present value of a European call option
blackscholes(amount, exercise, rate, nperiods, sdev, hedge)
current stock price
exercise price of the call option
risk-free interest rate per period, (continuously compounded)
number of periods
standard deviation per period of the continuous return on the stock
(optional name) hedge ratio
The function blackscholes computes the present value of an European call option without dividends under Black-Scholes model.
The function requires the value of the standard deviation. It can be calculated from the variance by taking the square root.
The hedge ratio is the ratio of the expected stock price at expiration to the current stock price.
There are strong assumptions on the Black-Scholes model. Use at your own risk. Refer to appropriate finance books for the list of assumptions.
Since blackscholes used to be part of the (now deprecated) finance package, for compatibility with older worksheets, this command can also be called using finance[blackscholes]. However, it is recommended that you use the superseding package name, Finance, instead: Finance[blackscholes].
There is a 49 units call option with 199 days to maturity on a stock that is selling at present at 50 units. The annualized continuously compounding risk-free interest rate is 7%. The variance of the stock is estimated at 0.09 per year. Using the Black-Scholes model, the value of the option would be
B ≔ blackscholes⁡50.00,49.00,0.07,199365,0.09:
which is about 5.85 units.
Let us examine how this result changes by changing the parameters. Increasing the stock price
the option value increases.
Increasing exercise price
the option value decreases.
Increasing the risk-free interest rate
Increasing the time to expiration
Increasing the stock volatility
the option value increases. Plot the value of the call with respect to the share price.
The upper bound: option is never worth more than the share. The lower bound: option is never worth less than what one would get for immediate exercise of the call.
f ≔ x→evalf⁡blackscholes⁡x,49.00,0.07,199365,0.09:
U ≔ x→x:
L ≔ x→max⁡x−49.00,0.00:
plot⁡'f⁡x','U⁡x','L⁡x',x=0..100,labels=`share price`,`value of call`
The Finance[blackscholes] command was introduced in Maple 15.
For more information on Maple 15 changes, see Updates in Maple 15.
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