1. Barrier option simulation with Monte-Carlo Senem Kaya & Martin Pettersson Computational methods in financial mathematics 2014-10-02

2. Pre-set barrier either springs the option into existence or extinguishes an already existing option.  Depends on type of option  Always cheaper than vanilla option, less premium  Applied to different type of options. American, European, Bermudan etc.

3. Up and Out European Call Option  Becomes worthless if price of underlying asset increases beyond barrier

4. Standard Monte-Carlo Trajectories: 100 R=0.1 Sigma=0.5 Time-steps=0.005 K=15 S 0 =12

5. Visualization of barrier and value of underlying asset

6. New Monte-Carlo  Takes into consideration that the value could pass the barrier between two time steps  Probability of passing the barrier:

7. Barrier far from stock price Trajectories: 10000 B=50 R=0.1 Sigma=0.1:0.1:1 T=0.5 K=15 S 0 =12

8. Barrier close to stock price Trajectories: 10000 B=17 R=0.1 Sigma=0.1:0.1:1 T=0.5 K=15 S 0 =12

9. Convergence comparison

10. If barrier far away, the payoff looks like European call B=17, 50, 200 R=0.1 Sigma=0.3 T=0.5 K=15 S 0 =12

11. Delta - change in the price of the underlying asset to the corresponding change in the price of the derivative B=17, 50, 200 R=0.1 Sigma=0.3 T=0.5 K=15 S 0 =12

12. Conclusions  Easy to implement  Beneficial to use new Monte-Carlo, less error  New Monte-Carlo is more computationally heavy  Further improvements: Study time efficiency and making it faster