Interest rates in Japan are quite low and the Nikkei225 index is currently trading at 12,000. Say, the one year implied volatility of Nikkei is 15% (an assumption). What then is the price of a** two and half year, at the money** (ATM) call option on the Nikkei?

Is it possible to estimate the value of such a call option with **no other information available**? The answer, surprisingly, is yes. Even if you have no computer, no calculator, no model but just a pencil and a paper you can calculate the value of this simple long term vanilla call option. You wouldn't need Black-Scholes or Monte Carlo or any other model. All you would need is to do a simple multiplication.

The reason we want to write about this issue is that one of our members recently had a very interesting conversation with a derivatives sales guy in a bank in Tokyo. Apparently, a customer called up the equity derivatives sales desk asked a quote on an ATM, two and a half year call option on Nikkei futures (we are adapting the case for Nikkei spot). The sales professional at first thought that he had heard wrong so he asked the customer again: a two and a half year or a two and a half month call on Nikkei? The customer replied two and a half year call.

This was a bit unusual as nobody is interested in such a long dated naked vanilla option on an equity index. One just buys short dated options until six months or max one year maturity or does long dated equity swaps or structured notes which may or may not have long term optionality embedded. But to do a vanilla call option on an index for two and a half years? That is not common at all! But the sales guy nevertheless asked his trader to price the call.

The trader was also a bit amused and a tad nervous. The trader's key concern was: what is the volatility measure that should go to value the option? He had the best of models in front of him, the best of software that existed which would allow him to calculate the value of the option provided he had the correct estimate of volatility.

The trader started to wonder how to model the volatility? Since there are no liquid option contracts that stretch out for two and half years there is no way to get an implied volatility of a two and half year contracts that could be plugged into an option pricing model; There are Nikkei index LEAPS that trade in America but the liquidity isn't great and even using volatility cones most of his strategy colleagues don't go further than a year. Prices of Nikkei index LEAPs (long term options) could be used to infer the volatility but he thought perhaps a GARCH model is required to make a forecast of long term volatility first and maybe adjust it for the implied vol from the LEAPs. Further, if the implied vols were used how the skew could be taken into account for such a long dated option.

Finally, he realized that the single most important thing to estimate in this is that what would be the realized volatility two and half years from now. But since that is impossible to know, one has to make a forecast of the volatility based on the implied vols from the LEAPs, which existed (but were not very liquid contracts), the implied vols of the short dated options (up to one year) the skew and some form of GARCH modelling.

While toying with this pricing problem he also realized that though his banks regularly sells long dated structured notes tied to stocks and stock indices they make long term volatility forecasts to price the options embedded in them.

The trader finally priced the option, supposedly using a finite difference method or Monte Carlo simulation (we couldn't be sure) and came up with the answer in slightly less than two days time. He apparently conferred with his quant team in Europe and then together they made a volatility estimate and then priced the option. Two days could be an exaggeration made by the sales guy to us, but it is clear that the trader took some time to price this option.

Two days or even a day is inordinately long time given the fact that most vanilla options are priced within minutes of the request! But this apparently simple product had a very intractable problem - to forecast or rather estimate the two and a half year volatility of Nikkei index.

Meanwhile, the customer called up the sales guy again within minutes of his initial request (first phone call) and said that he got a price for the call option and wanted to know what the bank's offer was. The sales guy said that his trader is pricing the option and will get back to him in a moment. After another ten minutes the customer called the sales guy again and gave him the value of the call option and said if he could compare that with what his trader gave him.

The sales guy thought that perhaps the customer was giving him a quote from another bank. But the customer said that it was from their internal models and also that it had taken them less than a minute to arrive at that value, and that too without any model.

We believe here is what the customer of that bank must have done.

They took the one year implied volatility of the Nikkei225 index and scaled it using the square root of time to get an estimate for 2.5 year volatility.

vol_{annual}=15%(annualized)

Obviously, the customer thinks that this is a fair estimate for 2.5 year volatility (whether implied or forecast doesn't matter to him) and he inputs this in the standard option pricing formula. Thus using a one year implied volatility number he gets the estimate for 2.5 years volatility. Whether this is accurate or not is another matter but theoretically and mathematically the procedure is correct.

However, since the interest rates in Japan are quite low, he is probably making an assumption that they are very close to zero (zero in a mathematical sense) and since the option is at the money (ATM), whereby the spot equals the strike, he is in all likelihood using the simplified option pricing formula that results from using arithmetic Brownian motion assumption for the asset price.

It is a matter of simple observation that if time to maturity (time in the above equation) is equal to 30 months or in other words if time = 30/12 = 2.5 years then the value of the call is simply the volatility times the spot.

The above follows from the fact that we have used an annualized volatility of 15% for the Nikkei225, i.e. We can also write the above as:

In other words, a quote for the 30 month call (two and a half year call) could simply be 23.72%. In index point terms of course, the price of the call will be 1,138.56 (assuming a spot value for Nikkei225 as 12,000) and if each point is equivalent to 500 Yen then the value of this long term option will be JPY 569,280 per contract or US$4,950 per contract (at an FX rate of 115.00).

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