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

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?

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).

• Thank you now i'll go and read!!! Thank you Commented Jan 3, 2021 at 20:54

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.

• aha i have read of Jensens inequality in Talebs book. Thank you for the link and explanation! I am currently reading Convex function in wiki and i m gonna check Jensens inequality. Commented Jan 3, 2021 at 21:14