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+Chapter 34: Futures and Futures Options 679 
+This will not work well near expiration, since the future expires one week prior to the 
+PHLX option. In addition, it ignores the early exercise value of the PHLX options. 
+However, except for these small differentials, the shortcut will give theoretical values 
+that can be used in strategy-making decisions. 
+Example: It is sometime in April and one desires to calculate the theoretical values 
+of the June deutsche mark physical delivery options in Philadelphia. Assume that one 
+knows four of the basic items necessary for input to the Black-Scholes formula: 60 
+days to expiration, strike price of 68, interest rate of 10%, and volatility of 18%. But 
+what should be used as the price of the underlying deutsche mark? Merely use the 
+price of the June deutsche mark futures contract in Chicago. 
+STRATEGIES REGARDING TRADING LIMITS 
+The fact that trading limits exist in most futures contracts could be detrimental to 
+both option buyers and option writers. At other times, however, the trading limit may 
+present a unique opportunity. The following section focuses on who might benefit 
+from trading limits in futures and who would not.. 
+Recall that a trading limit in a futures contract limits the absolute number of 
+points that the contract can trade up or down from the previous close. Thus, if the 
+trading limit in T-bonds is 3 points and they closed last night at 7 421132, then the high­
+est they can trade on the next day is 7721132, regardless of what might be happening 
+in the cash bond market. Trading limits exist in many futures contracts in order to 
+help ensure that the market cannot be manipulated by someone forcing the price to 
+move tremendously in one direction or the other. Another reason for having trading 
+limits is ostensibly to allow only a fixed move, approximately equal to the amount cov­
+ered by the initial margin, so that maintenance margin can be collected if need be. 
+However, limits have been applied in case~which they are unnecessary. For exam­
+ple, in T-bonds, there is too much liquidity for anyone to be able to manipulate the 
+market. Moreover, it is relatively easy to arbitrage the T-bond futures contract against 
+cash bonds. This also increases liquidity and would keep the future from trading at a 
+price substantially different from its theoretical value. 
+Sometimes the markets actually need to move far quickly and cannot because of 
+the trading limit. Perhaps cash bonds have rallied 4 points, when the limit is 3 points. 
+This makes no difference when a futures contract has risen as high as it can go for 
+the day, it is bid there (a situation called "limit bid") and usually doesn't trade again 
+as long as the underlying commodity moves higher. It is, of course, possible for a 
+future to be limit bid, only to find that later in the day, the underlying commodity 
+becomes weaker, and traders begin to sell the future, driving it down off the limit.