Risk Latte - Index Call with FX Barrier

Index Call with FX Barrier

Team Latte
Mar 14, 2005

This structure is prompted by our recent discussion with a buy side client who had a bullish view on the Japanese equity market but feels that the Japanese Yen can get much stronger than the present level. The client feels that if the Dollar-Yen got too strong then that will in general depress the level of Nikkei and more importantly, will wipe out most of the gains from the rise of the index. Keeping this view in mind he wanted to buy a structure that is long on Nikkei index but wanted it cheap and at the same time wanted an automatic closure of the position if the Japanese Yen became too strong.

We came up with this structure: the client buys an at the money (ATM) vanilla call on Nikkei 225 index with a knock-out barrier (KO) at USD/JPY level of 100.

The Nikkei is trading at 11,850 and the spot USD/JPY is 104.63. This structure is a knock-out option with the barrier being place at 100. That is if the USD/JPY gets strong and touches the level of 100 any time during the life of the option - in this case one year - then the option is knocked out, which means that the buyer of the option - our client - walks away from the position with nothing.

We had a bit of a difficulty naming this structure - option - in the common barrier option parlance. Is it a down-and-out or up-and-out? This was because the underlying is on the equity index but the barrier is on the FX rate.

Our client came back to us in a few days and told us that his prime broker or bank could not offer (sell) him this option as they said this structure did not exist in their pricing library. This was not part of the "standard" exotic structure and therefore not done in Asia. The prime broker suggested to the client that he buy a quanto option. But the client was adamant that he wanted a knock-out option which knocks out on a different underlying.

Apparently, there is also a problem of naming this structure within the common parlance of barrier option. Is it a "down-and-out" option or an "up-and-out option" or something else? This is because the knock out barrier is on an underlying that is different from the underlying on which the option payoff is calculated.

We then asked a few other traders in the banks in Hong Kong and Tokyo and they too said that they haven't seen this kind of a structure and their quants have not priced a structure such as this.

We found this very baffling. To us this seems a very logical structure - albeit a funky one - if one has a predicament such as our client. Also, the pricing of this structure seemed very easy using Monte Carlo simulation.

Actually, we have found out from our research that these kind of options were proposed by Bankers Trust long ago and what we need to define is an inside barrier and an outside barrier. For example, Bankers Trust proposed to investors in October 1993 a call on a basket of Belgian stocks where the knock-out level corresponds to an appreciation by more than 3.5% of the Belgian Franc.

The analytic pricing model is given by Heynen and Kat (1994a) and it is a bit daunting and complicated using integral equations.

We have priced this structure using Monte Carlo simulation and find is rather simple to do once the problem is well defined.

We present here the pricing mechanism for this structure using MC simulation:

Pricing of a one year ATM Call on Nikkei at 11,850 with Knock-out (KO) at USD/JPY level of 100:

You can use Visual Basic or C/C++ and run the following algorithm

  • Using historical analysis estimate the correlation of Nikkei225 and USD/JPY (or else use implied correlation);

  • Using historical vol, EWMA vol, GARCH vol or implied vol and the above correlation calculate the variance-covariance matrix;

  • Then do the cholesky decomposition on the VCV matrix;

  • Generate two sets of 252 random normal numbers and then do the Cholesky transformation on these random numbers;

  • Generate the price path of Nikkei225 and USD-JPY using the correlated random numbers;

  • Finally, calculate the value of the call using the knock-out level as the qualifier;

  • Discount the above back to get the PV;

  • Run the above procedure for 20,000 times and get an average value; this is the price of the option.

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