Question

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?

Answer 1

Black scholes and Binomial model:

Binomial determines future payoffs of a bond or stock using trees. Based on future cash flows and interest rates based on certain probability we derive the price of bond/option or other instrument using backward induction methodology where the payoffs of future are discounted to the present. Then the payoffs or interest rates are calculated in the future using size of movement or probability. And then these payoffs are discounted to present to get the value.

Black scholes model is used to value the European option. It is derived from delta hedged portfolio earning risk free yield. This is based on arbitrage principle. There are few assumptions like stock price which can vary from zero to infinity, constant volatility, constant risk free rate and no dividends etc.

For example, there may be 50/50 chance that the underlying asset price can increase or decrease by 30% in one period. For the second period however the probability that underlying asset price may increase can grow to 70/30. If we consider an oil well, there is a 50/50 chance that the price can go up. If the prices go up in period 1, making the oil well more valuable and the market fundamentals now point to continued increases in oil prices. The probability of further appreciation in price may be now 70%. The binomial model allows this flexibility whereas the black scholes model doesn’t.

No arbitrage on Binomial:

There are several assumptions used in deriving the black scholes equation. It is assumed that principle of no arbitrage is assumed to be satisfied in deriving the black scholes equation.

Let us denote p(B) the market price of contingent claim contract. Under the condition, we have,

P (B) = E^Q(Be^-t)

Hence in the context of binomial model, no arbitrage indicates a unique valuation rule. This is the no arbitrage price of contingent claim B.