mathematical finance Mathematical finance, also known as quantitative finance and financial mathematics, is a field of applied mathematics, concerned with mathematical modeling of financial markets. In general, there exist two separate branches of finance that require ...

, convexity refers to non-linearities in a

financial model Financial modeling is the task of building an abstract representation (a model) of a real world financial situation. This is a mathematical model designed to represent (a simplified version of) the performance of a financial asset or portfolio ...

. 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 In calculus, the second derivative, or the second order derivative, of a function is the derivative of the derivative of . Roughly speaking, the second derivative measures how the rate of change of a quantity is itself changing; for example, ...

(or, loosely speaking,

higher-order terms In mathematics and applied mathematics, perturbation theory comprises methods for finding an approximate solution to a problem, by starting from the exact solution of a related, simpler problem. A critical feature of the technique is a middle ...

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

Terminology

Strictly speaking, convexity refers to the second derivative of output price with respect to an input price. In derivative pricing, this is referred to as

Gamma Gamma (uppercase , lowercase ; ''gámma'') is the third letter of the Greek alphabet. In the system of Greek numerals it has a value of 3. In Ancient Greek, the letter gamma represented a voiced velar stop . In Modern Greek, this letter r ...

(Γ), one of the

Greeks The Greeks or Hellenes (; el, Έλληνες, ''Éllines'' ) are an ethnic group and nation indigenous to the Eastern Mediterranean and the Black Sea regions, namely Greece, Cyprus, Albania, Italy, Turkey, Egypt, and, to a lesser extent, ot ...

. In practice the most significant of these is

bond convexity In finance, bond convexity is a measure of the non-linear relationship of bond prices to changes in interest rates, the second derivative of the price of the bond with respect to interest rates ( duration is the first derivative). In general, th ...

, 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 referred to as a convexity correction.

Mathematics

Formally, the convexity adjustment arises from the Jensen inequality in probability theory: the expected value of a convex function is greater than or equal to the function of the expected value: :

E (X) \geq f(E .

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. 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).

Interpretation

The convexity can be used to interpret derivative pricing: mathematically, convexity is optionality – the price of an option (the value of optionality) corresponds to the convexity of the underlying payout. In Black–Scholes pricing of options, omitting interest rates and the first derivative, the Black–Scholes equation reduces to

\Theta = -\Gamma,

"(infinitesimally) the time value is the convexity". That is, the value of an option is due to the convexity of the ultimate payout: one has the ''option'' to buy an asset or not (in a call; for a put it is an option to sell), and the ultimate payout function (a

hockey stick A hockey stick is a piece of sports equipment used by the players in all the forms of hockey to move the ball or puck (as appropriate to the type of hockey) either to push, pull, hit, strike, flick, steer, launch or stop the ball/ puck during pla ...

shape) is convex – "optionality" corresponds to convexity in the 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 expected value of a convex function is higher than the function of the expected value (Jensen inequality). 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 In finance, a straddle strategy involves two transactions in options on the same underlying, with opposite positions. One holds long risk, the other short. As a result, it involves the purchase or sale of particular option derivatives that all ...

– purchasing an at-the-money straddle (whose value increases if the price of the underlying increases or decreases) has (initially) no delta: one is simply purchasing convexity (optionality), without taking a position on the underlying asset – one benefits from the ''degree'' of movement, not the ''direction''. 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 loses if prices move ''less'' than expected.

Convexity adjustments

From a modeling perspective, convexity adjustments arise every time the underlying financial variables modeled are not a martingale under the pricing measure. Applying

Girsanov's theorem In probability theory, the Girsanov theorem tells how stochastic processes change under changes in measure. The theorem is especially important in the theory of financial mathematics as it tells how to convert from the physical measure which des ...

allows expressing the dynamics of the modeled financial variables under the pricing measure and therefore estimating this convexity adjustment. Typical examples of convexity adjustments include: * Quanto options: the underlying is denominated in a currency different from the payment currency. If the discounted underlying is martingale under its domestic risk neutral measure, it is not any more under the payment currency risk neutral measure *

Constant maturity swap A constant maturity swap, also known as a CMS, is a swap that allows the purchaser to fix the duration of received flows on a swap. The floating leg of an interest rate swap typically resets against a published index. The floating leg of a consta ...

(CMS) instruments (swaps, caps/floors)P. Hagan (2003) Convexity Conundrums: Pricing CMS Swaps, Caps, and Floors, Wilmott Magazine
*

Option-adjusted spread Option-adjusted spread (OAS) is the yield spread which has to be added to a benchmark yield curve to discount a security's payments to match its market price, using a dynamic pricing model that accounts for embedded options. OAS is hence model ...

(OAS) analysis for mortgage-backed securities or other

callable bond A callable bond (also called redeemable bond) is a type of bond (debt security) that allows the issuer of the bond to retain the privilege of redeeming the bond at some point before the bond reaches its date of maturity. In other words, on the call ...

s * IBOR forward rate calculation from Eurodollar futures * IBOR forwards under

LIBOR market model The LIBOR market model, also known as the BGM Model (Brace Gatarek Musiela Model, in reference to the names of some of the inventors) is a financial model of interest rates. It is used for pricing interest rate derivatives, especially exotic deriv ...

(LMM)

References

* Benhamou, Eric, ''Global derivatives: products, theory and practices,'
pp. 111–120
5.4 Convexity Adjustment (esp. 5.4.1 Convexity correction) * {{Cite journal , last = Pelsser , first = Antoon , title = Mathematical Foundation of Convexity Correction , journal = , date = April 2001 , ssrn=267995 Mathematical finance Convex geometry