Explanation

Bank A will estimate a higher VaR value because:

• Bank A uses linear approximation (delta-normal method): This method approximates the option's price change using only the delta (0.6). It assumes a linear relationship between the option price and the underlying stock price, ignoring gamma effects.

• Bank B uses Monte Carlo simulation with full revaluation: This method captures the non-linear nature of options pricing, including gamma effects. For an at-the-money call option, gamma is positive, meaning the option's delta increases as the stock price rises and decreases as the stock price falls.

• Key insight: For a long call option position, the linear approximation (delta-normal) method overestimates losses when the stock price decreases because:

  • When stock price falls, the option's delta decreases (due to positive gamma)
  • The actual loss is less than what the linear approximation predicts
  • Therefore, the linear method produces a higher VaR estimate

• Mathematical reasoning:

  • Linear VaR = |delta| × Stock Price × Volatility × Z-score
  • Full revaluation accounts for the convexity (gamma) which reduces the actual price decline when stock prices fall

Since Bank A's linear method ignores this convexity benefit, it will compute a higher VaR estimate than Bank B's more accurate full revaluation approach.