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Finance[blackscholes] - present value of a call option
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Calling Sequence
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blackscholes(amount, exercise, rate, nperiods, sdev, hedge)
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Parameters
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amount
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current stock price
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exercise
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exercise price of the call option
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rate
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risk-free interest rate per period, (continuously compounded)
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nperiods
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number of periods
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sdev
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standard deviation per period of the continuous return on the stock
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hedge
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(optional name) hedge ratio
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Description
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The function blackscholes computes the present value of a call option under the hypotheses of the model of Black and Scholes.
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The function requires the value of the standard deviation. It can be calculated from the variance by taking the square root.
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The hedge ratio give ratio of the expected stock price at expiration to the current stock price.
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There are strong assumptions on the Black-Scholes model. Use at your own risk. Refer to appropriate finance books for the list of assumptions.
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The command with(Finance,blackscholes) allows the use of the abbreviated form of this command.
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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].
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Compatibility
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The Finance[blackscholes] command was introduced in Maple 15.
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Examples
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There is a 49 U call option with 199 days to maturity on a stock that is selling at present at 50 U. 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
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which is about 5.85 U.
Let us examine how this result changes by changing the parameters. Increasing the stock price
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the option value increases.
Increasing exercise price
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the option value decreases.
Increasing the risk-free interest rate
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the option value increases.
Increasing the time to expiration
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the option value increases.
Increasing the stock volatility
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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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![plot({'L(x)', 'U(x)', 'f(x)'}, x = 0 .. 100, labels = [`share price`, `value of call`])](/support/helpjp/helpview.aspx?si=8936/file01706/math225.png)
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