1. Lecturer :Jan Röman Students:Daria Novoderejkina,Arad Tahmidi,Dmytro Sheludchenko

2.  Asian options are options that are based on an average value over a certain time period  Mostly written on commodities and currencies  Monte Carlo method is a powerful tool for pricing such options

3.  Quanto means that investor has no currency risks  Min Lookback options are giving their holder a right to buy the underlying asset for its lowest value recorded during the options lifetime  Basket options are a type of Multi-asset options with the underlying security represented by not one, but several distinct assets with specified weights

4.  In our case the strike price is determined by the minimum value of the underlying asset over an initial time period.  Payoff is determined as a difference between average price on predetermined time period and the strike.

5.  In Mathematical terms:

6.  Where A(T) is the Asian price  K – minimum price over predetermined initial period  B - price of the basket at time t  V - corresponding weight of the underlying asset i in the basket, represents the price of the underlying asset i  N is the number of the reset dates  M is the amount of the lookback dates  d is the number of the underlying assets.

7.  From Black-Scholes world we know :  The first equation describes risk-free asset price (B(t)) dynamic. The second equation represents the risky asset price (S(t)) movement and is a stochastic differential equation.

8.  When solving the differential equation mentioned above and use Girsanov theorem, we end up with the following result:

9.  Here we will give a short description of how Monte-Carlo simulations work and how they can be used to price complex instruments.  The easiest approach would be to start with a plain European call option in the Black-Scholes world.  Risk-free interest rate is continuously compounded and price of the underlying is governed by the stochastic equation described before.

10.  If we consider natural logarithm of the stock price: x(t)=ln(S(t))  We will find that it can be described by dynamics below:

11.  In other words:  Z increment in the equation above is distributed with zero mean and ∆t variance. Considering this, we are able to simulate the random process with ∆t*ℰ and a normally distributed sigma. We obtain:

12.  We used all the information provided above to make a Matlab application.  We consider a portfolio consisting of three assets with its own usual parameters as well as its weight in the portfolio.  User can also set number of simulations and lookback dates.  User determines also length of initial and average period for an option