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European Option

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Overview

A European option is an option which has a payoff at maturity based on a function of the underlying security at expiration time T and Strike price K. Some examples of European options are:

  • European calls
  • European puts
  • Binary cash-or-nothing
These three options are also called vanilla options since they are very common. In general, options which cannot be exercised early are called European for this non-early-exercise property.

European Call

The European call option gives the right to buy the underlying security S at price K at time T. This can be simulated with a simple ThetaML as follows:

%% The model European_call computes of the price "P" of
%% European call option
model European_call
    import S	"Stock"
    import K	"Strike"
    import EUR	"Numeraire"
    import T	"Maturity time"
    export P	"Option Value"
 
    P = V!
 
    Theta T
 
    V = max(S-K,0)*EUR
end

European Put

The European put option grants the right to sell the underlying security S at price K at time T. This can be simulated with a simple ThetaML as follows:

%% The model European_put computes of the price "P" of
%% European put option
model European_put
    import S    "Stock"
    import K    "Strike"
    import EUR  "Numeraire"
    import T    "Maturity time"
    export P    "Option Value"
 
    P = V!
 
    Theta T
 
    V = max(K-S,0)*EUR
end

European Binary

The European Binary or Digital option pays 1 if the underlying has a price greater than the strike K at time T and pays 0 otherwise.

%% The model European_binary computes of the price "P" of
%% European binary option
model Binary
  import S "Underlying stock"
  import K "Strike"
  import T "Maturity time"
  export V "Option value"
 
  Theta 1
 
  if S>K
    V=1
  else
    V=0
  end
end

The binary cash-or-nothing option pays a fixed cash amount if the option expires in-the-money and pays nothing otherwise.



Bcono1.svg


Black-Scholes pricing formula for European Binary option


$ C_{cash} = Ke^{-rT}N(d) $

Where:

$ d = \frac {\ln \left (\frac{S}{K} \right )+(r-D+0.5\sigma^2)T}{\sigma\sqrt{T}} $

Option European Option
Underlying Common Stock
Underlying Price S
Start Date 0
Maturity Date T
Call True
Payoff C
Strike Price K
Divident Yield D
Cumulative Normal Distribution N(.)
Volatility $ \sigma $
Interest Rate $ r $




ThetaML graphic illustration of modelling European Binary option


Bcono2.svg




ThetaML implementation of pricing European binary option

%% The model European_binary computes price "P" of
%% a European binary option
model Binary
  import S "Underlying stock"
  import K "Strike"
  import T "Maturity time"
  export V "Option value"
 
  Theta 1
 
  if (S-K)>0
    V=1
  else
    V=0
  end
end


Numerical example for European binary option

A binary option with parameters given below has a price of P=35.8647. The graph shown below demonstrates the behaviour of option prices with respect to the number of paths in ThetaML n.

Parameter Symbol Value
Underlying Price S 100
Volatility $ \sigma $ 40,00%
Interest Rate $ r $ 5,00%
Maturity T 1 year
Numeraire EUR 1
Cash C 100
Strike K 110



Fsao3.svg Eo4.svg