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

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
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$B\u2254\mathrm{blackscholes}\left(50.00\,49.00\,0.07\,\frac{199}{365}\,\mathrm{sqrt}\left(0.09\right)\right)\:$

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

which is about 5.85 units.
Let us examine how this result changes by changing the parameters. Increasing the stock price
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$\mathrm{evalf}\left(\mathrm{blackscholes}\left(50.00+1.00\,49.00\,0.07\,\frac{199}{365}\,\mathrm{sqrt}\left(0.09\right)\right)\right)$

the option value increases.
Increasing exercise price
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$\mathrm{evalf}\left(\mathrm{blackscholes}\left(50.00\,49.00+1.00\,0.07\,\frac{199}{365}\,\mathrm{sqrt}\left(0.09\right)\right)\right)$

the option value decreases.
Increasing the riskfree interest rate
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$\mathrm{evalf}\left(\mathrm{blackscholes}\left(50.00\,49.00\,0.07+0.01\,\frac{199}{365}\,\mathrm{sqrt}\left(0.09\right)\right)\right)$

the option value increases.
Increasing the time to expiration
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$\mathrm{evalf}\left(\mathrm{blackscholes}\left(50.00\,49.00\,0.07\,\frac{199+1}{365}\,\mathrm{sqrt}\left(0.09\right)\right)\right)$

the option value increases.
Increasing the stock volatility
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$\mathrm{evalf}\left(\mathrm{blackscholes}\left(50.00\,49.00\,0.07\,\frac{199}{365}\,\mathrm{sqrt}\left(0.09+0.01\right)\right)\right)$

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.
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$f\u2254x\mapsto \mathrm{evalf}\left(\mathrm{blackscholes}\left(x\,49.00\,0.07\,\frac{199}{365}\,\mathrm{sqrt}\left(0.09\right)\right)\right)\:$

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$U\u2254x\mapsto x\:$

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$L\u2254x\mapsto \mathrm{max}\left(x49.00\,0.\right)\:$

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