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billyx524
Topic Author
Posts: 45
Joined: February 12th, 2016, 4:48 pm

### Interview Question on Scribd

Hi Everyone,

I saw this interview question on scribd.  I am not sure how to proceed with this question.  Can anyone please offer me some hints?  Thanks in advance.

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Consider a 1x2 swaption and a cap. The cap under consideration is a strip
of two one year caplets, one expiring a year from now, the other expiring
in two year’s time. Comment on the Black vol of the swaption relative
to the Black vol of the cap under the assumption that forward rates are
correlated. Which is higher?
---------------------------------------------------------------------------------------

bearish
Posts: 5518
Joined: February 3rd, 2011, 2:19 pm

### Re: Interview Question on Scribd

Hi Everyone,

I saw this interview question on scribd.  I am not sure how to proceed with this question.  Can anyone please offer me some hints?  Thanks in advance.

--------------------------------------------------------------------------------------
Consider a 1x2 swaption and a cap. The cap under consideration is a strip
of two one year caplets, one expiring a year from now, the other expiring
in two year’s time. Comment on the Black vol of the swaption relative
to the Black vol of the cap under the assumption that forward rates are
correlated. Which is higher?
---------------------------------------------------------------------------------------
The question is sufficiently ill posed that it is even hard to guess what they are after. One vaguely relevant observation is that if the strike is the same for the cap and swaption then the cap is almost always worth more than the swaption, with equality only arising in contrived corner cases. This falls under the general result that a portfolio of options is worth more than the corresponding option on the portfolio.

fyvr
Posts: 16
Joined: November 15th, 2015, 3:10 pm

### Re: Interview Question on Scribd

This is an interview oldie (and a famous hedge fund arb trade back in the day). To pose it correctly, you are asked to compare the vol of a 1Y2Y swaption with that of a 2yr cap out of a 1yr forward start (so the relevant forward period is identical).
Remembering that a swap rate is just an ‘average’ of forward LIBORs, think about how much the forward swap rate would move if the front LIBOR moved 10 bp and all the forward LIBORs were PERFECTLY CORRELATED.
Then consider the same question assuming something like 80% correlation among the forward LIBORs.
The rest I will leave to you.

bearish
Posts: 5518
Joined: February 3rd, 2011, 2:19 pm

### Re: Interview Question on Scribd

And I'll posit an interest rate model where the short rate is constrained to (a,b), and consider a cap with strike b and a swaption with strike (a+b)/2. I'll even be nice and say that a>0. If I wanted to be nasty, I'd make b<0.

billyx524
Topic Author
Posts: 45
Joined: February 12th, 2016, 4:48 pm

### Re: Interview Question on Scribd

Quick Question, when we say 1Y2Y swaption, is the 2Y relative to today, or to the 1Y point?  Meaning is the underlying swap tenor 2 years, or just 1 year?

Thanks!

fyvr
Posts: 16
Joined: November 15th, 2015, 3:10 pm

### Re: Interview Question on Scribd

It’s an option, expiring in 1year, into a 2 year swap (which starts at option expiry, if exercised)

billyx524
Topic Author
Posts: 45
Joined: February 12th, 2016, 4:48 pm

### Re: Interview Question on Scribd

Thanks for this clarification.  I still have a couple of questions:

1).  I am not sure if I understand why you say the "relevant forward period is identical".  For the swaption case, the vol is the forward swap rate in one years time (relative to today) for a future two year period [1, 3].  For the cap case, we have two forward rates, one expiring at one year for the period [1,2], the other expiring in two years for the period [2,3].  Is this right?

2).  I can intuitively understand why swap rate is an average of forward libors.  However, how can I show this mathematically.  I am asking because the swap rate formula is a ratio of zero coupon bond prices:

swap rate = ( P(t, Ta) - P(t, Tb) )   / sum ( tau * P(t, Ti) )

Thanks!

fyvr
Posts: 16
Joined: November 15th, 2015, 3:10 pm

### Re: Interview Question on Scribd

1) as you’ve demonstrated, the forward period for the cap is also two years, (1, 3) in your notation. It’s not necessarily two periods either; if the LIBOR in question is 3mo then you have eight periods.

2) don’t make the mistake of getting bogged down by formalism, you need to understand things too. The par swap rate is the fixed rate the market says is economically equivalent to receiving a string of LIBOR fixings. So, you can (should) think of it as just the average (appropriately defined) of the forward LIBORs that span the forward period.

If you understand this, the problem you posted is rather trivial.

fyvr
Posts: 16
Joined: November 15th, 2015, 3:10 pm

### Re: Interview Question on Scribd

And as an aside, you should be aware that the formula you wrote down (for the par swap rate) has, strictly speaking, not been true since about 2008.

billyx524
Topic Author
Posts: 45
Joined: February 12th, 2016, 4:48 pm

### Re: Interview Question on Scribd

Thanks!

For the swap rate formula, it would need a basis adjustment right?  But how you apply the basis is a matter of choice?

fyvr
Posts: 16
Joined: November 15th, 2015, 3:10 pm

### Re: Interview Question on Scribd

Not really, and definitely no.

The formula is only rigorously true if the discount curve is the same as the forward (projection) curve.

billyx524
Topic Author
Posts: 45
Joined: February 12th, 2016, 4:48 pm

### Re: Interview Question on Scribd

Sorry, I think the basis adjustment is between the forward (projection) curve and the discounting curve?  But how you want to model the basis is up to you, for example it could be a deterministic spread if you like.

Is this right?

fyvr
Posts: 16
Joined: November 15th, 2015, 3:10 pm

### Re: Interview Question on Scribd

Well, you can call it whatever you like, but it’s not really a basis adjustment in the sense that I would define the term. (For me a basis spread is just the difference between the forward curves of two near-identical observables: 3M vs 6M libor, or the delivery price of a bond future vs the forward price of the CTD, etc and so forth. The impact of OIS discounting is rather different, though I suppose if you wanted to you could model the effect as some kind of pseudo- basis spread).

cquasar
Posts: 5
Joined: February 25th, 2016, 2:03 pm
Location: Mumbai
Contact:

### Re: Interview Question on Scribd

Thanks for this clarification.  I still have a couple of questions:

1).  I am not sure if I understand why you say the "relevant forward period is identical".  For the swaption case, the vol is the forward swap rate in one years time (relative to today) for a future two year period [1, 3].  For the cap case, we have two forward rates, one expiring at one year for the period [1,2], the other expiring in two years for the period [2,3].  Is this right?

2).  I can intuitively understand why swap rate is an average of forward libors.  However, how can I show this mathematically.  I am asking because the swap rate formula is a ratio of zero coupon bond prices:

swap rate = ( P(t, Ta) - P(t, Tb) )   / sum ( tau * P(t, Ti) )

Thanks!
Perhaps, think of a swap like this:

Floating leg
$PV_{float}(t) = \sum_{i=1}^{n}{NF(t,T_{i-1}) \tau_{i} DF(t,T_{i})}$

where, $F(t,T)$ is the index forward rate(some IBOR rate) and $DF(t,T_{i})$ = the discount factor from $T_{i}$ to $t$. So, the swap rate is an economic equivalent of what the market thinks should be the value of a string of IBOR forwards.

Let's make the simplifying assumptions :

(1) The duration of the LIBOR deposit matches the coupon period.
(2) Index-forward rates are equal to discount forward rates.

Denoting the discount forward rate between the period $(T_{i-1},T_{i})$ by $R(t,T_{i-1},T_{i})$,

$F(t,T_{i-1})=R(t,T_{i-1},T_{i})=\frac{1}{\tau_{i}}\left(\frac{DF(t,T_{i-1})}{DF(t,T_{i})}-1\right)$

Substituting this, we would get a formula similar to what you stated:

$PV_{float} = NDF(t,T_{0})-NDF(t,T_{n})$

Maybe that helps. Instead of using any of the above formulae, it's nice to price a few swaps on Excel, make a schedule for the fixed leg and the floating legs and try to find the par swap rate, to get an intuitive feel.

Post 2008, the simplifying assumptions no longer hold. Swap pricing is far more complex.

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