# Barrier option on a basket with arbitrary stochastic process

Suppose I want to price a Down-and-out European call, barrier option. However, the stochastic process is not a gBm or any other Levy process with known structure. Practically, I want a barrier option on a "tailor made" portfolio. How could I price a barrier option on this portfolio, given that individual processes are gBm (for stocks), a Poisson-point process for loans and mean-reverting for exchange rates. I can simulate individually each trajectory via simulation and sum $$\forall t$$ the market value of each asset class. However, my intuition is that raw monte-carlo, even if high computational cost is not a problem, would lead to biased estimates for the option value. The barrier refers to the aggregate portfolio value and not to a certain class (to boil-down to a rainbow option).

• European or American? Knock in or knock out? – Bob Jansen Jun 23 at 7:48
• Why do you think you would get biased estimates from Monte Carlo? – Daneel Olivaw Jun 26 at 22:51
• Since one of the components follows a Levy process, hitting the barrier could lead to the "overshoot" problem. – alexbougias Jun 26 at 22:53

There are a few issues that need to be separated here.

Issue "zero" is whether your MC is able to correctly represent the dynamics you've chosen for your assets. If you implement your MC properly, by construction it should converge in distribution to the postulated dynamics. No bias there. Variance yes potentially, because of discretisation, but no systematic bias.

Issue one is whether your postulated asset dynamics make sense for the problem at hand. If you're pricing barrier options then at a minimum you should make sure your asset dynamics include skew, and perhaps stochastic volatility. Then you will need to come up with a reasonable codependence structure between the assets (likely a copula in this case, perhaps something else).

Issue two, once the dynamics are sorted, is whether your MC engine is able to correctly price your payoff in a theoretical sense. Now, if your barrier is purely European, then you only need distribution at maturity and you should be all set (remember you will still have variance but should have no systematic bias). If your barrier is continuous, then you're essentially screwed (from a theoretical perspective) because no matter what you do, you will only be able to represent discrete versions of your basket process -possibly augmented with Brownian bridge or other relevant technical device- but not an exact continuous process, only some approximation.

Issue three is accounting for practical trading and hedging considerations. In reality the dynamics you've chosen and the assumptions underpinning the model will never be "reality". Asset prices jump around, liquidity is not infinite, markets do not open continuously and so on. There is definitely bias there, but it's not MC specific. In barrier options, payoff discontinuities around the barrier make hedging tricky and prices reflect dealers' "adjustments" for that.

Actual option prices reflect these possible sources of bias, and no matter how sophisticated your model is, if you don't calibrate it to actual prices of something "relevant to" the actual instrument you are trading i.e. that you will use to hedge and that is demonstrably effective as a hedge, you may end up in trouble. This is why calibration is very important: because it tells you what to add or subtract from your model price to account for bias. You need to calibrate your model (ideally using your MC engine rather than another model implementing your dynamics, say PDE) to market observables relevant to the task at hand. For example vanilla option prices of various maturities and strikes on each asset, ideally other barrier prices if available (probably not).

If you don't have anything to calibrate to, no plausible hedge instruments, then the best tool at your disposal is that old friend the fat margin.