AFIN8013 Alternative Securities

Leverage and the Black-Scholes formula for option pricing are fascinating topics. The below questions seek to analyse historical data and recent interesting examples from the press using these theories. Historical data from Bloomberg is provided on ilearn.

Total assignment marks: 23; Page limit: 4.5 pages.

Question 1 (total of 3 marks, 1 page): These questions are about the VIX, the S&P 500 option-implied volatility index.

Question 1a (1 mark, half page): Find the implied volatility one month ahead using any recent (post-1st Jan 2024) S&P500 index option and briefly document your inputs and calculations in a table. You may use MS Excel goal seek or solver to calculate the implied volatility, or other means. Explain any difference between your estimate and the VIX on that same date. Use sources to justify your answer. State all assumptions. Please post a screenshot of the option price from the CME,barchart.comor other source. Question 1b (2 marks, half page): This question asks you to construct a single graph with 5 lines!

Graph of the S&P500 capital and accumulation indices with the two indices on the left hand side (LHS) axis using a logarithmic scale, where each is indexed to 100 on 4 Jan 1988.

On the same graph on the right hand side (RHS), add the United States' VIX (use the series that’s spliced with the old VXO) on the right hand side (RHS) axis;

Also add the rolling annual historical standard deviations (sd) of the S&P500 capital index over the past year at each point in time. Make one line for the equal-weighted (1/n where n is the days in the year) sd.

Add another line with the annual historical standard deviation based on an Exponentially Weighted Moving Average (EWMA) with a 97% lambda.

Do not index the VIX or sd’s. Make their RHS axis show a linear scale (not logarithmic). Make sure that the annual standard deviations are of the past year at each date, not the future year which is obviously unknown at that date.

Label each line with a colour-matched text box, do not use a legend.

Show the full history of each series in your graph, so your x-axis dates should begin on 30 Dec 1927. For an example of how an indexed graph with a log scale should look, see this graph:

http://www.rba.gov.au/chart-pack/share-markets.html

Question 2 (total of 9 marks, 1.5 pages): Read Angel’s (2021) interesting

account of the GameStop phenomenon:

https://papers.ssrn.com/sol3/papers.cfm?abstract_id=3782195

Question 2ai (2 marks, half page): Angel (2021) provides a detailed description of how an option market maker can hedge their short call position: "For example, for a $20 stock with an annualized volatility of 40%, a one-month “at-the-money” call option with an exercise price of $20 would sell for approximately $0.93 per share. Calculations are done using the Black-Scholes option pricing model assuming no dividends and an annual interest rate of 1.0%. In order to hedge the option, an option market maker would purchase 0.53 shares worth $10.52 and borrow $9.59. (The difference $10.52 - $9.59 gives the $0.93 value of the option.)" (Page 9, Angel 2013).

Make a table of each of the above figures used by Angel, along with an extra column using the current VIX as the standard deviation and current Federal Funds rate as the risk free rate. Briefly describe your sources or methods of calculation for each updated figure.

Question 2aii (3 marks, half page): Using your updated numbers, illustrate graphically how one call option plus delta shares less some bonds gives a portfolio position with zero delta (gradient) at the current share price (S0), based on the Black-Scholes formula.

Do this by showing 4 separate series on one graph. Show one straight or curvy line for the:

.    Short one call option curve;

.    Long delta stock line, where the delta is based on the current underlying asset spot price S0;

.    Short government bonds line, again where the delta is kept constant at the current S0; and

.    Portfolio of all 3 of the previous curves and lines summed together.

Note that the long delta stock and the short government bonds are supposed to comprise a long position in one synthetic call option, to delta-hedge the ‘physical’ short call option.

The graphs should all have the current payoff (f0, not the payoff at maturity or  profit at maturity) on the y-axis, and the current underlying stock price (S0, not the underlying stock price at maturity ST) on the x-axis.

See the picture in the first slide here which shows the current 'before maturity' red line value of a long call option which might help you figure out how the short call option should look:

http://www.fightfinance.com/resources/derivatives/7aBlackScholesOptionPrici

ngEuropeanNoDividends.pdf

Question 2bi (3 marks, half page): Show how a market maker can hedge the short-call position using put-call parity (not delta hedging) by making a graph   showing a curve for the short call option the same as in the previous question, together with curves or lines showing the other securities necessary to hedge that short call.

Note that put-call parity hedging is different to delta hedging. Put-call parity hedging involves whole stocks and options while delta hedging involves a fraction (delta) of a stock for each whole option hedged.

Your graph should look similar to those shown here:

http://www.fightfinance.com/resources/derivatives/5bPutCallParity.pdf

However, your graph should show all amounts before maturity (not at maturity) so your option lines should be curves, and your stock and bonds should be lines. Also, you should show only one graph with lots of lines for each security rather than a separate graph for each line.

Also show a ‘synthetic call’ portfolio line of the put, stock and bond too. See question 4B in the link here for an example of how put-call parity can be used to hedge an option:

https://www.fightfinance.com/resources/derivatives/WSU_Derivatives_200079 FinalExam_2015Sem2_Solutions.pdf

Question 2ci (0.5 marks, 1 line): Which of these two hedging strategies will require constant monitoring and adjusting to maintain the zero-delta hedge? The first one, the second one, both or neither? No need to explain, just answer in one word 'first, second, both or neither'.

Question 2cii (0.5 marks, 1 line): Which of these two hedging strategies will gamma hedge the transaction? The first one, the second one, both or neither?

No need to explain, just answer in one word 'first, second, both or neither'.

Question 3 (4 marks, half page): The crisis at Archegos Capital Management (ACM) in March 2021 highlighted the spectacular amount of leverage that Bill Hwang was able to secure using total return swaps on equities.

Total return swaps (https://en.wikipedia.org/wiki/Total_return_swap) marked-  to-market daily are similar to a long equity position funded with a margin loan.

In one graph, make 3 lines showing an investor’s wealth over time if she invested $1 of her wealth in the S&P500 on 30-12-1927. The S&P500 index data will be provided in a spreadsheet file on ilearn. The 3 lines should have initial LVR’s of 0%, 20% and 40% on 30-12-1927. Make the axis logarithmic rather than linear.

Margin calls occur when the portfolio’s loan-to-valuation ratio (LVR) exceeds the maximum. Note that the LVR is a debt-to-assets ratio (D/V). The margin  loan is the debt (D), the stocks are the assets (V) and the ‘equity’ (E) is your  wealth in the portfolio (so V=D+E).

Assume that:

.    Margin calls occur when the portfolio’s current LVR exceeds the maximum LVR;

.    The margin loan’s maximum LVR is 50%;

.    Margin calls only happen at the end of the day, a moment before the market closes;

.    All the investor’s wealth is invested into the portfolio so she can never meet margin calls. Therefore, in the event of a margin call, the broker will sell down her portfolio (share assets will be sold) and the margin   debt will be paid off until off until the portfolio LVR is equal to the maximum LVR. Note that these share sales at the close will not affect the wealth (equity E) in the portfolio, but V and D will both fall.

.    The margin loan interest rate is a constant 4% pa (simple interest) over the whole time period, and there are 250 days per year so the daily interest rate is 4%/250.

.    Margin interest is capitalised every day, which means that the interest expense is added to the margin loan debt liability at the end of each    day.

Question 4 (4 marks total, 1 page): This question attempts to quantify the risk of a margin call on a levered position in the S&P500 index using the Black-Scholes formula.

Question 4a (3 marks, half page): The Black-Scholes equation’s N(-d2) gives the risk-neutral probability that the stock price (S) will be below the option’s strike (K) at maturity. Use this equation and the (spliced) VIX to find the risk neutral probability of a margin call in the next year if the margin loan’s maximum LVR is 50%. Make one graph with 2 lines where the initial LVR at each date is 20% for the first line and 40% for the second line. Show the graph over the period from Jan 1986 to the most recent date available. Use the US 3 month Treasury Bill yield as the risk free rate. Note that the T-Bill   yield and VIX are given as per annum figures in percentage points, so you’ll need to divide them by 100.

References

Angel, James J., Gamestonk: What Happened and What to Do about It

(March 1, 2021). Available at SSRN:https://ssrn.com/abstract=3782195or http://dx.doi.org/10.2139/ssrn.3782195

Damodaran, A. (2005), The Promise and Peril of Real Options (July 2005).

NYU Working Paper No. S-DRP-05-02, Available at SSRN:

https://ssrn.com/abstract=1295849

Administrative Requirements:

• The page suggestions given in brackets are just indications, you’ll answer some questions in less space than others. You can go above or below the suggestion for each question. However, the total page limit must be adhered to.

• Submit a soft copy of your assignment to the Turnitin link on ilearn by the due date given on the front page. There is no need to submit a hard copy to your teacher. For every 24hr period after the time that your assignment is late, you will be penalized by 20%.

• Extra evidence such as tables and graphs are highly desirable. They must be placed in the text. The figures or tables should be placed directly above or wrapped in the paragraph that they are discussed in. Do not put figures in the appendix; they will not be marked. There’s no need for an appendix.

• The main body of your assignment should not exceed the page limit, typed with 12-point Arial font, 1.5 spaced with default margins. This page is formatted in this way. Exceeding the page count will incur a one mark penalty for each half page. The bibliography or reference list is not included in the word count. Please refrain from including an appendix since you should not require it, all tables and graphs should be included in-text next to where they are discussed.

• A good assignment shows research and clarity of argument.

• Assignments that are not proof-read will be marked down. Grammar and sentence structure is important.