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Valuing Options with Transaction Costs: Accounting for the Imperfection of the Reality

Kaaz Ninomiya
Management Science & Engineering
Stanford University
November 2002

The 1997 Nobel Prize in Economic Science was awarded to Robert Merton and Myron Scholes who along with the late Fisher Black invented a formula for pricing options, called the Black-Scholes formula, which revolutionized both the theory and practice of financial markets thereafter. But, just as everything else in life, this formula was not perfect. For one, it assumed that the world was a perfect place. My research is about finding a way to value options while accounting for the imperfections of the reality.

What are Options? In the simplest definition, options are the financial contract that gives the owner the right, but not the obligation, to buy or sell a specified asset for a specified price at a specified time in future. Options are also called 'derivatives' because their values always depend on the price of the specified asset that underlies the options contract. In other words, the value of options is derived from the price of the underlying assets - a Yahoo stock option from the price of Yahoo stock traded on NASDAQ, a lean hog option from the price of lean hog futures traded in Chicago Mercantile Exchange, An Euro/Dollar currency option from the Euro/Dollar forex rate, to take some examples of the options that exist in today's financial world.

Why are Options important? In today's financial world, the use of options is ubiquitous, and on any given day, the outstanding balance of all options around the globe is about $20trillion, which is roughly 10 times the size of the annual budget of the U.S. government. Investors and businesses worldwide depend on options for their financial risk management. For example, Dell Computer buys LCD screens, DRAM chips and many other components used in its computers from Asian manufacturers, and consequently it constantly faces foreign currency fluctuations. These currency fluctuations can easily hurt the company's profitability if remain unmanaged. To manage its currency exposure, Dell regularly uses currency options. For these investors and businesses who use options to manage risks, valuing and pricing options correctly is very important.

How does the Black-Scholes price Options? The Black-Scholes assumes a perfect world in which option's payoffs can be perfectly replicated by forming a portfolio of the underlying asset and a risk-free bond and by continuously adjusting it. Then you think, "Well, if this portfolio gives the same payoff as the option, people should be indifferent between the two. In fact, the option should be priced at exactly what it costs to form this portfolio, because otherwise people can make tons of money by either selling the option and buying the portfolio or vice-versa." The 'replication' and 'no-arbitrage' are the main principles that the Black-Scholes and almost every option pricing models are based upon.

Unfortunately, this elegant theory breaks down completely when there are transaction costs to trade underlying assets, and in reality, there always is some transaction cost - commissions and bid-ask spreads to name a few. I developed an alternate theory of option valuation called 'Optimal Hedge Pricing.' In finance, 'hedge' means entering into a transaction that reduces your existing investment risk. Optimal Hedge Pricing is based on the idea that the value of an investment is related to the cost of hedging it.

Under Optimal Hedge Pricing, the value of an option varies depending upon the transaction cost in the underlying asset as well as on the way an individual or a firm who is evaluating this option perceives the investment risk in general. Whereas the Black-Scholes predicts a unique value and market price for an option, Optimal Hedge Pricing specifies a unique value to a particular person. Given that there are different people with different values in the world, Optimal Hedge Pricing predicts a range of possible prices. However, if there is no transaction cost, the result of Optimal Hedge Pricing concurs with that of the Black-Scholes. In this way, the Black-Scholes can be thought of as a special case of Optimal Hedge Pricing.

When we in academia cannot find a solution to a problem, we often approximate the problem and solve it. This, in essence, is what has been done in the field of option pricing thus far. There is another approach, and that is to find the approximate solution to the original problem. This, in gist, is what my research is all about. Optimal Hedge Pricing provides a new and fuller perspective on the problem of option pricing to which no perfect solution exists.