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Finance

 blackscholes
 present value of a European call option

 Calling Sequence blackscholes(amount, exercise, rate, nperiods, sdev, hedge)

Parameters

 amount - current stock price exercise - exercise price of the call option rate - risk-free interest rate per period, (continuously compounded) nperiods - number of periods sdev - standard deviation per period of the continuous return on the stock hedge - (optional name) hedge ratio

Description

 • 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].

Examples

 > $\mathrm{with}\left(\mathrm{Finance}\right):$

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

 > $B≔\mathrm{blackscholes}\left(50.00,49.00,0.07,\frac{199}{365},\mathrm{sqrt}\left(0.09\right)\right):$
 > $\mathrm{evalf}\left(B\right)$
 ${5.849179520}$ (1)

Let us examine how this result changes by changing the parameters. Increasing the stock price

 > $\mathrm{evalf}\left(\mathrm{blackscholes}\left(50.00+1.00,49.00,0.07,\frac{199}{365},\mathrm{sqrt}\left(0.09\right)\right)\right)$
 ${6.511554996}$ (2)

the option value increases.

Increasing exercise price

 > $\mathrm{evalf}\left(\mathrm{blackscholes}\left(50.00,49.00+1.00,0.07,\frac{199}{365},\mathrm{sqrt}\left(0.09\right)\right)\right)$
 ${5.326914003}$ (3)

the option value decreases.

Increasing the risk-free interest rate

 > $\mathrm{evalf}\left(\mathrm{blackscholes}\left(50.00,49.00,0.07+0.01,\frac{199}{365},\mathrm{sqrt}\left(0.09\right)\right)\right)$
 ${5.994210290}$ (4)

the option value increases.

Increasing the time to expiration

 > $\mathrm{evalf}\left(\mathrm{blackscholes}\left(50.00,49.00,0.07,\frac{199+1}{365},\mathrm{sqrt}\left(0.09\right)\right)\right)$
 ${5.864587748}$ (5)

the option value increases.

Increasing the stock volatility

 > $\mathrm{evalf}\left(\mathrm{blackscholes}\left(50.00,49.00,0.07,\frac{199}{365},\mathrm{sqrt}\left(0.09+0.01\right)\right)\right)$
 ${6.072347530}$ (6)

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.

 > $f≔x↦\mathrm{evalf}\left(\mathrm{blackscholes}\left(x,49.00,0.07,\frac{199}{365},\mathrm{sqrt}\left(0.09\right)\right)\right):$
 > $U≔x↦x:$
 > $L≔x↦\mathrm{max}\left(x-49.00,0.\right):$
 > $\mathrm{plot}\left(\left\{'L\left(x\right)','U\left(x\right)','f\left(x\right)'\right\},x=0..100,\mathrm{labels}=\left[\mathrm{share price},\mathrm{value of call}\right]\right)$

Compatibility

 • The Finance[blackscholes] command was introduced in Maple 15.