An intuition behind Black–Scholes
In this article, we’ll talk about the most famous formula in finance. Perhaps some readers looked at the derivation of the Black-Scholes formula once, saw phrases like ‘continuous delta hedging’ and decided that all options are priced according to some dark magic. I want to provide a very intuitive derivation of the Black-Scholes formula without second-order differential equations and hardcore mathematics. In fact, I’d like to simply calculate an expected value of an option, and link it to the Monte-Carlo simulation of an Options price. In other words, I will not discuss the ‘P-measure’ and ‘Q-measure’ stories and just assume we operate in a ‘Q-measure’ space.
Note: In this article, I will consider the European call option. This is a type of contract that gives a right, but not an obligation to exchange security (S) at a given price (K) on a given date(T). We will need the following parameters:
This way, the payoff of the option is simply
For simplicity let’s talk about stock options (it’s more compact to write ‘stock price’ than ‘value of an underlying asset’).
What is the Monte-Carlo simulation for options?
Monte-Carlo simulations are numerical methods to study stochastic processes and the idea behind this options pricing is to create a lot of different paths for the random walk of stock price, calculate the value of an option for each path and take an average.
And now, we’ll make use of the Law of large numbers, which simply says an average of random variables converges to their mean.
Hence, Monte-Carlo simulations in fact try to price options as their expected value.
Assumptions of our model
We deal with random stock price and of course, we need to know what kind of random is it. As we want to end up with the Black-Scholes formula, we need the same assumption of Geometric Brownian motion:
This is the stochastic differential equation that describes a movement of the stock price as a random variable. It looks quite complicated in this form, but in fact, all it says is that log-returns of stock price are normally distributed at every point in time.
We also require an additional condition that says we don’t have particular knowledge about the value of the stock price at time T, hence the present value of future stock price equals the current value.
If we combine all these conditions in one (it’s not that hard, see appendix for derivation), we get
We also implicitly use a concept of risk-neutral probability measure: it allows us to set the price of an option equal to its expected payoff.
A bit of Math
Now all we need to do is to solve
Note that the negative exponent comes from the fact we calculate the present value of future cash flow
We already talked about the distribution of stock price, now let’s state it explicitly
Finally, we can calculate the expectation
Now, I claim that the last line is in fact Black-Sholes formula.
Wait, you’ll say: the Black-Scholes formula is
And these formulas are nothing alike! Well, they are. It can be shown analytically with some calculus, but we’ll do it numerically.
Numerical ‘proof’
To show that we have the same thing, let’s calculate this integral and original formula a lot of times and see whether the values are close to each other in Python.
We’ll need a method to calculate the Black-Scholes formula
And a method to perform our integration
Finally, let’s run both functions 100 times with random parameters and see if there is any set of parameters where the difference between the two functions exceeds 0.00001
We can make the comparison more intuitive by plotting values against each other
Indeed, we have verified that our formula shows results very similar to the Black-Scholes formula.
Concluding thoughts
We went through the intuitive proof of the Black-Sholes formula, and along the way showed that the limit of Monte-Carlo simulations also yields to the most famous formula in finance.
It’s interesting to compare the execution time of the BMS formula and our formula. BMS takes about 3 times less than our formula, however, our formula takes about 100 times less than the Monte-Carlo simulation with 30000 draws.
Our approach is flexible and it allows for interesting generalisations, for example, we can derive options pricing formulas for different distributions of returns. If we want to see how more fat-tailed returns affect the price of the contract we can just set, for example,
And go through similar calculations. This approach is applied and not rigorous mathematically as it is not easy to justify a different distribution than normal if we consider a Brownian motion-like process.
Appendix: From Geometric Brownian motion to pdf
We know two conditions mentioned above
and
We aim to solve these two equations for mu, and we can use a nice trick to do so
This way we can get the answer in two lines without using Itô calculus:)