Quanto Basket Min Lookback Asian.Matlab Application.

1. QuantoBasket Min LookbackAsian.Matlab Application. Lecturer :Jan Röman Students:DariaNovoderejkina,AradTahmidi,DmytroSheludchenko

2. Description of a Quanto Basket Min Lookback Asian Option • 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. Description of a Quanto Basket Min Lookback Asian Option • 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. Description of a Quanto Basket Min Lookback Asian Option • In ourcase 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. Description of a Quanto Basket Min Lookback Asian Option • In Mathematical terms:

6. Description of a Quanto Basket Min Lookback Asian Option • 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. Financial Background • From Black-Scholes world weknow : • 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. Financial Background • When solving the differential equation mentioned above and use Girsanov theorem, we end up with the following result:

9. Monte Carlo Simulations • 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. Monte Carlo Simulations • 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. Monte Carlo Simulations • 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. MatlabApplication • Weused all the information providedabove to make a Matlabapplication. • Weconsider a portfolio consisting of three assets with itsownusual parameters as well as itsweight in the portfolio. • Usercanalso set number of simulations and lookback dates. • Userdeterminesalsolength of initial and average period for an option