# 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

• 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

# 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