Python for Finance

Options and Futures
• Using the binomial tree method to price both European and American options
• Implied volatility, volatility smile and skewness
Options theory is an integral part of finance theory. It is difficult to image that
a finance student would not understand it. However, it is quite demanding to
comprehend the theory thoroughly. Many finance-major students view options
theory as rocket science, since it involves how to solve various differential equations.
In order to satisfy as many readers as possible, in this chapter we avoid complex
mathematical derivations.
An option would give the option buyer a right to buy or sell something in the future
with a fixed price determined today. If the buyer has a right to buy something in
the future, it is called a call option. If the option buyer is entitled to sell something,
it is called a put option. Since there are two persons (sides) for each transaction, the
buyer pays to acquire a right, while the seller receives cash inflow today to bear an
obligation. Unlike options, a futures contract would give the buyer and seller both
rights and obligations. Unlike options with an initial cash flow from buyer to seller,
for a futures contract, usually there is no initial cash flow. Forward contracts are
quite similar to future contracts with a few exceptions. In this chapter, these two
types of contracts (futures and forwards) are not distinguished. A forward contract
is easier to analyze than a future contract. If a reader wants a more in-depth analysis,
he/she should consult other related textbooks.
Introducing futures
Before discussing the basic concepts and formulas related to futures, let's review
the concept of continuously compounded interest rates. In Chapter 3, Time Value of
Money, we learned that the following formula could be applied to estimate the future
value of a given present value:
Here, FV is the future value, PV is the present value, R is the effective period rate and
n is the number of periods. For example, assume that the Annual Percentage Rate
(APR) is 8%, compounded semiannually. If we deposit $100 today, what is its future
value in two years? The following code shows the result:
import scipy as ps
pv=100
APR=0.08
rate=APR/2.0
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