The Jensen's inequality in layman’s terms…
In finance, convexity plays a pivotal role in areas like fixed income (where bond price movements with respect to interest rate changes are governed by convexity) and options pricing (where "gamma" represents the convexity of an option's price with respect to changes in the underlying asset).
At its core, for a function to be considered convex, its second-order derivative must be positive. This property ensures that as one variable changes, the change in another variable is non-linear and can offer advantages such as enhanced returns in finance.
Jensen's inequality is a powerful demonstration of convexity.
Jensen's inequality is a mathematical concept that relates to convex functions. It states that for a convex function, the expected value of the function applied to a random variable is greater than or equal to the function applied to the expected value of that random variable.
Let (X) be a random variable and (f) be a convex function. Then, Jensen's inequality states that:
E[f(X)] ≥ f(E[X])
In simpler terms, Jensen's inequality implies that if we have a convex function and a random variable, the expected value of the function applied to the random variable will be at least as large as the function applied to the expected value of the random variable.
This principle illuminates how averages behave differently for linear versus convex functions, with profound ramifications in financial contexts.
Imagine a European call option on stock XYZ with a strike price (K) of $100. Based on market analyses, a trader predicts that at expiration, there’s a 50% likelihood the stock will close at either $120 (S1) or $80 (S2).
Let’s consider the payoff function defined by:
f(x)= MAX(X-K,0) with X= stock price and K= strike price
1.Expected Option Payoff: E[f(X)]
• Payoff if stock ends at S1 = max(0, $120 - $100) = $20
• Payoff if stock ends at S2 = max(0, $80 - $100) = $0
• Expected Payoff = 0.5 x $20 + 0.5 x $0 = $10
2.Option Payoff at Expected Spot Price: f(E[X])
• Expected Spot Price = ($120 + $80) / 2 = $100
• Payoff at Expected Spot Price = max(0, $100 - $100) = $0
With the call option payoff’s convex nature, Jensen’s inequality indicates that the Expected Option Payoff ($10) will consistently be greater than or equal to the Option Payoff at Expected Spot Price ($0).
Even when the average expected spot price aligns with the strike price, convexity ensures a positive expected payoff across potential outcomes, emphasizing convexity’s pivotal role in shaping financial strategies.
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