Convexity of call options

Convexity of call options

Posted: Ewgen On: 24.06.2017
Boundary Conditions on Options

In mathematical finance , convexity refers to non-linearities in a financial model. In other words, if the price of an underlying variable changes, the price of an output does not change linearly, but depends on the second derivative or, loosely speaking, higher-order terms of the modeling function.

Geometrically, the model is no longer flat but curved, and the degree of curvature is called the convexity.

Strictly speaking, convexity refers to the second derivative of output price with respect to an input price. In practice the most significant of these is bond convexity , the second derivative of bond price with respect to interest rates. As the second derivative is the first non-linear term, and thus often the most significant, "convexity" is also used loosely to refer to non-linearities generally, including higher-order terms.

Refining a model to account for non-linearities is called "correcting for convexity" or adding a convexity correction. Formally, the convexity adjustment arises from the Jensen inequality in probability theory: Geometrically, if the model price curves up on both sides of the present value the payoff function is convex up, and is above a tangent line at that point , then if the price of the underlying changes, the price of the output is greater than is modeled using only the first derivative.

convexity of call options

Conversely, if the model price curves down the convexity is negative, the payoff function is below the tangent line , the price of the output is lower than is modeled using only the first derivative. The precise convexity adjustment depends on the model of future price movements of the underlying the probability distribution and on the model of the price, though it is linear in the convexity second derivative of the price function.

The convexity can be used to interpret derivative pricing: That is, the value of an option is due to the convexity of the ultimate payout: Thus, if one purchases a call option, the expected value of the option is higher than simply taking the expected future value of the underlying and inputting it into the option payout function: The price of the option — the value of the optionality — thus reflects the convexity of the payoff function. This value is isolated via a straddle — purchasing an at-the-money straddle whose value increases if the price of the underlying increases or decreases has initially no delta: From the point of view of risk management, being long convexity having positive Gamma and hence ignoring interest rates and Delta negative Theta means that one benefits from volatility positive Gamma , but loses money over time negative Theta — one net profits if prices move more than expected, and net lose if prices move less than expected.

convexity of call options

From a modeling perspective, convexity adjustments arise every time the underlying financial variables modeled are not martingale under the pricing measure. Applying Girsanov theorem [1] allows expressing the dynamics of the modeled financial variables under the pricing measure and therefore estimating this convexity adjustment.

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

Typical examples of convexity adjustments include:. From Wikipedia, the free encyclopedia. Hagan Convexity Conundrums: Pricing CMS Swaps, Caps, and Floors, Wilmott Magazine.

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