# 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? ## 2 Answers

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 – ExoticBirdsMerchant Jan 3 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. – ExoticBirdsMerchant Jan 3 at 21:14