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3FD3 – Financial Modeling

Assignment #3 – Options & Simulation (Winter 2022)

Due date: Sunday, April 10th, 2022 @11:59pm Eastern via Avenue to Learn

For each group:

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• Only one complete Excel file should be submitted

• Ensure your group number, name and student id for each group member are listed on

the cover page

Instructions:

The objective of this assignment is to familiarize you with the basics of options and simulation.

There are three (2) worksheets in the Excel file accompanying this assignment.

(1) There are several options data in worksheet “Options”, which will be used in Part 1 of this assignment.

(2) In worksheet “S&P500”, there are daily price data for S&P 500 index downloaded from Yahoo!Finance. The data will be used in Part 2 and Part 3 of this assignment.

Part 1: Option Basics

(1) Suppose you think Apple stock is going to appreciate substantially in value in the next year. Assume the current stock price (S0) is $200, and a European call option expiring in one year has an exercise price (X) of $210 and is selling at a price (C) of $20. With $10,000 to invest, you are considering three alternatives.

a. Invest all $10,000 in the stock, buying 50 shares.

b. Invest all $10,000 in 500 options (5 contracts)

c. Buy 200 options (2 contracts) for $4,000 and invest the remaining $6,000 in a money market fund paying 4% annual interest.

(2) What is your rate of return for each alternative for different stock prices in one year (ST)? Assume the possible stock prices in one year (ST) are from $160 to $260, with a step size of $10. Summarize your results in a table and plot them in a chart.

(3) Now consider a European put option with the same exercise price and expiration date on the same stock. This put option is currently sold with a price of $30. Do you think it is overpriced or underpriced? Support your conclusion.

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(4) Now assume you observe the following information on the market (the same as those in worksheet “Options” in the Excel file). Which options(s) do you think is (are) most probably wrongly priced? Justify your observation(s)?

GOOGX GOOGX GOOGX GOOGX GOOGX GOOGX GOOGX GOOGX GOOGX

Part 2: Option Pricing

Call or Put

Call Call Call Call Call Call Call Call Call

Expiration

June, 2022 June, 2022 June, 2022 July, 2022 July, 2022 July, 2022 August, 2022 August, 2022 August, 2022

Exercise Price

Option Price

$22.5 $20.0 $19.2 $22.8 $19.5 $19.8 $23.2 $21.1 $20.7

(1) There is a European call and a European put on AAPL (will not pay dividends in the near future) with the same exercise price and the same time to maturity. Assume the current stock price is $100, the exercise price is $110, the time to maturity is 9-month, the risk- free rate is 3% and the standard deviation of AAPL’s return is 30%.

a. Based on Black-Scholes formula, what is the call price and the put price?

b. Assume the risk-free rate might change from 0.25% to 5%, with a step size of 0.25%, what the call price and put price would be with different risk-free rates? Plot them on a chart with y-axis as option price and x-axis as risk-free rate. If the risk-free rate increases, will the option prices (call and put) increase or decrease?

c. Assume the stock price might change from $60 to $150, with a step size of $5, produce a chart comparing the put’s intrinsic value [=max(X-S,0)] and its Black- Scholes price.

d. In the class we have got a conclusion that early exercise is not valuable for American calls. Will you have a similar conclusion for American puts? Any reason for this?

(2) Today is Sep 28th, 2007. Seeing the crazy bubble in housing market, you expect the financial market might crash in the next 2 years due to the potential crisis from subprime mortgages. You want to profit from this opportunity. BMO provides a product called Hedge Against Subprime Crisis (HASC). It has the following features:

a. Initial investment $1,000, with time to maturity of 2 years.

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b. No interest paid until maturity.

c. In 2 years, if the S&P 500 index increases, you will get your $1,000 back; if the S&P 500 decreases, you will get the $1000 plus 75% of the decrease in S&P 500. Or shown in formula:

1000, 𝑖𝑓 𝑆! ≥ 𝑆”

1000+1000×75%×.1−𝑆!0, 𝑖𝑓𝑆! <𝑆" 𝑆"
Where 𝑆" is today’s S&P 500 index and 𝑆! is the S&P 500 index in 2 years. Assume the risk-free rate is 4% and you have gathered YTD (year-to-date) daily price
data for S&P 500 index (as shown in the Excel file).
a. Based on historical volatility of S&P 500 index, do you think this product is underpriced or overpriced? Show how you reach your conclusion.
b. Assume your conclusion is that the HASC is overpriced from previous question. A friend comes to you and tells you that you might be wrong. He tells you that usually the volatility would increase significantly during crisis and the product might not be overpriced if you take this into account. Curious about this, you are interested in what the implied volatility would be if HASC is fairly priced. Show what the implied volatility is in this case.
Part 3: Simulation
(1) One method for computing π is using the following formula: π# 11
6 = 1 + 2# + 3# + ⋯
Use this formula to approximate π. Show the difference between your approximated value and the true value of π. In this exercise, you should have enough terms to end-up with a difference less than 0.002.
(2) Now look at the given daily price data for S&P 500 index (as shown in the Excel file)
a. What is the historical mean return (μ) for S&P 500 index? What about the
standard deviation (σ)?
b. Now assume the S&P 500 index is lognormally distributed with the mean (μ) and standard deviation (σ$ = σ) as you just computed. In addition, assume the current S&P index is 𝑆" = 1000. Use Norm.S.Inv(Rand()) in Excel to simulate the price path of S&P 500 over 120 months. Let’s call this as scenario A.
c. Now assume the standard deviation increases 15% (σ% = σ+15%) in scenario B and 30% (σ& = σ+30%) in scenario C. Do the 120-month price simulation again, with SAME normal deviates as in scenario A (i.e., in the first month you may get a random number of 0.87, then the simulated prices for scenarios A/B/C are all
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based on this 0.87; in the second month the random number might be 0.06, then scenarios A/B/C all uses this 0.06).
d. Plot the simulated price paths in scenarios A/B/C on a chart and compare them. Considering the difference in standard deviations, do you think the chart makes sense?
e. Now do the 120-month price simulation again, with DIFFERENT normal deviates for scenarios A/B/C. Plot them on a chart. Do you get a similar graph? If not, any comment on this?
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