Assignment 3
Data-Driven Finance II
Rice University

Instructions

This assignment is due at midnight on Thursday, Feb 1. Submit a Jupyter notebook to Canvas. Copy each question into a Markdown cell and perform the analysis in the following cell(s).

  1. Consider a three-month (\(T=0.25\)) call option with a strike of 45 on a stock that is now trading at 50. Assume the annual continuously compounded risk-free rate is \(r_f = 0.04\) and the annual volatility of the stock is \(\sigma=0.3\). Assume the stock will not pay a dividend before the option matures (\(q=0\)).

    1. Price the option using a two-period binomial model calibrated to the data given.
    2. Compute the value of the option using the Black-Scholes formula.
    3. Using the Black-Scholes formula, plot the value of the option as a function of \(\sigma\) for \(\sigma\) varying between 0.1 and 0.6.
    4. Assume the market price (premium) of the option is 7.10. For what value of \(\sigma\) is the Black-Scholes formula equal to 7.10? (This is the implied volatility).

  2. Get CME gold futures settlement prices from the CME’s website. How much higher is the Feb 2025 futures price than the Feb 2024 futures price in percentage terms? It should be approximately the same as the one-year interest rate. Get the one-year interest rate from the U.S. Treasury’s website and check that they are roughly the same.

  3. Get adjusted closing prices for SPY, SPXL, and SPXS from Yahoo finance and compute daily returns. Drop NaNs so that you have the same time period for all three. SPXL is a three-times levered version of SPY and SPXS is a three-times levered short version of SPY. Both are created using futures contracts on the S&P 500.

    1. Create a scatter plot with SPXL returns on the y-axis and SPY returns on the x-axis and do the same for SPYXS. Run regressions of SPXL returns on SPY returns and of SPYXS returns on SPY returns. What are the slopes of the regression lines?
    2. Do the total compound returns \((1+r_1)\cdots(1+r_T)-1\) of SPXL compared to SPY and SPXS compared to SPY match the relationships you found in (a)?
    3. Plot the accumulations \((1+r_1)\cdots(1+r_T)\) for the three funds.