Question

In: Finance

Black Scholes and Binomial Trees are widely used by industry players. However, what makes these models...

Black Scholes and Binomial Trees are widely used by industry players. However, what makes these models far from perfect?

2-3 paragraphs please

Solutions

Expert Solution

Both models are based on the same theoretical foundations and assumptions (such as the geometric Brownian motion theory of stock price behaviour and risk-neutral valuation).  

The Black-Scholes model is used to calculate a theoretical call price (ignoring dividends paid during the life of the option) using the five key determinants of an option's price: stock price, strike price, volatility, time to expiration, and short-term (risk free) interest rate.

Advantages & Limitations

Advantage: The main advantage of the Black-Scholes model is speed -- it lets you calculate a very large number of option prices in a very short time.

Limitation: The Black-Scholes model has one major limitation: it cannot be used to accurately price options with an American-style exercise as it only calculates the option price at one point in time -- at expiration. It does not consider the steps along the way where there could be the possibility of early exercise of an American option.

The binomial model breaks down the time to expiration into potentially a very large number of time intervals, or steps. A tree of stock prices is initially produced working forward from the present to expiration. At each step it is assumed that the stock price will move up or down by an amount calculated using volatility and time to expiration. This produces a binomial distribution, or recombining tree, of underlying stock prices. The tree represents all the possible paths that the stock price could take during the life of the option.  

Advantages & Limitations

Advantage: The big advantage the binomial model has over the Black-Scholes model is that it can be used to accurately price American options.   This is because with the binomial model it's possible to check at every point in an option's life (ie at every step of the binomial tree) for the possibility of early exercise (eg where, due to eg a dividend, or a put being deeply in the money the option price at that point is less than its intrinsic value).

Limitation: The main limitation of the binomial model is its relatively slow speed. It's great for half a dozen calculations at a time but even with today's fastest PCs it's not a practical solution for the calculation of thousands of prices in a few seconds.

Relationship to the Black-Scholes model

The same underlying assumptions regarding stock prices underpin both the binomial and Black-Scholes models: that stock prices follow a stochastic process described by geometric brownian motion. As a result, for European options, the binomial model converges on the Black-Scholes formula as the number of binomial calculation steps increases. In fact the Black-Scholes model for European options is really a special case of the binomial model where the number of binomial steps is infinite. In other words, the binomial model provides discrete approximations to the continuous process underlying the Black-Scholes model.

Whilst the Cox, Ross & Rubinstein binomial model and the Black-Scholes model ultimately converge as the number of time steps gets infinitely large and the length of each step gets infinitesimally small this convergence, except for at-the-money options, is anything but smooth or uniform.  

The Binomial options pricing model approach has been widely used since it is able to handle a variety of conditions for which other models cannot easily be applied. This is largely because the BOPM is based on the description of an underlying instrument over a period of time rather than a single point. As a consequence, it is used to value American options that are exercisable at any time in a given interval as well as Bermudan options that are exercisable at specific instances of time. Being relatively simple, the model is readily implementable in computer software (including a spreadsheet).

Although computationally slower than the Black–Scholes formula, it is more accurate, particularly for longer-dated options on securities with dividend payments. For these reasons, various versions of the binomial model are widely used by practitioners in the options markets


Related Solutions

Black Scholes and Binomial Trees are widely used by industry players. However, what makes these models...
Black Scholes and Binomial Trees are widely used by industry players. However, what makes these models far from perfect?
The Black-Scholes Model and the Binomial Model are based on similar assumptions; however, there are some...
The Black-Scholes Model and the Binomial Model are based on similar assumptions; however, there are some important differences between the two models. Use a specific example to illustrate a difference between the two models. How does the concept of “no arbitrage” affect each model?
]Using either a Black Scholes or Binomial Tree option calculator on the internet, what is the...
]Using either a Black Scholes or Binomial Tree option calculator on the internet, what is the value of a 6 month put on the money (current price) Tesla Stock if we use volatility of .50, no dividend, and a risk free rate of 1%?
Please discuss the Black & Scholes model and the binomial model approach to option pricing. What...
Please discuss the Black & Scholes model and the binomial model approach to option pricing. What are the advantages and disadvantages of these two approaches? Determine the price of a call and put option assuming that the exercise price is $105, the value of the stock is $101, risk-free rate is 2.05%, standard deviation of returns on the stock is 28%, and the option has 6 months remaining to maturity. What is the price sensitivity of the call and put...
Post by Day 3 a 3- to 6-paragraph comparison of the Black-Scholes and Binomial Tree option...
Post by Day 3 a 3- to 6-paragraph comparison of the Black-Scholes and Binomial Tree option pricing models. Make sure to include responses to the following specific questions: What are the distinct advantages and disadvantages of each model? List 2–3 advantages and disadvantages for each. Who might use one model over the other more often? (Suggest specific users, such as banks, speculators, hedge fund managers, etc.). Why do you think that the selected preferred users of each model make such...
Post by Day 3 a 3- to 6-paragraph comparison of the Black-Scholes and Binomial Tree option...
Post by Day 3 a 3- to 6-paragraph comparison of the Black-Scholes and Binomial Tree option pricing models. Make sure to include responses to the following specific questions: (1) What are the distinct advantages and disadvantages of each model? List 2–3 advantages and disadvantages for each. (2) Who might use one model over the other more often? (Suggest specific users, such as banks, speculators, hedge fund managers, etc.). (3) Why do you think that the selected preferred users of each...
When using binomial approach and Black-Scholes formula for pricing options, do you expect the results to...
When using binomial approach and Black-Scholes formula for pricing options, do you expect the results to be the same? (3)Why or why not? (2)  Price a put and a call with data offered below using both methods and show the prices. Do your results support initial expectations? (5) Present stock price $30, exercise price $40, interest rate 5%, option expires one year from now, volatility 27%, stock will either move up by 40% or down by 27%.
Can Black-Scholes formula be used in pricing executive stock options? Explain
Can Black-Scholes formula be used in pricing executive stock options? Explain
Why Binomial option price model and Black-Scholes model give different results? Which one is better to...
Why Binomial option price model and Black-Scholes model give different results? Which one is better to use for the option valuation and why?
7. Black-Scholes model shares common intuitions with risk-neutral option pricing model (also known as the binomial...
7. Black-Scholes model shares common intuitions with risk-neutral option pricing model (also known as the binomial option pricing model). One of the biggest underlying assumptions of risk-neutral (binomial) model is that we live in a risk-neutral world. In a risk-neutral world, all investors only demand risk-free return on all assets. Although the risk-neutral assumption is counterfactual, it is brilliant and desirable because the prices of an option estimated by risk-neutral approach are exactly the same with or without the risk-neutral...
ADVERTISEMENT
ADVERTISEMENT
ADVERTISEMENT