Pricing Option Approaches
Valuation of options
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Today's plan
Discussion of a simple business-ethics case Review of what we have learned about options We discuss two ways of valuing options
Binomial tree (two states)
Simple idea Risk-neutral valuation
The Black-Scholes formula (infinite number of states)
Understanding the intuition How to apply this formula
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What have we learned in the last lecture?
Options
Financial and real options European and American options Rights to exercise and obligations to deliver the underlying asset Position diagrams Draw position diagrams for a given portfolio Given position diagrams, figure out the portfolio No arbitrage argument Put-call parity How to write (sell) options
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The basic idea behind the binomial tree approach
Suppose we want to value a call option on IBM with a strike price of K and maturity T. We let C(K,T) be the value of this call option.
Remember C(K,T) is the price for the call or present value of the call option.
Let the current price of IBM is S and there are two states when the call option matures: up and down. If the state is up, the stock price for IBM is Su; if the state is down, the price of IBM is Sd.
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The stock price now and at maturity
Su S Sd Now maturity uS
S Now
dS maturity
If we define:
u = Su/S and d = Sd/S. Then we have Su=uS and Sd=dS
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The risk free security
The
price now and at maturity
Rf Here Rf=1+rf
1 Rf now maturity
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The call option payoff
Cu=Max(uS-K,0)
C(K,T)
Cd=Max(dS-K,0) Now matuirty
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Now form a replicating portfolio
A
portfolio is called the replicating portfolio of an option if the portfolio and the option have exactly the same payoff in each state of future. By using no arbitrage argument, the cost or price of the replication portfolio is the same as the value of the option.
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Now form a replicating portfolio (continue)
Since
we have three securities for investment: the stock of IBM, the riskfree security, and the call option, how can we form this portfolio to figure out the price of the call option on IBM?
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Now form a replicating portfolio (continue)
Suppose
we buy shares of stock and borrow B dollars from the bank to form a portfolio. What is the payoff for the this portfolio for each state when the option matures? What is the cost of this portfolio? How can we make sure that this portfolio is the replicating portfolio of the option?
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How can we get a replicating portfolio?
Look
at the payoffs for the option and the uS+BRf portfolio Cu=max(uS-K,0) Portfolio
Option
C(K,T)
B+S
dS+BRf Cd=max(dS-K,0) now Maturity Now maturity
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Form a replicating portfolio
From the payoffs in the previous slide for the call option and the portfolio, to make sure that the portfolio is the replicating portfolio of the option, the option and the portfolio must have exactly the same payoff in each state at the expiration date. That is,
uS+BR = C dS+BR = C
f f
u d
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Form a replicating portfolio
Use the following two equations to solve for and B to get the replicating portfolio:
uS+BR = C dS+BR = C
f f
u d
The solution is Cu - Cd uCd - dCu = and B = (u - d ) S (u - d ) R f
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To get the value of the call option
By no arbitrage argument, the value or the price of the option is the cost of the replicating portfolio, B+S. Can you believe that valuing the option is so simple? Can you summarize the procedure to do it? This procedure walks you through the way of understanding the concept of no arbitrage argument.
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Summary
Using
the no arbitrage argument, we can see the cash flows from investing in a call option can be replicated by investing in stocks and risk-free bond. Specifically, we can buy shares of stock and borrow B dollars from the bank. The value of the option is
*S+B ( the number of shares *stock price
borrowed money), where B is negative
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Example of valuing a call
Suppose
that a call on IBM has a strike price of $55 and maturity of six-month. The current stock price is $55. At the expiration state, there is a probability of 0.4 that the stock price is $73.33, and there is a probability of 0.6 that the stock price is $41.25. The risk-free rate is 4%. Can you calculate the value of this call option? (the value is $8.32)
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How to value a put using the similar idea
We
can use the similar idea to value a European put. Before you look at my next two slides, can you do it yourself?
Still try to form a replicating portfolio so that
the put option and the portfolio have the exactly the same payoff in each state at the expiration date.
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How can we get a replicating portfolio of a pot option?
Look
at the payoffs for the put option and the portfolio Portfolio
uS+BRf Cu=max(K-uS,0)
Put option
P(K,T) Cd=max(K-dS,0)
B+S dS+BRf Now
maturity
now
Maturity
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Form a replicating portfolio
From the payoffs in the previous slide for the put option and the portfolio, to make sure that the portfolio is the replicating portfolio of the option, the put option and the portfolio must have exactly the same payoff in each state at the expiration date. That is,
uS+BR = C dS+BR = C
f f
u d
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Form a replicating portfolio
Use the following two equations to solve for and B to get replicating portfolio:
uS+BR = C dS+BR = C
f f
u d
The solution is
C - Cd uC - dCu = u and B = d (u - d ) S (u - d ) R f
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What happens?
You can see that the formula for calculating the value of a put option is exactly the same as the formula for a call option? Where is the difference?
The difference is the calculation of the payoff or cash flows in each state.
To get this, please try the valuation of put option in the next slide.
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Example of valuing a put
Suppose that a European put on IBM has a strike price of $55 and maturity of six-month. The current stock price is $55. At the expiration state, there is a probability of 0.5 that the stock price is $73.33, and is there a probability of 0.5 that the stock price is $41.25. The risk-free rate is 4%. Can you calculate the value of this put option? (the value is $7.24)
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Example of valuing a put option (continue)
Recall
that the value of call option with the same strike price and maturity is $8.32.
Can you use this call option value and the
put-call parity to calculate the value of the put option? Do you get the same results? ( if not, you have trouble)
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Can you learn something more?
Everybody knows how to set fire by using match. Long, long time ago, our ancestors found that rubbing two rocks will generate heat and thus can yield fire, but why don't we rub two rocks to generate fire now?
It is clumsy, not efficient
What have you learned from this example?
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What can we learn?
Using
the idea in the last slide, to value a call option, we don't need to figure out the replicating portfolio by calculating the number of shares and the amount of money to borrow. Instead we can jump to calculate the value of the call option using the way in the next slide.
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Risk-neutral probability
The
price of call option is
Cu - Cd uCd - dCu C = S + B = + u-d (u - d ) R f u - Rf 1 Rf -d = Cu + Cd Rf u -d u-d
let
p=(Rf-d)/(u-d) < 1. Then
1 [ pCu + (1 - p)Cd ] C= Rf
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Risk-neutral probability (continue)
Now we can see that the value of the call option is just the expected cash flow discounted by the risk-free rate. For this reason, p is the risk-neutral probability for payoff Cu, and (1-p) is the risk-neutral probability for payoff Cd. In this way, we just directly calculate the riskneutral probability and payoff in each state. Then using the risk-free rate as a discount rate to discount the expected cash flow to get the value of the call option.
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Examples for risk-neutral probability
Using
the risk neutral probability approach to calculate the values of the call and put options in the previous two examples.
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Two-period binomial tree
Suppose
that we want to value a call option with a strike price of $55 and maturity of six-month. The current stock price is $55. In each three months, there is a probability of 0.3 and 0.7, respectively, that the stock price will go up by 22.6% and fall by 18.4%. The riskfree rate is 4%. Do you know how to value this call?
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Solution
First
Stock price 67.43 C(K,T)=? 55 55 44.88 36.62 Now Three month Sixth month Now Three month Six month 1-p 1-p 0 p
draw the stock price for each period and option payoff at the expiration 27.67
82.67 Option p p 1-p 0
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Solution
Risk-neutral probability is
p=(Rf-d)/(u-d) =(1.01-0.816)/(1.226-0.816)=0.473
The probability for the payoff of 27.67 is 0.473*0.473, the probability for other two states are 2*0.473*527, and 0.527*0.527. The expected payoff from the option is 0.473*0.473*27.67= The present value of this payoff is 6.07 So the value of the call option is $6.07
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How to calculate u and d
In the risk-neutral valuation, it is important to know how to decide the values of u and d, which are used in the calculation of the risk-neutral probability. In practice, if we know the volatility of the stock return of , we can calculate u and d as following:
u = e h d = 1/ u
Where h the interval as a fraction of year. For example, h=1/4=0.25 if the interval is three month.
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Example for u and d
Using
the two-period binomial tree problem in the previous example. If is 40.69%,
Please calculate u and d? Please calculate the risk-neutral probability p? Please calculate the value of the call option?
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The mo...