| The Royal Swedish Academy of Sciences has | | | | Merton and Scholes: that the investor can trade |
| decided to award the Bank of Sweden Prize in | | | | continuously without any transaction costs (though |
| Economic Sciences in Memory of Alfred Nobel | | | | others amended the formula later). |
| 1997, to Professor Robert C. Merton, Harvard | | | | According to their formula, the value of a call |
| University, and to Professor Myron S. Scholes, | | | | option is given by the difference between the |
| Stanford University, jointly. The prize was | | | | expected share price and the expected cost if |
| awarded for a new method to determine the | | | | the option is exercised. The value of the option is |
| value of derivatives. | | | | higher, the higher the current share price, the |
| This sounds like a trifle achievement - but it is not. | | | | higher the volatility of the share price (as |
| It touches upon the very heart of the science of | | | | measured by its standard deviation), the higher |
| Economics: the concept of Risk. Risk reflects the | | | | the risk-free interest rate, the longer the time to |
| effect on the value of an asset where there is an | | | | maturity, the lower the strike price, and the |
| option to change it (the value) in the future. | | | | higher the probability that the option will be |
| We could be talking about a physical assets or a | | | | exercised. |
| non-tangible asset, such as a contract between | | | | All the parameters in the equation are observable |
| two parties. An asset is also an investment, an | | | | except the volatility , which has to be estimated |
| insurance policy, a bank guarantee and any other | | | | from market data. If the price of the call option is |
| form of contingent liability, corporate or not. | | | | known, the formula can be used to solve for the |
| Scholes himself said that his formula is good for | | | | market's estimate of the share volatility. |
| any situation involving a contract whose value | | | | Merton contributed to this revolutionary thinking |
| depends on the (uncertain) future value of an | | | | by saying that to evaluate stock options, the |
| asset. | | | | market does not need to be in equilibrium. It is |
| The discipline of risk management is relatively old. | | | | sufficient that no arbitrage opportunities will arise |
| As early as 200 years ago households and firms | | | | (namely, that the market will price the share and |
| were able to defray their risk and to maintain a | | | | the option correctly). So, Merton was not afraid |
| level of risk acceptable to them by redistributing | | | | to include a fluctuating (stochastic) interest rate in |
| risks towards other agents who were willing and | | | | HIS treatment of the Black and Scholes formula. |
| able to assume them. In the financial markets this | | | | His much more flexible approach also fitted more |
| is done by using derivative securities options, | | | | complex types of options (known as synthetic |
| futures and others. Futures and forwards hedge | | | | options - created by buying or selling two |
| against future (potential - all risks are potentials) | | | | unrelated securities). |
| risks. These are contracts which promise a future | | | | Theory and Practice |
| delivery of a certain item at a certain price no | | | | The Nobel laureates succeeded to solve a |
| later than a given date. Firms can thus sell their | | | | problem more than 70 years old. |
| future production (agricultural produce, minerals) in | | | | But their contribution had both theoretical and |
| advance at the futures market specific to their | | | | practical importance. It assisted in solving many |
| goods. The risk of future price movements is | | | | economic problems, to price derivatives and to |
| re-allocated, this way, from the producer or | | | | valuation in other areas. Their method has been |
| manufacturer to the buyer of the contract. | | | | used to determine the value of currency options, |
| Options are designed to hedge against one-sided | | | | interest rate options, options on futures, and so |
| risks; they represent the right, but not the | | | | on. |
| obligation, to buy or sell something at a | | | | Today, we no longer use the original formula. The |
| pre-determined price in the future. An importer | | | | interest rate in modern theories is stochastic, the |
| that has to make a large payment in a foreign | | | | volatility of the share price varies stochastically |
| currency can suffer large losses due to a future | | | | over time, prices develop in jumps, transaction |
| depreciation of his domestic currency. He can | | | | costs are taken into account and prices can be |
| avoid these losses by buying call options for the | | | | controlled (e.g. currencies are restricted to move |
| foreign currency on the market for foreign | | | | inside bands in many countries). |
| currency options (and, obviously, pay the correct | | | | Specific Applications of the Formula: Corporate |
| price for them). | | | | Liabilities |
| Fischer Black, Robert Merton and Myron Scholes | | | | A share can be thought of as an option on the |
| developed a method of correctly pricing | | | | firm. If the value of the firm is lower than the |
| derivatives. Their work in the early 1970s | | | | value of its maturing debt, the shareholders have |
| proposed a solution to a crucial problem in | | | | the right, but not the obligation, to repay the |
| financing theory: what is the best (=correctly or | | | | loans. We can, therefore, use the Black and |
| minimally priced) way of dealing with financial risk. | | | | Scholes to value shares, even when are not |
| It was this solution which brought about the rapid | | | | traded. Shares are liabilities of the firm and all |
| growth of markets for derivatives in the last two | | | | other liabilities can be treated the same way. |
| decades. Fischer Black died in August 1995, in his | | | | In financial contract theory the methodology has |
| early fifties. Had he lived longer, he most definitely | | | | been used to design optimal financial contracts, |
| would have shared the Nobel Prize. | | | | taking into account various aspects of bankruptcy |
| Black, Merton and Scholes can be applied to a | | | | law. |
| number of economic contracts and decisions | | | | Investment evaluation Flexibility is a key factor in |
| which can be construed as options. Any | | | | a successful choice between investments. Let us |
| investment may provide opportunities (options) to | | | | take a surprising example: equipment differs in its |
| expand into new markets in the future. Their | | | | flexibility - some equipment can be deactivated |
| methodology can be used to value things as | | | | and reactivated at will (as the market price of the |
| diverse as investments, insurance policies and | | | | product fluctuates), uses different sources of |
| guarantees. | | | | energy with varying relative prices (example: the |
| Valuing Financial Options | | | | relative prices of oil versus electricity), etc. This |
| One of the earliest efforts to determine the value | | | | kind of equipment is really an option: to operate |
| of stock options was made by Louis Bachelier in | | | | or to shut down, to use oil or electricity). |
| his Ph.D. thesis at the Sorbonne in 1900. His | | | | The Black and Scholes formula could help make |
| formula was based on unrealistic assumptions | | | | the right decision. |
| such as a zero interest rate and negative share | | | | Guarantees and Insurance Contracts |
| prices. | | | | Insurance policies and financial (and non financial) |
| Still, scholars like Case Sprenkle, James Boness | | | | guarantees can be evaluated using option-pricing |
| and Paul Samuelson used his formula. They | | | | theory. Insurance against the non-payment of a |
| introduced several now universally accepted | | | | debt security is equivalent to a put option on the |
| assumptions: that stock prices are normally | | | | debt security with a strike price that is equal to |
| distributed (which guarantees that share prices | | | | the nominal value of the security. A real put |
| are positive), a non-zero (negative or positive) | | | | option would provide its holder with the right to |
| interest rate, the risk aversion of investors, the | | | | sell the debt security if its value declines below |
| existence of a risk premium (on top of the | | | | the strike price. |
| risk-free interest rate). In 1964, Boness came up | | | | Put differently, the put option owner has the |
| with a formula which was very similar to the | | | | possibility to limit his losses. |
| Black-Scholes formula. Yet, it still incorporated | | | | Option contracts are, indeed, a kind of insurance |
| compensation for the risk associated with a stock | | | | contracts and the two markets are competing. |
| through an unknown interest rate. | | | | Complete Markets |
| Prior to 1973, people discounted (capitalized) the | | | | Merton (1977) extend the dynamic theory of |
| expected value of a stock option at expiration. | | | | financial markets. In the 1950s, Kenneth Arrow |
| They used arbitrary risk premiums in the | | | | and Gerard Debreu (both Nobel Prize winners) |
| discounting process. The risk premium | | | | demonstrated that individuals, households and |
| represented the volatility of the underlying stock. | | | | firms can abolish their risk: if there exist as many |
| In other words, it represented the chances to find | | | | independent securities as there are future states |
| the price of the stock within a given range of | | | | of the world (a quite large number). Merton |
| prices on expiration. It did not represent the | | | | proved that far fewer financial instruments are |
| investors' risk aversion, something which is | | | | sufficient to eliminate risk, even when the number |
| impossible to observe in reality. | | | | of future states is very large. |
| The Black and Scholes Formula | | | | Practical Importance |
| The revolution brought about by Merton, Black | | | | Option contracts began to be traded on the |
| and Scholes was recognizing that it is not | | | | Chicago Board Options Exchange (CBOE) in April |
| necessary to use any risk premium when valuing | | | | 1973, one month before the formula was |
| an option because it is already included in the price | | | | published. |
| of the stock. In 1973 Fischer Black and Myron S. | | | | It was only in 1975 that traders had begun |
| Scholes published the famous option pricing Black | | | | applying it - using programmed calculators. |
| and Scholes formula. Merton extended it in 1973. | | | | Thousands of traders and investors use the |
| The idea was simple: a formula for option | | | | formula daily in markets throughout the world. In |
| valuation should determine exactly how the value | | | | many countries, it is mandatory by law to use the |
| of the option depends on the current share price | | | | formula to price stock warrants and options. In |
| (professionally called the "delta" of the option). A | | | | Israel, the formula must be included and explained |
| delta of 1 means that a $1 increase or decrease | | | | in every public offering prospectus. |
| in the price of the share is translated to a $1 | | | | Today, we cannot conceive of the financial world |
| identical movement in the price of the option. | | | | without the formula. |
| An investor that holds the share and wants to | | | | Investment portfolio managers use put options to |
| protect himself against the changes in its price can | | | | hedge against a decline in share prices. Companies |
| eliminate the risk by selling (writing) options as the | | | | use derivative instruments to fight currency, |
| number of shares he owns. If the share price | | | | interest rates and other financial risks. Banks and |
| increases, the investor will make a profit on the | | | | other financial institutions use it to price (even to |
| shares which will be identical to the losses on the | | | | characterize) new products, offer customized |
| options. The seller of an option incurs losses when | | | | financial solutions and instruments to their clients |
| the share price goes up, because he has to pay | | | | and to minimize their own risks. |
| money to the people who bought it or give to | | | | Some Other Scientific Contributions |
| them the shares at a price that is lower than the | | | | The work of Merton and Scholes was not |
| market price - the strike price of the option. The | | | | confined to inventing the formula. |
| reverse is true for decreases in the share price. | | | | Merton analysed individual consumption and |
| Yet, the money received by the investor from | | | | investment decisions in continuous time. He |
| the buyers of the options that he sold is invested. | | | | generalized an important asset pricing model called |
| Altogether, the investor should receive a yield | | | | the CAPM and gave it a dynamic form. He applied |
| equivalent to the yield on risk free investments | | | | option pricing formulas in different fields. |
| (for instance, treasury bills). | | | | He is most known for deriving a formula which |
| Changes in the share price and drawing nearer to | | | | allows stock price movements to be |
| the maturity (expiration) date of the option | | | | discontinuous. |
| changes the delta of the option. The investor has | | | | Scholes studied the effect of dividends on share |
| to change the portfolio of his investments | | | | prices and estimated the risks associated with the |
| (shares, sold options and the money received | | | | share which are not specific to it. He is a great |
| from the option buyers) to account for this | | | | guru of the efficient marketplace ("The Invisible |
| changing delta. | | | | Hand of the Market"). |
| This is the first unrealistic assumption of Black, | | | | |