Tag Archives: Black Scholes

The Binomial Model: An Alternative To Black Scholes

May 25, 2010   admin   No comments

Without a doubt the Black Scholes formula dominates the competition when it comes to options pricing models. However, the binomial model has a loyal following and is often used in conjunction with the Black Scholes formula by option traders when analyzing option prices. In this post we take a closer look at the binomial model from its inception to its current place in the options market.

The Inception of the Binomial Model

Introduced in 1979 by John Cox, Stephen Ross, and Mark Rubinstein, the binomial model uses a “discrete time” model of the varying underlying stock price in the valuation of options. Because the binomial model is based on the description of the underlying stock over a period of time (binomial tree) rather than a single point (Black Scholes formula) the computation of option prices become very complex and time consuming. This was a big hindrance back in the day but with the advancements in computational power and speed over the last few decades this is no longer a real concern. The binomial model for options pricing yields a more accurate value for an option’s worth than the Black Scholes formula especially for longer-dated options on dividend paying stocks.

The Binomial Tree

The binomial model follows the evolution of the option’s key underlying variables in discrete time. So an option’s  price is determined for each time increment (let’s use 1 day for simplicity sake) node. Option valuation using the binomial model is performed iteratively, starting at each of the final nodes (at expiration) and then working backwards until we reach the initial date. As you can imagine, the number of calculations needed to apply the binomial model can easily reach into the millions depending on the number of time increments. An example binomial tree is included below:

Binomial Model Assumptions

As in the case of the Black Scholes formula, there are assumptions underlying the binomial model that may or may not agree with practitioners in the options space. Two of the more prominent assumptions of the binomial model are:

  1. Discrete Time: The underlying stock is assumed to move in a discrete manner, which is more in line with reality (at least for now) given securities generally open and close every weekday and are typically closed on the weekend. The Black Scholes formula assumes a continuously traded market for the underlying security. One has to wonder how long it will be before worldwide exchanges work on a continuous time basis. Seems to be a likely evolution doesn’t it?
  2. Binomial Distribution: The binomial model assumes price movements follow a binomial distribution which when tested over many trials converges on the normal distribution assumed in the Black Scholes formula.

As we mentioned in the previous post “Why The Black Scholes Formula Dominates the Competition”, the Black Scholes formula maintains its dominance over other more accurate options pricing models like the binomial model primarily due to its simplicity. I’m sure the ultra advanced high-powered computers of 2010 have yielded the time difference in calculating the value of an option using the two methods virtually insignificant, but I guess there’s something to be said for good ‘ol comfort level.

For further information on the Binomial Model and other Options Trading Strategies visit OptionsUniversity.com.

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Why The Black Scholes Formula Dominates the Competition

May 25, 2010   admin   2 Comments

It goes without question that the most popular options pricing model used by scholars and option traders alike when analyzing the theoretical price of a call and put is the Black Scholes formula. But why does the Black Scholes formula dominate the competition in spite of its obvious flaws? One has to look back to the inception of the Black Scholes formula to really understand.

The Black Scholes Formula — The Early Years

The original Black Scholes formula was introduced to the world by Fischer Black and Myron Scholes in 1973. The two combined to publish a white paper for the Journal of Political Economy outlining an analytic model that would determine the fair market value for European type call options on non-payout assets. After an initial rejection, Black and Scholes sought feedback from professors Merton Miller and Eugene Fama who were able to help the duo improve the initial model and resubmit to the Journal of Political Economy for ultimate approval. The finance community’s acceptance of the model helped jump start options trading on the first options trading exchange, the Chicago Board Options Exchange (CBOE), in 1975. The original Black Scholes formula is highlighted below:

Black Scholes

Black Scholes

Key Drivers of the Black Scholes Formula

There are six major inputs needed to calculate the value of a call and put option according to the Black Scholes formula:

  1. Stock Price (P): The current price of the underlying security.
  2. Strike Price (K or s in the model above): The price at which the holder of an option can buy (in the case of a call option) or sell (in the case of a put option) the underlying security upon exercise.
  3. Time remaining until expiration (t): expressed as a % of a year.
  4. Interest rate (r): risk free interest rate expected over the option contract term.
  5. Dividend yield (d): not included in the original Black Scholes formula, but added later to account for its impact on stock price.
  6. Volatility (v): expected volatility (standard deviation) of stock price change out to expiration.

Of these six characteristics, volatility is the only true, price-moving, unknown value (interest rate and dividend yield typically have minimal impact over short time frames), thus when calculating the value of a call and put the expected volatility (implied volatility) calculation will be the key determinant in whether the option trader believes the option is underpriced or overpriced.

Assumptions Underlying the Black Scholes Formula

You know what they say about assumptions? Well the Black Scholes formula is full of them, leading some to question the validity of the formula in determining the true price of an option. Three of the bigger assumptions of the Black Scholes formula are:

  1. Constant Volatility: Probably the most significant of all the assumptions, the Black Scholes formula assumes the expected value of the stock’s movement until expiration is constant. The likelihood of this ever happening, volatility remaining constant over any time frame, is less than 0.0000000001%. (EPIC FAIL)
  2. Stock Returns are Normally Distributed: The Black Scholes formula assumes stock returns are normally distributed or in other words that the underlying stock prices themselves are lognormally distributed.  A lognormal distribution has a longer right tail compared with a normal distribution. The lognormal distribution allows for a stock price distribution of between zero and infinity (no negative prices) and has an upward bias (representing the fact that a stock price can only drop 100% but can rise by more than 100%). In practice underlying asset price distributions often depart significantly from the lognormal distribution.  (FAIL)
  3. European Exercise: Options may only be exercised on the day of expiration. The majority of options traded on American exchanges have American exercise terms, meaning options can be exercised at any time. (FAIL)

With so many known shortcomings why do most option traders still prefer to utilize the Black Scholes formula in calculating the price of a call and put option? One word: simplicity. Over the years alternative, more advanced models have been introduced in hopes of finding the ‘truer’ price of options. However, through all the attempts and machinations the values calculated rarely differ more than a few pennies from the output of the Black Scholes formula, thus market makers and option traders / investors alike continue to utilize the Black Scholes formula when pricing call and put options. Will the popularity of alternative option pricing models grow as the bid-ask spread of option prices continue to shrink? I guess time will tell.

For further information on the Black Scholes formula and other Options Trading Strategies visit OptionsUniversity.com.

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