In the last decades there has been a growing interest in the behavior of financial markets. Due to the increasing globalization of markets, they began to play a central role in international business and economic decision making. Thus, the meaning of ``risk'' became the central theme in this context. Risk management is essential in a modern market economy. Financial markets enable firms and households to select an appropriate level of risk in their transactions, by redistributing risks towards other agents who are willing and able to assume them. Markets for options, futures and other so-called derivative securities -- derivatives, for short -- have a particular status. Futures allow agents to hedge against upcoming risks; such contracts promise the future delivery of a certain item at a certain price. As an example, a firm might decide to engage in copper mining after determining that the metal to be extracted can be sold in advance at the futures market for copper. The risk of future movements in the copper price is thereby transferred from the owner of the mine to the buyer of the contract. Due to their design, options allow agents to hedge against one-sided risks; options give the right, but not the obligation, to buy or sell something at a prespecified price in the future.
In avoiding the risk of long positions one could for example try to hedge the risk by going short in options on the corresponding asset and adapting the proportion held in assets and short-selled options according to the underlying price process of that asset. Therefore, formulas for the pricing of those derivative securities generated a lot of practical and theoretical interest.
Already in the year 1900, Bachelier introduced Brownian motion as a model for price fluctuations on a speculative market. In 1973, Black and Scholes founded their famous option pricing formula which calculates the ``fair price'' of an option (which means that there is no arbitrage). This has generated a lot of theoretical work relying on that basic model.
The valuation of derivatives has a long history. One of the earliest endeavors was undertaken by Louis Bachelier (thesis at Sorbonne, 1900). But his formula was based on such assumptions as zero interest rate, and a process that allowed for a negative share price.
This formula was improved by Case Sprenkle, James Boness and Paul Samuelson. They assumed that stock prices are log-normally distributed, guaranteeing that share prices are positive, and allowed for a nonzero interest rate. They also assumed that investors are risk averse and demand a risk premium additionally to the interest rate. In 1964, Boness suggested a formula that came close to the Black-Scholes formula, but still relied on an unknown interest rate, which included compensation for the risk associated with the stock.
Further attempts at valuation (before 1973) basically determined the expected value of a stock option at expiration and discounted its value back to the time of evaluation. Unfortunately, those approaches require taking a stance on which risk premium to use in the discounting. But assigning a risk premium is not straightforward, since it should reflect not only the risk for changes in the stock price, but also the investorsĘ attitude towards risk. The latter is hard or impossible to observe in reality.
A commonly used model for the description of fluctuations
of asset prices is the following. X(.) denotes the price
process which is assumed to be the solution of the
stochastic differential equation
Fischer Black, Robert Merton and Myron Scholes developed a new method of determining the value of derivatives. Their work (in the early 1970s) solved a longstanding problem in financial economics and has provided ways of dealing with financial risk, both in theory and in practice. Further, their methodology has proven general enough for a wide range of applications. It can thus be used to value not only the flexibility of physical investment projects but also insurance contracts and guarantees.
In the press release, when Scholes and Merton were awarded the Nobel
Prize in 1997, was given the following example: Consider a
European call option at a strike price of $100 in three months.
(A European option gives the right to buy or sell only at a
certain date, whereas a so-called American option gives the
same right at any point in time up to a certain
date.)
Clearly, the
value of this call option depends on the current share price;
the higher the share price today the greater the probability
that it will exceed $100 in three months, in which case it
will pay to exercise the option. A formula for option valuation
should thus determine exactly how the value of the option
depends on the current share price. How much the value of the
option is altered by a change in the current share price is
called the ``delta'' of the option -- see also the
greeks.
Assume that the value of the option increases by $1 when the current share price goes up $2 and decreases by $1 when the stock goes down $2. Assume also that an investor holds a portfolio of the underlying stock and wants to hedge against the risk of changes in the share price. He can then construct a risk-free portfolio by selling twice as many options as the number of shares he owns. For reasonably small increases in the share price, the profit the investor makes on the shares will be the same as the loss he incurs on the options, and vice versa for decreases in the share price. As the portfolio thus constructed is risk free, it must yield exactly the same return as a risk-free three-month treasury bill. If it did not, arbitrage trading would begin to eliminate the possibility of making risk-free profits. As the share price is altered over time and as the time to maturity draws nearer, the delta of the option changes. In order to maintain a risk-free stock-option portfolio, the investor has to change its composition.
Black, Merton and Scholes assumed that such trading can take
place continuously without any transaction costs. The condition
that the return on a risk-free stock-option portfolio yields
the risk-free rate, at each point in time, implies a partial
differential equation, the solution of which is the Black-Scholes
formula for a call option:
According to this formula, the value of the call option C
is given by the difference between the expected share price --
the first term on the righthand side -- and the expected cost
-- the second term -- if the option is exercised.
The higher the option value, the higher the current share price S, the
higher the volatility of the share price ,
the higher the risk-free interest rate r, the longer the
time to maturity t, the lower the strike price L, and the
higher the probability that the option will be
exercised -- see also the quantlet influence. All the
parameters in the equation can be observed except sigma,
which has to be estimated from market data. Alternatively,
if the price of the call option is known, the formula can be used
to solve for the market-implied volatility. Market equilibrium
is not necessary for option valuation; it is sufficient that there
are no arbitrage opportunities. The method described in the
example above is based precisely on the absence of
arbitrage. It generalizes to valuation of other types of
derivatives. MertonĘs 1973 article included the Black-Scholes
formula and some generalizations, for instance, he allowed
the interest rate to be stochastic. The theory of Merton, Black and
Scholes can also be used for many other or related fields such as:
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