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blackscholes

  

present value of a European call option

 

Calling Sequence

Parameters

Description

Examples

Compatibility

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

withFinance:

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

Bblackscholes50.00,49.00,0.07,199365,0.09:

evalfB

5.849179520

(1)

which is about 5.85 units.

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

evalfblackscholes50.00+1.00,49.00,0.07,199365,0.09

6.511554996

(2)

the option value increases.

Increasing exercise price

evalfblackscholes50.00,49.00+1.00,0.07,199365,0.09

5.326914003

(3)

the option value decreases.

Increasing the risk-free interest rate

evalfblackscholes50.00,49.00,0.07+0.01,199365,0.09

5.994210290

(4)

the option value increases.

Increasing the time to expiration

evalfblackscholes50.00,49.00,0.07,199+1365,0.09

5.864587748

(5)

the option value increases.

Increasing the stock volatility

evalfblackscholes50.00,49.00,0.07,199365,0.09+0.01

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.

fx→evalfblackscholesx,49.00,0.07,199365,0.09:

Ux→x:

Lx→maxx49.00,0.00:

plot'fx','Ux','Lx',x=0..100,labels=`share price`,`value of call`

Compatibility

• 

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

• 

For more information on Maple 15 changes, see Updates in Maple 15.

See Also

Finance

Finance[BlackScholesBinomialTree]

Finance[LatticeMethods]

 


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