FRM Fun 23, Market Risk (guest submission)

[David Harper CFA FRM]

jcjc0602 posted two market risk practice questions (I don't think they are ours, i hope not) because he/she perceived an issue with each of them. I happen to agree with jcjc0602. But I thought I'd make it a "fun" problem because I think each is instructive. Here they are:

1. A zero-coupon bond with a maturity of 10 years has an annual effective yield of 10%. what is its modified duration?
A. 9
B. 10
C. 100

2. Which type of option produces discontinuous payoff profiles?
A. Choose options
B. Barrier options
C. Binary options
D. Lookback options

Question: does either/neither/both of the questions have a problem or imprecision?

 

[trabala38]

Hi David,

According to me both of them have a problem or imprecision.

1. "Effective" is not applicable for the yield. "Effective" is usually associated with effective duration: "effective" is used when you reprice (2 times) a bond after shocking the interest rate and compute the price sensitivity thanks to this re-computation of prices. On the other hand, the yield is the IRR of the bond and cannot compute it on a "effective" way. You need to resolve the equation or to use an iterative method that finds the interest rate that is required to retrieve the (market) bond price.

2. Payoff graph (* = usual payoff graph) is most of the time represented with the "payoff" on the vertical axis and the asset price as the horizontal axis (S). Moreover, the payoff is usually associated with the final delta final asset price and strike price at the exercise date: max (S-K;0) for a call; max (K-S; 0) for a put.

The problems/imprecision are the following:

- Chooser option: the payoff is linked the underlying: a call and a put. The payoff is dependent on the difference between the call price and the put price (and not a difference between a strike price and an asset price).
- Barrier option: the issue is linked to the fact that the payoff function is dependent on observation period (that can be continuous or not) in which the asset price (S) needs to cross a certain barrier to come into existence or to cease to exist. Therefore, the payoff profile is conditional on a certain event happening in a certain time frame. The profile can't be regarded as discountinuous as a function of strike price as the discontinuity is caused by a event (and not only the asset price at the end of the period). Note: the occurrence of the event is time dependent.
- Binary: OK can exhibit a jump (=discontinuity !!!) if cash-or-nothing call/put (either 0 or Q depending solely on the final asset price at expiration time). Payoff profile similar (no discontinuity!!!) to a regular call/put for a an asset-or-nothing call/put.
!!! The answer continuity vs non-continuity depends thus on the type of binary option (asset-or-nothing is continuous; cash-or-nothing is discontinuous) !!!
- Lookback option:
a) For a floating loopback call => Payoff=Max ( <delta final asset price - min. asset price during a certain period>; 0) => the horizontal axis should be delta price between final asset price and the minimum asset price during the period (no discontinuity if represented that way). Note: the horizontal axis value is the difference between 2 variables!
b) For a floating loopback put => Payoff=Max ( <delta max. asset price - final asset price during a certain period>; 0) => the horizontal axis should be delta price between maximum asset price during that period and the final asset price (no discontinuity if represented that way). Note: the horizontal axis value is the difference between 2 variables!
c) For a fixed loopback call => Payoff=Max ( <delta maximum asset price during a certain period - strike price>; 0) => the horizontal axis should be the maximum asset price during a certain period (no discontinuity if represented that way)
d) For a fixed loopback put => Payoff=Max ( <strike price - minimum asset price during a certain period>; 0) => the horizontal axis should be the minimum asset price during a certain period (no discontinuity if represented that way)


What do you you guys think?

Cheers,

trabala38

 

[PL]

In regards the first one ....I'm also have an issue with the "effective" since effective duration is the yield of a bond assuming that coupons are reinvested (source investopedia....http://www.investopedia.com/terms/e/effectiveyield.asp#axzz27gfzB8II)
But what coupons in a zero bond?
Additionally a better wording would be "bond equivalent yield" have a look on the following link (David's video)

Also the compounding should be mentioned in order to transform the mac duration to modified.

In regards to second q...trabala38 gave an excellent and detailed explanation!

 

[David Harper CFA FRM]

Hi trabala & PL, that was quick, thank you!

I think they both do have a problem, but I only currently have high confidence in regard to the 1st (duration), where I think PL identifies it with "Also the compounding should be mentioned in order to transform the mac duration to modified." Specifically,

Effective annual yield, EAY, (aka, effective annual rate, EAR) is the rate which converts from a stated (aka, nominal) rate, from any compound frequency, so it can correspond to any of several compound frequencies.

The point of "effective" is to return an annual rate given (corresponding to) to a combination of stated rate + compound frequency; it is exactly NOT to convey that the compound frequency is annual. For example,

• if the stated rate = 10% per annum with annual compounding, then the EAY = 10.00%;
• if the stated rate = 10% per annum with semi-annual compounding, the EAY = (1+10%/2)^2 - 1 = 10.25%
  ... there is a reason Hull writes the questions like this!

In this case,

• EAY of 10% (i.e., annual discrete) corresponds to continuous LN[1+10%] = 9.531% continuous; i.e., exp(9.531%)= 1.10
• EAY of 10% also corresponds to semi-annual 2*[EXP(LN[1+10%]/2)-1] = 9.7618%; i.e., (1+9.7618%/2)^2 = 1.10

In which case, this 10-year zero coupon bond can have various modified durations, all with EAY = 10%, as the compound frequency is not given (compound/discount frequency impacts the pricing function):

• If annual frequency, mod duration = 10/(1+10%) = 9.0909
• If semi-annual, mod duration = 10/(1+9.7618%/2) = 9.5346
• If continuous, as k = inf, this is the special case where mod = mac and mod duration = 10/(1+y/k), where k--> inf, and mod duration = 10
  
  I conclude, mod duration varies from ~9 to 10, all for the same 10% EAY, depending on compound (discount) frequency

In regard to the second, my tentative problem (i basically agree with trabala) is:

• The payoff function, unless otherwise stated, is payoff (not including premium paid, which would be profit) on Y-axis plotted against stock price on X-axis, and
• "Discontinuous" has a very specific definition, which generally comports with our intuition: http://en.wikipedia.org/wiki/Continuous_function
• And, unless I miss something (?), the plot of a barrier payoff versus asset price includes a discontinuous jump down to the zero X-axis at the point (X = barrier, Y = 0). The barrier looks discontinuous to me, as i do not see how the path dependency of the barrier nullifies what appears to be a discontinuous payoff at maturity or earlier?

 

[ShaktiRathore]

Hello David and all,
Regarding second part, continuous function is one where small changes in stock price will always result in corresponding small change in payoff. And discontinuous function is where above does not happens that is small changes in stock price will not always result in corresponding small change in payoff so sometimes there can be good shifts in payoff for small changes in sock price.
Applying the above definitions to our payoffs of the options mentioned,
Chooser option is to choose b/w call and put option, before choosing there are no payoffs but after choosing the call/put option comes into existence so that chooser option is nothing but a call or put option after sometime whose payoffs are continuous.
The barrier options are special where options comes into existence when stock price hits a particular price barrier are called the In options. Whereas when options dies when stock price hits a particular price barrier are called the Out options.So when options are In they have continuous payoffs but when options dies the barrier options have continuous payoffs before they die. So there is no question of discontinuity.
Binary options are horizontal payoffs at maturity if St>k and zero if St<k. So here there is a point of discontinuity at St=k where there is large change in options payoff for very small change in stock price around St. We can say that binary has discontinuous payoff.
Lookback options has payoff S(T)-S(min) which is continuous payoff as it is determined at time T not before T but after T as a horizontal P/off.So its payoff is continuous which varies from zero to infinity as stock price varies from zero to infinity.
Regarding the first part,
David I agree with your compounding part that if effective annual yield is given as 10% and the compounding period of Z-coupon bond is not given that is it is uncertain whether the compounding occurs every 3 month or every six months or yearly or continuous. Otherwise it is also possible (or might be a possibility, i may not be correct)that Question has given only the yearly effective yield so that we can assume that compounding is annually, so that there are no error in calculation. If we assume other compounding other than a year than the values of modified duration can come incorrect as we need to change the given value of yield to some other value,based on our assumption of compounding. So it is safe to assume yearly compounding given that only yearly effective yield is given so Question wants us to assume yearly compounding.So duration=10/1.1=9.09~9.

thanks

 

[ShaktiRathore]

double post

thanks

 

[David Harper CFA FRM]

The barrier options are special where options comes into existence when stock price hits a particular price barrier are called the In options. Whereas when options dies when stock price hits a particular price barrier are called the Out options.So when options are In they have continuous payoffs but when options dies the barrier options have continuous payoffs before they die. So there is no question of discontinuity.

Thanks ShaktiRathore, that is exactly my view too, if the plot is payoff versus asset price, I cannot see how the barrier option can be viewed as continuous (to definite it continuously only for certain X values is to contradict the definition!). Thanks for supporting my view!