valuing options using binomial trees

DrakeDRAKE UNIVERSITY

Fin 288

Valuing Options Using Binomial Trees

DrakeDrake University

Fin 288Binomial Trees

A binomial tree is designed to represent the possible paths the stock price might follow.To begin we want to assume that the stock will be held for one period of time with two possible outcomes for the price at the end of the period, either an increase or decrease (following a binomial probability distribution)The model will then be extended to account for multiple price movements through multiple periods of time.


Fin 288A simple Example

Assume that the stock is currently selling for $20 and at the end of three months it will be worth either $22 or $18.Assume we want to value a European call option with an exercise price of $21 at the end of the three months. There are two possible outcomes:

Stock Price = $22 and intrinsic value of option = $1Stock Price = $18 and intrinsic value of option = $0


Fin 288

Possible Outcomes Shown on Binomial Tree Model

Stock Price =$20

Stock Price = $22

Stock Price = $18

Time 0 Time 1

Option Price = $1

Option Price = $0


Fin 288A no arbitrage solution

To price the option we want to assume that we can use it to eliminate risk.Combining a short position in the call option with a spot position in the underlying stock produces a result where gains on the stock are offset by option. We want to establish a riskless portfolio so that the value is the same regardless of whether the price of the stock increases to $22 or decreases to $18 in 3 months. Given that the portfolio is riskless, the portfolio should earn a rate equal to the risk free rate.


Fin 288

Value of Portfolio in 3 months

Consider a portfolio of shares of the stock and one option.If the stock price increases to $22 and the option has a $1 intrinsic value the portfolio is worth:

$22-$1If the stock price decreases to $18 and the option has a $0 intrinsic value the portfolio is worth:

$18


Fin 288

Determining the number of shares in the portfolio

The portfolio has two possible outcomes in three months: $22 - $1 and $18 Since the two possible outcomes have equal values it is possible to solve for

$22 - $1 = $18 $22 - $18 $1 ($22 - $18) $1

$1/$4 = .25


Fin 288Time Value

If the stock price increases to $22 the portfolio is worth $22 $1 = $22(.25) -1 = $4.5If the stock price decreases to $18 the portfolio is worth $18 $18(.25) = $4.5Since the portfolio is risk free it should have a present value at time 0 equal to $4.5 discounted at the risk free rate for 3 months.Assuming a risk free rate of 12% the PV is:

$4.5e-0.12(.25)=$4.367


Fin 288Pricing the option

Let the price of the option be fGiven the current share price of $20, the value of the portfolio at time zero should be equal to

$20(.25) – fWhich must equal the value of either portfolio

4.367Setting the two equal to each other an solving for f finds the value of the call.

$20(.25) – f = 4.367f =.633


Fin 288An overpriced call

If the option price was greater than 0.633 you could make a risk free return above the risk free rate.What if the value of the call was .80 today?The current value of the portfolio = $20(.25)-.80 = $4.20 In either outcome the value of the portfolio after three months is 4.5 this implies a risk free return of

4.20er(.25)=4.5r = .2759 = 27.59%


Fin 288An overpriced call

Since you can earn a return greater than the risk free rate the demand for the stock will increase causing its price to increase. Additionally the supply of the option will increase (since everyone will want to write an option to take advantage of the higher price) decreasing its price.


Fin 288An underpriced call

What if the price of the call is less than .633?Assume it is .50, since it is cheap, you buy it and sell the stock short receiving $25(.25) = $5 your total cash flow is $5 - .50 = $4.50In three months you need to buy the stock to close the short sale. If the price increased to $22 you need to pay

-$22(.25) + 1 = -$4.50If the price decreased to $18 you need to pay

-$18(.25) = -$4.50


Fin 288An underpriced call

In either case you essentially borrowed $4.50 at time zero and repaid $4.50 in three months. You have borrowed at a rate much less than the risk free rate.The market will react to this increasing the call price (as demand for the option increases) and decreasing the share price (as there are more people shorting the stock).


Fin 288Generalizing the Model

Let the stock price at time 0 be S0 with two possible outcomes:

The stock price increases by a factor greater than 1, u The resulting stock price is then S0uThe stock price decrease by a factor less than 1, d. The resulting stock price is then S0d

If the stock price increases the value of the option is fu

If the stock price decreases the value of the option is fd


Fin 288Binomial Tree Model

S0

f

S0u

fu

S0d

fd

Time 0 Time 1


Fin 288The generalization

Setting the value of the two portfolios at time 1 equal produces

S0u-fu = S0d-fd

Solving for produces:

Delta is thus the change in the option price divided by the change in the stock price

dSuS

ff du

00


Fin 288The PV of the portfolio

Since the value of the portfolio at time 1 is the same regardless of whether the price increased or decreased either price can be used to find the value of the portfolio at time 0Using a stock price increase the value of the portfolio at time 0 is

(S0u-fu)e-rT


Fin 288The value of the option

The PV of the portfolio has to equal the cost of setting up the option at time 0:

S0 f = (S0u -fu)e-rT The goal is to solving for f, the current value of the option.

f = S0 (S0u -fue-rT

f = S0 S0u e-rT-fue-rT

f = S0 ue-rT)-fue-rT


Fin 288

Solving for the price of the option, f

substitutef = S0 ue-rT)-fue-rT

Simplifying producesf = e-rT[pfu+(1-p)fd]

Where :

du

dep

rT

dSuS

ff du

00


Fin 288Returning to the Example:

In the previous example we had a current price of $20 with the possibility of an increase to $22 or $20u=$22 solving for u, u =1.1Similarly $20d=$18, solving for d, d=.9Given the inputs from beforeu = 1.1 and d=.9 r=.12, T = .25, fu=1, and fd=0Plugging into the equations on the previous slide

p = .6523 and f = .6333Which is the same as before


Fin 288A Quick Question

Does the probability of an increase in the stock price impact the value of the option?NO! regardless of the probability the value of the option is the same – the probability of an increase or decrease is already incorporated in the price of the stock. Since the option price is based on the stock price, the probability does not impact the option price.


Fin 288Risk Preferences Review

Risk Loving vs. Risk Neutral vs. Risk AdverseAssume you are faced with two equally likely outcomes:

A gain of $10 and a loss of $5.

How much would you be willing to pay to accept the risk of the possible loss?


Fin 288Risk Preferences Review

Risk Neutral: If you are willing to pay $2.50, you are willing to pay a “fair price” to accept the risk. (If you repeated the event over and over on average you would receive your $2.50)Risk Averse: If you are willing to pay less than 2.50 lets say 2.00, you are risk averse. The $.50 represents a risk premium, the additional return you expect to earn for accepting the risk. The lower the amount you are willing to pay the more risk averse you are.


Fin 288Risk - Neutral Valuation

Now assume that p is the probability of an increase in the stock price and 1-p is the probability of a decrease in the price of the stock. This would imply that the option price

f = e-rT[pfu+(1-p)fd]

is the expected payoff from owning the option discounted at the risk free rate.


Fin 288Risk Neutral Valuation

Using p in the same way produces the Expected value of the stock at time t

E(ST)=pS0u+(1-p)S0d

E(ST)=pS0(u-d)+S0d

E(ST)=S0erT

The stock price grows on average at the risk free rate!

du

dep

rT

Substituting


Fin 288Risk Neutral Valuation

In a risk neutral world investors do not get compensated for risk. Implying the expected return on all securities is the risk free rate of interest. In the case of assuming p = the probability of an increase in the stock price, the option is equal to its expected payoff discounted at the risk free rate. This is based upon our portfolio assumptions that both return the risk free return. The risk neutral valuation is the same as the no arbitrage valuation.


Fin 288No Arbitrage

Stock Price =$20

Stock Price = $22

Stock Price = $18

Time 0 Time 1

Option Price = $1

Option Price = $0


Fin 288No arbitrage

Previously we found that the no arbitrage value of the option was .633 assuming that the we used the risk free rate of 12% in the valuation.


Fin 288No arbitrage vs. Risk Neutral

In the risk neutral world the expected value of the stock price in one period would be also equal to the risk free return of 12%Therefore, if p is the probability of a stock price increase, the expected value of the stock in the future must equal the value of the stock today assuming it grows at the risk free rate or:

22p+18(1-p) = 20e.12(.25)

22p-18p = 20e.12(.25)-184p=2.6092p=0.6523


Fin 288

Expected Value of Option in 3 months

Given the probability of a stock price increase is 0.6523, if the price increases to $22 in our example the option is worth $1. If the price decreased to $18 the option is worth $0The expected value of the option in three months in our example would then be:

0.6523($1)+(1-0.6523)$0 =0.6523


Fin 288Value of Option at Time 0

In a risk neutral setting the value of the option should also increase at the risk free rate of 12%By discounting the value of the option at time = 3 months back to time =0 at the risk free rate we can find the value of the option at time 0

0.6523e-(.12)(.25)= 0.633

Which is the same value of the option in the No Arbitrage solution


Fin 288

Risk Neutral vs. “Real World”

In the “real world” the probability of an increase in the stock price is not equal to p. This occurs when the expected payoff on the stock is equal to the risk free rate, which is not usually the case.What if expected return on the stock is 16%?Let q = the probability of an increase and solve for q assuming a 16% expected return on the stock.

22q+18(1-q) = 20e.16(.25)

q = .7041


Fin 288

Real World return on the Option

If q = .7041 the expected payoff on the option is

.7041(1)+(1-.7041)0 = .7041

Given a risk neutral value of the call at time 0 of 0.633 the implied discount rate on the option is

.633=.7041e-r(.25)

R = .4258 = 42.58%


Fin 288Risk Neutral

While the implied discount rate for the option could be found it is based on the assumption that the value of the option at time 0 is based on the risk neutral value of the option.Without the risk neutral value of the option it would not be possible to compare how much extra return the option should pay based on its increased risk compared to the stocks expected return of 16%.


Fin 288Extending the tree

Now assume that we want to look another 6 months out.In the previous example it was assumed that the price increased by10% or decreased by 10%.Assume that at time t = 3 months this is again the case. You can build the tree out from the possible payoffs at time t= 3 months.


Fin 288


Stock Price =$20

Stock Price = $22

Stock Price = $18

Time 0Time 1

(3 months)


Fin 288

Possible Outcomes after an initial price increase

Stock Price =$22

Stock Price = $24.2

Stock Price = $19.8

Time 1(3 months)

Time 2(6 months)


Fin 288

Possible Outcomes after an initial price decrease

Stock Price =$18

Stock Price = $19.8

Stock Price = $16.2

Time 1(3 months)

Time 2(6 months)


Fin 288

Combining the possible outcomes

$20

Time 0

Time 1(3 months)

Time 2(6

months)

Stock Price =$22

Stock Price

= $18

Stock Price

= $24.2

Stock Price

= $19.8

Stock Price

= $16.2


Fin 288Calculating the Option price

The option price at time zero now depends upon two different time periods. Assume we want to look at an option that has a strike price of $21 at time t = 6 months.The price of the option at time 6 months will equal its intrinsic value as before, but the price at time t = 3 months will need to be calculated based upon the possible outcomes at time 6 months.


Fin 288

Possible Outcomes after an initial price increase

Stock Price =$22

Stock Price = $24.2Option Price = $3.2

Stock Price = $19.8Option Price = $0Time 1

(3 months)Time 2

(6 months)


Fin 288Option price

Using the option price equation from before the price of the option assuming an initial increase to time t=3 can be found.


f = e-0.12(.25)[(.6523)(3.2)+(.3477)(0)]=2.2057


Fin 288

Possible Outcomes after an initial price decrease

Stock Price =$18

Stock Price = $19.8Option Price = $0

Stock Price = $16.2Option Price = $0Time 1

(3 months)Time 2

(6 months)


Fin 288Option price

Using the option price equation from before the price of the option assuming an initial decrease to time t-3 can be found.


f = e-0.12(.25)[(.6523)(0)+(.3477)(0)]

=0


Fin 288


$20

Time 0

Time 1(3 months)

Time 2(6

months)

Stock Price =$22

Option Price =$2.0257

Stock Price = $18

Option Price =$0

Stock Price = $24.2

Option Price = $3.2

Stock Price=$19.8Option Price

=$0

Stock Price = $16.2

Option Price =$0


Fin 288Option price at time 0

Using the option price equation from before the price of the option at time 0 can be found using the option prices at time t= 3 months


f = e-0.12(.25)[(.6523)(2.0257)+(.3477)(0)]=1.2823


Fin 288Note:

In our example the time period was fixed for each step of the process and the proportional up and down movement were the same for each step.Therefore the risk neutral probability p is the same at each step.


Fin 288

Generalizing the possible outcomes

f

Time 0

Time 1(3 months)

Time 2(6

months)

Stock Price =S0u

Option Price =fu

Stock Price = S0d

Option Price =fd

Stock Price = S0u2

Option Price = fuu

Stock Price=S0ud

Option Price =fud

Stock Price = S0d2

Option Price =fdd


Fin 288

Substituting for generalization of f

The original pricing equation gave usf = e-rT[pfu+(1-p)fd]

When applied to time t=3 it becomesfu = e-rT[pfuu+(1-p)fud] and fd = e-rT[pfud+(1-p)fdd]


f = e-2rT[p2fuu+2p(1-p)fud+(1-p)2fdd]

Substituting


Fin 288Put Options

The same procedure can be used to calculate the value of a put option. Using the same problem as before but to find the price of a put option we would need to look at the tree to find the value of the option at time t = 6 months then work back.


Fin 288Valuing a European Put

$201.059

Time 0

Time 1(3 months)

Time 2(6

months)

Stock Price =$22


Stock Price = $18


Stock Price = $24.2

Option Price = $0


=$1.2

Stock Price = $16.2

Option Price =$4.8


Fin 288

Generalization of f in the put option

The original pricing equation gave us


f = e-2(.12)(.25)[(.6523)20+2(.6523)(.3477)1.2+(.3477)24.8] =1.059136


Fin 288American Put Options

So far we have assumed that the option can only be exercised at the end of the six months. For a put option sometimes it is favorable to exercise the option early. If the option is exercised early, the payoff will equal the exercise price minus the stock price.


Fin 288


$201.268

Time 0

Time 1(3 months)

Time 2(6

months)

Stock Price =$22


Stock Price = $18

Option Price =$2.3793 $3

Stock Price = $24.2

Option Price = $0


=$1.2

Stock Price = $16.2

Option Price =$4.8


Fin 288Option Pricing Equation

Now the generalization for two periods does not hold. fu and fd will need to be found independently

fu = e-rT[pfuu+(1-p)fud] and fd = e-rT[pfud+(1-p)fdd]

fu = e-.12(.25)[.6523(0)+(.3477)1.2]= .4049

fd = 3

f = e-.12(.25)[(.6523)0.4049+(.3477)3]=1.268


Fin 288Delta

In the Black Scholes introduction we defined delta as a measure of the sensitivity of the option value to a change in the price of the underlying asset. Earlier today, we defined delta to be the change in the option price divided by the change in the stock price or

dSuS

ff du

00


Fin 288Delta

In either case delta is essentially a hedge ratio.Today we found delta by solving for the number of shares of stock needed to produce a risk free return in our portfolio of a short call option and a long stock. In other words, Delta is the number of shares we should hold to produce a riskless hedge (known as delta hedging). This is consistent with the Black Scholes definition.


Fin 288Delta

The delta of a call will be positive, the delta of a put will be negative and the delta will change over time.


Fin 288

Delta in our earlier 2 period call example

$20Delta=2.0257/(22-18)=0.5964

Time 0Time 1

(3 months)

Time 2(6

months)

Stock Price =$22Option Price

=$2.0257Delta

=3.2/(24.2-19.8)=.7273

Stock Price = $18

Option Price =$0Delta = 0

Stock Price = $24.2

Option Price = $3.2


=$0

Stock Price = $16.2

Option Price =$0


Fin 288Practice Problem

Assume we have a European Put option with a strike price of $52 on a stock currently selling for $50.Assume that there are 2 years prior to expiration and each year the stock price either moves up by 20% or down by 20% and the risk free rate of interest is 5%

Find the value of the option at time 0 Find the delta at each nodeReprice the option assuming it is an American Option


Fin 288

Increasing the number of steps

So far we have presented extremely simple examples of the binominal tree. The example can be extended to shorter time frames and an increased number of steps. Looking at the two step model there were 4 possible paths the stock price could take.


Fin 288

Generalizing the possible outcomes

$20

Time 0

Time 1(3 months)

Time 2(6

months)

Stock Price =S0u

Option Price =fu

Stock Price = S0d

Option Price =fd

Stock Price = S0u2

Option Price = fuu

Stock Price=S0ud

Option Price =fud

Stock Price = S0d2

Option Price =fdd


Fin 288A three step model

In a thee step model there are 23=8 possible paths


Fin 288Increasing steps

With 30 time steps there would be 31 terminal stock prices resulting in 230 or about 1 billion possible price paths. However the equations defining the payouts do not change as the number of steps increases.The key is the ability to define, u, d, p and the size of the step


Fin 288Determining u and d

u and d should be determined by the stock price volatility. Let t be one time step. Then:

tr

t

eadu

dap

udeu

where

1


Fin 288

Increasing the number of intervals

The number of intervals assumed to occur prior to the expiration of the option plays a key role in price obtained from the binomial tree model. As the number of periods increases the model will converge to a single price.


Fin 288An Example: one step

Assume as before that we have a European call option on a share of stock and the current price of the share of stock is $20, the option expires in one year and has a strike price of $21. Let the annual volatility of the stock price be .40 and the risk free rate be 10%


Fin 288

We need to calculate u, d and p

5192.7536.3269.1

7536.0513.1

where

7536.3269.1

11

3269.15.4.

p

eadu

dap

ud

eeu

tr

t


Fin 288Su and Sd

The values of the stock at the end of the 6 month period would be

Su=S0(1.3269) = 26.5379

Sd=S0(.7536) = 15.0728


Fin 288


Stock Price =$20

Stock Price = $ 26.5379Option Price = 5.5379

Stock Price = $15.0728Option Price = $ 0

Time 0Time 1

(6 months)


Fin 288

It is then easy to solve for the call option price.


f = e-.1(.5)[.5192(5.5379)+(1-.816)0]=2.735


Fin 288Using two steps

The same problem could be done assuming a two step tree over the 6 month period. Assume as before that we have a European call option on a share of stock and the current price of the share of stock is $20, the option expires in one year and has a strike price of $21. Let the annual volatility of the stock price be .40 and he risk free rate be 10%


Fin 288The two step model

Again we need to solve for u, d, and p

51303.8187.2532.1

8187.0253.1

where

8187.3269.1

11

2214.125.4.

p

eadu

dap

ud

eeu

tr

t


Fin 288The Two Step Model

$20

Time 0

Time 1(3 months)

Time 2(6

months)

Stock Price =24.4381

Option Price =fu

Stock Price =16.3746

Option Price =fd

Stock Price = 29.8365

Option Price = 8.8365

Stock Price=20Option Price =0

Stock Price = 13.4064

Option Price =0


Fin 288

We can use the generalized approach to solve for the value of the option at time 0 since there will not be any early exercise


f = e-2(.1).25[.5132(8.8365)+2(.513)(1-.513)0+(1-.513)20]

=2.2124


Fin 288The three period model

You could extend the model again to the three period model.

Now the change in each step is equal to 2 months or .2563

valuing options using binomial trees

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