SERVING THE QUANTITATIVE FINANCE COMMUNITY

 
User avatar
lePiddu
Topic Author
Posts: 20
Joined: March 17th, 2015, 2:15 pm

Hedging with a different underlying - bond options case

January 30th, 2020, 11:36 am

Hello everyone,

I'm working on government-bond options pricing (Black-Scholes world, nothing fancy). In EUR, that's pretty much a "non market" in the sense that there's pretty much no quotes, so no implied vols, no bid-offer spreads, no term structure... Well, nothing on bonds, but something on govies bond futures. Some standard option maturities (2, maybe 3 when you are lucky) on some standard future maturities (mainly the first 2) with some standard underlying (10yrs or 5yrs).

But some customers would like to trade options on government bonds. So I was wondering: if I were to trade an otc bond option, I would end up hedging my position using those futures (for delta) and maybe those options on futures (for vega).

Since I'm hedging with those instruments, they will make up for my "replicating portfolio"... and therefore they must also make up for the price of my otc option.

Hence my question: how would you price an option on an underlying that you cannot hedge with?
 
User avatar
frolloos
Posts: 1621
Joined: September 27th, 2007, 5:29 pm
Location: Netherlands

Re: Hedging with a different underlying - bond options case

February 5th, 2020, 3:39 pm

As you know I am no IR expert but I think the question is applicable to any asset class so I will hazard an answer:

You say that you can't hedge it, but if there are govie bond futures then at the very least you can delta hedge the option, is this correct? Or are the govie futures not on the same underlying govie as your option?

Assuming that the govie futures are on the same govie as your option then why don't you set a realistic upper bound on the vol of the govie bond, charge that vol to your customer and delta hedge the option using that vol?
 
User avatar
DavidJN
Posts: 1770
Joined: July 14th, 2002, 3:00 am

Re: Hedging with a different underlying - bond options case

February 5th, 2020, 7:18 pm

If the underlying bonds do not trade at all it would be a challenge to find highly correlated tradeable bonds as hedge candidates because without trade how would you ascertain the correlation? Methinks more market specifics are required.
 
User avatar
lePiddu
Topic Author
Posts: 20
Joined: March 17th, 2015, 2:15 pm

Re: Hedging with a different underlying - bond options case

February 6th, 2020, 11:23 am

Thank you all for your replies, I'll try to cover some questions:

@frolloos
Or are the govie futures not on the same underlying govie as your option?
That's precisely the problem. Govie futures underlying is the so called "cheapest to deliver" (CTD) bond. I'd say that, if the underlying of my otc bond option is the CTD (or extremely similar in terms of duration) then I'm pretty much fine. The difference is just going to be represented by the "conversion factor" of the future and the CTD. In general the premise is not true: customers would like to underwrite options on whatever bonds they like. Such bonds may differ wildly (in terms of duration) from the CTD.

@DavidJN
without trade how would you ascertain the correlation?
I completely agree. Thing is, all the bonds are "correlated" in the sense that they share the same yield curve (let's leave aside problems like on-the-run or off-the-run bonds). And the sensitivity of the bond price to the yield curve is the duration (and the convexity for 2nd order). Let's just say "I cannot trade in the bond, but everyone else can. I can trade only in futures and options on futures".

So I was wondering how to replicate an option on a bond BUT delta-hedging it with futures (underlied by a probably different bond) and (possibly) vega-hedging it with options on futures. Of course we are talking about futures on bonds on the same yield curve (but as I said, different duration and yield than the underlying of my option).

EXTRA -> Let me add some real-life examples: Italy Govies market
Only one future is really liquid (CTD is some 10yr benchmark bond Italian govy). This future is liquid only at 2 IMM dates (say March and June). Options on this future are liquid only at the same dates (say March/March and June/June). Now, if you ask me to write a bond option on the CTD, I got everything I need. Maybe I'll have to interpolate here and there for the exact maturity and strike, but still. What if you ask me to write an option on the 5yr benchmark? No futures on that bond (therefore no options). But I would like to come up with a price based on a replication strategy (as I said, straight Black-like assumptions) involving only those liquid futures.

  • Advertisement

ABOUT WILMOTT

PW by JB

Wilmott.com has been "Serving the Quantitative Finance Community" since 2001. Continued...


Twitter LinkedIn Instagram

JOBS BOARD

JOBS BOARD

Looking for a quant job, risk, algo trading,...? Browse jobs here...


GZIP: On