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:
- 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 = 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?
- 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