Introduction
Suppose you would like to develop a new option trading
strategy. You wrote a computer program or developed a
database to analyze history prices for backtesting. The
option history files are expensive, large, and complicated.
It is better and much simpler to analyze stock history
files and calculate the option prices using some model.
The most popular is the Black-Scholes option pricing model.
It can be written for call options as
where
p -stock price
s - striking price
t - time remaining until expiration, expressed as a
part of a year (3 months = 0.25)
r - current risk free interest rate (if it is 6.5% then
r = 0.065)
v - volatility measured by annual standard deviation
for the stock prices
N(x) - cumulative normal density function
Visual Basics program for
calculation of the option (call and puts) prices using
this model can be written as
Public
Function BlackScholes(CallPutIndicator As String,
p As Double, s As Double, _
t As Double, r As Double, v As Double) As Double
Dim d1 As Double, d2 As Double
d1 = (Log(p / s) + (r + v ^ 2 / 2) * t) / (v *
Sqr(t))
d2 = d1 - v * Sqr(t)
If CallPutIndicator = "call" Then
BlackScholes = p * CND(d1) - s * Exp(-r * t) *
CND(d2)
Else If CallPutIndicator = "put" Then
BlackScholes = s * Exp(-r * t) * CND(-d2) - p
* CND(-d1)
End If
End Function
Public Function CND(X As Double) As Double
Dim L As Double, K As Double
Const a1 = 0.31938153: Const a2 = -0.356563782:
Const a3 = 1.781477937:
Const a4 = -1.821255978: Const a5 = 1.330274429
L = Abs(X)
K = 1 / (1 + 0.2316419 * L)
CND = 1 - 0.39894228 * Exp(-L ^ 2 / 2) * (a1 *
K + a2 * K ^ 2 + a3 * K ^ 3 + a4 * K ^ 4 + a5
* K ^ 5)
If X < 0 Then
CND = 1 - CND
End If
End Function |
How accurate is this model?
There is a long discussion about it. A crucial question
is calculation of the stock volatility. A short popular
review one can find in the book of L. McMillan Options
as a Strategic Investment. Here, we will perform
a simple analysis of the Black-Scholes equation to analyze
importance of some parameters (including stock volatility)
for the calculation of the option prices.
Risk
free interest rate
Suppose the interest rate
is lowed by 0.5%. Is it important for the option prices?
The answer is: for short-term option it is not important.
Let us show some examples. Consider call options with
t < 90 days. The next plot shows the option prices
for the rates = 1 and 5%. One can see that for the in
money calls (stock price > strike price) the influence
of the rates is very small.
The nest picture
shows the differences between option prices calculated
(t = 30 days) for r = 1 and 3% depending on stock volatility
and the differences between stock prices (P) and strike
prices (S)
One can see that influence
of r is important only for out of money options (S >
P) and only for stocks with low values of volatility.
Volatility
The next figure illustrates the influence of the stock
volatility on the option price.
One can see that for "well
in-money" options the influence of volatility is
not important. However, even small mistakes in calculating
of volatility give a large error in option prices when
a stock price is close or less than an option striking
price.
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