437 Option Valuation

8/10/2019 437 Option Valuation

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Notes on Option Valuation

1. The Binomial Option Pricing Model

2. How to Choose the Step Sizes

3. The Black-Scholes Formula

4. Monte Carlo Simulation

5. Sensitivities for Option Value

6. Common Types of Exotic Options

7. Options on Futures


2/49

The Binomial Option Pricing Model

S = spot price of underlying asset

K = striking price of option

r = annualized continuously

compounded interest rate

y = annualized continuously

compounded yield from income

component

h = length of one period (in years)

With reinvestment of income component, one unit at

beginning of period grows tohye units at end of

period.

equity dividend yield

commodity convenience yield

currency foreign interest rate


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Over each period,

u S

S

d S

For a call expiring in one period,

Cu = max [ u S - K, 0 ]

C

Cd = max [ d S - K, 0 ]

For a portfolio with units of the underlying asset

and B dollars in bonds,

BeSue hrhy

BS

BeSde hrhy


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Choose and B so that

uhrhy CBeSue

dhrhy CBeSde

Solving these equations gives

Sdue

CChy

du

)(

)( due

CdCuB

hr

ud

With this choice of and B, the value of a European

call must be

BSC

Otherwise, there would be an arbitrage opportunity.


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Writing and B in terms of Cu and Cd gives the

fundamental valuation equation

hrdu eCpCpC /])1([

where

du

dep

hyr

)(

The parameter p is the probability of an upward

move that would make the expected rate of return on

the underlying asset equal to the interest rate.

hrhyhy edepuep )1(

It is the risk-neutral probability of an upward move,

not the actual probability.

For an American call, the current value must be

],[max BSKSC .


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The value of the option depended only on

Spot price of underlying asset

Striking price

Volatility (through u and d)

Interest rate

Income yield

Time to expiration

It did not depend on

Forecasting direction of future moves

Probabilities of upward and downward moves

Beta of the underlying asset

Investors attitudes toward risk


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For a call with two periods until expiration,

u2Su S

S u d S

d S

d2S

Cuu = max [u2S K, 0]

Cu

C Cud = Cdu= max [u d S K , 0]

Cd

Cdd = max [d2S K, 0]

To replicate a multiperiod call, we need a dynamic

portfolio. The portfolio's composition will be different

at each node on the tree.

A call with any number of periods until expiration can

be valued by working backward from the end one step

at a time. Each step provides the information needed

for the next step.


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For a European call,

Sudue

CCe/CpCpC

hy

duuu

u

hr

duuuu

)(

])1([

Sddue

CC

e/CpCpC

hy

ddud

d

hrddudd

)(

])1([

Sdue

CC

e/CpCpC

hy

du

hrdu

)(

])1([

By substitution we could write C as

dd

hr

ud

hr

duhr

uuhr

CepCepp

CeppCepC

222

222

)1()1(

)1(

This says that we could value the European call bycalculating the discounted payoff along each path,

weighting each result by the risk-neutral probability of

that path occurring, and then summing across all of

the paths.


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For an American call,

hr

duuuu e/CpCpKSuC ])1([,max

hrddudd e/CpCpKSdC ])1([,max

hrdu e/CpCpKSC ])1([,max

The value of h is determined by dividing the time to

expiration T by the number of periods chosen for the

calculation.

The particular payoff of a call option played no special

role. The same arguments apply to other derivativeswith payoffs that depend only on the value of the

underlying asset (but not on its previous path).

If V is the value of a general derivative receiving

periodic payouts of Xu or Xd , then the fundamental

equation becomes

hrdduu e/XVpXVpV ])()1()([


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Paths for Stock and Option Values

270

180

120 90

80 60

40 30

20

10

190

107.27(1.00)

60.4610

(.848)

34.07 5.45

(.719) (.167)

2.97

0(.136)

0

0.00

0


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Step III - Stock Goes Down to 60

1) Sell .848 - .167 = .681 shares, taking in

.681 (60) = 40.860

2) Use this to repay part of the borrowing. This

reduces the total borrowing to

41.281 (1.1) - 40.860 = 4.549

Step IV u - Stock Goes Up to 90

1) The shares owned are now worth

.167 (90) = 15

2) Total borrowing is

4.549 (1.1) = 5

3) Total value of the portfolio is 10, which is exactly

the value of the call.


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Step IV d - Stock Goes Down to 30

1) The shares owned are now worth

.167 (30) = 5

2) Total borrowing is

4.549 (1.1) = 5

3) Total value of the portfolio is 0, which is exactly the

value of the call.


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STOCK-BOND PORTFOLIOS

EQUIVALENT TO OPTIONS

Long stock

(less than one share)

Short stock

(less than one share)

+ + + +

Long

bonds

(lending)

Short

bonds

(borrowing)

Long

bonds

(lending)

Short

bonds

(borrowing)

Asstockp

rice

rises

Buy

stock

and

sell

bonds

Longstock

(one

share)

+

Long

one put

Long

one

call

Long

one

put

Shortstock

(one

share)

+

Long

one call

Sell

stock

and

buy

bonds

Asstock

price

falls

Sell

stock

and

buy

bonds

Long

stock

(one

share)

+

Short

one call

Short

one

put

Short

one

call

Short

stock

(one

share)

+

Short

one put

Buy

stock

and

sell

bonds


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Futures contracts can be used as a substitute for holding

the underlying asset. Under the assumptions, the

proper futures price for a contract with time t untildelivery is

.e )( tyrSF

Over each period, holding one unit of the underlying

asset with reinvestment of income is equivalent toholding er(t - h)

eyt

futures contracts and placing the

amount S in a fixed-income account. At the end of the

period,

hyhrtyrhtyrtyhtr SuSSSu ee)ee(ee )()()()(

hyhrtyrhtyrtyhtr SdSSSd ee)ee(ee )()()()(

Instead of holding units of the underlying asset and

the amount B in bonds, one would hold

number of futures = e

yt r(t- h)

amount in bonds = S+ B


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How to Choose the Step Sizes

To get good results, h should be small and u and dshould reflect the volatility of the underlying asset, .

The parameter is the annualized standard deviation

of the continuously compounded rate of return of the

underlying asset.

Three different methods for specifying u and d may

be used. When h is very small, they all produce

nearly identical results, but each has some advantages

and disadvantages.

A. The method used in the software package providedwith the text is

hh edeu

ee

eep

hh

hh)yr(


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This method is easiest for sensitivity calculations, but

h must be small enough to have

eee hh)yr(h

In the software, the termhyre )( is referred to as

the growth factor per step.

B.

Another method is

hhyreu )(

hhyred )(

ee

ep

hh

h

1

Here sensitivity calculations are more awkward, but

there is no restriction on inputs.


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C. A third possibility is

eee

euh

hh

h)yr(

2

eee

edh

hh

h)yr(

2

2

1p

This method is similar to B, but with the additionaladvantage that p is 1/2 and the relation between

the tree and standard Monte Carlo simulation

becomes clearer.


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The Black-Scholes Formula

For a European call, as h becomes small the results

produced by the binomial tree with all three methods

for choosing u and d converge to the Black-Scholes

formula,

,)(Ne)(N 2

1 dKdeSCTrTy

where N is the standard normal distribution function

and

T

)e/e(log

12

2

2

1

1

dd

TTKSd

rTyT


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The fundamental valuation equation linking beginning

and end-of-period values becomes the Black-Scholes

partial differential equation

,0)(2221 CrCCSyrCS TSSS

where the subscripts indicate derivatives of C with

respect to S and T.

By using the put-call parity relation,

,TrTy eKeSPC

and basic properties of the standard normal

distribution function, we can find the Black-Scholes

formula for a European put,

)(N)(N 12 deSdeKPTyTr

Analytic formulas are also available for some exotic

options.


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Monte Carlo Simulation

The payoffs of some derivatives depend on the entire

path followed by the price of the underlying asset.

One example would be a security that pays on the

expiration date the maximum price achieved by the

underlying asset during the life of the contract. For

this derivative, a binomial tree cannot be used in the

usual way because the value at any node will depend

on the previous path followed in reaching that node.

Securities like this can still be valued by calculating the

discounted payoff along each path, weighting each

result by its risk-neutral probability of that path

occurring, and summing across all of the paths. Therecursive procedure we used earlier provided a

powerful way to effectively evaluate all of the paths,

but it cannot be used here because of the path-

dependent nature of the payoffs. Without this help,

evaluating every path individually would become

prohibitively time-consuming.


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The solution is to limit the scenarios considered to a

randomly chosen subset of the complete possibilities.

For example, using Method C of choosing u and d ,we could construct a random path by letting the price

moves over each period be

,2 ~)( bh

hh

hyr e

ee

eSS

where b~

is a binomial random variable taking on the

value +1 if a coin flip is heads and - 1 if it is tails. If

the time to maturity T is divided into n periods, then

constructing each path would require n coin flips. Bybuilding and using a sufficient number of paths, all of

which would have the same risk-neutral probability of

occurring, we can reach a reasonable estimate of the

derivative value.

Since there is now no advantage in having paths that

recombine, the binomial random variable is usuallyreplaced with a normal random variable and the

middle term becomes2/2 he . We then have


25/49

,][~2/)( 2 nhhhyr eeeSS

where n~ is a normal random variable with a mean ofzero and a standard deviation of one.

This description can be generalized in several ways.

The periods do not need to be of equal length, and r,

y , and can be different in each period. In fact, y

and can also depend on current and past values of

the price of the underlying asset.

Monte Carlo simulation is well-suited for derivatives

with path-dependent payoffs, but it cannot handle

securities with American features. When making

recursive calculations on a binomial tree, we knew at

each node the value contributed by all of the future

possibilities that could occur, so it was easy to decide

about early exercise. With Monte Carlo simulation, we

would know at each point the value of only one of the

future possibilities.


26/49

A number of techniques are available to improve the

efficiency of a Monte Carlo simulation. One of the

simplest is the antithetic variable method. For eachrandom path constructed, a second path is built using

the same random drawings with their signs reversed.


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Sensitivities for Option Value

It is often helpful to know how options will respond tosmall changes in the inputs. These sensitivies are

normally defined as the derivatives with respect to the

inputs. The most widely used sensitivities are referred

to by Greek letters.

delta value asset price

gamma delta asset price

theta value time

kappa value volatility

rho value interest rate

Kappa is also known as vega.

With an analytic formula, the derivatives can be

calculated explicitly. For example, with the Black-

Scholes formula the deltas are


28/49

delta of call = )(N 1deTy

delta of put = )(N 1deTy

With the binomial model, using method A for choosing

u and d , the sensitivities for calls are usually

calculated as

delta =Sdue

CChy

du

)(

gamma =Sdu

du

)(

theta =hCC du

2

Rho and kappa (or vega) are calculated by running the

model a second time with a small change in the

corresponding input. Similar procedures are used forother securities and when u and d are defined

differently.


29/49

A slightly different way of calculating delta and gamma

is used in the text and software,

delta =Sdu

CC du

)(

gamma =2/)(

22 Sdu

du

With Monte Carlo simulation, all of the sensitivities are

calculated by running the process again with a small

change in the corresponding input. The same set of

random drawings is used in both runs.


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Barrier Options

Terms of option change when the underlying assetprice reaches a designated level called the barrier.

1) What happens at the barrier?

knock-out option option is cancelled

knock-in option option is activated

mandatory exercise

optionoption is exercised

2) Where is the barrier in relation to the current

asset price?

down-and-out option

option is cancelled at a

barrier below thecurrent asset price


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up-and-out option

option is cancelled at a

barrier above the

current asset price

down-and-in option

option is activated at a

barrier below the

current asset price

up-and-in option

option is activated at a

barrier above thecurrent asset price

mandatory exercise call

option is exercised at a

barrier above the

current asset price

mandatory exercise putoption is exercised at abarrier below the

current asset price

In some variations, the terms of the option change

only if the underlying asset price remains above (or

below) the barrier for a specified period of time.


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Asian Options

Payoff depends on average price of underlying asset

over some period.

Upon exercise, an average price call pays

average price striking price

and an average price put pays

striking price average price

Terms will specify

1) period over which average is calculated

2) frequency of observations used in calculating

average


38/49

Binary Options

1) Asset-or-nothing call

Payoff at expiration: S*

if S*K

0 if S*K

Current value:)(N 1deS

Ty

2) Asset-or-nothing put

Payoff at expiration: S*

if S*K

0 if S*K

Current value: )(N 1deSTy

3) Cash-or-nothing call

Payoff at expiration: K if S*

K0 if S

*K

Current value: )(N 2deKTr


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Lookback Options

Payoff depends on maximum or minimum price of

underlying asset over some period.

Upon exercise, a lookback call pays

final asset price minimum price

and a lookback put pays

maximum price final asset price

Terms will specify

1) period over which maximum or minimum is

calculated

2) frequency of the observations used in calculating

maximum or minimum


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Exchange Options

An exchange option gives its holder the right toexchange one asset for another. The current value of

a European option to exchange B for A is

222

212

2

2

1

21

2

)2

1()/(log

where

)(N)(N

BBAA

ABBA

Ty

B

Ty

A

Txx

T

TyySSx

xeSxeSC BA

and is the correlation between A and B.


42/49

A European exchange option can be evaluated with

the Black-Scholes formula by replacing K with SB, r

with yB, and with .

An ordinary option is a special case of an exchange

option where B is a zero coupon bond with a principal

of K and a time to maturity of T.


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Compound Options

A compound option is an option on an option.

Both the underlying option and the compound option

may be either a call or a put.

The terms of the contract specify the expiration dates

and strike prices of both options.

An analytic formula is available for European

compound options, but it is quite complicated.

The software package includes European compound

options.


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Options on Futures

A futures option pays upon exercise the difference

between the futures price and the striking price.

A futures price is not an asset price and does not

behave in the same way as an asset price. Valuing

options on futures is fundamentally different from

valuing options on assets.

If F is the futures price of a contract with time t until

delivery, then with our assumptions

tyreSF )(

Given the movement in the price of the underlying,

over each period

FueeSu hyrhtyr )()()(

F

FdeeSd hyrhtyr )()()(


45/49

A portfolio with futures contracts and B dollars in

bonds requires a current investment of only B dollars.

This is because the futures position can be established

with no initial cost. The value of the portfolio at theend of the period will be

BeFFue hrhyr )( )( B

BeFFde hrhyr )( )(

The contribution of each futures contract is simply the

change in the futures price over the period. We want

the end-of-period values of the portfolio to match the

end-of-period values for the option, so

d

hrhyr

u

hrhyr

CBeFFde

CBeFFue

)(

)(

)(

)(

Solving these equation gives

FdueCC

hyrdu

)(

)(


46/49

hrd

hyr

u

hyr

e

du

CeuCdeB

)(

)()( )()(

With this choice of and B, the beginning-of-period

value of the call must be the same as the beginning-of-

period value of the portfolio. The value of the

portfolio is B, so

hrdu eCpCpC /])1([ ,

where

du

de

p

hyr

)(

.

For an American call, we would have

hrdu eCpCpKFC ])1([,max .

At this point, we could use any of the three methods

for choosing u and d . With our assumptions, the

volatility of a properly priced futures is the same as


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that of the underlying asset, so the same applies to

both.

Although all three methods converge to the sameconclusions when h becomes small, there is a special

advantage in using Methods B or C for a futures

option. The benefit is that y does not have to be

specified separately as an input. All of the relevant

information is contained in the futures price. For

example, if Method B is used, with

hhyreu )( hhyred )(

then we have

FeFue hhyr )( F

FeFde hhyr )(

and

hh

h

ee

e

p

1

The calculations made for a futures option using

Method B are identical to those for an option on an


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asset when F replaces S, r replaces y, and Method

A is used to specify u and d . This is the procedure

used in the software.

For European options, as h becomes small the results

produced by the binomial tree converge to the Black

formulas for calls and puts on futures,

)N()N( 21 aeKaeFCTrTr

)N()N( 12 aeFaeKPTrTr ,

where N is again the standard normal distribution

function and

T

TKFa

2

12

1)/(log

Taa 12

Since pricing a futures option is formally identical to

pricing an option on an asset with an income yield

equal to the interest rate, both American calls and


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American puts will be worth more than their European

versions.

Documents

437 Option Valuation