Equations to Test of local linearity of a derivative security [closed]

Question

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 1

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 2

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.