FIN 819-lecture 7

Option Pricing Approaches

Valuation of options

Today’s plan Review of what we have learned about options We first discuss a simple business ethics case. 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

Business ethics

Suppose that you have applied to two jobs: A and B. Now you have received the offer letter for job A and have to make a decision now about whether or not to take job A. But you like job B much more and the decision for job B will be made one week later.

What is your decision?

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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

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The basic idea behind the binomial tree approach

Suppose we want to value a call option on ABC stock 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 the value of the call option at time zero.

Let the current price of ABC is S and there are two states when the call option matures: up and down. If the state is up, the stock price for ABC is Su; if the state is down, the price of ABC is Sd.

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The stock price now and at maturity

S

uS

dS

Su

Sd

Now maturity

If we define:

u = Su/S and d = Sd/S. Then we have

Su=uS and Sd=dS

S

Nowmaturity

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The risk free security

The price now and at maturity

Rf

Rf

1

nowmaturity

Here Rf=1+rf

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The call option payoff

Cu=Max(uS-K,0)

Cd=Max(dS-K,0)

C(K,T)

Now

maturity

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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 ABM, the risk-free security, and the call option, how can we form this portfolio to figure out the price of the call option on ABC?

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Now form a replicating portfolio (continue)

Suppose we buy Δ 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 portfolio

Option

Portfolio

C(K,T)

Cd=max(dS-K,0)

Cu=max(uS-K,0)

ΔdS+BRf

ΔuS+BRf

B+ΔS

now MaturityNow 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,

• ΔuS+BRf = Cu

• ΔdS+BRf = Cd

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Form a replicating portfolio

Use the following two equations to solve for Δ and B to get the replicating portfolio:

• ΔuS+BRf = Cu

• ΔdS+BRf = Cd

The solution is

f

uddu

Rdu

dCuCBand

Sdu

CC

)()(

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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+ΔS.

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 Δ shares of stock and borrow B dollars from the bank.

The value of the option is• Δ*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 ABC has a strike price of $55

and maturity of six-month. The current stock price for ABC 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) (u=1.33,d=0.75, Cd=0, Cu= $18.33, Δ=0.57, B=-$23.1)

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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 put option?

Look at the payoffs for the put option and the portfolio

Put option

Portfolio

P(K,T)

Cd=max(K-dS,0)

Cu=max(K-uS,0)

ΔdS+BRf

ΔuS+BRf

B+ΔS

now Maturity

Nowmaturity

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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,

• ΔuS+BRf = Cu

• ΔdS+BRf = Cd

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Form a replicating portfolio

Use the following two equations to solve for Δ and B to get replicating portfolio:

• ΔuS+BRf = Cu

• ΔdS+BRf = Cd

The solution is

f

uddu

Rdu

dCuCBand

Sdu

CC

)()(

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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 there is 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) (u=1.33,d=0.75, Cu=0, Cd=$13.75 Δ=-0.43, B=$30.8)

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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

let p=(Rf-d)/(u-d) < 1. Then

df

uf

f

f

uddu

Cdu

RuC

du

dR

R

Rdu

dCuC

du

CCBSC

1

)(

duf

CppCR

C )1(1

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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 risk-neutral 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.

Call

( u=1.33,d=0.75, Rf=1.02, p=0.47, C1=0.47*18.33,

PV(C1) = C1/Rf= $8.37. Put. ( p=0.47, C1=0.53*13.75, PV(C1)=C1/Rf=$7.14)

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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 risk-free rate is 4%.

Do you know how to value this call?

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Solution

First draw the stock price for each period and option payoff at the expiration

55

67.43

82.67

55

36.6244.88

C(K,T)=?

27.67

0

0

p

1-p

p

1-p

p

1-p

Stock price Option

Now Three month

Sixthmonth

NowThreemonth

Sixmonth

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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 σ, we can calculate u and d as following:

Where h the interval as a fraction of year. For example, h=1/4=0.25 if the interval is three month.

ud

eu h

/1

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Example for u and d

Using the two-period binomial tree problem in the previous example. If σ is 40.69%, • Please calculate u and d?

• Please calculate the risk-neutral probability p?

• Please calculate the value of the call option?

• (u=1.17,d=0.85, p= 0.5)

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The motivation for the Black-Scholes formula

In the real world, there are far more than two possible values for a stock price at the expiration of the options. However, we can get as many possible states as possible if we split the year into smaller periods. If there are n periods, there are n+1 values for a stock price. When n is approaching infinity, the value of a European call option on a non-dividend paying stock converges to the well-known Black-Scholes formula.

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A three period binomial tree

S

u3S

u2dS

ud2S

d3S

There are three periods. We have four possible values for the stock price

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The Black-Scholes formula for a call option

The Black-Scholes formula for a European call is

Where

)()(),,,,( 11 tdNKedSNrtKSC rt

tt

rtKSd

2

1)/ln(1

optiontheofpricestrikeK

pricestockstodayS 'periodpervolatilityreturnstock

ratefreeriskcompoundedlycontinuousr irationtotimet exp

functionondistributinormalcumulativedN )(

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The Black-Scholes formula for a put option

The Black-Scholes formula for a European put is

Where

)()(),,,,( 11 dSNdtNKertKSP rt

tt

rtKSd

2

1)/ln(1

optiontheofpricestrikeK

pricestockstodayS 'periodpervolatilityreturnstock

ratefreeriskcompoundedlycontinuousr

irationtotimet expfunctionondistributinormalcumulativedN )(

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The Black-Scholes formula (continue)

One way to understand the Black-Scholes formula is to find the present value of the payoff of the call option if you are sure that you can exercise the option at maturity, that is, S-exp(-rt)K.

Comparing this present value of this payoff to the Black-Scholes formula, we know that N(d1) can be regarded as the probability that the option will be exercised at maturity

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An example

Microsoft sells for $50 per share. Its return volatility is 20% annually. What is the value of a call option on Microsoft with a strike price of $70 and maturing two years from now suppose that the risk-free rate is 8%?

What is the value of a put option on Microsoft with a strike price of $70 and maturing in two years?

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Solution

The parameter values are

Then50,70

08.0,2,2.0

SK

rt

27.12$;63.2$

22.0)765.0()(

315.0685.01)(1)(

4825.02

1)/ln(

1

11

1

PC

NtdN

dNdN

tt

rtKSd