Page 242 - DCOM504_SECURITY_ANALYSIS_AND_PORTFOLIO_MANAGEMENT

Unit 8: Derivatives

C = 20 [N(d )] – 20 e (–0.12 × 0.25) [N(d )] Notes
1 2
= 20 [N(0.25)] – 20(0.9704)[N(0.05)]
= 20(0.5987) – 19.41 (0.5199)
= 11.97 – 10.09

= 1.88

Lognormal Distribution

The model is based on a normal distribution of underlying asset returns, which is the same thing
as saying that the underlying asset prices themselves are lognormally distributed. A lognormal
distribution has a longer right tail compared with a normal, or bell-shaped, distribution. The
lognormal distribution allows for a stock price distribution between zero and infinity (i.e. no
negative prices) and has an upward bias (representing the fact that a stock price can only drop
100% but can rise by more than 100%).
In practice, underlying asset price distributions often depart significantly from the
lognormal. For example, historical distributions of underlying asset returns often have fatter
left and right tails than a normal distribution indicating that dramatic market moves occur with
greater frequency than would be predicted by a normal distribution of returns – i.e. more very
high returns and more very low returns.

A corollary of this is the volatility smile – the way in which at-the-money options often have a
lower volatility than deeply out-of-the-money options or deeply in-the-money options.
Modified Black-Scholes and binomial pricing models (using implied binomial trees) are deployed
for European and American option pricing with non-lognormal distributions. These models can
be used to gauge the impact on option prices of non-lognormal price distributions (as measured
by coefficients of skewness (symmetry) and kurtosis (fatness of distribution tails and height of
peaks)), and to calculate and plot the volatility smile implied by these distributions.

Measuring the degree to which historical asset price distributions diverge from the lognormal
(as measured by coefficients of skewness and kurtosis).

Task The stock option has 120 days until expiration and the strike price is 85.
The simple rate of interest is 6% p.a. The underlying asset value is 80
and the volatility (standard deviation) is 0.30.Calculate the value of the
stock option.

Relationship with Black-Scholes Model

The same underlying assumptions regarding stock prices underpin both the binomial and Black-
Scholes models: that stock prices follow a stochastic process described by geometric Brownian
motion. As a result, for European options, the binomial model converges on the Black-Scholes
formula as the number of binomial calculation steps increases. In fact the Black-Scholes model
for European options is really a special case of the binomial model where the number of binomial
steps is infinite. In other words, the binomial model provides discrete approximations to the
continuous process underlying the Black-Scholes model.

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