European Option

From ThetaWiki

1 Overview
2 European Put
3 European Call
4 European Binary
- 4.1 Black-Scholes Pricing
5 Modelling Using ThetaML
6 ThetaScript
7 Numerical Example

Overview

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

European calls
European puts
Binary cash-or-nothings

These three options are also called vanilla options since they are very common. In general, options which cannot be exercised early (like American Options) are often called European to emphasize this property.

European Put

The European put option grants the right to sell the underlying security S at price K at time T. This can be modeled in a simple ThetaScript 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 Call

Similar to the European put, the European call option is the the right to buy the underlying security S at price K at time T. This can be modeled in a simple ThetaScript 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 Binary

The European Binary or Digital option pays 1 if the underlying has a price greater 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.

Black-Scholes Pricing

$C c a s h = K e - r T N (d)$

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	$σ$
Interest Rate	$r$

Modelling Using ThetaML

ThetaScript

%% 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

A binary option with parameters given below has a price of P=35.8647. The graph shown below demonstrates the properties of ThetaScript with different numbers of paths n.