Facts About Delta - dV/dX (almost no math!)

Delta Generally

1) Pure Delta is defined as the change in the value of an asset, typically a financial derivative (Future, Swap, Option, Call, Put, Range Accrual Note, etc) or group of financial derivatives managed in a Risk Management Book  to a given change in value or level of the underlying asset (Interest Rate, Zero Rate, Stock Price, FX rate, Commodity Price, etc).

2) A pure first order sensitivity (first derivative) might be described as dV/dX the rate of change of the one or a collection of financial instruments (V) for a change in the Value of something else (X). For the calculation of Delta (X) is usually the, or one of the, underlying assets.

3) In finte form delta may be defined as,

V_{(@ high underyling value)}-V_{(@ low underlying value)}

2) For any individual trade at any time

3) The "perfect" Delta hedge (excluding credit related issues) for a trade done with counterparty X would be to enter the exactly the opposite trade with counterparty Y. E.g. To Sell an Call on the FTSE struck at 4500 for JUly Maturity from X and simultaneouly to enter into a Buy a Call on the FTSE STRUCK AT 4500 from counterparty Y. In general this is rarely/never feasible in exotic derivative markets. In vanilla market this may be possible but may result in any profits made on the Sale to countrerparty X being paid to counterparty Y. 

2) It may also be heard call "sensitivity"
2) If a risk management book/trade has got Delta of other than zero then its definitely exposed to P&L moves by asset/underlying moves.
3) Even if a risk management book/trade that currently has zero Delta may display a P&L move as the asset/underlying moves due to Gamma (the rate of change of Delta).
4) Not all models give good Delta numbers.

Delta With Regard To Fixed Income

5) Becasue the numberical value of a true Delta derivative dPV/dX would be very large (implying a 100% move in an interest rate e.g. a rate move from 7% to 107%) and offer no intuitive feel, Deltas for fixed income markets are usually quoted in terms of the trade of Risk Managements PV change per basis point (a basis point in 1/100th of a percent = 0.01%).
4) There are commonly two types of Delta discussed. Market Rate Delta "Market Risk" and Zero Delta "Zero Risk".
5) Market Rate Delta, as the name suggests, is the risk on associated with a trade or Rism Management Book in terms of Market Rates.
5) A Long position in an interest rate market (whcih may appear someshat counter intuitively) is a position where a rate must fall for the trade/risk management book to make money
6) Zero Delta is
5) A Parallel Delta (The sum of the Delta Ladder deltas i.e. Over Night, 1 Week, 1 Month, 3 Months, 6 months, 12 months, Futures, ) that sums to zero does not imply mean that delta risk is zero. What it does mean is that instantaneously the risk management book/trade exhibits no sensitivity to a parallel move of the yeild curve.
6) A trade or Risk Management Book that is/has a steepener
7) Like a trade or Risk management Book that is/has a flattner
7) Invariably, with the exception of very small trades or trades which happen to offset some exisitng risk, a risk management book which is not taking propriatry trade, a risk management book which puts on a new trade with a client (Bank, Corporate, National (e.g. Bank of England), Hedge Fund, etc.) will also and near simultaneously do offsetting trades with other risk management books internally

Delta Generation

1) Deltas may be calculated one in two ways, analytically or by changing the value of the given underlying asset by a given amount a measuring the effect on value (V) known as known as bumping since the value of the underlying asset is incremented and a new (V) calculted. 

2) Analytic deltas are feasinle for an ever smaller proportion of trade types (as new and generally ever more complex derivative instruments appear for which analytical risk numbers are not feasible) so analytics are only used in the most vanilla of risk management books (futures, swaps and possibly caps floors and swaptions.

Delta With Regard To Models

1) Every trade has an exact delta, but some derivatives models are better than others at producing accurate or even good delta numbers.

2) Given that not all pricing models cannot generate analytic risk numbers bumping is the methodology most commonly used to generate the risk for many trades.

3) Gerenally the models which produce the best bump risk delta will be purely analkytic models sice they operate in a "continuouis" as opposed "discretized" model of time assetvalue space, however as previosuly states few derivative types can be evalauted using analytical mdoels.

4) Binomial and Trinomial (otherwise known as Finite Difference) grids can produce good deltas if the modelling has been performed with sufficient rigor. Typically option/trade value and delta and estimates improve as the number of discretizations increases and y averaging the number of steps taken (See Hull)

5) Typically the worst type of model from which to extract bump deltas is a Monte Carlo model. Please see the section on Monte Carlo models for a discussion of obtaining delta estimates from this model.

 resembling a  risk number (including Delta) by bumping the underlying asset value and measuring the change in generally the best bumprisk numbers come from purely analytical models since these models don't usually have any