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Mark up of the Week (or two...)

This week ... The Black-Scholes Equation.

Below is the Black-Scholes Equation, used for pricing derivatives. Its creators were honoured with the Nobel Prize in Economics in 1997. Following the equation is an excerpt from the Commendation Letter.

\[C = SN(d) - L e^{-rt} N(d - \sigma \sqrt{t}), \]

where the variable $d$ is defined by:

\[d = \frac{\ln \frac{S}{L} + {(r + \sigma^2)}t}{ \sigma \sqrt{t}} .\]

Excerpt from Commendation:
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 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 pre-specified price in the future. An importing British firm that anticipates making a large payment in US dollars can hedge against the one-sided risk of large losses due to a future depreciation of sterling by buying call options for dollars on the market for foreign currency options. Effective risk management requires that such instruments be correctly priced.

Fischer Black, Robert Merton and Myron Scholes made a pioneering contribution to economic sciences by developing a new method of determining the value of derivatives. Their innovative work in the early 1970s, which solved a longstanding problem in financial economics, has provided us with completely new ways of dealing with financial risk, both in theory and in practice. Their method has contributed substantially to the rapid growth of markets for derivatives in the last two decades. Fischer Black died in his early fifties in August 1995.
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The idea behind the new method developed by Black, Merton and Scholes can be explained in the following simplified way. Consider a so-called European call option that gives the right to buy a certain share at a strike price of USD100 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 100USD 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.

Assume that the value of the option increases by USD1 when the current share price goes up USD2 and decreases by USD1 when the stock goes down USD2 (i.e. delta is equal to one half). 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, in fact, construct a risk-free portfolio by selling (writing) 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 (transaction costs were later introduced by others). 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 as above.

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 right-hand side - and the expected cost - the second term - if the option is exercised. The option value is higher, the higher the current share price $S$, the higher the volatility of the share price (as measured by its standard deviation) $\sigma$, 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 (this probability is, under risk neutrality, evaluated by the standardized normal distribution function $N$ ). 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's estimate of $\sigma$.