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Which interest rate to use when calculating greeks?
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I have a quick suggestion for you. Plug 0.25% and 8% into your model, see what the range on the greeks you care about is. With most of the stuff I have worked with (deltas on equities, indexes, commodities) it usually does not matter - the first few decimal places of the delta are the same.
Also I have a different opinion than DMO on which rate to use. When calculating risk numbers you probably want to use the rates that the market will use, not your personal rate. Do you want to hedge the market's value or your own private value? I think using your personal rate might be okay for holding something to maturity with no margin call possibility, but not if you care about hedging changes in the market price. But I think this is probably just an academic option geek debate unless the measures you care about actually do vary significantly with interest rates.
Also I have a different opinion than DMO on which rate to use. When calculating risk numbers you probably want to use the rates that the market will use, not your personal rate. Do you want to hedge the market's value or your own private value? I think using your personal rate might be okay for holding something to maturity with no margin call possibility, but not if you care about hedging changes in the market price. But I think this is probably just an academic option geek debate unless the measures you care about actually do vary significantly with interest rates.
Quote from black diamond:
I have a quick suggestion for you. Plug 0.25% and 8% into your model, see what the range on the greeks you care about is. With most of the stuff I have worked with (deltas on equities, indexes, commodities) it usually does not matter - the first few decimal places of the delta are the same.
Also I have a different opinion than DMO on which rate to use. When calculating risk numbers you probably want to use the rates that the market will use, not your personal rate. Do you want to hedge the market's value or your own private value? I think using your personal rate might be okay for holding something to maturity with no margin call possibility, but not if you care about hedging changes in the market price. But I think this is probably just an academic option geek debate unless the measures you care about actually do vary significantly with interest rates.
I agree that for most people in most situations, this is an academic argument. The differences will probably be too small to matter. But if you happen to be in a situation where it DOES matter, then you really, truly, absolutely need to use your own subjective interest rate.
Let me give you a nuts-and-bolts practical demonstration why.
Imagine you're trading T-bond futures and options. You are long 1000 conversions - long 1000 130 puts, long 1000 futures, and short 1000 130 calls. The options have exactly 1 year until expiration.
Are you delta neutral? NO!!!! Why not? Because if the futures drop a point, you will have to come up with $1,000,000 in variation margin (1 point in T-bond futures = $1000). If they drop 10 points, you will have to come up with $10,000,000 in variation margin.
Your clearing firm will probably be more than happy to lend you that - at an interest rate. If that interest rate is 5%, then if the futures drop 1 point and stay there for a year, it will cost you 5% of $1,000,000, or $50,000.
So in order to TRULY be delta neutral, you will need to be long 950 futures against those 1000 long puts and 1000 short calls - not 1000 futures. That way, if the futures drop a point and stay there until expiration, you will have made $1,000,000 on your options and lost $950,000 on your futures. The extra $50,000 you made will just pay for your interest on the variation margin. Overall, you'll break even. That's the true meaning of delta neutrality.
To make that calculation of course you have to know exactly what interest rate you will have to pay. If your interest rate is 6% then you would need to be long 940 futures, not 950. Big difference. And it doesn't matter what Coach's interest rate is, or Spin's, or the 10-year T-note rate. All that matters is what YOU will pay.
Now, it may surprise you to learn that your BS model prices all that in beautifully. If you go to http://www.sitmo.com/live/OptionVanilla.html and use their calculator (I chose that one because it lets you specify options on futures, which is consistent with our example), enter a futures price and strike price of 130, a volatility of .2, a time to maturity of 1 (1 year) and an interest rate of .05, you will see the following deltas for the 130 put and the 130 call:
130 call - .5135
130 put -.4377
Multiply those out by 1000 short calls (-1000 x .5135 = -513.5) and 1000 short puts (1000 x -.4377 = -437.7), add them together (-513.5 - 437.7 = -951.2) and you'll see that your BS model told you pretty much what I did - that you're going to have to be short 951 futures against those 1000 puts and calls in order to be delta neutral.
Now try it using an interest rate of 6% (.06). This time it tells you that your delta on the 1000 long puts and 1000 short calls combined is 941.8. Again, big difference depending on the interest rate.
And again, all that matters is YOUR interest rate.
Quote from dmo:
The options have exactly 1 year until expiration.
...
if the futures drop a point, you will have to come up with $1,000,000 in variation margin ...(1 point in T-bond futures = $1000). If they drop 10 points, you will have to come up with $10,000,000 in variation margin. ...
So in order to TRULY be delta neutral, you will need to be long 950 futures against those 1000 long puts and 1000 short calls - not 1000 futures. That way, if the futures drop a point and stay there until expiration, you will have made $1,000,000 on your options and lost $950,000 on your futures. The extra $50,000 you made will just pay for your interest on the variation margin. Overall, you'll break even. That's the true meaning of delta neutrality.
... it doesn't matter what Coach's interest rate is, or Spin's, or the 10-year T-note rate. All that matters is what YOU will pay.
... And again, all that matters is YOUR interest rate.
I have to think about that a little more, maybe your personal interest rate matters in your example, but I think I can convince you that the market rate matters too.
First, delta hedging is to hedge the market value of your option against small moves in the underlying over short time periods. Mark-to-market P/L, not hold-to-maturity P/L. Your example with a full year horizon is not really delta hedging - you would have to rebalance a real delta hedge many times over the period. And without doing any math I am pretty sure delta hedging does not cover a 1 pt move in treasury futures and definitely not a 10 pt move, you are talking gamma there.
Second, if you are really delta hedged, you are not going to have to meet a margin call after the next small move (a tick, not a point) in the underlying, so the interest on the margin over the next year argument is not relevant.
So for actual delta hedging, you care about the market price move of the option for the next small move in the underlying. What determines the updated option price after the underlying moves? The bid is the personal value of the option for the trader that values it most, and the ask is the personal value of the trader that values it least, your personal values don't matter unless you are one of those two. If you are one of those two and your personal value is much different from everybody else's, your impact on the price is very temporary. If everybody agrees on vol and there are no temporary liquidity or supply/demand issues (strong assumptions I admit), then the bid for a call is determined by the trader who can invest cash at the highest "safe" rate and the ask is determined by the trader who can borrow at the lowest rate. Vice-versa for a put. These are the guys who determine your mark-to-market performance and whether you get a margin call.
Also as an aside - interest rate options are not a very good example because the Black/Black Scholes models assume flat term structure and constant discount rates. They also give biased deltas when the discount rate is correlated with the price of the underlying. If you are really bored google "tailing the hedge". People use them anyway because they are usually close enough for normal people. But I get your point - you were looking for an example where interest rates mattered a lot.
I was smoking crack last night on the tailing the hedge part, I think the rest of that is right. Hedge tailing is only for when you get cash flows before expiration, like with futures marking to market.
It is pretty esoteric and doesn't really matter, but I have a funny reason for (sort of) remembering it. There was a guy I worked with a few years ago who was an derivatives know-it-all and we suspected he would just make stuff up if he didn't know something. We wanted to come up with something we were sure he didn't know to see what he did, and somebody asked him what tailing the hedge was and whether we did it or not and he made up something ridiculous and hilarious. After that it became an interview question.
It is pretty esoteric and doesn't really matter, but I have a funny reason for (sort of) remembering it. There was a guy I worked with a few years ago who was an derivatives know-it-all and we suspected he would just make stuff up if he didn't know something. We wanted to come up with something we were sure he didn't know to see what he did, and somebody asked him what tailing the hedge was and whether we did it or not and he made up something ridiculous and hilarious. After that it became an interview question.