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Finance[EuropeanOption] - create a new European option

Calling Sequence

EuropeanOption(payoff, exercise, opts)

Parameters

payoff

-

payoff function

exercise

-

a non-negative constant, a string containing a date specification recognized in a format recognized by Finance[ParseDate], or a date data structure; the maturity time or date

opts

-

(optional) equation(s) of the form option = value where option is one of referencedate or daycounter; specify options for the EuropeanOption command

Description

• 

The EuropeanOption command creates a new European-style option with the specified payoff and maturity. This option can be exercised only at the time or date specified by the exercise parameter. This is the opposite of an American-style option, which can be exercised at any time before the expiration.

• 

The parameter payoff is the payoff function for the option. It can be either an algebraic expression or a procedure. A procedure defining a payoff function must accept one parameter (the value of the underlying) and return the corresponding payoff. This procedure will be called with floating-point arguments only and must return floating-point values. If payoff is given as an algebraic expression it must depend on a single variable. This expression will be converted to a Maple procedure using the unapply function.

• 

The parameter exercise specifies the time or date when the option can be exercised. It can be given either as a non-negative constant or as a date in any of the formats recognized by the Finance[ParseDate] command.

• 

The LatticePrice command can be used to price a European-style option using any given binomial or trinomial tree.

Examples

withFinance:

Set the global evaluation date to January 3, 2006.

SetEvaluationDateJanuary 3, 2006:

Settingsdaycounter=Thirty360European

Historical

(1)

Maturity:=AdvanceDateEvaluationDate,6,Months,output=formatted

Maturity:=July 3, 2006

(2)

Construct a binomial tree approximating a Black-Scholes process with an initial value of 100, a risk-free rate of 10% and a constant volatility of 40%. We will assume that no dividend is paid. Build the tree by subdividing the time period 0..0.6 into 1000 equal time steps.

T:=BlackScholesBinomialTree100,0.1,0.0,0.4,0.6,1000:

Consider a European call option with a strike price of 90 that matures in 6 months.

P1:=S→maxS90,0

P1:=S→maxS90,0

(3)

plotP1,80..120,color=blue,thickness=2,axes=BOXED,gridlines=true

E1:=EuropeanOptionP1,0.5:

E2:=EuropeanOptionP1,Maturity:

Calculate the price of this option using the tree constructed above. Use the risk-free rate as the discount rate.

LatticePriceE1,T,0.1

19.16307346

(4)

LatticePriceE2,T,0.1

19.16307346

(5)

Consider a European call option with a strike price of 110 that matures in 6 months.

P2:=S→maxS110,0

P2:=S→maxS110,0

(6)

plotP2,80..120,color=blue,thickness=2,axes=BOXED,gridlines=true

E3:=EuropeanOptionP2,0.5:

E4:=EuropeanOptionP2,Maturity:

Calculate the price of this option using the tree constructed above. Use the risk-free rate as the discount rate.

LatticePriceE3,T,0.1

9.341739175

(7)

LatticePriceE4,T,0.1

9.341739175

(8)

Finally, consider a call option with a strike price of 100 maturing in 6 months.

P3:=S→maxS100,0

P3:=S→maxS100,0

(9)

plotP3,80..120,color=blue,thickness=2,axes=BOXED,gridlines=true

E5:=EuropeanOptionP3,0.5:

E6:=EuropeanOptionP3,Maturity:

Calculate the price of this option using the tree constructed above. Use the risk-free rate as the discount rate.

LatticePriceE5,T,0.1

13.58412136

(10)

LatticePriceE6,T,0.1

13.58412136

(11)

Consider a more complicated payoff function.

P4:=S&rarr;piecewiseS<90&comma;0&comma;S<100&comma;S90&comma;S<110&comma;110S&comma;0

P4:=S&rarr;piecewiseS<90&comma;0&comma;S<100&comma;S90&comma;S<110&comma;110S&comma;0

(12)

plotP4&comma;80..120&comma;color&equals;blue&comma;thickness&equals;2&comma;axes&equals;BOXED&comma;gridlines&equals;true

E7:=EuropeanOptionP4&comma;0.5&colon;

E8:=EuropeanOptionP4&comma;Maturity&colon;

Calculate the price of this option using the tree constructed above. Use the risk-free rate as the discount rate.

LatticePriceE7&comma;T&comma;0.1

1.336569904

(13)

LatticePriceE8&comma;T&comma;0.1

1.336569904

(14)

Note that the payoff of the options E8 and E7 can be replicated using the payoffs of the options E1, E2, E3, E4, E5, and E6.

plotP4&comma;P1&plus;P22P3&comma;80..120&comma;color&equals;blue&comma;red&comma;thickness&equals;2&comma;axes&equals;BOXED&comma;gridlines&equals;true

This means that the prices should also match.

LatticePriceE1&comma;T&comma;0.1&plus;LatticePriceE3&comma;T&comma;0.12LatticePriceE5&comma;T&comma;0.1&equals;LatticePriceE7&comma;T&comma;0.1

1.33656992&equals;1.336569904

(15)

LatticePriceE2&comma;T&comma;0.1&plus;LatticePriceE4&comma;T&comma;0.12LatticePriceE6&comma;T&comma;0.1&equals;LatticePriceE8&comma;T&comma;0.1

1.33656992&equals;1.336569904

(16)

See Also

Finance[AmericanOption], Finance[BermudanOption], Finance[BinomialTree], Finance[BlackScholesBinomialTree], Finance[BlackScholesTrinomialTree], Finance[GetDescendants], Finance[GetProbabilities], Finance[GetUnderlying], Finance[ImpliedBinomialTree], Finance[ImpliedTrinomialTree], Finance[LatticeMethods], Finance[LatticePrice], Finance[SetProbabilities], Finance[SetUnderlying], Finance[StochasticProcesses], Finance[TreePlot], Finance[TrinomialTree]

References

  

Glasserman, P., Monte Carlo Methods in Financial Engineering, Springer-Verlag, 2004.

  

Hull, J., Options, Futures and Other Derivatives, 5th. edition. Prentice Hall, 2003.

  

Jackel, P., Monte Carlo Methods in Finance, John Wiley & Sons, 2002.

  

Joshi, M., The Concepts and Practice of Mathematical Finance, Cambridge University Press, 2003.

  

Wilmott, P., Paul Wilmott on Quantitative Finance, John Wiley and Sons Ltd, 2000.

  

Wilmott, P., Howison, S., and Dewyne, J., The Mathematics of Financial Derivatives, New York: Cambridge University Press, 1995.


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