>

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

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 riskfree interest rate is 7%. The variance of the stock is estimated at 0.09 per year. Using the BlackScholes model, the value of the option would be
>

$B:=\mathrm{blackscholes}\left(50.00\,49.00\,0.07\,\frac{199}{365}\,\sqrt{0.09}\right)\:$

>

$\mathrm{evalf}\left(B\right)$

which is about 5.85 units.
Let us examine how this result changes by changing the parameters. Increasing the stock price
>

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

the option value increases.
Increasing exercise price
>

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

the option value decreases.
Increasing the riskfree interest rate
>

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

the option value increases.
Increasing the time to expiration
>

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

the option value increases.
Increasing the stock volatility
>

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

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.
>

$f:=x\→\mathrm{evalf}\left(\mathrm{blackscholes}\left(x\,49.00\,0.07\,\frac{199}{365}\,\sqrt{0.09}\right)\right)\:$

>

$L:=x\→\mathrm{max}\left(x49.00\,0.00\right)\:$

>

$\mathrm{plot}\left(\left\{\'f\left(x\right)\'\,\'U\left(x\right)\'\,\'L\left(x\right)\'\right\}\,x\=0..100\,\mathrm{labels}\=\left[\mathrm{`share\; price`}\,\mathrm{`value\; of\; call`}\right]\right)$
