finance - turning finance into science - risk management and the black-scholes options pricing model

Turning Finance into Science: Risk Managem e nt and the Black -Scholes Options Pricing Model Albert Kim Mary Frauley Writing for the Sciences English ENG­LBE 09... Recently, the people

Turning Finance into Science:

Risk Managem e nt and the Black

-Scholes Options Pricing Model

Albert Kim Mary Frauley Writing for the Sciences English ENG­LBE 09

Recently, the people behind the fa m e d Black - Scholes Options Pricing Model received the Nobel Prize in Econo mics Scientific Ame rica n delves into this for m ula that has its share of praise,

and criticism

A New Breed of $cience

In recent years, a new discipline called financial engineerin g has emerge d

in order to atte m p t to un der s t a n d finance using a scientific appro ac h Mathe m a tician s, physicists and trade r s work toget he r in this discipline in order

to incorp o r a t e the use of advanced mat he m a tics with everyday finance (Stix, 1998)

Althou g h financial engineeri ng deals with many aspect s of finance, the main applicatio n of this discipline is risk man ag e m e n t within the stock marke t Regardles s of what type of stock mark et trans actio n one perfo r m s, risk is always prese n t However, it is the man age m e n t of this risk that is stu die d by these “financial engineer s” People need a fast and reliable way to calculate and contr ol the risk involved in all their stock tradin g

This is where the Black - Scholes Option Pricing Model comes in This ideas behin d this for m ula, create d by Prof Robert C Merton, Prof Myron S Scholes and the late Fisher Black, has been describe d by one econo mis t as “the mos t succes sf ul theory not only in finance but in all of econo mics.” (Stix, 1998)

Option s

The functioning of the Black - Scholes Model is based on the use of stock optio n s Stock optio n s are a for m of financial derivative (an item that is not a stock in itself, but is an offsh o o t of one) It consists of a contract that gives

one the right, but not the obligation , to buy stocks later at a fixed price (known

as the exercise or strike price) The exercise price does not change, regar dles s

of all changes in the stock’s value These optio n s are purc h a s e d at a fee known as the premi u m

To illustr at e, let’s say someo n e obtaine d the option to purc ha s e 100 shares a year fro m now (a date known as the call date) for $100 each If the stock were to rise to $120 by the call date, it would be feasible for this perso n

to exercise his / h e r optio n, becau se the shares would still only cost you $100, even thoug h they are worth $120

However, if the value of the stock were $80 at the call date, then it would not be feasible to purch a s e these shares, becau se you would be paying

$100 for shares that are wort h $80 each (a loss of $20 a share) Thus, this perso n proba bly would not exercise his / h e r option, and would only lose the pre mi u m he / s h e paid for the optio n s a year ago

Thus, stock option s are a for m of insur a n ce policy What makes stock optio n s so appealing is that the purc h a s e r knows that the limit of his / h e r

losses can only be the pre miu m price However, there are no limits to his / h e r gains, becau se the limit of the value of the stock is theor etically limitless (Devlin, 1997)

The ques tio n is, what is the fair price for an optio n on a particular stock? In other word s, what is the optio n wort h? When a stock has a call price

of $100 and a value of $120, the optio n is worth at least $20 ($120 - $100 =

$20) The value of the optio n clearly depen d s on the value of the stock Thus,

if there were a for m ula that could tell you the fair price for an optio n while taking into accou n t all necess a ry factor s, it would come of great use to the financial world This is what the Black - Scholes Optio n s Pricing Model does

The Math Behind It

Optio n pricing requires five inpu t s: the option’s exercise price, the time

to expiratio n, the price of the stock at the time of evaluatio n, curren t intere s t rates and the volatility of the stock (Dam m e r s, 1998) The only unreliable factor is the volatility of the stock This nu m b e r can be estima te d fro m marke t data (Stix, 1998) The for m ula is as follows:

where the variable d is defined by:

According to this form u la, the value of the optio n C, is given by the

difference betwee n the expecte d share price (the first ter m) on the right - han d side, and the expecte d cost (the secon d ter m) if the option is exercise d The

higher the curren t share price S, the higher the volatility of the share price (Greek letter) sigma , the higher the intere s t rate r, the longer the time until the call date t and the lower the strike price L, the higher the value of the optio n C

will be

Limitations of the Model

As consist en t as the model, there are limitatio n s to the mod el One limitatio n is that it assu m e s that the option s can only be exercise d on the call date In other word s, it cann o t be exercised earlier This mod el involves

“Europ ea n Style” option s, rather than “American Style” optio n s “American -Style” optio n s can be exercised anytim e (Damm e r s, 1998) American option s are more flexible, thus more valuable The mod el only takes European - style optio n s into accou n t Thus, the mo del und er e s ti m a t e s the value of option s Most optio n s are not exercised until the call date anyways, but this law is not written in sto ne

Anoth er limitatio n is that it assu m e s that the interes t rate (deter mi n e d

by the U.S Govern m e n t) is known and will remain more - or - less const an t Many researc h er s have conclu d e d that this is a safe assu m p t i o n to make But there are times where the interes t rate can change rapidly (Rubash, 1998), thu s putti ng the results of the mod el into questio n

However, these limitatio n s are consid er e d insignifican t, becau se they do not affect the value of the optio n unless in extre m e circu m s t a n c e s (such as a sud d e n raise in interes t rates or even a mar ket crash)

The Birth of the Model

This for m ula did not create itself out of nowher e Its roots lie deep in the branc h e s of mat h e m a tic s known as proba bility and statistics The combina tio n of these two do mai n s of mat h e m a tic s deals with the collectio n, organi za tio n, and analysis of nu me rical data in order to assist decision -making In shor t, statistics let you “predict the futu r e”, not with 100% accur acy, but well enoug h so that you can make a wise decisio n as to your next course of action (Devlin, 1997)

It all star te d when Charles Castelli wrote a book called “The Theory of Optio ns in Stocks and Shares” in 1877 Castelli’s book was the first to deal with the use of option s However, this book lacked the theoretical basis neede d for actual applicatio n (Rubas h, 1998)

Twenty - three years later, a grad u a t e stu d e n t by the name of Louis Bachelier publish e d his thesis paper “La Théorie de la Spéculatio n” (The Theory of Speculatio n) at the Sorbon n e, in Paris (Rubash, 1998) In this paper, Bachelier dealt with the “struct u r e of ran d o m n e s s” in the marke t He com p a r e d the behavior of buyer s and sellers to the rand o m move m e n t s of particles susp e n d e d in fluids (NOVA Online, 2000) Remar ka bly, this paper anticipat e d key insight s develope d later on by famed physicist Albert Einstein and futu r e theories in the field of proba bility

He created the first comple te mat he m a tical mo del of option s trading

He believed the move m e n t s of stock prices were rand o m and could never be predicte d, but risk could be manag e d (NOVA Online, 2000) He created a for m ula that yielded an out p u t that could help protect mar ke t investo r s from excessive risk by mean s of pricing optio n s However, this for m ula containe d financially unrealistic assu m p t i o n s, such as the existen ce of negative values for stock prices and a zero intere s t rate (Stix, 1998) His pape r was shelved and went unno tice d for decad es

It wasn’t until 1955 that the idea of optio n s pricing resu rf ace d, when a profess o r at the Massach u s e t t s Instit u t e of Techn ology name d Paul Samuelso n browse d thro ug h the Sorbon n e library He began develo ping a for m ula of his own Other math e m a t ician s, such as Case Sprenkle and James Boness began

toying with Bachelier’s ideas as well (Royal Swedish Acade my of Sciences, 1997) But all of their effort s went fruitless

A Revolutio n

Then in 1968, a 31 - year - old indep e n d e n t finance contr act o r name d Fisher Black and a 28 - year - old assista n t profess o r of finance at MIT name d Myron Scholes (Rubash, 1998) began their work on optio n s pricing They were dissatisfie d with all the for m ulas that had preced e d the m, becau se they were overly complicate d and mad e assu m p t i o n s that didn’t make sense They wante d to find a for m ula that would calculate the fair price of an option at any

mo m e n t in time just by knowing the curren t price of the stock, but they could n’t see their way thro ug h the mass of equa tio n s they had inherite d (NOVA Online, 2000)

Then they decided to try somet hi n g differen t They decided to strip previo usly derived for m ulas to their bare - boned state They dro p p e d everythin g that repre se n t e d somet h in g un - meas u r a b le (NOVA Online, 2000) They were left with the vitals of calculating an optio n: the optio n’s exercise price, the time to expiratio n, the price of the stock at the time of evaluatio n, curren t intere st rates and the volatility of the stock But they were stuck with one problem: one could n’t meas u r e volatility, or in other word s, risk

So they decide d if they could n’t meas u r e the risk of an optio n, they sho uld make it less significan t (NOVA Online, 2000) Their solutio n to this proble m was to beco m e of the most celebra te d discoveries of the 20th centu ry The solutio n was rooted in the old gambler s’ practice of hedging When one makes a risky bet, one hedges his / h e r bet by also betting in the oppo si te directio n

To illustr a te, let’s say one were to bet that the favored Detroit Red Wings would beat the Colora d o Avalanche in a 2nd roun d playoff series If one already bet $50 on the Red Wings, one would hedge that bet by betting $45 on the Avalanche Althoug h Detroit is favore d, by hedging this bet with a slightly smaller bet on Colora d o, we are protecting ourselves in the event of a Colorad o a n upset We minimi ze risk at the cost of lowering our possible winning s However, since Detroit is favored, the chances of winning $5 are sub st a n ti al

A more busines s - orien te d exam ple would be as such: Let’s say a British com p a n y is expecting to make several large pay me n t s in US dollars in a few

mo n t h s They can hedge against a huge drop in the Sterling Poun d (thus making it more expen sive to buy US dollars) by purch a si n g option s for US dollars on a foreign curren cies mar ket Effective risk man age m e n t require s that such option s be correctly priced (Royal Swedish Acade my of Sciences, 1997)

Fig 1: Hedging Cash Flows [Stix, G (1998, May) A Calculus of Risk Scientific Ame rica n ,

p.94.]

To hedge against risks in changes in share price, the investo r can buy two optio n s for every share he or she owns; the profit will then counter the loss Hedging creates a risk free portfolio (Stix, 1998) As the share price changes over time, the investo r mus t alter the com p o sitio n of the portfolio, the ratio of nu m b e r of shares to the nu m b e r of optio n s, to ensu r e that the holding s remain witho u t risk (Stix, 1998)

They made up a theor etical portfolio of stock s and optio n s Whenever either fluctuat e d up or down, they tried to hedge against the movem e n t by

making anot h er move in the oppo site directio n Their aim was to keep the overall value of the portfolio in perfect balance In other word s, they tried to mini mi ze risk

They discovere d that they could indeed reduce risk by creating a balance

in which all move m e n t s in the marke t s cancelled each other out Black and Scholes had foun d a theo re tical way to neut r ali ze risk (NOVA Online, 2000) With risk now virtually eliminate d from their equatio n, they had a mat h e m a tical form ula that could give them the price of any option This was a marvelou s achievem e n t

There was a practical problem with their for m ula It assu m e d that marke t s were always in equilibriu m, that sup ply equals dema n d (NOVA Online, 2000) They neede d a way to insta n tly rebalance a portf olio of stock s and optio n s to keep coun te ri ng all their move m e n t s

A Harvar d grad u at e by the nam e of Robert Merto n solved this proble m

by intro d u ci ng the notion of contin u o u s time This idea is roote d in rocket science A Japane se mat h e m a t ician by the name of Kiyosi Ito theo ri ze d that when you plot the trajecto ry of a rocket, knowing where the rocket was secon d - by - secon d was not enoug h You neede d to know where the rocket was contin u o u sly So he broke time down into infinitely small incre me n t s,