They say that Taiwanese are gamblers. We say they are calculated risk takers. Here is an idea that originated while one of us was having lunch with one of the Taiwanese bankers, who happens to be the Head of Equity Derivatives of a bank in Taiwan . The structure that the banker proposed to us was a bit complicated as we shall see below, but first let us illustrate the basics using a simpler structure (he actually wanted us to do a valuation model so that he can validate his own internal results which were done by quants elsewhere).

The Nikkei225 index has been hammered in recent days (along with all other Asian stock indices). Today it stands at 14,780. An investor is bullish on the Nikkei (Japanese equities in general) and because of the corrected levels he thinks it is a good buy in the medium term (6 months) at this level. So what can he do? He can buy a European style 6 month at the money (ATM) call option on the Nikkei225.

But, he also thinks that the upside may be limited and the Nikkei may not scale great heights from here on. He believes that the Nikkei will go up from here but again within the next six months there will be another correction. So he wants to profit from the limited upside and get out quickly as soon as his call make a decent profit. So what can he do? He can frequently trade the European style ATM call that he has bought (the liquidity in Nikkei is pretty good), but doing so will make his transactions costs quite high and defeat his purpose of investing in the call option. Or, he can buy an American style ATM call option on the Nikkei225 index. But given the high levels of implied volatility he fears the cost of buying an American style 6 month ATM call will be too high, and rightly so, his bankers tell him that. He can then mull over his options and choose one.

Or, he can buy a 6-month ATM Call Shout option on the Nikkei225. What the hell is a "shout" option? The investor can, at one time during the life of the option, shout to the seller of the option. What will take achieve? What this act of shouting achieves is that at maturity of the option (after 6 months) the option holder will get *either * the usual payoff from the European style 6 month ATM call *or * the intrinsic value at the time of the shout, *whichever is greater *.

Let us say that the strike price of the call is 14,780. The investor enters into a 6-month ATM call shout option and the holder shouts after 3 months (i.e. the shout time period is t = 3 months) when the Nikkei225 is at 16,000 (this is the "shout price"). Then if the final price of Nikkei225 after 6 months (maturity of the option at t = 6 months) is less than 16,000 then the investor will get the difference between the "shout price" and the "strike price", i.e. 16,000 and 14,780 points, i.e. 1,220 points (or Yen 61,000 per contract). But if he is pleasantly surprised by a rising Nikkei and after 6 months the Nikkei225 is higher than the "shout price" then the investor receives the excess of the spot price at maturity over the strike price, in other words, the payoff from a vanilla European style call option. The payoff of a call shout option is given by:

The above can be very easily valued using a Monte Carlo simulation algorithm. The point is that the shout feature is itself a European style feature, i.e. the investor (the holder of the call shout option) can shout only once during the life of the option, that is at the end of 3 months

What the banker was proposing to us was that he wants to design a call shout option on an equity index (of course, not Nikkei225) whereby the holder of the option, the investor, can shout any time during the life of the option. This gives great flexibility to the investor and makes the option a very attractive product. But looking closely at this structure one realizes that it is very similar in nature (almost same from a valuation point of view) to an American style option. In fact, most quants, value such options exactly like an American style option. A binomial or a trinomial tree of the underlying is created and as we roll back through the tree, at each node we check the value of the call if the holder shouts and the value if he does not shout. The option's price at each node is the greater of the two. This procedure is exactly like valuing an American option.

**Questions: **

- A shout option shares a lot of properties with another kind of exotic option. Which one is it?

- Between the two which would be cheaper and why?

Send you answers, as well as comments, to team@risklatte.com and the first three winners will get a free copy of Shout option pricing model using MC simulation. (Both answers should be correct and full name, email address, location and job details should be included)

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