I read somewhere that the full revaluation VaR will be underestimated compared to linear approximation. How is this possible? I thought delta normal VaR underestimates due to the convex nature of options
Here is the full quote I read, dont' remember the source:
"The option’s return function is convex with respect to the value of the underlying; therefore the linear approximation method will always underestimate the true value of the option for any potential change in price. Therefore the VaR will always be higher under the linear approximation method than a full revaluation conducted by Monte Carlo simulation analysis. The difference is the bias resulting from the linear approximation, and this bias increases in size with the change in the option price and with the holding period."
Hi @bpdulog I can illustrate. Assume that while Rf = 3%, Term = 1 year, σ = 40%, K = 100: When the Stock (S) = $110, the delta, N(d1) = 0.70 and then the call price is $23.67
Imagine the stock price drops by $10.00 to $100.00
If we just use delta (the linear approximation), we estimate a drop in the call price of $10.00 * 0.70 = $7.00; i.e, from $23.67 to 16.67
However, the actual drop in the call price is only $6.54, from $23.67 to $17.14 (if i had more time, I would paste the picture: visualize the tangent line to the convex call/stock price curve)
Your quote is correct. Confusion can arise if we aren't careful about the change in option price versus the VaR which is the loss. In my example, the linear approximation underestimatesthe true value (price) of the option because it predicts a new price of $16.67 versus the true (i.e., full revaluation) price of $17.14; the difference due to convexity. At the same time, the linear approximation overestimates the VaR because it predicts a loss of $7.00 versus the true loss of $6.54. I hope that helps. Good luck tomorrow!!