Local Volatility - Development

Development

The concept of a local volatility was developed when Bruno Dupire and Emanuel Derman and Iraj Kani noted that there is a unique diffusion process consistent with the risk neutral densities derived from the market prices of European options.

Derman and Kani described and implemented a local volatility function to model instantaneous volatility. They used this function at each node in a binomial options pricing model. The tree successfully produced option valuations consistent with all market prices across strikes and expirations. The Derman-Kani model was thus formulated with discrete time and stock-price steps. The key continuous-time equations used in local volatility models were developed by Bruno Dupire in 1994. Dupire's equation states

\frac{\partial C}{\partial T} = \frac{1}{2} \sigma^2(K,T; S_0)K^2 \frac{\partial^2C}{\partial K^2}-(r - q)K \frac{\partial C}{\partial K} - qC

Read more about this topic:  Local Volatility

Famous quotes containing the word development:

    “Such condition of suspended judgment indeed, in its more genial development and under felicitous culture, is but the expectation, the receptivity, of the faithful scholar, determined not to foreclose what is still a question—the “philosophic temper,” in short, for which a survival of query will be still the salt of truth, even in the most absolutely ascertained knowledge.”
    —Walter Pater (1839–1894)

    “On fields all drenched with blood he made his record in war, abstained from lawless violence when left on the plantation, and received his freedom in peace with moderation. But he holds in this Republic the position of an alien race among a people impatient of a rival. And in the eyes of some it seems that no valor redeems him, no social advancement nor individual development wipes off the ban which clings to him.”
    —Frances Ellen Watkins Harper (1825–1911)

    “The proper aim of education is to promote significant learning. Significant learning entails development. Development means successively asking broader and deeper questions of the relationship between oneself and the world. This is as true for first graders as graduate students, for fledging artists as graying accountants.”
    —Laurent A. Daloz (20th century)