Part 1: Fundamental Exercises
Exercise 1: Differentiation in Option Pricing
The Black-Scholes price for a European call option is given by:
\[ C = S_0 N(d_1) - K e^{-rT} N(d_2), \]
where \( d_1 \) and \( d_2 \) are:
\[
d_1 = \frac{\ln(S_0/K) + (r + \sigma^2/2)T}{\sigma\sqrt{T}},
d_2 = d_1 - \sigma\sqrt{T}.
\]
Compute \( \frac{\partial C}{\partial S_0} \) and interpret its meaning in the context of financial Greeks.
Exercise 2: Integration in Continuous Compounding
Calculate the present value of a continuously compounded bond that pays $1000 in 5 years, given a risk-free rate of \( r = 5\% \) per annum.
Exercise 3: Eigenvalues in Portfolio Optimization
Consider the covariance matrix of asset returns:
\[
\Sigma = \begin{bmatrix}
0.04 & 0.02
0.02 & 0.03
\end{bmatrix}.
\]
Find the eigenvalues and eigenvectors of \( \Sigma \) and discuss their significance in risk factor analysis.
Exercise 4: Constrained Optimization in Portfolio Theory
Using Lagrange multipliers, determine the optimal portfolio weights in a two-asset portfolio under the constraints:
\[ w_1 + w_2 = 1, \quad E[R_p] = 10\%, \quad \sigma_p = \min. \]
Exercise 5: Probability in Asset Returns
Assume daily stock returns follow a normal distribution with a mean \( \mu = 0.001 \) and standard deviation \( \sigma = 0.02 \). Calculate the probability that a stock will experience a return below \(-2\%\) on a given day.
Part 2: Advanced Exercises
Exercise 6: Stochastic Processes in Stock Price Modeling
Given that stock prices follow a geometric Brownian motion:
\[ dS_t = \mu S_t \, dt + \sigma S_t \, dW_t, \]
derive the expected value of \( S_T \) given \( S_0 \) at \( t=0 \).
Exercise 7: Hypothesis Testing in Market Efficiency
A hedge fund claims to have developed a trading strategy that consistently outperforms the market. Design a statistical hypothesis test to evaluate whether their results are statistically significant.
Exercise 8: GARCH Modeling in Volatility Forecasting
Consider the GARCH(1,1) model:
\[ \sigma_t^2 = \alpha_0 + \alpha_1 \epsilon_{t-1}^2 + \beta_1 \sigma_{t-1}^2. \]
Explain the procedure to estimate \( \alpha_0 \), \( \alpha_1 \), and \( \beta_1 \) using maximum likelihood estimation.
Exercise 9: Monte Carlo Simulation in Option Pricing
Simulate the price of a European call option using Monte Carlo methods with the following parameters: \( S_0 = 100 \), \( K = 100 \), \( r = 5\% \), \( \sigma = 20\% \), and \( T = 1 \) year. Perform 10,000 simulations.
Exercise 10: Markov Chains in Credit Ratings
A company’s credit rating follows a Markov process with the transition matrix:
\[P = \begin{bmatrix} 0.9 & 0.1 & 0 \\\\ 0.05 & 0.9 & 0.05 \\\\ 0 & 0.1 & 0.9 \end{bmatrix}.\]If the company starts with a rating of 2, determine the probability of being in rating 1 after two transitions.
See the solutions at (Solutions).