page_title: Financial economics
section_title: Derivative pricing
hierarchical_section_title: Financial economics / Extensions / Derivative pricing
context_page_description: Financial economics is a branch of economics which concerns trade in which some type of money appears on both sides of the transaction, as opposed to situations in which money is traded for a good or service.
  Its concern is thus the interrelation of financial variables, such as prices, interest rates and shares, as opposed to those concerning the real economy.
  It has two main areas of focus: asset pricing and corporate finance; the first being the perspective of providers of capital, i.e. investors, and the second of users of capital.
  It thus provides the theoretical underpinning for much of finance.
  The subject is concerned with "the allocation and deployment of economic resources, both spatially and across time, in an uncertain environment". It therefore centers on decision making under uncertainty in the context of the financial markets, and the resultant economic and financial models and principles, and is concerned with deriving testable or policy implications from acceptable assumptions. It is built on the foundations of microeconomics and decision theory.
context_section_description: As regards derivative pricing, the binomial options pricing model provides a discretized version of Black–Scholes, useful for the valuation of American styled options. Discretized models of this type are built—at least implicitly—using state-prices (as above); relatedly, a large number of researchers have used options to extract state-prices for a variety of other applications in financial economics. For path dependent derivatives, Monte Carlo methods for option pricing are employed; here the modelling is in continuous time, but similarly uses risk neutral expected value. Various other numeric techniques have also been developed. The theoretical framework too has been extended such that martingale pricing is now the standard approach.
  Drawing on these techniques, models for various other underlyings and applications have also been developed, all based on the same logic (using "contingent claim analysis"). Real options valuation allows that option holders can influence the option's underlying; models for employee stock option valuation explicitly assume non-rationality on the part of option holders; Credit derivatives allow that payment obligations or delivery requirements might not be honored. Exotic derivatives are now routinely valued. Multi-asset underlyers are handled via simulation or copula based analysis.
  Similarly, the various short rate models allow for an extension of these techniques to fixed income- and interest rate derivatives. (The  Vasicek and CIR models are equilibrium-based, while Ho–Lee and subsequent models are based on arbitrage-free pricing.) 
  The more general HJM Framework describes the dynamics of the full forward rate curve - as opposed to working with short rates - and is then more widely applied.
  The valuation of the underlying bonds -  additional to their derivatives - is relatedly extended, particularly for hybrid securities, where credit risk is combined with uncertainty re future rates; 
  see Bond valuation § Stochastic calculus approach and Lattice model (finance) § Hybrid securities.
  (Oldrich Vasicek developed his pioneering short rate model in 1977. 
  The HJM framework originates from the work of David Heath, Robert A. Jarrow, and Andrew Morton in 1987.)
  Following the Crash of 1987, equity options traded in American markets began to exhibit what is known as a "volatility smile";  that is, for a given expiration, options whose strike price differs substantially from the underlying asset's price command higher prices, and thus implied volatilities, than what is suggested by BSM. (The pattern differs across various markets.) Modelling the volatility smile is an active area of research, and developments here — as well as implications re the standard theory — are discussed in the next section.
  After the financial crisis of 2007–2008, a further development: (over the counter) derivative pricing had relied on the BSM risk neutral pricing framework, under the assumptions of funding at the risk free rate and the ability to perfectly replicate cashflows so as to fully hedge. This, in turn, is built on the assumption of a credit-risk-free environment — called into question during the crisis. 
  Addressing this, therefore, issues such as counterparty credit risk, funding costs and costs of capital are now additionally considered when pricing, and a Credit Valuation Adjustment, or CVA—and potentially other valuation adjustments, collectively xVA—is generally added to the risk-neutral derivative value.
  A related, and perhaps more fundamental change, is that discounting is now on the Overnight Index Swap (OIS) curve, as opposed to LIBOR as used previously. This is because post-crisis, the overnight rate  is considered a better proxy for the "risk-free rate". (Also, practically, the interest paid on cash collateral is usually the overnight rate; OIS discounting is then, sometimes, referred to as "CSA discounting".) Swap pricing - and, therefore, yield curve construction - is further modified: previously, swaps were valued off a single "self discounting" interest rate curve; whereas post crisis, to accommoda
caption: Stylized volatility smile: showing the (implied) volatility by strike-price, where the Black-Scholes formula returns market prices.