Question

Question about Financial risk management and derivative products
Explain why the binary model will allow the option price to converge to a specific value as the number of periods increases? What is that convergent option value? When n approaches infinity, what famous model converges to the binary model?

Answer 1

EXPLAIN WHY THE BINARY MODEL WILL ALLOW THE OPTION PRICE TO CONVERGE TO A SPECIFIC VALUE AS THE NUMBER OF PERIODS INCREASES? WHAT IS THAT CONVERGENT OPTION VALUE? WHEN N?

USING THE BINOMIAL PRICING MODEL, ,IT IS BY NO MEANS VITAL FOR A TRADER TO UNDERSTAND THE BINOMIAL PRICING MODEL AND USE IT FOR TRADING DECISIONS. IT DOES HAVE ITS USES, AND IT CAN BE BENEFICIAL FOR FORECASTING THEORETICAL VALUES OF OPTIONS BASED ON HOW THE UNDERLYING SECURITY MOVES IN PRICE AND THE AMOUNT OF TIME THAT PASSES. HOWEVER, IT'S NOT SOMETHING THAT IS ABSOLUTELY ESSENTIAL AND IT'S PERFECTLY POSSIBLE TO BE A SUCCESSFUL OPTIONS TRADER WITHOUT USING IT.

FOR THOSE TRADERS THAT PREFER TO USE A PRICING MODEL, THE BIGGEST ADVANTAGE OF THE BINOMIAL MODEL IS THAT IT'S FAR MORE ACCURATE IN CALCULATING THEORETICAL VALUES FOR AMERICAN STYLE OPTIONS AND TAKING EARLY EXERCISE INTO ACCOUNT. IT'S ALSO MORE FLEXIBLE FOR CALCULATING HOW THE THEORETICAL VALUES WILL CHANGE BASED ON DIFFERENT VARIABLES.

CONVERGENT TRADING STRATEGIES LOOK FOR PRICES TO REVERT BACK TO A MEAN. THIS CAN MEAN THAT TWO ASSETS THAT ARE EXTENDED AWAY FROM THEIR HISTORICAL CORRELATION WILL EVENTUALLY REGAIN THEIR PAST RELATIONSHIP. CONVERGENT CAN ALSO DESCRIBE A TRADING STYLE OF SELLING SHORT AN OVERBOUGHT ASSET OR BUYING AN OVERSOLD ASSET BELIEVING THAT THE PRICE IS TOO HIGH OR TOO LOW AND WILL REVERT BACK TO AN AVERAGE PRICE DURING THE TIME OF THE TRADE.

WHEN APPROACHES INFINITY , THE FAMOUS MODEL CONVERGES TO THE BINARY MODEL IS WELL KNOWN THAT THE BINOMIAL MODEL CONVERGES TO THE BLACK-SCHOLES MODEL WHEN THE NUMBER OF TIME PERIODS INCREASES TO INFINITY AND THE LENGTH OF EACH TIME PERIOD IS INFINITESIMALLY SHORT. THIS PROOF WAS PROVIDED IN COX, ROSS AND RUBINSTEIN (1979). THEIR PROOF, HOWEVER, IS UNNECESSARILY LONG AND RELIES ON A SPECIFIC CASE OF THE CENTRAL LIMIT THEOREM. THIS IS SEEN ON PP. 250 AND 252 OF THEIR ARTICLE, WHICH REFERS TO THE FACT THAT SKEWNESS BECOMES ZERO IN THE LIMIT. ALSO, THEIR RESULTS ARE DERIVED ONLY FOR THE SPECIAL CASE WHERE THE UP AND DOWN FACTORS ARE GIVEN BY SPECIFIC FORMULAS THEY OBTAIN THAT ALLOW THE DISTRIBUTION OF THE STOCK RETURN TO HAVE THE SAME PARAMETERS AS THE DESIRED LOGNORMAL DISTRIBUTION IN THE LIMIT. EFFECTIVELY, THEIR DISTRIBUTION CONVERGES TO THE LOGNORMAL IN THE LIMIT. THEY GO ON TO SHOW THAT THE BINOMIAL MODEL THEN CONVERGES TO THE BLACKSCHOLES MODEL UNDER THEIR ASSUMPTIONS.1 THE COX-ROSS-RUBINSTEIN PROOF IS ELEGANT BUT FAR TOO SPECIFIC. A MORE GENERAL PROOF OF THE CONVERGENCE OF THE BINOMIAL TO THE BLACK-SCHOLES MODEL IS PROVIDED BY HSIA (1983). HIS PAPER, PUBLISHED IN THE JOURNAL OF FINANCIAL RESEARCH, GOT VIRTUALLY NO ATTENTION; YET, IT IS CLEARLY THE BEST OVERALL PROOF. IT IMPOSES NO RESTRICTIONS ON THE CHOICE OF UP AND DOWN PARAMETERS. MOREOVER, THE PROOF IS MUCH SHORTER, EASIER TO FOLLOW, AND REQUIRES FEW CASES OF TAKING LIMITS.