Actuarial Outpost ASM PE 9 Question 2 - error?
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 Investment / Financial Markets Old Exam MFE Forum

#1
07-31-2012, 05:13 PM
 Math Member Join Date: Jan 2011 Posts: 296
ASM PE 9 Question 2 - error?

- no dividends
- S0 = 50
- alpha = .13
- sigma = .25
- r=.04

A knockout barrier call option has strike 50 and barrier 60. It expires in 6 months. Value is determined using simlation within 3 month periods. Barrier can ONLY be crosse at the end of a period (i.e. it has two chances to cross the barrier, at 3 months and 6 months time)

Use the following pairs of numbers to perform three trials and get the average value of the price of the option.

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The solution of the manual uses the RISK FREE rate (r) to get mu and calculate the prices at 3 months and 6 months, in order to determine whether the call has knocked out or not.

My concern:

Shouldn't you use actual return (alpha) to see whether the option has knocked out? Pretending it is risk neutral is only used in risk neutral pricing, but you cannot tell whether it has knocked out using risk neutral scenario. You should only use true scenarios to see if the stock has actually knocked out. Then afterwards, go back to risk neutral to actually price the option.

What is wrong with what I am saying above? Any help is great. Thank you.
#2
07-31-2012, 05:30 PM
 Abraham Weishaus Member SOA AAA Join Date: Oct 2001 Posts: 7,253

How is this different from a simulation question pricing a standard option? There too, the real probabilities of payoff and the amounts of payoff depend on the true rate of return of the stock.

Using risk-neutral probabilities and risk-free rates is equivalent to using true probabilities and true rates of return, and saves you the difficulty of determining the true rate of return of an option.
#3
07-31-2012, 06:29 PM
 Math Member Join Date: Jan 2011 Posts: 296

Thank you Dr. Weishaus for the response.

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In determining the value of an option, yes using risk neutral is fine, but I'm talking about whether the barrier of the call option is broken and not its value. The value of the option is not being considered, only whether the stock price ever reaches a certain value. The theory of risk neutral pricing cannot be applied, since there's nothing actually being priced here, only whether a barrier is being broken, which is equivalent to saying whether the stock ever reaches a certain number.

---

A similar example involves binomial trees.

Say you get So, p, p*, u, d, etc and the question asks for the probability that a call option pays off in the end. You would get your final prices at the end of the tree, see which ones make your option pay, and calculate the probability using p (true p), and not p*.

But if you want to price the option, you will go ahead with p* and r instead of p.

I think the same would apply here. If you want to see whether your option is knocked out or not, it depends on the true price of the stock, and not in the risk neutral world. Hence you need alpha instead of r.

Now once you determined that your option is not knocked out, then you revert back to r and price accordingly. Is there anything wrong with this understanding Dr. Weishaus?
#4
07-31-2012, 06:56 PM
 Abraham Weishaus Member SOA AAA Join Date: Oct 2001 Posts: 7,253

My response stands. With a standard option, risk-neutral pricing may result in the option not paying off when true pricing results in it paying off. It makes no difference. Risk-neutral pricing always works.

If you want true probabilities, use true pricing. If you want to price an option, use risk-neutral pricing.
#5
07-31-2012, 10:33 PM
 Math Member Join Date: Jan 2011 Posts: 296

Thanks Dr. Weishaus

In the example above, if you use alpha instead of r, the payoff knocks out in every case (Stock is something like 61.88 at the end for all 3 trials)

Wouldn't the value of the option be 0, regardless of the true return of the option?
#6
08-01-2012, 02:20 PM
 Abraham Weishaus Member SOA AAA Join Date: Oct 2001 Posts: 7,253

Quote:
 Originally Posted by Math Thanks Dr. Weishaus In the example above, if you use alpha instead of r, the payoff knocks out in every case (Stock is something like 61.88 at the end for all 3 trials) Wouldn't the value of the option be 0, regardless of the true return of the option?
You should run the simulation with a couple thousand trials. Then discount with the true rate of return on the option (I have no idea how you'll get that). If you can do that, you'll see the result is pretty close to the result using risk-neutral valuation.

If you just do 3 trials, different methods will lead to different results, most of them far from the real value.