Question

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 assumption. Use your words to explain why that is the case, and how risk-neutral assumption greatly simplifies the calculations of risk-neutral option pricing approach.

Answer 1

Significantly, the expected rate of return of the stock (which would incorporate risk preferences of investors as an equity risk premium) is not one of the variables in the Black-Scholes model (or any other model for option valuation). The important implication is that the value of an option is completely independent of the expected growth of the underlying asset (and is therefore risk neutral).

Thus, while any two investors may strongly disagree on the rate of return they expect on a stock they will, given agreement to the assumptions of volatility and the risk free rate, always agree on the fair value of the option on that underlying asset.

The fact that a prediction of the future price of the underlying asset is not necessary to value an option may appear to be counter intuitive, but it can easily be shown to be correct. Dynamically hedging a call using underlying asset prices generated from Monte Carlo simulation is a particularly convincing way of demonstrating this. Irrespective of the assumptions regarding stock price growth built into the Monte Carlo simulation the cost of hedging a call (ie dynamically maintaining a delta neutral position by buying & selling the underlying asset) will always be the same, and will be very close to the Black-Scholes value.

Putting it another way, whether the stock price rises or falls after, eg, writing a call, it will always cost the same (providing volatility remains constant) to dynamically hedge the call and this cost, when discounted back to present value at the risk free rate, is very close to the Black-Scholes value.

Which is hardly surprising given that the Black-Scholes price is nothing more than the amount an option writer would require as compensation for writing a call and completely hedging the risk. The important point is that the hedger's view about future stock prices is irrelevant.

This key concept underlying the valuation of all derivatives -- that fact that the price of an option is independent of the risk preferences of investors is called risk-neutral valuation. It means that all derivatives can be valued by assuming that the return from their underlying assets is the risk free rate.

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.  

At the end of the tree -- ie at expiration of the option -- all the terminal option prices for each of the final possible stock prices are known as they simply equal their intrinsic values.

Next the option prices at each step of the tree are calculated working back from expiration to the present. The option prices at each step are used to derive the option prices at the next step of the tree using risk neutral valuation based on the probabilities of the stock prices moving up or down, the risk free rate and the time interval of each step. Any adjustments to stock prices (at an ex-dividend date) or option prices (as a result of early exercise of American options) are worked into the calculations at the required point in time.

Relationship between 2 models :

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 . This lets you examine graphically how convergence changes as the number of steps in the binomial calculation increases as well as the impact on convergence of changes to the strike price, stock price, time to expiration, volatility and risk free interest rate.