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

Turning Finance into Science: Risk Management and the Black- Scholes Options Pricing Model Albert Kim Mary Frauley Writing for the Sciences English ENG-LBE 09 Monday, May 1 st 2000 Recently, the people behind the famed Black- Scholes Options Pricing Model received the Nobel Prize in Economics. Scientific American delves into this formula that has its share of praise, and criticism A New Breed of $cience In recent years, a new discipline called financial engineering has emerged in order to attempt to understand finance using a scientific approach. Mathematicians, physicists and traders work together in this discipline in order to incorporate the use of advanced mathematics with everyday finance (Stix, 1998). Although financial engineering deals with many aspects of finance, the main application of this discipline is risk management within the stock market. Regardless of what type of stock market transaction one performs, risk is always present. However, it is the management of this risk that is studied by these “financial engineers”. People need a fast and reliable way to calculate and control the risk involved in all their stock trading. This is where the Black- Scholes Option Pricing Model comes in. This ideas behind this formula, created by Prof. Robert C. Merton, Prof. Myron S. Scholes and the late Fisher Black, has been described by one economist as “the most successful theory not only in finance but in all of economics.” (Stix, 1998) Options 2 The functioning of the Black- Scholes Model is based on the use of stock options. Stock options are a form of financial derivative (an item that is not a stock in itself, but is an offshoot 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, regardless of all changes in the stock’s value. These options are purchased at a fee known as the premium. To illustrate, let’s say someone obtained the option to purchase 100 shares a year from 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 person to exercise his/her option, because the shares would still only cost you $100, even though they are worth $120. However, if the value of the stock were $80 at the call date, then it would not be feasible to purchase these shares, because you would be paying $100 for shares that are worth $80 each (a loss of $20 a share). Thus, this person probably would not exercise his/her option, and would only lose the premium he/she paid for the options a year ago. Thus, stock options are a form of insurance policy. What makes stock options so appealing is that the purchaser knows that the limit of his/her 3 losses can only be the premium price. However, there are no limits to his/her gains, because the limit of the value of the stock is theoretically limitless (Devlin, 1997). The question is, what is the fair price for an option on a particular stock? In other words, what is the option worth? When a stock has a call price of $100 and a value of $120, the option is worth at least $20 ($120 - $100 = $20). The value of the option clearly depends on the value of the stock. Thus, if there were a formula that could tell you the fair price for an option while taking into account all necessary factors, it would come of great use to the financial world. This is what the Black- Scholes Options Pricing Model does. The Math Behind It Option pricing requires five inputs: the option’s exercise price, the time to expiration, the price of the stock at the time of evaluation, current interest rates and the volatility of the stock (Dammers, 1998). The only unreliable factor is the volatility of the stock. This number can be estimated from market data (Stix, 1998). The formula is as follows: where the variable d is defined by: 4 According to this formula, the value of the option C, is given by the difference between the expected share price (the first term) on the right- hand side, and the expected cost (the second term) if the option is exercised. The higher the current share price S, the higher the volatility of the share price (Greek letter) sigma , the higher the interest rate r, the longer the time until the call date t and the lower the strike price L, the higher the value of the option C will be. Limitations of the Model As consistent as the model, there are limitations to the model. One limitation is that it assumes that the options can only be exercised on the call date. In other words, it cannot be exercised earlier. This model involves “European- Style” options, rather than “American- Style” options. “American- Style” options can be exercised anytime (Dammers, 1998). American options are more flexible, thus more valuable. The model only takes European- style options into account. Thus, the model underestimates the value of options. Most options are not exercised until the call date anyways, but this law is not written in stone. 5 Another limitation is that it assumes that the interest rate (determined by the U.S. Government) is known and will remain more- or- less constant. Many researchers have concluded that this is a safe assumption to make. But there are times where the interest rate can change rapidly (Rubash, 1998), thus putting the results of the model into question. However, these limitations are considered insignificant, because they do not affect the value of the option unless in extreme circumstances (such as a sudden raise in interest rates or even a market crash). The Birth of the Model This formula did not create itself out of nowhere. Its roots lie deep in the branches of mathematics known as probability and statistics. The combination of these two domains of mathematics deals with the collection, organization, and analysis of numerical data in order to assist decision- making. In short, statistics let you “predict the future”, not with 100% accuracy, but well enough so that you can make a wise decision as to your next course of action (Devlin, 1997). It all started when Charles Castelli wrote a book called “The Theory of Options in Stocks and Shares” in 1877. Castelli’s book was the first to deal with the use of options. However, this book lacked the theoretical basis needed for actual application (Rubash, 1998). 6 Twenty- three years later, a graduate student by the name of Louis Bachelier published his thesis paper “La Théorie de la Spéculation” (The Theory of Speculation) at the Sorbonne, in Paris (Rubash, 1998). In this paper, Bachelier dealt with the “structure of randomness” in the market. He compared the behavior of buyers and sellers to the random movements of particles suspended in fluids (NOVA Online, 2000). Remarkably, this paper anticipated key insights developed later on by famed physicist Albert Einstein and future theories in the field of probability. He created the first complete mathematical model of options trading. He believed the movements of stock prices were random and could never be predicted, but risk could be managed (NOVA Online, 2000). He created a formula that yielded an output that could help protect market investors from excessive risk by means of pricing options. However, this formula contained financially unrealistic assumptions, such as the existence of negative values for stock prices and a zero interest rate (Stix, 1998). His paper was shelved and went unnoticed for decades. It wasn’t until 1955 that the idea of options pricing resurfaced, when a professor at the Massachusetts Institute of Technology named Paul Samuelson browsed through the Sorbonne library. He began developing a formula of his own. Other mathematicians, such as Case Sprenkle and James Boness began 7 toying with Bachelier’s ideas as well (Royal Swedish Academy of Sciences, 1997). But all of their efforts went fruitless. A Revolution Then in 1968, a 31- year- old independent finance contractor named Fisher Black and a 28- year- old assistant professor of finance at MIT named Myron Scholes (Rubash, 1998) began their work on options pricing. They were dissatisfied with all the formulas that had preceded them, because they were overly complicated and made assumptions that didn’t make sense. They wanted to find a formula that would calculate the fair price of an option at any moment in time just by knowing the current price of the stock, but they couldn’t see their way through the mass of equations they had inherited (NOVA Online, 2000). Then they decided to try something different. They decided to strip previously derived formulas to their bare- boned state. They dropped everything that represented something un- measurable (NOVA Online, 2000). They were left with the vitals of calculating an option: the option’s exercise price, the time to expiration, the price of the stock at the time of evaluation, current interest rates and the volatility of the stock. But they were stuck with one problem: one couldn’t measure volatility, or in other words, risk. 8 So they decided if they couldn’t measure the risk of an option, they should make it less significant (NOVA Online, 2000). Their solution to this problem was to become of the most celebrated discoveries of the 20 th century. The solution was rooted in the old gamblers’ practice of hedging. When one makes a risky bet, one hedges his/her bet by also betting in the opposite direction. To illustrate, let’s say one were to bet that the favored Detroit Red Wings would beat the Colorado Avalanche in a 2 nd round playoff series. If one already bet $50 on the Red Wings, one would hedge that bet by betting $45 on the Avalanche. Although Detroit is favored, by hedging this bet with a slightly smaller bet on Colorado, we are protecting ourselves in the event of a Coloradoan upset. We minimize risk at the cost of lowering our possible winnings. However, since Detroit is favored, the chances of winning $5 are substantial. A more business- oriented example would be as such: Let’s say a British company is expecting to make several large payments in US dollars in a few months. They can hedge against a huge drop in the Sterling Pound (thus making it more expensive to buy US dollars) by purchasing options for US dollars on a foreign currencies market. Effective risk management requires that such options be correctly priced (Royal Swedish Academy of Sciences, 1997). 9 Fig 1: Hedging Cash Flows [Stix, G. (1998, May). A Calculus of Risk. Scientific American , p.94.] To hedge against risks in changes in share price, the investor can buy two options 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 investor must alter the composition of the portfolio, the ratio of number of shares to the number of options, to ensure that the holdings remain without risk (Stix, 1998). They made up a theoretical portfolio of stocks and options. Whenever either fluctuated up or down, they tried to hedge against the movement by 10 [...]...making another move in the opposite direction Their aim was to keep the overall value of the portfolio in perfect balance In other words, they tried to minimize risk They discovered that they could indeed reduce risk by creating a balance in which all moveme n t s in the markets cancelled each other out Black and Scholes had found a theoretical way to neutralize risk (NOVA Online, 2000) With risk now... accepting models without carefully questioning them,” says Joseph A Langsa m, a former mathe m a tician who develops and tests models for fixed - income securities at Morgan Stanley (Stix, 1998) Thus, the Black- Scholes is not the culprit for all derivative losses, but trader s’ blind faith in them Numbers vs Instinct Many traders still use the ideas behind the Black- Scholes Options Pricing Model, if not the. .. despite the simple nature of its use Traders began using their ideas imme diately incorp o r ate d their formula into their latest Texas Instru m e n t s had calculator, annou ncing their feature in the Wall Street Journal (Devlin, 1997) The options market exploded soon after 12 So overwhelming was the sudde n mass use of the Black- Scholes Model, that when the stock market crashed in 1978, the influential... sciences for their efforts Their colleague Fisher Black had unfort u na tely passed away in 1994 (Royal Swedish Academy of Sciences, 1997) The History of the Model In 1973, the Chicago Options Exchange was launche d, one month before the Black- Scholes model was publishe d (Stix, 1998) When these three men had published their paper in 1973 in the Journal of Political Economy, trader s, acade mics and economis... Options Pricing Model, if not the model itself equation forever changed the stock The funda m e n t al ideas behind the market Today, trader s use many principles of the Black- Scholes Model as guides through the treachero u s waters of the stock market laureates For this, Scholes and Fisher became Nobel But the lessons of putting all of one’s eggs in the same “Black - Scholes Model basket have been learned... the Models” (Stix, 1998) This group reflects the recent backlash against financial 13 models Many figures in the financial indus try question whether models can match trader s’ skill and gut intuition about market dyna mics (Royal Swedish Academy of Sciences, 1997) Derivatives make the news because, like an airplane crash, their losses can dramatic and chaotic Enormou s losses by Proctor & Gamble and. .. says: “If a rando m bolt of lightning hits you when you’re stan ding in the middle of the field, that feels like a rando m event But if your business is to stand in rando m fields during lightning stor m s, then you should anticipate, perha p s a little more robustly, the risks you’re taking on.” (NOVA 15 Online, 2000) This formula is a metho d to calculate these risks, not a risk neutralizer “There is... graduate by the name of Robert Merton solved this problem by introd ucing the notion of continuo u s time science This idea is rooted in rocket A Japanese mathe m a tician by the name of Kiyosi Ito theorize d that when you plot the trajectory of a rocket, knowing where the rocket was second - by- second was not enough was continuou sly You neede d to know where the rocket So he broke time down into infinitely... smoothe ning the graphing of its path out until it became a continu u m so that the trajectory could be consta ntly update d (NOVA Online, 2000) Merton applied this idea to the Black- Scholes model so that the value of an option could be constantly recalculated and risk eliminate d continually (NOVA Online, 2000) In 1997, Robert Merton and Myron Scholes were awarde d the Nobel prize in economic sciences... Greetings and the bankr u p tcies of Barings Bank and Orange County, California have been attribute d to the use of models (Stix, 1998) Fig 3: Derivative s Debacles [Stix, G (1998, May) A Calculus of Risk Scientific American , pp 91.] 14 However, Scholes says that it was not so much the formula itself that caused these losses, rather its misuse by market traders Every statistician and mathe m a tician . Turning Finance into Science: Risk Management and the Black- Scholes Options Pricing Model Albert Kim Mary Frauley Writing for the Sciences English ENG-LBE 09 Monday, May. letter) sigma , the higher the interest rate r, the longer the time until the call date t and the lower the strike price L, the higher the value of the option C will be. Limitations of the Model As. (the first term) on the right- hand side, and the expected cost (the second term) if the option is exercised. The higher the current share price S, the higher the volatility of the share price (Greek