## Barrier Option

Pernah dengar barrier option? Agak beda dengan option yang konvensional. Valuasinya juga tentunya beda dari sekedar menggunakan formula Black-Scholes. Berikut artikel mengenai produk ini.

A Probability Conundrum: First Passage time of a Brownian Motion
Rahul Bhattacharya
Jan 25, 2005

Recently we encountered a bit of a problem with the use of first passage time in our forecasting models. The problem appeared more fundamental in terms of the mathematical formulation. I asked a trader who trades currency options in a large bank in Hong Kong as to what is the probability that an asset, say a currency or a stock, which is currently trading at \$100 with a historical volatility of 15% and a mean historical return of 10% will hit a barrier at \$95. It took him ten seconds to plug it in his option pricing calculator the values and he gave me an answer of 57.3% (assuming the implied vol for 3 months is 15%).

I am sure this is correct given the fact that he had calculated the probability of hitting the barrier by using the pricing model of a one touch digital option. However, when we use a different formula for the probability of the first passage time, as the problem is known, in asset forecasting models we get a different result for the probability of the asset hitting the barrier.

First passage time is a central concept in the analysis of Brownian motions with absorbing barriers. Consider a Brownian particle undergoing a one-dimensional random walk within a domain between x=0 and x=L, the boundaries being such that the particle will be reflected at x=0 and absorbed at x=L. If the particle starts from some position within the domain when will it hit the absorbing boundary for the first time? This first-passage problem in stochastic processes, in various spatial dimensions and with a variety of geometries and boundary conditions, is of longstanding interest and has ramifications and applications in diverse areas.

In simple English, First Passage Time refers to the probability of a random particle hitting and being absorbed by a barrier or a boundary. This barrier or a boundary could be the barrier of an asset price as in a barrier option, or a predetermined band such as a government determined peg rate of a currency.

Mathematically, the probability that an asset following a geometric Brownian motion will hit a barrier is given by: Let us assume that an asset, for example a stock or a stock index, follows a geometric Brownian motion. The stock is currently at \$100, the annualized volatility of the stock is 15% and the mean historical return of the stock is 10%. Then the probability that in three months time the stock will hit a barrier at \$95 will be given by the above formula and is simply equal to 38.46%.

This could explain the difference – fairly large – between our result and the result that my friend got from his digital option pricing formula. Later on he used Monte Carlo simulation and yet he came up with a probability number much greater than 40%.

We believe that the formula mentioned above gives the correct probability of a stock hitting a barrier provided it follows a geometric Brownian motion with a drift equal to the mean return. This could be viewed as the real probability of hitting the barrier as opposed to a risk neutral probability measure.