The asset pricing kernel**,** also known as the stochastic discount factor (SDF), is the random variable that satisfies the function used in computing the price of an asset.

### Pricing Kernel and Asset Pricing

The pricing kernel, or stochastic discount factor, is an important concept in mathematical finance and financial economics. The term *kernel *is a common mathematical term used to represent an operator, whereas the term *stochastic discount factor* has roots in financial economics and extends the concept of the kernel to include adjustments for risk.

The fundamental theorem of asset pricing in finance suggests that the price of any asset is its discounted expected value of future payoff specifically under risk-neutral measure or valuation. Risk-neutral valuation can only exist if the market is free of arbitrage opportunities, or opportunities to exploit price differences between two markets and profit from the difference. This relationship between an asset's price and its expected payoff is considered the underlying concept behind all asset pricing. This expected payoff is discounted by a unique factor that depends upon the framework set forth by the market. In theory, risk-neutral valuation (in which there is an absence of arbitrage opportunities in the market) implies the existence of some positive random variable or the stochastic discount factor. In risk-neutral measure, this positive stochastic discount factor would theoretically be used to discount the payoff of any asset.

Additionally, the existence of such a pricing kernel or stochastic discount factor is equivalent to the law of one price, which presumes that an asset must sell for the same price in all locales or, in other words, an asset will have the same price when exchange rates are taken into consideration.

### Real-Life Applications of Pricing Kernels

Pricing kernels have numerous uses in mathematical finance and economics.

For instance, pricing kernels can be used to produce contingent claim prices. If we were to know the current prices of a set of securities in addition to the future payoffs of those securities, then a positive pricing kernel or stochastic discount factor would provide an efficient means of producing contingent claim prices assuming an arbitrage-free market. This valuation technique is particularly helpful in an incomplete market, or a market in which total supply is not sufficient to meet the demand.

### Other Applications of Stochastic Discount Factors

Apart from asset pricing, another use of stochastic discount factor is in the evaluation of the performance of hedge funds managers. In this application, however, the stochastic discount factor would not strictly be considered the equivalent to a pricing kernel.