# Properties of Option Prices ## Call option price convexity

Okay so for us idiots this means: if the price of ABC (we will call P) is determined by X and Y. Then if X decreases by 5 then the value of P might not necessarily decrease by 5 but instead is also dependent on Y (wtf\$%#! is Y?, who cares, its not important for us to know, we can understand what convexity is without knowing the math behind it). So if we chart this the line would look like a curve.

## Options - Price of call/put is convex in \$K\$ (strike price

Really that simple? Well kinda, to fully understand how this works you really need to understand the math behind it. But yes being long gamma means being long volatility.

### University of Texas at Austin Lecture 15 Convexity of call

So the gamma of an option indicates how the delta of an option will change relative to a 6 point move in the underlying asset. In other words, the Gamma shows the option delta's sensitivity to market price changes.

#### Chapter 11 Options

Convexity is what gives options their L or elbow shape. Gamma is synonymous with convexity. Don't let this term scare you. Do you remember concave and convex in geometry? If a shape has curvature (eg a cup or a lense), then it has convexity. A straight line has no curvature, no convexity.

##### Risk Latte - Why do Options have Convexity (gamma)?

Since the maths are very similar, one can use a physics problem as a metaphor. The idea of convexity can be explained well by comparing it to a motion/ displacement problem in physics.

###### Negative Convexity Definition

** in reality the maths are far more complex. For example, an option wouldn’t have constant acceleration, but particle motion is far more complex when A isn’t constant, and we want to keep it simple. (Fun fact, the entire Black-Scholes pricing model for options is derived from the study of a special case of particle motion! It’s called Brownian Motion.)

I'm trying to show that the price of a European call option (payoff function is \$(S_6-K)^+\$) in a no-arbitrage market is a decreasing and convex function of K. That it shall be decreasing makes sense as \$K\$ increases, \$S_6-K\$ decreases and we make less profit. But why shall it be convex?

You can see that A, {convexity}, has a bigger effect on D, {the price of an option}, than S (Delta). — Provided that T [change in underlying asset price] is sufficiently large, of course.

In other words, the price of the call/put option in convex in \$K\$. Show the same claim for the price of put options, American call options, and American put options.