Testing Derivative Linearity and Convexity with Jensen’s Inequality
Summary
The document explains how to classify a derivative’s value as a function of its underlying asset price. For two underlying prices, the value at an intermediate weighted price can be compared with the same weighted average of the endpoint values. Equality indicates linear behavior over that interval; a value below the average indicates convexity, while a value above it indicates concavity.
The answers connect this inequality test to the usual definitions of convex and concave functions and to Jensen’s inequality. They also relate convexity and concavity, under suitable smoothness conditions, to the sign of the second derivative. The discussion is conceptual rather than a worked derivative-pricing example, and the calculus connection depends on regularity assumptions. It describes local behavior on an interval, so a derivative need not retain the same shape across its entire range of underlying prices.
Key ideas
- Linearity over an interval means the function’s value at a weighted intermediate point equals the weighted endpoint values.
- Convexity places the function’s intermediate value below or equal to the weighted endpoint average.
- Concavity reverses that inequality.
- Jensen’s inequality provides the general mathematical framework for these comparisons.
- For sufficiently regular functions, convexity and concavity correspond to the sign of the second derivative.
Tags
Full text
# Equations to Test of local linearity of a derivative security
# Equations to Test of local linearity of a derivative security
Friends any hint as to why is this set of equations a test of linearity of a derivative security?
```
From Taleb - Dynamic Hedging pg. 11
```
,, Derivatives are not always linear, convex, or concave across all moves (See Figures 1.2A-D, I did put a picture excerpt from the book down ). A test of local linearity of a derivative security (that is a function of the underlying asset) between asset prices $S_1$ and $S_2$ with $0<λ<1$, will satisfy the following equality: $$V(λS_1 + (1-λ)S_2) = λV(S_1) + (1 - λ)V(S_2)$$
It will be convex between $S_1$ and $S_2$ if: $$V(λS_1 + (1-λ)S_2) ≤ λV(S_1) + (1 - λ)V(S_2)$$ It will be concave if: $$V(λS_1 + (1-λ)S_2) ≥ λV(S_1) + (1 - λ)V(S_2)$$
,,
Any hint which mathematical theorem is behind this equation. $V$ i assume means Value of the derivative security. Is there somewhere to read to hone some insight about this for the unenlightened scholar?
## Answer by d_797 (score 1, accepted)
https://quant.stackexchange.com/a/60310
The definition of linear is just the usual definition, it would imply that $V(S) = aS$ for some constant $a$ (on an interval like $[S_1,S_2]$).
The definitions of convexity, concavity are more the "first-principles" definition and are equivalent to conditions related to positivity or negativity of $V''(S)$ that one sees in introductory calculus. (Again on $[S_1,S_2]$ under suitable regularity, see https://en.wikipedia.org/wiki/Convex_function#Properties).
## Answer by Pleb (score 2)
https://quant.stackexchange.com/a/60311
To extend @d_797's answer, then this stems from Jensens inequality (see this):
For a function $V: I \rightarrow \mathbb{R}$ for $I$ being an interval in $\mathbb{R}$, then V is convex if it satisfies:
$$ V(S_1 \lambda + (1-\lambda)S_2) \leq \lambda V(S_1)+(1-\lambda)V(S_2)$$
for any two points $S_1,S_2 \in \mathbb{R}$ and $\lambda\in [0,1]$. Now, if $V(\cdot)$ is convex then $-V(\cdot)$ is concave and therefore for any concave function $\bar{V}=-V(\cdot)$, Jensens inequality becomes (we change the direction of the inequality due to multiplying with $-1$):
$$ \bar{V}(S_1 \lambda + (1-\lambda)S_2) \geq \lambda \bar{V}(S_1)+(1-\lambda)\bar{V}(S_2)$$
Remember that a linear function is both convex and concave, thus giving you an equality in the above formulation.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.