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# European Option

## Contents

## 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

**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.

### 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

### 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 |