Using numbers to explain Convexity and why derivatives have value - II
April 27, 2011
Continuing on from the previous article, now letís talk about Jensenís inequality. Jensenís inequality is perhaps the most famous theorem in quantitative finance (note that it is a "theorem" and not a model or a formula) and it is the reason why financial derivatives have value. Concept of convexity, Jensenís inequality, randomness and volatility of an asset price are intricately linked.
In our opinion, study of financial derivatives should start with an understanding and the explanation of convex functions and the Jensenís inequality. If we can get an intuition behind this concept then the rest of the discipline is simply complex and boring math.
Paul Wilmott is perhaps the only person that we know of who has actually talked about this seminal concept in his textbook and has used a mathematical derivation (using Taylor series expansion) to show by how much the left hand side of the above equation is greater than the right hand side. And therein lies the fundamental intuition behind the concept of gamma (convexity) and randomness (coupled with volatility / variance of the asset).
Anyway, we would try to understand Jensenís inequality by using numbers.
Jensenís inequality states that if is a random variable and is a function of that random variable then the following identity holds:
That is, the expected value of the function, is always greater than or equal to the function of the expected value of the variable, .
Like the previous example, letís take two functions, and which are given by and respectively. Here, we should note that is a random variable (drawn from a certain probability distribution). For example, can be the price of S&P500 stock index. As again, is the payoff of a call option, say on SP500, with a constant strike price . The function would also give the payoff of a financial derivative which is given by the square of , or in this case, the square of the final price of SP500 stock index. Both functions are convex.
In the above context, we should write the Jensenís inequality as:
The subscript, denotes the final price of the variable (SP500); for example, could be equal to one year if the maturity of the derivative is one year.
We will now take a very simple and highly stylized example, using numbers, to prove the Jensenís inequality. Say, SP500 is currently trading at 100 (normalized value). Now, to keep things extremely simple letís say that SP500 index can only take 6 (final) values at the end of one year. The final price of SP500, can be any one of these values: 100, 110, 120, 130, 140 and 150.
The first derivative, given by the function, , is the square of the final SP500 index values. The expected value of the function is given by:
The function of the expected value is given by:
Therefore, from the above we see that .
Now consider the second function, , which is a one year call option on SP500 index. Say, the strike price, the constant, in the function is the same as before, i.e. 102. The expected value of this function is given by:
The function of the expected value would be given by:
Therefore, once again we see that . Hence we observe that given these values of (all of which are assumed to be random) both the functions, and satisfy the Jensenís inequality.
One can take any values and a large enough set of numerical values for the Jensenís inequality for both the above functions will hold. We would urge our readers to experiment with any random set (no matter how large is the set) of numerical values for to test the Jensenís inequality.
Since, the expected value of the function, , is always greater than the function of the expected value, a call option on SP500 (or a call option on any other asset that is modeled as a random walk) will have an inherent, fundamental value at inception (time, t = 0). And that fundamental value is given by the (discounted) expected value of the function .
In our simple and stylized example above we assumed only six possible values for the variable (SP500). In reality, there will a large Ė infinite Ė number of values for all drawn from a certain probability distribution (because is random). Thus, in real life, the fair value of a call option on SP500 stock index at inception (t = 0) would be given by:
Where, is the probability measure under which the expectation, (or, in laymanís terms, the averaging) is being carried out. In fact, as Wilmott shows
The "higher order terms" capture the gamma (convexity) of the option and the variance of the randomness. All financial derivatives, whether they are given by functions, , or any other convex functions (of a random variable) share the above property. Convexity terms (higher order terms) are added to the expected value (mean) of the function to give the financial derivative a fundamental (inherent) fair value at inception (t = 0).
Reference: For an excellent explanation of Jensenís inequality see Paul Wilmottís excellent textbook on Quantitative Finance
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