Mapping Positions to Risk Factors

Mapping Positions to Risk Factors part I

http://financetrain.com/var-mapping-positions-to-risk-factors/

VaR：Mapping Positions to Risk Factors

Posted on August 16, 2012 by Manish 

While working with Value at Risk, we are often looking at calculating the P/L of individual positions, and then calculating the portfolio P/L based on the individual positions. However, there are a few problems in doing so.

First, it is not possible to model each and every position in the portfolio, as there could be many positions with different complexities or lacking historical data to support the data model. Second, the portfolios of these positions can become quite complex with many dependencies. For example, the covariance matrix of risk factors in all the instruments (say n instruments) can become extremely large. The number of volatilities (n), and data on correlations (n(n-1)/2) grows as n grows. This also poses challenge in terms of the computational power required to process the data.

The process can be significantly simplified by mapping these positions to a small set of risk factors. Instead of trying to calculate VaR for each instrument, what we need to do is to decompose these instruments into building blocks, or primitive instruments, which are further mapped to a small set of risk factors. The four common building blocks are: equity positions, zero-coupon bonds, futures/forwards, and spot foreign exchange.

The generic VaR mapping process is as follows：

1. Choose a set of elementary risk factors and estimate their probability distributions

2. Mapping: Decompose financial instruments into exposures on these risk factors. Each position is marked to market. Market value of each instrument is allocated to risk factors.

3. Aggregate the exposure for all positions and build the distribution of P/L on portfolio. Arrive at the portfolio VaR.

-----------------------------------------------------------------------------------------------------

Mapping Positions to Risk Factors part 2

http://financetrain.com/mapping-spot-fx-positions/

Mapping Spot FX Positions

Posted on August 18, 2012 by Manish

In the previous article, we learned the need for VaR Mapping, and how financial instruments can be broken down into simpler building blocks/primitive instruments, which are mapped to a small set of risk factors.

One such building block instrument is spot FX positions, where you hold a foreign currency instrument whose value is fixed in terms of the foreign currency.
The mapping system of the financial institution will have a set of core currencies for which various historical and estimated data such as volatilities and correlations will already be available in the system.

If our FX position is based on one of these core currencies then our work is even easier. However, if our position is on a currency not from this core currency list, then we will have derive a proxy of our position in terms of one of the core currencies.
Assuming the exchange rates follow a normal distribution, the VaR of an FX spot position will be calculated as follows：

VaR = -Z_α * σ_x * P * X
Where,
P is the value of our position if foreign currency units
X is the exchange rate

Let’s say that you have an FX spot position of EUR10,000,000 and the domestic currency is USD. Exchange rate is 1 EUR = 1.23 USD.
The estimated volatility is 20%
As 99% confidence level, the daily VaR for this position will be：

VaR = -2.32635 * 0.20/Sqrt(250) * 12,300,000 = USD 361,943

This method can be applied to other spot positions also, such as commodities.
-----------------------------------------------------------------------------------------------------

Mapping Positions to Risk Factors part 7

http://financetrain.com/var-mapping-for-options-positions/

VaR Mapping for Options Positions

Posted on September 20, 2012 by Manish

In the previous article we learned about how to map various complex positions such as coupon paying bonds, FRAs, and swaps. All these complex positions had linear payoffs and were fairly straightforward to map. However, there are other positions such as options where the payoff is non-linear. In such a case the mapping process becomes a bit more complex.

For portfolios and positions with non-linear payoff, the mapping is done with the help of first-order and second-order Taylor approximation, i.e., the Delta and Delta-Gamma approach.

Let’s look at these two approaches.

Delta Approach

Let’s say that we have a call option with a value of c. The value of this option will depend on many variables such as the underlying stock price (S), the strike price, and the volatility. Ignoring other factors and considering the underlying stock price as the only factor, we can use the first order approximation, in which case：

Δc = δΔS

Where, δ is the option’s delta, which is readily available for actively traded options.

With this information, we can calculate the option’s VaR as follows：

VaR = δZασS

This approach however assumes that we have a short holding period and the delta remains constant. Also, it may not be very reliable for positions with high optionality or additional nonlinear features.

Delta-Gamma Approach

Our approximation can be made more accurate by using the second order Taylor approximation (Delta-Gamma), in which case the call value will be represented as follows：

Δc = δΔS +g/2(ΔS)²

Note that for a long call option, both Delta and Gamma will be positive, and for a short position both will be negative. A positive gamma favourably impacts the option value, and vice verse.

The VaR of the position can be calculated as follows：

VaR = δZασΔS – γ/2(ZασS)²

Note the minus sign in the formula. This is because positive gamma reduces VaR and vice verse.

Example：

Let’s say we have the following information about a European call option:

S = 100

r = 6%

Option maturity = 3 months

S = 25% (Annual)

D = 0.6

G = 2.2

A 10-day VaR at 99% confidence interval can be calculated as follows:

VaR = 0.6*2.33*Sqrt(10/250)*0.25 –(2.2)/2*(2.33* Sqrt(10/250)*0.25)^2

VaR = 0.0549

Note that VaR without incorporating gamma would have been higher.