La inclusión de la interdisciplina en las propuestas curriculares

It is assumed by the Black-Scholes-Merton model that the underlying contract price changes are normally distributed over small periods of time, leading to a lognormal distri- bution of underlying prices at expiration. However, many statistical studies have proven that this hypothesis is not necessarily correct. Moreover, they seem to show that extremity tails, representing huge variation in the underlying asset price are underestimated and seem to happen more often than predicted by the Black-Scholes-Merton model.

Even though the normal distribution tails are relatively low compared to the extreme events’ real probability in the financial markets, the normal distribution tails are sym- metric which might not reflect the reality as well. In fact, financial markets might have more extreme downward movements than extreme upward movement or vice versa. This hypothesis is not taken into consideration in the normal distribution as it predicts few big downward and upward movements in the underlying prices but mainly a huge amount of small variations of the underlying contract price overtime.

Finally, under the hypothesis of a normal distribution, the underlying price can have upward and downward moves of the same magnitude. This means that if a stock is cur- rently pricing $40 and increases by $60 to $100, the stock price can also decrease by -$60 and therefore pricing -$20. Obviously, stocks, commodities or futures and other financial instruments cannot take negative values and this proves that the assumption of a normal distribution in the Black-Scholes-Merton model is clearly flawed.

4.17

Conclusion

Even though it is possible to find theoretical pricing models that better applies to specific financial derivative instruments, like the Black Model (1976) to evaluate options on futures contracts, the Garman-Kohlhagen Model (1983) to evaluate options on foreign currencies or the model of Constant-Elasticity of Variance (CEV) which is based on the connection between price level and volatility of the underlying market, the Black-Scholes- Merton pricing formulas remain the most widely used of all options pricing models for its easy and efficient math computations.

Also, the Black-Scholes-Merton model, which assumes no early exercise of options, poorly suits most of the financial markets where most of the options are American. How- ever, in some markets and especially in the futures option markets,“the early exercise value is so small that there is virtually no difference between values obtained from the Black-Scholes-Merton model and values obtained from an American option pricing model” (Natenberg, 1994, p44) which require more inputs and estimations and therefore less chance to closely approximate real world conditions.

Chapter 5

The Greeks

When selling options in the over the counter market, financial institutions must face the problem of managing the options’ risks. If similar options are traded at the same time on an exchange, financial institutions can buy the same option on exchange that was sold by it in the over the counter market in order to neutralize the risks. But when the option is not available on an exchange due to particular characteristics of the derivative to sat- isfy a client in the over the counter market, like an unusual maturity date or strike price, it becomes more difficult to hedge the risk exposures for the financial institutions. The Greek letters measure different dimensions to the option position risk present and deriva- tive trader’s objective is to make the risk he is encountering acceptable by managing the Greeks.

One strategy is that the financial institutions do nothing. This strategy is called as a naked position and the financial institution which sold the options hope that they will finish out-of-the-money. At maturity of the options, the financial institution makes a profit equal to the premium of one option times the quantity w of options sold in the over the counter market. However, if the options are in-the-money at expiration, the long position exercises the options, by paying Kw to the financial institution, which in return, must deliver a certain quantity of the underlying stock. The financial institution will have to buy the securities in the underlying market at a higher price than the call option’s strike price ST > K, or at a price which is less as compared to the put option’s strike price

ST < K. This will generate a loss for the financial institution equivalent to w(ST − K) for

Financial institutions can use another strategy apart from the naked position that is to buy directly the underlying stocks when selling the call or put options. If the call or put options are in-the-money at expiration and therefore exercised by the long positions, this strategy, referred as a covered position works well. However, if call and put options are out-of-the-money, a covered position can generate a significant loss for the financial institution.

Thus, a good hedge is neither provided by a covered position nor a naked position and this is the reason why most traders and financial institutions use hedging procedures that are more improved involving steps such as gamma, delta, vega, rho and theta. These data, also called the Greeks, allow traders to be aware and anticipate risks that could potentially affect their positions.