Important issues. 723g28. The Term Structure of Interest Rates. The term structure is the set of interest rates on the same risk instrument for various time to maturity. The longer the time to maturity of a bond, the higher the expected return of the bond in normal time. (YTM)
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A upward sloping yield curve is the normal shape of a yield curve: liquidity premium theory.
The longer time to maturity, the more sensitive the bond price to the interest rate changes. To compensate the risk of interest rate changes.
If the expected future rate is up, investors investing in longer term bonds need to be compensated on the yield now in order to be tied up in the bond contract.
The risks of future uncertainty needs to be compensated: a risk premium rp. (1+r2)2 = (1+r1)(1+f2) + rp2 = (1+r1)(1+E(r2)) + rp2
(1+r2)2 = (1+r1)(1+f2) = (1+r1)(1+E(r2)).
Efficient Market – well functioning capital market in which prices reflect all available information.
In an efficient market, one cannot expect to make persistent abnormal return over the risk adjusted return. The risk adjusted return is determined by the CAPM.
Discounted value of t1
Cycles disappear once identified
Mean: weighted average value.
Variance - Average value of squared deviations from mean. A measure of volatility.
Standard Deviation - Average value of squared deviations from mean. A measure of volatility.
Unique Risk - Risk factors affecting only that firm. Also called “diversifiable risk.”
Market Risk - Economy-wide sources of risk that affect the overall stock market. Also called “systematic risk.”
Efficient portfolio provides the highest return for a given level of risk, or least risk for a given level of return. The market portfolio is the one that has the highest Sharpe ratio with the expected return and risk. Sharpe ratio=(ri-rf)/σ
Long Position in an Option Contract
The value of a put option at expiration is
Where S is the stock price at expiration, K is the exercise price, P is the value of the put option, and max is the maximum of the two values in the bracket.
Clearly the value of the put is high, when K-S is high.
Equity Value of the firm
Consider the two different ways to construct portfolio insurance discussed above.
Purchase the stock and a put
Purchase a bond and a call
Because both positions provide exactly the same payoff, the Law of One Price requires that they must have the same price.
Where K is the strike price of the option (the price you want to ensure that the stock will not drop below), C is the call price, P is the put price, and S is the stock price
This relationship between the value of the stock, the bond, and call and put options is known as put-call parity.
You want to buy a one-year call option and put option on Dell.
The strike price for each is $15.
The current price per share of Dell is $14.79.
The risk-free rate is 2.5%.
The price of each call is $2.23
Using put-call parity, what should be the price of each put?
Put-Call Parity states:
If the stock pays a dividend, put-call parity becomes
Binomial Option Pricing Model
A European call option expires in one period and has an exercise price of $50.
The stock price today is equal to $50 and the stock pays no dividends.
In one period, the stock price will either rise by $10 or fall by $10.
The one-period risk-free rate is 6%.
We can construct the binomial tree based on these information.
is a portfolio consisting of a stock and a risk-free bond that has the same value and payoffs in one period as an option written on the same stock
The Law of One Price implies that the current value of the call and the replicating portfolio must be equal.
The payoffs can be summarized in a binomial tree.
Let D be the number of shares of stock purchased, and let Bbe an investment in bonds.
To create a call option using the stock and the bond, the value of the portfolio consisting of the stock and bond must match the value of the option in the two possible state (u, d).
Create a call: Hold the stock and short the bond.
In the up state, the value of the portfolio must be $10.
In the down state, the value of the portfolio must be $0.
Two equations, two variables, D and B can be solved for.
D = 0.5
B = –18.8679
The call payoff at period 1
Note that by using the Law of One Price, we are able to solve for the price of the option without knowing the probabilities of the states in the binomial tree.
A portfolio that is long 0.5 share of stock and short approximately $18.87 worth of bonds will have a value in one period that exactly matches the value of the call.
60 × 0.5 – 1.06 × 18.87 = 10
40 × 0.5 – 1.06 × 18.87 = 0
By the Law of One Price, the price of the call option today must equal the current market value of the replicating portfolio
The value of the portfolio today is the value of 0.5 shares at the current share price of $50, less the amount borrowed (the short position). Thus, the call premium is c=
The result is the same as in the risk neutral method.
S is the current stock price, and S will either go up to Su or go down to Sd next period.
The risk-free interest rate is rf .
Cu is the value of the call option if the stock goes up and Cd is the value of the call option if the stock goes down.
Given the above assumptions, the binomial tree would look like:
The payoffs of the replicating portfolios could be written as:
Solving the two replicating portfolio equations for the two unknowns D and B yields the general formula for the replicating formula in the binomial model.
Replicating Portfolio in the Binomial Model
The value of the option is the value of the portfolio today: Option Price in the Binomial Model
The replicating portfolio methods should give the same result, try yourself!