
The Path Generator
We are pricing an **Arithmetic Asian Call Option** with strike \(K\) and maturity \(T\). The payoff is \(\max(0, \text{Avg}(S) - K)\).
A) Write the pseudo-code logic for simulating a single price path of \(N\) steps using the discrete GBM formula. Identify which variables must be updated at each time step.
B) In the context of risk-neutral pricing, explain why the "drift" term in your simulation must be the risk-free rate \(r\) and not the historical expected return \(\mu\).
2
Numerical Convergence
Suppose you run a simulation with \(M = 10,000\) paths. The mean discounted payoff is \(\$4.52\) and the sample standard deviation of the discounted payoffs is \(\$12.80\).
A) Calculate the Standard Error (SE) of the price estimate.
B) Provide a 95% Confidence Interval for the option price.
C) How many total paths (\(M\)) would be required to reduce the Standard Error to \(\$0.01\)? Show your work.
3
Antithetic Variates
To speed up convergence, we can use **Antithetic Variates**: for every path generated with random shocks \(\{Z_1, Z_2, ...\}\), we also calculate the path generated by \(\{-Z_1, -Z_2, ...\}\).
Why does this technique reduce the variance of the estimator? Under what market conditions (high or low volatility) would you expect this technique to be most effective?
Computational Trade-off
If your simulation takes 2 minutes to run and has a 2% error, but the trader needs a price in 5 seconds with a 0.5% error, what adjustments (besides antithetic variates) could you make to the model parameters (\(N\), \(M\), or the stochastic process itself)?
