Aug 02

Value at Risk (VaR): 2007 FRM. Part 3

by David Harper, CFA, FRM, CIPM

FRM | Risk |

var_nonlinear

2007 Learning Outcomes

  • LO 7.7 Differentiate between linear and non-linear derivatives.
  • LO 7.8 Describe the calculation of VAR for a linear derivative.
  • LO 7.9 Explain how the addition of second-order terms through the Taylor approximation improves the estimate of VAR for non-linear derivatives.
  • LO 7.10 Discuss why the Taylor approximation is ineffective for certain types of securities.

 

Value at risk (VaR) for a linear derivative

The classic linear derivative is a forward contract. It is linear because the price of the forward is a constant function of the underlying (i.e., the stock). For a linear derivative, the value at risk (VaR) is delta multiplied by the VaR of the underlying:

linearvar

 

Nonlinear derivative: derivative price is not a constant function of underlying

The classic nonlinear derivative is a call option. In the diagram below, the price of the call option (as by the Black-Scholes) is plotted in green. Note the plot of underlying (the stock price; x-axis) versus the derivatives (the call option; y-axis) is not a straight line. That's a nonlinear derivative.

optionDelta2

The key problem, from a risk measurement perspective, of this non-linear derivative is that the sensitivity is hard to figure. That is, a change in the option price, as a function of the change in the stock price, changes. We would like to use delta. Delta is a first-order derivative: the percentage change in option price for a given percentage change in the (underlying) stock. Delta is the slope of the tangent line.

But note the problem: delta is only locally accurate. Convexity (or curvature) in the true line creates a gap; the gap is the "error" produced if we only use delta!

We fill this gap (created by convexity) with the addition of second-order terms in the Taylor approximation

Remember we said delta is a first derivative? The Taylor approximation adds additional derivative orders; e.g., it adds a second-order derivative which is called the "convexity correction":

taylor

So that's the fine job of the Taylor approximation: it adds curvature to our blunt, linear delta-based line. It gets us nearer to the true curve.

But Taylor likes "well-behaved" lines, he doesn't cope well with extreme non-linearities

The Taylor works if the true derivative relationship is like the green line above: well-behaved. But if the derivative exhibits extreme non-linearities, the Taylor does not work fine. The classic is a callable bond:

a callable bond (a bond with an embedded option): the price-yield curve in this case is partly convex and, partly, exhibits negative convexity (the curve bends back down at lower yields). The Taylor approximation cannot cope with such capricious behavior.

Key takeaways

  • LO 7.7: linear/nonlinear depends on the relationship between the underlying (e.g., stock) and the derivative (e.g., stock option). If the relationship is constant such that we can say price of derivative = (k)(price of underlying), it's a linear derivative
  • LO 7.8: For a linear derivative, VaR = (delta)(VaR of underlying)
  • LO 7.9: The addition of second-order terms through the Taylor approximation improves on the linear delta by accounting for (some of) the convexity
  • LO 7.10: On the other hand, the Taylor approximation is ineffective when the derivative exhibits extreme non-linearities.

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