# Delta Normal VAR: Linear and Non-Linear Assets

**Linear:**When the value of the**delta is constant**for any change in the underlying- Primarily in the case of
**forwards and futures**we have linear assets - The method to calculate VAR for linear assets is called Delta Normal method
- Delta Normal method assumes that the variables are normally distributed

- Primarily in the case of
**Non Linear:**When the value of the**delta keeps on changing**with the change in the underlying asset- Options are non-linear assets, where delta-normal method cannot be used as they assume the linear payoff of the assets
- To calculate the VAR for non-linear assets, full revaluation of the portfolio needs to be done
- Monte Carlo methods or Historical Simulation are commonly used to fully revaluate the portfolio

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# Delta Normal VAR: VaR for Linear Derivatives

**Linear Derivatives:**Payoff diagrams that are linear or almost linear:- Forwards, futures

- Delta of Derivative: Change in price of Derivative to change in underlying asset
**For example:**- The permitted lot size of S&P CNX Nifty futures contracts is 200 and multiples thereof.
- So VAR of Nifty Futures contract is 200 * VAR of Nifty Index

# Delta Normal VAR: VaR for Non-Linear Derivatives

- Main reason for difference is the shape of the payoff curve
- For
**Delta Normal VAR**- A linear approximation is created
- Approximation is an imperfect proxy for the portfolio
- Computationally easy but may be less accurate.
- The delta-normal approach (generally) does not work for portfolios of nonlinear securities.
- E.g.; Options VAR = Delta of Option * (VaR at Zx%)

- Consider a portfolio of options dependent on a single stock price, S. Define:

- Approximately:
- For Many Underlying variables:

**Question 6 (Linear Assets)**

- If the daily VaR at 5%of Nikkei is USD 0.8 mn and you have 100 lots of Nikkei contract.
- Calculate annual VaR at 95% confidence for your portfolio assuming 250 days?

**Solution**

- Solution = 0.8 x 100 x (250)
^{0.5}= USD 1264.911 mn - Here, delta = 100, because for every 1 unit change in the Index Nikkei, the futures price will change by 100 units because the lot size is 100

**Question 7 (Non-Linear Assets)**

- If the value of stock is 100 and the value of the put option at 110 is 20. 10 units change in the underlying brings in change of 4 units change in the option premium. If the annual volatility is 0.25. Calculate daily VaR at 97.5% assuming 250 days?

**Solution**

- Daily volatility = 0.25/(250)
^{0.5}= 0.0158; Daily VaR = 100 x 0.0158 x 1.96 = 3.099; - Daily VaR of option = 0.4 x 3.099 = 1.239