Market Risk Practice Questions

[jyothi1965]

Hi David

I was working on the market risk practice questions, one of which is reproduced below:

Q: A Taylor approximation mitigates the curvature problem exhibited by nonlinear derivatives. However, is which case is the Taylor approximation ill-advised?

1 Long duration bond

2. MBS portfolio

3 Portfolio with small Gamma

4 out of the money call

(choices are reproduced from memory since they can't be copied)

Taylor's approx is "ill-advised" in portfolios with marked non linearities e.g. if you have "at the money" options, long dated bonds, etc I would choose 4 above, as an out of the money call is likely to exhibit very little non linearity- meaning they would be linear { essentially the straight lines of the pay-off diagram}

The non linearities in the MBS portfolio is different because of the concavity (prepayment problem). However, Non linearities can be either concave or convex. (Remember the short straddle/strangle option trading strategies can exhibit concavity (a inverted V or a inverted U), where in Gamma is very relevant and Taylor approx is not ill-advised

So my choice is 4 and not 2.

Can you please clarify

Jyothi

 

[David Harper CFA FRM]

Jyothi,

I would stand by MBS as the correct answer, although I do think you have a point about the imprecision of saying, as the Allen reading does, that Taylor is inappropriate for extreme nonlinearites.

Whether we refer to options or bonds, we are always talking about non-linear relationships. The Taylor is helpful when the price/yield (or in the case of options, the call price/stock price curve) relationship is, as Allen says, "well behaved." But what she really means is: Taylor works when the first derivative isn't changing abruptly and when there is no *inflection point* (technically: where the second derivative is zero, as when concave changes to convex or vice-versa). That's why the MBS is the correct answer here, as you point out, the convexity switches over (at lower yields) to concave (or negative convexity).

But Taylor is okay for the out of money call because you have a positive delta throughout the function (without inflection point). I see your point, the deep out of money has delta approaching zero and gamma approaching zero, so that's converging to a straight line. But that's okay for Taylor. For a straight line, at worse Taylor is overkill but not inappropriate. I think frankly the Allen reading is imprecise. Rather than extreme nonlinearities or marked non linearities, technically better to say Taylor is appropriate to monotonic functions or functions without inflection points or functions that are strictly concave or strictly convex.

(I am aware of the esoteric objections to Taylor series expansion applied to vanilla calls, but it is well beyond the scope of FRM. It occurs to me you could select D on these grounds; i.e., that the power series does not converge for the Black-Scholes. But I don't want to go down that rabbit hole. For our purposes, Taylor series expansion is appropriate to the delta-gamma approximation in the case of options and the duration-convexitity approximation in the case of bonds).

Thanks, David

 

[rahul.goyl]

Hi david,

That's contradicting statement. Keeping exam point of view what should we choose as a right opton. Is it B the right answer or we should select D.

Thanks
Rahul

 

[David Harper CFA FRM]

Hi Rahul - Definitely the exam is looking for (B). This is under Linda Allen, where she says Taylor is not good for derivatives with extreme non-linearities and the classic example is MBS because of the negative convexity (where the price/yield curve is "bending back down" at lower yields due to prepayments) - David