Barrier Option Pricing Introduction Barrier Options and Monte - PowerPoint PPT Presentation

Monte Carlo Simulations Jessica Radeschnig Barrier Option Pricing Introduction Barrier Options and Monte Carlo Simulations The Discretized − A Monte Carlo Simulation Approach Black-Scholes Model Simulation The Algorithm Example Jessica Radeschnig Example 1 Others The End October 21, 2013

Overview Monte Carlo Simulations Jessica 1 Introduction Radeschnig Barrier Options and Monte Carlo Simulations Introduction The Discretized Black-Scholes Model Barrier Options and Monte Carlo Simulations The Discretized Black-Scholes 2 Simulation Model The Algorithm Simulation The Algorithm Example 3 Example Example 1 Others Example 1 The End 4 Others The End

Barrier Options and Monte Carlo Simulations Monte Carlo Simulations Jessica The price should be set as to be the value V of the option, Radeschnig where Introduction Barrier Options and Monte Carlo n Simulations V = 1 The Discretized � V i (1) Black-Scholes Model n i =1 Simulation The Algorithm where n is the number of simulations and Example Example 1 Others � C i , The End for call options V i = P i , for put options

Barrier Options and Monte Carlo Simulations Monte Carlo Simulations Jessica C i = 1 e − rT max { S i ( T ) − K , 0 } Radeschnig Introduction P i = 1 e − rT max { K − S i ( T ) , 0 } Barrier Options and Monte Carlo Simulations The Discretized Black-Scholes where r is the risk-free interest rate, T is the time to maturity, Model Simulation S ( T ) is the stock price at maturity and K is the strike price. The Algorithm Example 1 is the indicator function: Example 1 Others The End � 1 , if in the contract 1 = 0 , if out of the contract

The Discretized Black-Scholes Model Monte Carlo Simulations Jessica Radeschnig S ( t j ) e ( r − σ 2 / 2) T / m + σ √ Introduction S ( t j +1 ) = ˆ ˆ T / mZj . (2) Barrier Options and Monte Carlo Simulations The Discretized Black-Scholes Model where t is the current time and S ( t ) represents the stock price Simulation The Algorithm at that time. T is the time of maturity, r is the annualised Example risk-free interest rate, σ is the annualized volatility and W ( t ) is Example 1 √ Others the Brownian Motion with distribution of TZ , where Z is an The End independently distributed standard normal variable.

The Algorithm Monte Carlo for i = 1 to n do Simulations if Knock-In Option Jessica Radeschnig set I = 0 elseif Knock-Out Option Introduction Barrier Options set I = 1 and Monte Carlo Simulations for j = 0 to m − 1 do The Discretized Black-Scholes Model if j = 0 Simulation generate Z i ( t 1 ) The Algorithm calculate S i ( t 1 ) by (2) Example Example 1 if Down-and-In Option and S i ( t 1 ) ≤ B , Others or Up-and-In Option and S i ( t 1 ) ≥ B The End I = 1 elseif Down-and-Out Option and S i ( t 1 ) ≤ B , or Up-and-Out Option and S i ( t 1 ) ≥ B I = 0 end if

The Algorithm Monte Carlo Simulations Jessica else j � = 0 Radeschnig generate Z i ( t j +1 ) Introduction calculate S i ( t j +1 ) by (2) Barrier Options and Monte Carlo if Down-and-In Option and S i ( t 1 ) ≤ B , Simulations The Discretized Black-Scholes or Up-and-In Option and S i ( t 1 ) ≥ B Model I = 1 Simulation The Algorithm elseif Down-and-Out Option and S i ( t 1 ) ≤ B , Example or Up-and-Out Option and S i ( t 1 ) ≥ B Example 1 Others I = 0 The End end if end if end for

The Algorithm Monte Carlo Simulations Jessica Radeschnig Introduction if Call-Option Barrier Options and Monte Carlo set C i = e − rT max { S i ( T ) − K , 0 } × I Simulations The Discretized Black-Scholes elseif Put Option Model set P i = e − rT max { K − S i ( T ) , 0 } × I Simulation The Algorithm end if Example end for Example 1 Others set V by (1) The End

Example 1 Monte Carlo Simulations Jessica Radeschnig Introduction Barrier Options and Monte Carlo Simulations The Discretized Black-Scholes Model Coming up.... Simulation The Algorithm Example Example 1 Others The End

Conclusions Monte Carlo Simulations Jessica Radeschnig Introduction Barrier Options × Good for solving complicated stochastic differential and Monte Carlo Simulations The Discretized equations Black-Scholes Model × Different results each trial Simulation The Algorithm × Increased precision = ⇒ increased computational time Example Example 1 × When does benefits outweigh costs? Others The End

Monte Carlo Simulations Jessica Radeschnig Introduction Barrier Options and Monte Carlo Simulations The Discretized Black-Scholes Model Thank You! Simulation The Algorithm Example Example 1 Others The End