Page 244 - DCOM504_SECURITY_ANALYSIS_AND_PORTFOLIO_MANAGEMENT

Unit 8: Derivatives

Binomial Option Pricing Model Notes

The binomial model has proved over time to be the most flexible, intuitive and popular approach
to option pricing. It is based on the simplification that over a single period (of possibly very
short duration), the underlying asset can only move from its current price to two possible levels.
Among other virtues, the model embodies the assumptions of no riskless arbitrage opportunities
and perfect markets. Neither does it rely on investor risk aversion or rationality, nor does its use
require estimation of the underlying asset expected return. It also embodies the risk-neutral
valuation principle, which can be used to shortcut the valuation of European options. In addition,
we show later, that the Black-Scholes formula is a special case applying to European options
resulting from specifying an infinite number of binomial periods during the time-to-expiration.
Nonetheless, a binomial tree has several curious, and possibly limiting, properties. For example,
all sample paths that lead to the same node in the tree have the same risk-neutral probability.
The types of volatility – objective, subjective and realized – are indistinguishable; and, in the
limit, its continuous-time sample path is not differentiable at any point.
Another way to approach binomial option pricing is through the inverse problem, implied
binomial trees. Instead of presuming we know the underlying asset volatility in advance to
construct the up and down moves in the tree, we use the current prices of related options to infer
the size of these moves.
Binomial trees can also be used to determine the sensitivity of option values to the underlying
asset price ( and ) , to the time-to-expiration ( ) , to volatility (vega), to the riskless return
(rho), and to the payout return (lambda). Of these, gamma is particularly important because it
measures the times in the life of the option when replication is likely to prove difficult in
practice. Fugit measures the risk-neutral expected life of the option and can also be calculated
from a binomial tree.

The standard binomial option pricing model for options on assets can easily be extended to
options on futures and options on foreign currencies. In addition, the model continues to work
even if its parameters are time-dependent, asset price-dependent, or dependent on the prior
path of the underlying asset price. But it fails if its parameters depend on some other random
variable. A more difficult task is to extend the binomial model to value options on bonds.

8.6 Summary

Derivatives are a new invention of the international financial markets, which are traded
both in Over-the-Counter and Exchanges.
Derivatives are very much useful for the concerns whose potential profits are volatile due
to changes in weather conditions like agricultural products processing, power generation,
oil exploration, tourism, insurance, etc.
Credit derivatives have been invented to hedge the risk of banks, financial institutions.

Credit derivatives allow users to isolate price and trade firm-specific risk into its component
parts and transfer each risk to those best suited or more interested in managing it.
Margin money is to be kept with the exchange for entering into futures contracts, as this
aims to minimise the risk of default by the counter party.
The futures contracts overcome the problems faced by forward contracts, since futures
contracts are entered into under the supervision and control of an organised exchange.
The futures contracts are entered into for a wide variety of instruments like agricultural
commodities, minerals, industrial raw materials, financial instruments etc.

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