GARP Practice Exam 2017

[Unusualskill]

Hi David,

This is Q40 From GARP 2017.

40. A portfolio contains a long position in an option contract on a US Treasury bond. The option exhibits positive convexity across the entire range of potential returns for the underlying bond. This positive convexity:

A. Implies that the option’s value increases at a decreasing rate as the option goes further into the money.
B. Makes a long option position a superior investment compared to a long bond position of equivalent duration.
C. Can be effectively hedged by the sale of a negatively convex financial instrument.
D. Implies that the option increases in value as market volatility increases.

Explanation: The relationship between convexity and volatility for a security can be seen most clearly through the second-order Taylor approximation of the change in price given a small change in yield. The resulting change in price can be estimated as: Δ/P≈ −Δ + /2 Δ^

where d is equal to the duration, c is the convexity and y is the change in the interest rate. Since Δ is always positive, positive convexity will lead to an increase in return as long as interest rates move, with larger interest moves in either direction leading to a greater return benefit from the positive convexity. Therefore, a position in a security with positive convexity can be considered a long position in volatility.

This relationship can also be explained graphically. The price curve of a security with positive convexity will lie above and tangentially to the price curve of the underlying. If volatility of the underlying increases, then so will the volatility of either a long call or a long put, but the deviation from the price of the underlying will be positive when there is positive convexity, and negative with negative convexity. Therefore, the expected terminal value over the in-the-money region will increase while the expected terminal value over the out-of-the-money region will remain zero, an aggregate effect of increasing the total expected value of the option

In the explanation, why does change in y always positive? and I am also quite loss at the second part of the explanations."This relationship can also..."

Appreciate your help, thank you!

 

[Nicole Seaman]

Hi David,

This is Q40 From GARP 2017.

40. A portfolio contains a long position in an option contract on a US Treasury bond. The option exhibits positive convexity across the entire range of potential returns for the underlying bond. This positive convexity:

A. Implies that the option’s value increases at a decreasing rate as the option goes further into the money.
B. Makes a long option position a superior investment compared to a long bond position of equivalent duration.
C. Can be effectively hedged by the sale of a negatively convex financial instrument.
D. Implies that the option increases in value as market volatility increases.

Explanation: The relationship between convexity and volatility for a security can be seen most clearly through the second-order Taylor approximation of the change in price given a small change in yield. The resulting change in price can be estimated as: Δ/P≈ −Δ + /2 Δ^

where d is equal to the duration, c is the convexity and y is the change in the interest rate. Since Δ is always positive, positive convexity will lead to an increase in return as long as interest rates move, with larger interest moves in either direction leading to a greater return benefit from the positive convexity. Therefore, a position in a security with positive convexity can be considered a long position in volatility.

This relationship can also be explained graphically. The price curve of a security with positive convexity will lie above and tangentially to the price curve of the underlying. If volatility of the underlying increases, then so will the volatility of either a long call or a long put, but the deviation from the price of the underlying will be positive when there is positive convexity, and negative with negative convexity. Therefore, the expected terminal value over the in-the-money region will increase while the expected terminal value over the out-of-the-money region will remain zero, an aggregate effect of increasing the total expected value of the option

In the explanation, why does change in y always positive? and I am also quite loss at the second part of the explanations."This relationship can also..."

Appreciate your help, thank you!

Hello @Unusualskill

I'm not sure if this will help to answer any of your questions regarding the second part of the explanation, but this question was discussed in the GARP practice questions section, and I think it is important to point out what David noted there:

@urna nandi I agree that #40 Q&A suffers from some imprecisions. First, it refers to an option rather than a bond with an embedded option. The graph shape of the call option (on the bond) would resemble the graph shape of a stock option put; i.e., bond option price as a non-linear decreasing function of increasing yield (the call becomes less valuable as yield increases because bond price decreases). The graph shape of a put option (on the bond) would resemble the graph shape of a stock option call; i.e., bond option price as an increasing function of increasing yield (the put becomes more valuable as yield increases and price drops). But the answer is more simply relying on the fact that option value increases with volatility, and the fact that a long position (in either a call or put) is generally long volatility per Tuckman below. I hope that helps, there are several nuances at play here, which IMO require more precision in the Q&A (I have added this question to my feedback list for GARP). Can i ask if you want to follow-up on this question, that we start a new thread in the PQ forum so we can tag it specifically? Thank you!

Thanks @Nicole Seaman I don't think I will spend further current time on this question @urna nandi because I just tend to agree with you that the phrasing could be improved especially in the answer (where I think some of the challenge is the potential confusion between the price/yield curve of a bond, or even a bond with an embedded option, versus the price/yield curve of an option on a bond). A related issue, I think, is that we actually have no FRM reference for naked options on bonds: all references are to bonds with embedded options, which makes precision really important. But that's improvement feedback. I wouldn't want to lose sight of what seems to me to be an essentially correct Q&A where the key concept is something like: if you are long an option (call or put), you are effectively long volatility and the convexity/gamma (both being the same 2nd partial derivative in the Taylor Series approximation) are responsible for (or reflect) the gain associated with increases in yield volatility. Thanks!

For those who have access to that section, the entire discussion is here: https://forum.bionicturtle.com/threads/garp-2017-p1-40-garp17-p1-40.10485/

Nicole