18.S096 | Fall 2013 | Undergraduate

# 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 dftT = μtT dt + σtTdBt and derive the no-arbitrage condition for drift μtT
• Derive drift at for the short rate Ho-Lee Model drt = atdt + σdBt. Next, show that the Ho-Lee model can be written in the HJM form. Remember that rt = ftt.
• Add a mean reversion to the Ho-Lee model drt = (at - κrt*)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

## Course Info

Fall 2013
##### Learning Resource Types
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Lecture Notes
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