I have a knock-out option with barrier $L>0$ and strike $K$ that pays at maturity $(S-K)_+$. So, positive payoff occurs only in case the price stays below the barrier over life of the option.

I am analyzing estimation of the price of this option using Monte Carlo simulation and came across with the below definition of the payoff:

$$\Pi_0=e^{-rT} E^Q [S_T * S_{T/2}]= e^{-rT} \frac{ \Sigma \ S_T(\omega_i) S_{T/2}(\omega_i)}{N}$$

Can this be correct? If yes, could you please explain the logic behind the multiplication $S_T S_{T/2}$?

  • $\begingroup$ Where did you find that formula? Please cite your reference and provide a link if possible $\endgroup$ – SRKX May 19 '16 at 1:19
  • $\begingroup$ sorry but I don't have any specific reference for this , because this formula comes from my notes, but I couldn't understand the idea here, I am not sure if this is correct and if yes then why the payoff is presented this way, I know it looks odd, that's why I wanted to verify this, not sure maybe I am missing some point here $\endgroup$ – Michal May 19 '16 at 1:38
  • $\begingroup$ This formula looks plain wrong IMHO. Just the fact that there's no dependency on the barrier level $L$ (nor the strike $K$) is extremely peculiar. What if $S_0 > L$ for an up-and-out-call as you describe ? We know that the option premium should be zero (it is immediately knocked out). I don't see that in your formula. I wonder where you got that from though. $\endgroup$ – Quantuple May 19 '16 at 8:10
  • $\begingroup$ Yeah I mean I agree with @Quantuple that's why I was asking for some context because this seems so off-topic there must be an explanation. $\endgroup$ – SRKX May 19 '16 at 9:28
  • $\begingroup$ Does it resemble you any other optionlike payoff? $\endgroup$ – Michal May 19 '16 at 10:11

Your valuation is NOT for the knock-out option that you have specified. Let \begin{align*} \tau = \inf\{t \mid 0 \le t \le T, S_t \ge L\}. \end{align*} Here, we set the infimum of an empty set to $\infty$. Then, the payoff of the knock-out option is of the form \begin{align*} (S_T-K)^+ 1_{\tau = \infty}. \end{align*} Under the Black-Scholes setting, this option can be valued analytically.

Monte Carlo simulation is not always good for such type of payoff. For example, you may need to have a particularly fine time slices, and set the payoff to $0$ on each path once the simulated asset value is at or above the barrier level $L$; otherwise, simulate until the maturity $T$, and set the payoff to $(S_T(\omega)-K)^+$, where we use $\omega$ to denote a path.


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