Financial Options
Drawing the payoffs for complicated securities is an excellent way to understand how the securities work and how they are constructed. With practice you will be able to look at a payoff diagram and write down the fundamental building blocks (a risk free bond, the asset, and calls and puts on the asset) which were used to construct the portfolio. In this question, I will describe the portfolio and you will draw the payoff diagram.
1) Portfolio 1 break-even. Draw the gross and net payoff diagrams for a portfolio which is constructed from buying one call option with a strike price of 45 and selling one call option with a strike price of 50. The cost of the first option is 7.75 and the cost of the second option is 3.50. Using your diagram, at what stock price will you break-even (have a zero profit on your investment)? Assume the options are European and they each have a maturity of a year.
2) Portfolio 1 market view. In an efficient market, purchasing a security or portfolio is a zero NPV investment. This means the price of the security is equal to its value. The assumption is the investors views are the same as the market. If an investor's expectations are different than the market, then the investor's value of the portfolio will differ from the price of the portfolio. The price of the portfolio is determined by the market's expectations. For portfolio 1 to be a good investment the investor must believe which of the following? (2)
- The expected future stock price is higher than the market believes.
- The expected future stock price is lower than the market believes.
- The volatility of stock returns is higher than the market believes.
- The volatility of stock returns is lower than the market believes.
- None of the above.
3) Portfolio 2 break-even: Low end. Draw the gross and net payoff diagrams for a portfolio which is constructed from buying one call option with a strike price of 80 (the option premium is 8.4), long one call option with a strike price of 100 (the option premium is 1.9), and short two call options with a strike price of 90 (the option premium is 4.2). Based on your diagram, this portfolio makes money for a range of stock prices. What is the bottom end of the stock price range over which this portfolio makes money? This is the stock price below which you lose money and above which you make money. Assume the options are European and they each have a maturity of a year.
4) Portfolio 2 break-even: High end. What is the top end of the stock price range over which this portfolio makes money? This is the stock price below which you make money and above which you lose money.
5) Portfolio 2 market view. For portfolio 2 to be a good investment the investor must believe which of the following
- The expected future stock price is higher than the market believes.
- The expected future stock price is lower than the market believes.
- The volatility of stock returns is higher than the market believes.
- The volatility of stock returns is lower than the market believes.
- None of the above.
6) Portfolio 2 returns. If you purchase portfolio 2 and the stock price at the end of the year is 90, what is your (net) rate of return?
7) Portfolio 2 returns. If you purchase portfolio 2 and the stock price at the end of the year is 103.40 what is your rate of return?
8) Option pricing: low stock price ($). Although we don't learn option pricing in this class, that is left to FIN465: Derivatives I, I find that if you have seen options prices and have an intuitive feel for what causes options prices to change you will understand them better. I want you to use the Black-Scholes formula in Excel to price a call option. The call option has a strike price of 100 and one year to maturity. Assume the risk-free rate is 8 percent, the dividend yield is 4 percent, and the annual volatility is 30 percent. If the stock price rises from 80 to 81, how much does the option price change in dollars terms?
9) Option pricing: low stock price (%). If the stock price rises from 80 to 81, how much does the call option price change in percent?1 For example if the option price rises from 10 to 11 this is an increase of $1.00 and an increase of 10.0% ([11-10]/10-1)?
10) Option pricing: high stock price ($). If the stock price rises from 120 to 121, how much does the call option price change in dollars terms? Continue to assume that the strike price of the option is 100, the option has one year to maturity, the risk-free rate is 8 percent, the dividend yield is 4 percent, and the annual volatility is 30 percent.
11) Option pricing: high stock price ($). If the stock price rises from 120 to 121, how much does the call option price change in percent? Continue to assume that the strike price of the option is 100, the option has one year to maturity, the risk-free rate is 8 percent, the dividend yield is 4 percent, and the annual volatility is 30 percent.
Using the Black-Scholes Option Pricing Function in Excel
The Excel function:
BSCall(stock price, exercise price, volatility, risk-free rate, expiration, dividend yield)
calculates the value of a call option using Black-Scholes. You use it like any other Excel function. It takes six parameters or inputs. The stock price is the value of the underlying today and the exercise price is the price at which you can buy the underlying at maturity. Volatility is the standard deviation of the underlying asset's return per year. The risk-free rate and the dividend yield are expressed as percent per year. The time until expiration are also expressed in years.
When valuing options using Black-Scholes we assume that the expected return on all assets and thus the correct discount rate is the risk free rate. Thus the expected return on the stock is the risk free rate. The price of the stock, however, is expected to rise at the risk free rate minus the dividend yield.
To see why this makes sense, take a stock whose expected return (and the risk free rate) is 10%. The stock has a dividend yield of 10%. If the stock price today is $100, then the stock is expected to rise to $110 by year end. It will then pay a $10 dividend and the stock price will end the year at 100. This is a zero percent capital gain return (10% - 10%). The actual value of the stock may be greater or less than this expected value. Black-Scholes assumes that the stock price is log normally distributed around this expected value. This is where the volatility assumption comes in. Thus Black-Scholes calculates all the possible values of:
Max Stock Price-Exercise Price, 0(1)
based on all the possible values of the stock price. The expected cash flow to the option is then discounted back at the risk free rate.
The excel function:
BSPut(stock price, exercise price, volatility, risk-free rate, expiration, dividend yield) calculates the value of a put using Black-Scholes.
Attachment:- quiz-8.xlsx