Very robust. You can drop quite a few of the assumptions underpinning Black-Scholes and it won't fall over.

Example

Transaction costs? Simply adjust volatility. Time-dependent volatility? Use root-mean-square-average volatility instead. Interest rate derivatives? Black '76 explains how to use the Black-Scholes formula in situations where it wasn't originally intended.

Long answer

Here are some assumptions that seems crucial to the whole Black-Scholes model, and what happens when you drop those assumptions.

Hedging is continuous

If you hedge discretely it turns out that Black-Scholes is right on average. In other words sometimes you lose because of discrete hedging, sometimes you win, but on average you break even. And Black-Scholes still applies.

There are no transaction costs

If there is a cost associated with buying and selling the underlying for hedging this can be modelled by a new term in the Black-Scholes equation that depends on gamma. And that term is usually quite small. If you rehedge at fixed time intervals then the correction is proportional to the absolute value of the gamma, and can be interpreted as a simple correction to volatility in the standard Black-Scholes formula. So instead of pricing with a volatility of 20%, say, you might use 17% and 23% to represent the bid-offer spread dues to transaction costs. This little trick only works if the contract has a gamma that is everywhere

the same sign, i.e. always and everywhere positive or always and everywhere negative.

Volatility is constant

If volatility is time dependent then the Black-Scholes formula are still valid as long as you plug in the 'average' volatility over the remaining life of the option. Here average means the root-mean-square average since volatilities can't be added but variances can. Even if volatility is stochastic we can still use basic Black-Scholes formula provided the volatility process is independent of, and uncorrelated with, the stock price. Just plug the average variance over the option's lifetime, conditional upon its current value, into the formula.

There are no arbitrage opportunities

Even if there are arbitrage opportunities because implied volatility is different from actual volatility you can still use the Black-Scholes formula to tell you how much profit you can expect to make, and use the delta formula to tell you how to hedge. Moreover, if there is an arbitrage opportunity and you don't hedge properly, it probably won't have that much impact on the profit you expect to make.

The underlying is lognormally distributed

The Black-Scholes model is often used for interest-rate products which are clearly not lognormal. But this approximation is often quite good, and has the advantage of being easy to understand. This is the model commonly referred to as Black '76.

There are no costs associated with borrowing stock for going short

Easily accommodated within a Black-Scholes model, all you need to do is make an adjustment to the risk-neutral drift rate, rather like when you have a dividend.

Returns are normally distributed

Thanks to near-continuous hedging and the Central Limit Theorem all you really need is for the returns distribution to have a finite variance, the precise shape of that distribution, its skew and kurtosis, don't much matter.

Black-Scholes is a remarkably robust model.

References and Further Reading

Wilmott, P 2006 Paul Wilmott on Quantitative Finance, second edition. John Wiley & Sons Ltd

Why is the Lognormal Distribution Important?

Short answer

The lognormal distribution is often used as a model for the distribution of equity or commodity prices, exchange rates and indices. The normal distribution is often used to model returns.

Example

The stochastic differential equation commonly used to represent stocks,

results in a lognormal distribution for S, provided x and a are not dependent on stock price.

Long answer

A quantity is lognormally distributed if its logarithm is normally distributed, that is the definition of lognormal. The probability density function is

where the parameters a and b > 0 represent location and scale. The distribution is skewed to the right, extending to infinity and bounded below by zero. (The left limit can be shifted to give an extra parameter, and it can be reflected in the vertical axis so as to extend to minus infinity instead.)

If we have the stochastic differential equation above then the probability density function Figure 2.15 for S in terms of time and the parameters is

where So is the value of S at time t = 0.

Figure 2.15: The probability density function for the lognormal random walk evolving through time.

You would expect equity prices to follow a random walk around an exponentially growing average. So take the logarithm of the stock price and you might expect that to be normal about some mean. That is the non-mathematical explanation for the appearance of the lognormal distribution.

More mathematically we could argue for lognormality via the Central Limit Theorem. Using Ri to represent the random return on a stock price from day i — 1today i we have

the stock price grows by the return from day zero, its starting value, to day 1. After the second day we also have

After n days we have

the stock price is the initial value multiplied by n factors, the factors being one plus the random returns. Taking logarithms of this we get

the logarithm of a product being the sum of the logarithms.

Now think Central Limit Theorem. If each Ri is random, then so is ln(1 + Ri). So the expression for ln(5n) is just the sum of a large number of random numbers. As long as the Ri are independent and identically distributed and the mean and standard deviation of ln(1 + Ri) are finite then we can apply the CLT and conclude that ln(5n) must be normally distributed. Thus Sn is normally distributed. Since here n is number of 'days' (or any fixed time period) the mean of ln(Sn) is going to be linear in n, i.e. will grow linearly with time, and the standard deviation will be proportional to the square root of n, i.e. will grow like the square root of time.

References and Further Reading

Wilmott, P 2006 Paul Wilmott on Quantitative Finance, second edition. John Wiley & Sons Ltd

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