Topics in Mathematics with Applications in Finance

Projects

Sample Topics for the Final Paper

Twenty-five percent of the course grade is based upon a final paper on a math finance topic of the student’s choice. Below are some sample topics. Students may propose other topics as well.

Portfolio Management

Based on what you learned in class, research further and come up with your own views in portfolio risk management.

Regime-Shift Modeling

Detail one or more approaches to regime-shift modeling, addressing the statistical modeling methodology and its use in a specific, real-world application.

Low-Volatility Investing

Critically review the rationales of low-volatility investing strategies in the U.S. equity market and their connection to the portfolio theory covered in class; evaluate the performance of such strategies as implemented in exchange-traded funds and / or mutual funds.

Modeling Financial Bubbles

Detail one or more approaches to modeling asset bubbles; e.g., the work of Didier Sornette.

Relationship between Black-Scholes and Heat Equations

• Go through the change of variables to get from Black-Scholes PDE to Heat Equation.
• Go through calculations verifying that a European call option price for a lognormaly distributed stock is in fact a discounted expected value of the pay-off under risk neutral measure.
• Explore possible numerical methods for the solution with various boundary conditions.
• Go through computations showing that Black-Scholes price of a digital option is a partial derivative of the call option price with respect to strike.

Hybrid products

• Price zero coupon bonds in USD and EUR in this jump–diffusion model.
• Determine the dynamic hedging strategy. There are two sources of risk, so need at least 2 hedging instruments. FX forwards are a great candidate.

HJM vs Short-Rate Interest Rate Models. 

• Start from the equation for forward rates df_tT = μ_tT dt + σ_tTdB_t and derive the no-arbitrage condition for drift μ_tT. 
• Derive drift at for the short rate Ho-Lee Model dr_t = a_tdt + σdB_t. Next, show that the Ho-Lee model can be written in the HJM form. Remember that r_t = f_tt.
• Add a mean reversion to the Ho-Lee model dr_t = (a_t - κr_t*)dt + σdB*_t and write it in the HJM form.

Ross Recovery

• Try to offer financial intuition for the Perron Forbenius theorem for positive matrices.
• Try to extend Ross recovery to a countable state space for a Markov chain.

A Few Topics Chosen by Students Last Year

• Transformation of Black-Scholes PDE to Heat Equation
• From Black-Scholes-Merton model to heat equation: Derivations and numerical solutions
• Solving Black-Scholes equation with Initial conditions by change of variables
• Derive HJM no arbitrage condition
• HJM model and Ho-Lee model
• Pricing zero-coupon USD and EUR bonds in the FX jump diffusion model
• Pricing Asian options
• On the Minimal Entropy Martingale Measure in Finite Probability Financial Market Model
• Principal Component Analysis on Oil, Gas, Power and Currency Swap Curves before and after the 2008 Financial Crisis
• A review of finite grid summation method and Monte-Carlo method for a three-legged spread option integration
• Monte-Carlo option pricing using the heston model for stochastic volatility