Note 9 BlackScholes

8/3/2019 Note 9 BlackScholes

1/54

Valuing Stock Options: The

BlackScholes Model

Chapter 12.

Lecture Notes # 9


2/54

Lecture Notes#9 The Black-Scholes Model 2

Outline

1. Introduction

2. Option Prices in the Black-Scholes Model

3. Plotting Option Prices

4. Dividends in the BlackScholes Model

5. Options on Indices & Currencies

6.

Implied Volatility


3/54


1. Introduction

The BlackScholes model is unambiguously thebest known model of option pricing.

Also one of the most widely used: forms thebenchmark model for pricing options on

Equities. Stock indices. Currencies. Futures.

Moreover, it forms the basis of the Black model thatis commonly used to price some interest-ratederivatives such as caps and floors.


4/54


Introduction (Contd)

Technically, the Black-Scholes model is much morecomplex than the Binomial model.

The Black-Scholes model assumes time is continuous.

Thus, all returns, interest rates, etc are quoted incontinuously-compounded terms.

Moreover, prices in the model are allowed to change

continuously.

Modeling the continuous evolution of uncertainty requiresthe use of very sophisticated mathematics, includingtechniques drawn from the field of stochastic calculus.


5/54


Introduction (Contd)

What is gained by all this sophistication?

Option prices in the Black-Scholes model can be expressed inclosed-form, i.e., as particular explicit functions of theparameters.

This makes computing option prices very easy.

More importantly, it makes computing option sensitivities very

easy.

These sensitivities are much more difficult to compute in thebinomial model.


6/54


Main Assumptions of the Model

The Black-Scholes model involves three mainassumptions:

1. Time is continuous. In particular:

Asset prices change continuously, not at discrete points intime like the binomial model. All returns (including interest rates) are quoted in continuously

compounded terms.

2. The risk-free rate of interest, denoted r, is constant.

3. The price of the risky asset (which we continue to call astock) evolves according to a geometric Brownianmotion.


7/54


Main Assumptions of the Model (Contd)

What is a geometric Brownian motion?

Essentially, two conditions must be satisfied:

1. Returns on the stock over any period are log-normally distributed:

if Sdenotes the current stock price, and STthe price at a futuretime t, then

The parameters and are assumed constant.

2. Stock prices must evolve smoothly: they cannot jump (the marketcannot gap).

2ln ,TS

N t tS


8/54


Main Assumptions of the Model (Contd)

These assumptions appear unreasonably restrictive:

Volatility of markets is typically not constant over time.

Market prices do sometimes jump.

In particular, the no-jumps assumption appears to rule outdividends.

Dividends (and similar predictable jumps) areactually easily handled by the model as we will see

later. The other issues are more problematic, and not

easily resolved.


9/54


Order of Analysis

We proceed in the following order:

1. Basic Black-Scholes model: European

options/no dividends.2. Extension to dividends.

3. Options on indices and currencies.

4. Implied volatility and the volatility smile orskew.


10/54


2. Option Prices in the Black-Scholes

Model

Option prices in the BlackScholes model may berecovered using either of the procedures weexamined in the Binomial model: Using a replicating/riskless hedge portfolio.

Using risk-neutral pricing methods.

Both methods are significantly more technicallycomplex than in the Binomial model.

We focus here on the final option prices that resultand the intuitive content of these prices.


11/54


Notations

t: current time.

T: Horizon of the model. (So time-left-to-maturity: T t)

K: strike price of option.

St: current price of stock.

ST: stock price at T.

, : Expected return and volatility of stock (annualized).

r: risk-free rate of interest.

C, P: Prices of call and put (European only).


12/54


Call Option Prices in the BlackScholes

Model

We have seen that, in general, to replicate acall, we must

Take a long position incunits of the underlying,

and BorrowBcat the risk-free rate.

In notational terms, we can express this as:C = c St Bc.


13/54


Call Prices in the BlackScholes Model

(Contd) In general, cand Bcwill depend on many factors including

1. Depth-in-the-moneyof the option. For example, c increasesfrom near-zero for deep OTM options to near unity for deep ITMoptions.

2. Volatility of the underlying. For deep ITM call, delta willdecrease as volatility increases. For deep OTM call, delta willincrease as volatility increases.

3. Time-to-maturityof the option. OTM call with very short time tomaturity will have delta close to zero. Same call with longermaturity will have larger delta.

4. Interest rate.Higher interest rate moves the call more into-the-money, raising its delta.

Complex nature of dependencies make closed-forms hard toobtain in general.


14/54



(Contd) Near-unique feature of BlackScholes: precise expressions

for cand Bc !

N():cumulative standard normal distribution: N(x)is theprobability under a standard normal distribution of anobservation less than or equal to x.

tTdtT

tTrKSd

tT

tTrKSd

dNKeB

dN

t

t

tTr

c

c

12

1

2

)(

1

))(2/2

()/ln(

))(2/2()/ln(where

)(.

)(


15/54



(Contd)

Thus, we have

This is the BlackScholes formula for a call optionon a non-dividend paying stock.

The proof is technically involved and needs someknowledge of stochastic calculus.

)(..)( 2)(

1 dNKedNSCtTr

t


16/54


Notice the similarity to the binomial

pricing:

* *

* *

n n

k k k k

n n

k k k k

1 1( ) ( )

1 1= ( ) ( )

k n k

n n

k n k

n n

C k u d S K k R R

C S k u d K k R R

)(..)( 2)(1 dNKedNSC tTrt


17/54



(Contd) However, these closed-forms do have the intuitive

properties we expect:

For example: behavior of the delta as depth-in-the-

money changes. For deep OTM options (S > K): c +1. c increases as Sincreases.

Similarly, at a more complex level: For deep OTM calls: c increases as increases. For deep ITM calls: cdecreases as increases.


18/54


Put Option Prices in the BlackScholes

Model Recall that to replicate a put in general, we must

Take a short position in |p|units of the underlying, and

Lend Bpat the risk-free rate.

Thus, in general, we can write the price of the put asP = Bp+ St. p

The Black-Scholes formula identifies the exact composition of pand Bp:

p= N(d1) Bp= PV (K)N(d2)

where N(), d1, and d2are all as defined above.


19/54


Put Prices in the BlackScholes Model

(Contd) Therefore, the price of the put is given by

P = PV (K) N(d2) St N(d1)

This is the Black-Scholes formula for a European put option.

Equivalently, since N(x) + N(x) = 1 for any x, we can also write

P = St [N(d1) 1] + PV (K) [1 N(d2)]

Of course, the put price may also be recovered from the callprice using put-call parity.


20/54


Black-Scholes via Risk-Neutral

Probabilities Alternative way to derive BlackScholes formulae: use risk-

neutral probabilities.

To identify option prices in this approach: take expectation ofterminal payoffs under the risk-neutral probability measure and

discount at the risk-free rate.

The terminal payoffs of a call option with strike K are given by

max{St K, 0}.

Therefore, the arbitrage-free price of the call option is given byC = er(T-t)Et [max{St K, 0}]

where Et[]denotes timetexpectations under the risk-neutral

measure.


21/54


Black-Scholes via Risk-Neutral Probs

(Contd) Equivalently, this expression may be written as

C = er(T-t)Et [St K || ST K]

Splitting up the expectation, we have

C = er(T-t)Et [St || ST K]- e

r(T-t)Et [ K || ST K]

From this to the BlackScholes formula is simply amatter of grinding through the expectations, whichare tedious, but not otherwise difficult.


22/54


Black-Scholes via Risk-Neutral Probs

(Contd)

Specifically, it can be shown that

er(T-t)Et [St || ST K] =StN(d1)

erTEt [ K || ST K] =e

r(T-t)K. N(d2)

Note, in particular, that N(d2)is the risk-neutral probability that the option finishes in-the-money (i.e., that ST K).


23/54


Remarks

Two remarkable features of the BlackScholes formulae:

Option prices only depend on five variables: S, K, r, T, and .

Of these five variables, only onethe volatility is not directlyobservable.

This makes the model easy to implement in practice.

Closed-form expressions exist in the Black-Scholes frameworkonly for European-style options.

For example, closed-forms do not exist for American put options. However, it is possible to obtain closed-form solutions for certain

classes of exotic options (such as compound options or barrieroptions).


24/54


Plotting Option Prices (Contd)

Assume the following:

The strike price is K = 100.

The time-to-maturity is T t = 0.5years.

The annualized volatility of the stock price is 20% ( =0.20).

The risk-free interest rate is r=5%.

What is the price of call option using B-S formula at: At S = 72, the call is deep OTM, while the put is deep

ITM.


25/54


1

2 1

( )1 2

2ln( / ) ( / 2)( )where

ln(72 /100) (0.05 0.04 / 2)(0.5)= 3.33820.2* 0.5

3.3382 (0.2* 0.5) 3.4382

( ) . . ( )

= 72*N( 3.3382)

t

r T tt

S K r T t d

T t

d d T t

C S N d e K N d

e

0.05*0.5

0.05*0.5

1 2

0.05*0.5

*100* ( 3.4382)

= 72*0.000422 *100*0.000293 0.0018

[ ( ) - 1] ( ) [1 - ( )]

72*[0.000422 1] *100*[1 0.000293] $25.5328

t

N

e

P S N d PV K N d

P e


26/54


Table for N(x) when 0 X

x 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09

0 0.5 0.504 0.508 0.512 0.516 0.5199 0.5239 0.5279 0.5319 0.5359

0.1 0.5398 0.5438 0.5478 0.5517 0.5557 0.5596 0.5636 0.5675 0.5714 0.5753

0.2 0.5793 0.5832 0.5871 0.591 0.5948 0.5987 0.6026 0.6064 0.6103 0.6141

: : : : : : : : : : :

Table for N(x) when X 0

x 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09

0 0.5 0.496 0.492 0.488 0.484 0.4801 0.4761 0.4721 0.4681 0.4641-0.1 0.4602 0.4562 0.4522 0.4483 0.4443 0.4404 0.4364 0.4325 0.4286 0.4247

-0.2 0.4207 0.4168 0.4129 0.409 0.4052 0.4013 0.3974 0.3936 0.3897 0.3859

: : : : : : : : : : :

: : : : : : : : : : :

-3.4 0.0003 0.0003 0.0003 0.0003 0.0003 0.0003 0.0003 0.0003 0.0003 0.0002

( 3.4382) ( 3.43) 0.82[ ( 3.43) ( 3.44)]

0.0003 0.82[0.0003 0.0003]

N N N N


27/54


Plotting Option Prices (Contd)

Four parameters are held fixed in the figure: The strike price is K = 100. The time-to-maturity is T t = 0.5years. The annualized volatility of the stock price is 20% ( = 0.20). The risk-free interest rate is r=5%.

The figures plot call and put prices as Svaries from 72to 128. At S = 72, the call is deep OTM, while the put is deep ITM. At S = 128, the call is deep ITM, while the put is deep OTM.

Observe non-linear reaction of option prices to changes in stockprice. Evident visually in the curvature of the option prices as Svaries. Can also be directly checked using the numbers.


28/54



29/54


4. Dividends in the BlackScholes Model

There are two possibilities to be considered here: Discrete (lumpy) dividends of sizes D1,D2, . . . at times T1,

T2, . . . During the life of the option.

A continuous dividend yield at rate m.

The two have different technical implications andneed to be handled differently. In particular, lumpy dividends cause a discontinuity in the

stock price, but a continuous dividend yield will not.

We consider lumpy dividends first.


30/54


Lumpy Dividends in the BlackScholes

Model For simplicity, we consider the case of a single dividend Dat time TD, t

< TD< T.

Let PV (D)denote the present value at tof an amount Dreceivable atTD.

Then, the stock price Stat time tcan be regarded as composed of twocomponents: A riskless component of PV (D). A risky component of St PV (D).

Only the risky component is relevant for pricing options.

Therefore, call can be priced simply by replacing the term St in theBlackScholes formula by (St PV (D)).


31/54


Lumpy Dividends in the BlackScholes

Model, contd

1

( )

2

1

2 1

( )

. ( )

2ln(( ( )) / ) ( / 2)( )

where

2ln(( ( )) / ) ( / 2)( )

c

r T t

c

t

t

N d

B e K N d

S PV D K r T t d

T t

S PV D K r T t d d T t

T t

( )

1 2

( ( )) ( ) . . ( )r T tt

C S PV D N d e K N d

P = (St-- PV(D)) [N(d1) 1] + PV (K) [1 N(d2)]


32/54


Lumpy Dividends (Contd)

The figure on the next page illustrates the impact of dividends onBlack-Scholes prices.

It considers the same range of parameter values as the earlierplot, and compares option prices under three dividend levels:

Low: D = 0.

Medium: D = 2.

High: D = 5.

The timing of the dividend payment is assumed to be 3 months(i.e., the halfway point of the option life).


33/54



34/54


Lumpy Dividends (Contd)

Note that dividends lower call prices and raise putprices.

This is intuitive: dividends lower the price of the underlying.

Thus, they push OTM calls further OTM, and make ITMcalls less ITM.

The impact is:

Maximal for deep ITM options (nearly one-for-one). Least for deep OTM options (near zero).


35/54


Continuous Dividend Yields in the

BlackScholes Model Suppose the stock pays a continuous dividend yield at rate m.

Compare two stocks that are identical except:

Stock 1: Continuous dividend yield at rate m.

Stock 2: No dividend payments.

Stock 1 will clearly grow at a rate mslower than Stock 2.

Thus, under the following conditions, the prices of the two stocks will

be identical at time T: Current price of Stock 1: St.

Current price of Stock 2: em(Tt) St.


36/54


Continuous Dividend Yields (Contd)

Therefore, it must also be that the following options have the samepayoffs at T (and, hence the same price today):

A Tmaturity European option on Stock 1 written at t whenthe stock price is St.

A Tmaturity European option on Stock 2 written at t whenthe stock price is em(Tt) St.

The second option is a European option on a non-dividend payingstock.

The price of this option can be found from the BlackScholesformula by using em(Tt) St. for the current stock price.


37/54



tTdtT

tTmrKSd

tT

tTmrKSd

dNSedNKeP

dNKedNSeC

t

t

ttTmtTr

tTr

t

tTm

12

1

1)(

2)(*

2

)(

1

)(*

))(2/2()/ln(

))(2/2

()/ln(where

)(.-)(.

)(.)(.

:pricesopionResulting


38/54



The following figure illustrates the impact of a continuous dividend yield.

It considers the same range of parameter values as the earlier plots,and compares option prices under three dividend yield levels: Low: m = 0. Medium: m = 0.025(i.e., a 2.5% annualized yield). High: m = 0.10(i.e., a 10% annualized yield).

The higher the yield, the lower the growth rate of the stock price.

Thus, higher mimplies lower call prices and higher put prices.

Again: impact is maximal for deep ITM options and least for deep OTMoptions.


39/54


mm

m

mmm


40/54


Options on Indices & Currencies

Using the Black-Scholes formula adjusted fora continuous dividend yield, we can extendthe pricing formulae to also cover:

Stock indices.

Currencies (sometimes called the Garman-Kohlhagen model).


41/54


Options on Indices

Many exchange-traded options exist on stock indices.

Both European- and American-style index options exist.

S&P 500 index options: European

S&P 100 index options: American

Like index futures, index options are also cash settled. If ST is theindex level at close of the trading day, then at maturity:

Holder of call receives max{ST K, 0} Holder of put receives max{K ST , 0}


42/54


Index Options (Contd) An index can be treated as an asset paying a continuous dividend

yield.

Therefore, the earlier formula can be used with mdenoting the yieldon the index.

This results in:

tTdtT

tTmrKSd

tT

tTmrKS

d

dNSedNKeP

dNKedNSeC

t

t

t

tTmtTr

tTr

t

tTm

**

*

1

)(

2

)(*

2

)(

1

)(*

12

1

))(2/2()/ln(

))(2/2

()/ln(

where

)(.-)(.

)(.)(.


43/54


Options on Currency

Options on foreign currencies are traded on thePhiladelphia Exchange and on the OTC market.

The underlying asset here is the foreign currency. Thus, the underlying is an interest-paying asset.

Let rfdenote the (continuously-compounded) interest rateon foreign currency.

Note an important symmetry: A call option to purchase DMwith $at a given exchange

rate is a put option to sell $for DMat that rate.


44/54


Currency Options (Contd)

Foreign currency = asset with a continuous dividend yield.

Earlier formulae apply with mreplaced by rfand S = exchange rate.

This results in:

tTdtT

tTrrKSd

tT

tTrrKSd

dNSedNKeP

dNKedNSeC

ft

ft

t

tTrtTr

tTr

t

tTr

f

f

**

*

1

)(

2

)(*

2

)(

1

)(*

12

1

))(2/2()/ln(

))(2/2()/ln(where

)(.-)(.

)(.)(.


45/54


6. Implied Volatility

Given an option price, one can ask the question:what level of volatility is implied by the observedprice?

This level is the implied volatility.

Formally, implied volatility is the volatility level that

would make observed option prices consistent withthe Black-Scholes formula, given values for theother parameters.


46/54


Implied Volatility (Contd)

For example, suppose we are looking at a call on a non-dividend-paying stock.

Let Kand T tdenote the calls strike and time-to-maturity, andlet Cbe the calls price.

Let St be the stock price and rthe interest rate.

Then, the implied volatility is the unique level for which

Cbs(S, K, T t, r, ) = C ,

where Cbs is the Black-Scholes call option pricing formula.

Note that implied volatility is uniquely defined since Cbs

is strictlyincreasing in .


47/54


Implied Volatility (Contd)

Implied volatility represents the markets

perception of volatility anticipated over theoptions lifetime.

Implied volatility is thus forward looking.

In contrast, historical volatility is backwardlooking.


48/54


The Volatility Smile/Skew

In theory, any option (any Kor T) may be used for measuringimplied volatility.

Thus, if we fix maturity and plot implied volatilities against strikeprices, the plot should be a flat line.

In practice, in equity markets, implied volatilities for low strikes(corresponding to out-of-the-money puts) are typically muchhigher than implied volatilities for ATM (At-of-the-money)options.

This is the volatility skew.

In currency markets: volatility smile.


49/54


The Source of the Volatility Skew

The Black-Scholes model assumes log-returns are normallydistributed.

However, in every financial market, extreme observations are farmore likely than predicted by the log-normal distribution.

Extreme observations = observations in the tail of the distribution.

Empirical distributions exhibit fat tails or leptokurtosis.

Moreover, empirical log-returns distributions are often skewed, incontrast to the normal distribution underlying BlackScholes.


50/54

50

Normal versus non-normal

distribution

0

1% of the normal distribution

normal distribution

actual distribution

1% of the actual

distribution

Underestimation of risk


51/54


Source of the Volatility Skew (Contd)

Fat tails Black-Scholes model with a constant volatility willunderprice out-of-the-moneyputs relative to those at-the-money.

Thus, in practice, when the Black-Scholes model is used for

pricing OTM puts, a higher volatility is used than that used forATM options.

This is inconsistent, of course.

However, it is a quick-fix that generates the necessary extraweight in the tails to raise the option price.


52/54


Information in the Volatility Smile/Skew

The volatility skew is evidence not only that the lognormal modelis not a completely accurate description of reality but also thatthe market recognizes this shortcoming.

As such, there is valuable information in the smile/skew

concerning the actual (more accurately, the marketsexpectation) of the return distribution:

For example:

More symmetric smile Less skewed distribution.

Flatter smile/skew Smaller kurtosis.


53/54


Generalizing Black-Scholes

Obvious question: why not generalize the log-normaldistribution?

Indeed, there may even be a natural generalization.

The Black-Scholes model makes two uncomfortableassumptions:

No jumps.

Constant volatility.

If jumps are added to the log-normal model or if volatility isallowed to be stochastic, the model will exhibit fat tails and evenskewness.


54/54

Generalizing Black-Scholes (Contd)

So why dont we just do this?

Tremendously increased complexity.

Jumps & stochastic volatility have very differentdynamic implications.

Reality is more complex than either model(indeed, substantially so).

Ultimately: better to use a simple model withknown shortcomings?

Note 9 BlackScholes

Documents

Transcript of Note 9 BlackScholes