The Nobel Prize in Economics was awarded to Robert Merton, Fischer Black
and Myron Scholes for their pioneering work in establishing the foundation for
the financial engineering that has revolutionized contemporary finance. They
developed an intellectually elegant model that enables traders to take a set of
observable prices and to calculate mathematically a price for an option. While
some may criticize it for the assumptions that it makes, the model has been
instrumental in allowing traders to understand intuitively the behavioral
characteristics of the product and it has facilitated the exponential growth of
derivatives worldwide. Indeed, one could argue that because of its assumptions
the model has forced traders to understand every aspect of options pricing.
This article will explain intuitively the way in which the options pricing
model was derived and it will illustrate the assumptions implicit in the model.
Let's take a look at an investor who wants to buy IBM stock. The buyer of
an IBM call option has the right but not the obligation to buy IBM stock at the
maturity date at a pre-set strike price. He will exercise this option if the
option is in-the-money. That is to say, the buyer will exercise his right to
purchase IBM stock if the strike price is less than the prevailing cash price
for IBM stock at the time the option expires. The question the modeler has to
answer is this. How do we evaluate what the stock price is going to be at
maturity, an event that could take place months from now? The first assumption
has to do with the way in which we model the process that characterizes the
movement of the underlying price of IBM. We know that the price of IBM stock
fluctuates. Perhaps it follows a trend. We have observed the volatility of the
stock price. If we have a way of describing the statistical process underlying
the IBM price, then we have taken the first critical step towards pricing the
option.
Let us assume that the stock price follows a Markov process. This is simply
a process in which only the current value of a variable is useful in
forecasting what the future value of the variable could be.
ASSUMPTION 1: THE UNDERLYING PRICE IS LOGNORMALLY DISTRIBUTED. In
fact, let us assume that the logarithm of the price is normally distributed
such that it has a trend and a volatility that we can specify. Doing this
enables us to predict what the expected value of the stock price will be and it
also makes the model mathematically tractable. Using the normal distribution
makes it easier to find a closed-form solution to the problem. A closed-form
solution is simply an equation that we can use to determine the price of the
option. Note that this means that volatility for the underlying price is
constant and the same for all maturities. As we shall see in subsequent
articles, there is usually a term structure to volatility in which different
maturities have different volatilities. Remember that an option is not an
option on the spot price but an option on the forward price. Different
maturities will trade with different volatility, in practice, because of cash
flow events, expectations of political instability, political events,
management changes, etc.
ASSUMPTION 2: THE SHORT SELLING OF SECURITIES WITH THE FULL USE OF
PROCEEDS IS PERMITTED.When we talked about delta hedging in
"Derivatives Explained," we assumed that we could buy and sell stock
against our options position in order to capture the effects of a volatile
underlying price, thereby paying for the option premium over the life of the
instrument. In order to sell stock, there can be no restriction on short sales.
ASSUMPTION 3: THERE ARE NO TRANSACTION COSTS OR TAXES. ALL SECURITIES ARE
PERFECTLY DESIRABLE. Transaction costs, such as brokerage, and taxes would
distort the simple problem of trying to understand how to price an option. In
practice, the investor or the options professional accounts for these factors
in the course of doing business. Transaction costs and taxes will distort the
delta hedging decision, providing a disincentive for delta hedging and altering
the way in which we determine the option's value.
ASSUMPTION 4: THERE ARE NO DIVIDENDS DURING THE LIFE OF THE DERIVATIVE
SECURITY. Again, we ignore dividends in the derivation of the Black-Scholes
model because of the distortionary effects these can have on our delta hedging
decision. If we buy a call option and need to delta hedge it by short selling
securities, we may be hesitant to short sell securities that pay a dividend.
ASSUMPTION 5: THERE ARE NO RISKLESS ARBITRAGE OPPORTUNITIES. This is
an assumption of efficient markets theory. An arbitrage opportunity exists when
one can buy and sell the same instrument (or virtually the same instrument)
simultaneously for different prices, thereby locking in a price. Because the
transaction is instantaneous, there is no risk to the individual. A market may
be said to be efficient if there are no such opportunities. Put another way, as
soon as such opportunities arise, they are immediately realized by some astute
investor. You would not expect them to last for long, as markets will correct
themselves rationally. Or, so says the theory. Ironically, Long Term Capital
Management claimed to be engaging in arbitrage by buying and selling similar
(but not identical) instruments with holding periods in excess of a few weeks.
This is ironic because some of the leaders in financial research were part of
this firm. It turned out that the similarity between those instruments was not
as solid or as durable as the rocket scientists originally thought.
ASSUMPTION 6: SECURITY TRADING IS CONTINUOUS. Prices of stocks on
North American exchanges move in discrete increments, such as 1/32. By assuming
that prices can trade in a mathematically continuous fashion, the model is more
mathematically tractable. For example, if IBM's stock traded at 189.8975, there
would be no reason why the next price could not be infinitesimally higher, say
189.8975001.
ASSUMPTION 7: THE RISK-FREE RATE OF INTEREST, r, IS CONSTANT AND THE SAME
FOR ALL MATURITIES. This is a big assumption. It states that the government
yield curve is flat. We know that is not true, just from common sense. But, in
order to solve the model for a wasting asset, it is important to model rates as
constant.
INTUITION Having made all of these assumptions, how did Black,
Scholes and Merton apply them to the pricing of options? They modeled a
portfolio that consisted of one unit of the option and a fraction (the delta)
of shares in the underlying instrument of the option, choosing the delta so
that the portfolio did not change in value for small movements in the price of
the underlying price. By doing so, they mathematically removed the uncertain
element of the Wiener process. There was no longer any risk in the portfolio.
Therefore, they reasoned, the test portfolio must have the same return as a
riskless portfolio, r.
Solving this differential equation using boundary conditions fixed by
whether or not the derivative was a call or a put resulted in the closed-form
solution for the options price. Now, when you want to price an option, you
input the following parameters into your Black-Scholes calculator and you have
the price:
- Stock's current price
- Strike price
- Today's date
- Maturity Date
- Delivery Date
- Risk-free interest rate
- Stock price implied volatility
- Call or a Put
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Note that one of the items here is "implied volatility." Let's say
you're the IBM investor and you observe everything in the market, including the
price of the call option you want to buy, except for the volatility. You can
use the Black-Scholes calculator to "back-out" the volatility that
the market is using. Doing so gives you the so-called "implied
volatility." Subsequent articles will discuss the ways in which options
pricing has evolved, in addressing the shortcomings of the Black-Scholes
assumptions and in extending the Black-Scholes approach to other markets,
including currencies, dividend-paying equities, futures, etc.
Article by Chand Sooran, Principal Victory
Risk Management Consulting, Inc. |
