Delta-normal method to calculate VaR for Linear Derivatives

[frmpart2dan]

I would appreciate if someone could explain in layman terms what is the Delta-normal method.

Also could someone explain how the following 2 positions are equivalent:

1. A 1 year forward contract to purchase pounds for dollar

2. A combination of 3 positions: a) A short position in a US Treasury bill b) A long position in a 1 year UK bond c) A long position in the British Pound spot market

 

[brian.field]

I never really understood this (delta-normal var) very well.

 

[brian.field]

Regarding the two positions, they are equivalent (I believe) as a result of put call parity for options on currencies.

Recall Put-Call Parity is as follows:
C(pounds, $, T) - P(pounds, $, T) = [ exchange rate (in $/pounds) ] * e^(r_pounds*T) - K * e^(r_dollars*T)

where C(pounds, $, T) represents a $-denominated call option on pounds or to purchase pounds at the strike K (both the premium for C and the K are $ denominated as well).

Restated, we can say the following:

C(pounds, $, T) = [ exchange rate (in $/pounds) ] * e^(r_pounds*T) - K * e^(r_dollars*T) + P(pounds, $, T)

 

[David Harper CFA FRM]

Hi @frmpart2dan Re delta-normal, there are three approaches to VaR: analytical (aka, parametric), historical simulation, and Monte Carlo simulation. Some authors (including FRM authors Jorion and L. Allen) refer to the analytical approach as "delta-normal" because it's probably the most common analytical VaR. In an analytical (parametric) VaR approach, while data certainly informs the parameters such as the elements of a covariance matrix, the VaR is retrieved from a distribution or function, not from an effectively sorted empirical (Monte Carlo or historical) distribution.

It's called delta-normal because delta is a first partial derivative with respect to the risk factor and so "delta" signifies sensitivity to the risk factor; "normal" because we typically assume (in market risk) the distribution of the risk factor is normal. So, delta-normal makes perfect sense for an option where option VaR = S*σ*Δ*α; i.e., the asset price is the risk factor, sensitivity to the risk factor is Δc/ΔS, and we assume the risk factor is normal distributed, is why we can use 1.65 for a 95% confident VaR. So, put another way, 95% option VaR is an analytical function of the option's delta and a normally distributed risk factor (stock price). In the case of a bond, it could still be delta-normal VaR but it's really really "duration-normal VaR:" the risk factor is the yield and we assume it is normally distributed, and duration is risk sensitivity. So, delta refers to any linear approximation given by the first partial derivative.

Carol Alexander prefers "parametric linear VaR" as more accurate than "delta-normal" and maybe you can see why. Parametric as opposed to simulation, and "linear" because the VaR is a linear function of risk factor shocks. This allows for non-linear
functions, and also does not constrain the risk factors to normal distribution. In this way, delta-normal would be a specific type of parametric linear VaR which in turn is linear case of any analytical (parametric) approach. I hope that's helpful, it took me a long time to sort this out simply because different authors use different terms.