Delta of forward and future
- Thread starter Ened
- Start date
The futures settles daily, but the forward does not.
See Hull 5.2 for why the futures trader (who settles daily) is slightly more volatile than the forward trader, for equivalent position: If you go long a futures contract in #X contracts @ Y$ and i do the same but in #X forward contracts @ Y$, a price gain on the spot of Y + Z$ implies you will gain by present value (PV) of +Z$ but I will gain by future value (FV) of +$Z. My forward gain, FV(+Z) is slightly less than < your futures gain, PV (+Z). The difference here is due (really) to time value of money: 1 versus exp[rT] is difference between today's dollar and FV (dollar).
Two other thoughts that occur to me, if they are helpful:
1. Did i mention the first derivative is helpful in the FRM? I am grateful you have found yet another application!
you can also figure this with first derivative:
Consider forward:
price[forward] = Spot - [Delivery]*EXP[(-r)(T)]
now take derivative with respect to Spot:
p'[f] = 1.0; i.e., change in price with respect to change in spot
Consider future:
price[futures, per cost of carry] = Spot*EXP[(r)(T)]
now dPrice/dSpot =
EXP[(r)(T)]
2. This daily settlement also arises in INTEREST RATE futures versus forward. See the AIM about "convexity bias" which speaks to difference btwn i rate forward/futures. A SIMILAR dynamic: the futures (e.g., Eurodollar futures) settles daily so that trader, in receiving excess margin or having margin calls, is slightly more volatile than a forward position without daily settlement.
David
See Hull 5.2 for why the futures trader (who settles daily) is slightly more volatile than the forward trader, for equivalent position: If you go long a futures contract in #X contracts @ Y$ and i do the same but in #X forward contracts @ Y$, a price gain on the spot of Y + Z$ implies you will gain by present value (PV) of +Z$ but I will gain by future value (FV) of +$Z. My forward gain, FV(+Z) is slightly less than < your futures gain, PV (+Z). The difference here is due (really) to time value of money: 1 versus exp[rT] is difference between today's dollar and FV (dollar).
Two other thoughts that occur to me, if they are helpful:
1. Did i mention the first derivative is helpful in the FRM? I am grateful you have found yet another application!
you can also figure this with first derivative:Consider forward:
price[forward] = Spot - [Delivery]*EXP[(-r)(T)]
now take derivative with respect to Spot:
p'[f] = 1.0; i.e., change in price with respect to change in spot
Consider future:
price[futures, per cost of carry] = Spot*EXP[(r)(T)]
now dPrice/dSpot =
EXP[(r)(T)]
2. This daily settlement also arises in INTEREST RATE futures versus forward. See the AIM about "convexity bias" which speaks to difference btwn i rate forward/futures. A SIMILAR dynamic: the futures (e.g., Eurodollar futures) settles daily so that trader, in receiving excess margin or having margin calls, is slightly more volatile than a forward position without daily settlement.
David
Ened,
Yes, because in John Hull, the PRICE of future/forward is same = S0*EXP([rate + storage - convenience - yield](T))
In terms of the cost of carry model, Hull gives no difference. You are correct.
For pricing forward/futures, don't worry about the difference. I was trying to explain why delta is different.
The VALUE of a forward = S - K*EXP[(-r)(T)]
(I think I have that in the notes, but under orange AIM b/c it was an AIM last year but dropped this year. You don't necessarily need to worry about this and my explain above, I was trying to help with the "why")
David
Yes, because in John Hull, the PRICE of future/forward is same = S0*EXP([rate + storage - convenience - yield](T))
In terms of the cost of carry model, Hull gives no difference. You are correct.
For pricing forward/futures, don't worry about the difference. I was trying to explain why delta is different.
The VALUE of a forward = S - K*EXP[(-r)(T)]
(I think I have that in the notes, but under orange AIM b/c it was an AIM last year but dropped this year. You don't necessarily need to worry about this and my explain above, I was trying to help with the "why")
David
Thanks for the effort, but I am still confused. I worry about the difference because I feel I'm losing grip understanding the basics.
Please follow my train of thoughts:
delta = d price / d S
price for both is (in Hull and in your notes):
price = S*EXP[(r)(T)]
So both have delta of EXP[(r)(T)]. End (for me)... But why is this wrong?
(I don't like if formulas are correct in some cases, but not in other cases. There should be a unifying framework in my opinion).
Further, I don't see where this is coming from:
price[forward] = Spot - [Delivery]*EXP[(-r)(T)]
Is that somewhere in the notes? Why does it change? It is not consistent with other notation?You always mention that forwards and futures are identical except that futures are standardized contract and futures trade on markets. If that is the only difference why should the formula be that much difference? What do you mean with Delivery in the formula above? Why do you relate it to value of forward? That is not related to my problem, right?
Finally, your very first explanation is confusing to me too. But maybe first you can answer the questions above.
I'm really sorry to bother you with this!!!
Please follow my train of thoughts:
delta = d price / d S
price for both is (in Hull and in your notes):
price = S*EXP[(r)(T)]
So both have delta of EXP[(r)(T)]. End (for me)... But why is this wrong?
(I don't like if formulas are correct in some cases, but not in other cases. There should be a unifying framework in my opinion).
Further, I don't see where this is coming from:
price[forward] = Spot - [Delivery]*EXP[(-r)(T)]
Is that somewhere in the notes? Why does it change? It is not consistent with other notation?You always mention that forwards and futures are identical except that futures are standardized contract and futures trade on markets. If that is the only difference why should the formula be that much difference? What do you mean with Delivery in the formula above? Why do you relate it to value of forward? That is not related to my problem, right?
Finally, your very first explanation is confusing to me too. But maybe first you can answer the questions above.
I'm really sorry to bother you with this!!!
Ened,
No problem. You are in control of the basics. You are just hovering on a subtle point (convexity bias) and I am too cute with the 1st derivative.
Not this:
price[forward] = Spot - [Delivery]*EXP[(-r)(T)]
But rather:
VALUE[forward] = Spot - [Delivery]*EXP[(-r)(T)]
which is from
VALUE (not price) of Forward = (F0 - K)*EXP[(-r)(T)]
PRICE = F0, which to your point = S0*EXP[(r)(T)]
e.g., you go long an oil contract with delivery (K) at $150. You enter a fair contract, so net VALUE to you at inception is about 0. Time passes. Oil spot goes up +$10, so figure Forward Price goes up $10*EXP[whatever]. As price - delivery (K) is something like 160-150, the *VALUE* of your contract is +$10 (+/-)
The VALUE of the forward is not an 2008 AIM, so it is included only as orange AIM
(was tested last year, not this year. I fear i confused by introducing it)
Back to your train:
"delta = d price / d S
price for both is (in Hull and in your notes):
price = S0*EXP[(r)(T)]
So both have delta of EXP[(r)(T)]. End (for me)… But why is this wrong?"
This is not wrong. This is correct. And, more importantly, the cost of carry model you site is the controlling equation. It is the testable equation. Before getting into the weeds, let's not forget 1.0 and EXP(r) are nearly the same! This convexity bias is irrelevant for short periods.
you hold corn futures at $7, futures price goes up +$1. You have EXCESS MARGIN; an extra dollar in your pocket today. Invest at r and you have ($1)*EXP(rT) at the end of (T)
now say you hold corn forward at $7, forward price goes up $1. No dollar today. Go forward (t) and you get +$1 at time +T. See the difference? Because forward doesn't daily settle, you make slightly more with the futures contract [+1$*exp(rT)] because your future is "a bird in the hand" but your forward must wait for its gain. But it cuts both ways, so, best is: future is more volatile due to mark-to-market.
Again, i offer it is the same point as the "Convexity bias" in the context of a Eurodollar future versus forward: the Eurodollar futures holder is immediately receiving/losing cash due the DAILY SETTLEMENT - he/she is *slightly* more volatile literally "at the margin" due to the impact on the margin account - this makes for slightly higher delta (but both are basically 1.0, right?) for the futures contract. Actually this daily settlement has two implications:
1. Delta slightly higher for future vs. forward
2. Prices technically are not the same (i.e., the Eurodollar futures rate will be higher than a forward by the convexity bias; a commodity future will be higher/lower than the forward, probably higher, b/c the future holder invests margin dollars)
My (1) and (2) violate our premise, that future and forward are identical. But it is okay:
1. The cost of carry is a pricing framework, which links the spot to the forward/future price. In the context of an unrealistic, theoretical model, we can treat the future = forward .
2. At the same time, the introduction of exogenous impacts (supply/demand) or features (daily settlement) will change the basic pricing model.
2a. as we've established on the "normal contango" thread, supply/demand can cause the forward price to trade cheap/rich relative to the model - we can do one of two things knowing supply/demand is real: try to get it into the model, or realize it will create a difference between our price and observed
2b. daily settlement is a technical feature that gives rise to a difference. The COC that you write is CORRECT and *implicitly* assumes daily settlement. But the forward does not settle daily and this "technical feature" creates a small difference between the two. Hull somewhere writes (i don't have texts with me, i am remote today) that it doesn't matter in the short run only for longer terms (which is the same point to be made about Eurodollar convexity bias: irrelevant in the short run). Note this difference in delta goes to zero for shorter (T).
It occurs to me, this moment, that another way to view this "convexity adjustment" is: it accounts for the *liquidity* difference between a future and forward contract. Maybe it is not justified by the formulas, but daily settlement gives the future more liquidity. Sorry for length, I am without my tools today!
David
No problem. You are in control of the basics. You are just hovering on a subtle point (convexity bias) and I am too cute with the 1st derivative.
Not this:
price[forward] = Spot - [Delivery]*EXP[(-r)(T)]
But rather:
VALUE[forward] = Spot - [Delivery]*EXP[(-r)(T)]
which is from
VALUE (not price) of Forward = (F0 - K)*EXP[(-r)(T)]
PRICE = F0, which to your point = S0*EXP[(r)(T)]
e.g., you go long an oil contract with delivery (K) at $150. You enter a fair contract, so net VALUE to you at inception is about 0. Time passes. Oil spot goes up +$10, so figure Forward Price goes up $10*EXP[whatever]. As price - delivery (K) is something like 160-150, the *VALUE* of your contract is +$10 (+/-)
The VALUE of the forward is not an 2008 AIM, so it is included only as orange AIM
(was tested last year, not this year. I fear i confused by introducing it)
Back to your train:
"delta = d price / d S
price for both is (in Hull and in your notes):
price = S0*EXP[(r)(T)]
So both have delta of EXP[(r)(T)]. End (for me)… But why is this wrong?"
This is not wrong. This is correct. And, more importantly, the cost of carry model you site is the controlling equation. It is the testable equation. Before getting into the weeds, let's not forget 1.0 and EXP(r) are nearly the same! This convexity bias is irrelevant for short periods.
you hold corn futures at $7, futures price goes up +$1. You have EXCESS MARGIN; an extra dollar in your pocket today. Invest at r and you have ($1)*EXP(rT) at the end of (T)
now say you hold corn forward at $7, forward price goes up $1. No dollar today. Go forward (t) and you get +$1 at time +T. See the difference? Because forward doesn't daily settle, you make slightly more with the futures contract [+1$*exp(rT)] because your future is "a bird in the hand" but your forward must wait for its gain. But it cuts both ways, so, best is: future is more volatile due to mark-to-market.
Again, i offer it is the same point as the "Convexity bias" in the context of a Eurodollar future versus forward: the Eurodollar futures holder is immediately receiving/losing cash due the DAILY SETTLEMENT - he/she is *slightly* more volatile literally "at the margin" due to the impact on the margin account - this makes for slightly higher delta (but both are basically 1.0, right?) for the futures contract. Actually this daily settlement has two implications:
1. Delta slightly higher for future vs. forward
2. Prices technically are not the same (i.e., the Eurodollar futures rate will be higher than a forward by the convexity bias; a commodity future will be higher/lower than the forward, probably higher, b/c the future holder invests margin dollars)
My (1) and (2) violate our premise, that future and forward are identical. But it is okay:
1. The cost of carry is a pricing framework, which links the spot to the forward/future price. In the context of an unrealistic, theoretical model, we can treat the future = forward .
2. At the same time, the introduction of exogenous impacts (supply/demand) or features (daily settlement) will change the basic pricing model.
2a. as we've established on the "normal contango" thread, supply/demand can cause the forward price to trade cheap/rich relative to the model - we can do one of two things knowing supply/demand is real: try to get it into the model, or realize it will create a difference between our price and observed
2b. daily settlement is a technical feature that gives rise to a difference. The COC that you write is CORRECT and *implicitly* assumes daily settlement. But the forward does not settle daily and this "technical feature" creates a small difference between the two. Hull somewhere writes (i don't have texts with me, i am remote today) that it doesn't matter in the short run only for longer terms (which is the same point to be made about Eurodollar convexity bias: irrelevant in the short run). Note this difference in delta goes to zero for shorter (T).
It occurs to me, this moment, that another way to view this "convexity adjustment" is: it accounts for the *liquidity* difference between a future and forward contract. Maybe it is not justified by the formulas, but daily settlement gives the future more liquidity. Sorry for length, I am without my tools today!
David
Ened,
Apologies to add a philosophical twist to a tactical quesiton, but this all remind me of how limited is the cost of carry model. It is ironic but:
F0 = S0 * EXP[(r)(T)]
from a practical standpoint is everywhere wrong, do you know what i mean?
It's wrong for corn: doesn't incorporate seasonality
It's wrong for oil futures: doesn't incoporate the long term backwardation historically typical given supply/demand
It implicity makes an assumption about daily settlement
So,
on the one hand, know the model. Like CAPM, Black-Scholes, it it will be tested
on the other hand, all of these models are "broken" because they willfully omit either exogenous impacts like supply/demand, taxes, behavioral biases or feature differences. So, consider them gross simplifications. Take corn futures, we should probably try and "fix" the model for seasonality, right? Somebody has probably done that...and, strictly strictly speaking, it is a "bit off" in regard to a forward, as the forward is less liquid. But we round them into one as it doesn't matter in the short.
for what it's worth...David
Apologies to add a philosophical twist to a tactical quesiton, but this all remind me of how limited is the cost of carry model. It is ironic but:
F0 = S0 * EXP[(r)(T)]
from a practical standpoint is everywhere wrong, do you know what i mean?
It's wrong for corn: doesn't incorporate seasonality
It's wrong for oil futures: doesn't incoporate the long term backwardation historically typical given supply/demand
It implicity makes an assumption about daily settlement
So,
on the one hand, know the model. Like CAPM, Black-Scholes, it it will be tested
on the other hand, all of these models are "broken" because they willfully omit either exogenous impacts like supply/demand, taxes, behavioral biases or feature differences. So, consider them gross simplifications. Take corn futures, we should probably try and "fix" the model for seasonality, right? Somebody has probably done that...and, strictly strictly speaking, it is a "bit off" in regard to a forward, as the forward is less liquid. But we round them into one as it doesn't matter in the short.
for what it's worth...David
Very clear, David!
One thing remains unclear. You write:
"Not this:
price[forward] = Spot - [Delivery]*EXP[(-r)(T)]
But rather:
VALUE[forward] = Spot - [Delivery]*EXP[(-r)(T)] "
So that means your "proof" under point 1 in your first answer was wrong. Right?
If so, is there a similar "proof" which is correct?
One thing remains unclear. You write:
"Not this:
price[forward] = Spot - [Delivery]*EXP[(-r)(T)]
But rather:
VALUE[forward] = Spot - [Delivery]*EXP[(-r)(T)] "
So that means your "proof" under point 1 in your first answer was wrong. Right?
If so, is there a similar "proof" which is correct?
Hi Ened,
I wrote that 1st proof wrong, that was sloppy, should be:
Value [forward] = Spot - Discounted Delivery
So the proof itself is still okay; i.e., V'[forward] = dV/dS = 1.0. And, you will rightly ask, how is the future different. And we are down the rabbit hole because (i) the forward/future are treated the same in these Hull readings (yet technically there is a tiny theoretical difference btwn them, which Hull full knows, and is ignoring for purposes of cost of carry] but yet (ii) he acknowledges the difference in regard to delta (a 1st derivative). Put another way, he could have given either 1.0 or EXP[rT] for a single delta and he'd have been "correct" and consistent with the simplifying assumption (i.e., to treat them identically). But he acknowledges their subtle difference at delta, not having acknowledged it before that.
I think the first point is simply to see that a future settles daily so the +$1 is a future change (FV change, a TVM term not an instrument term!) of +$1*exp[rT] while the forward does not settle daily, so it is a FV change of +$1. If you see that fundamental difference, you probably are okay to entertain they have tiny price/delta differences that, in the short run, don't impact cost of carry.
David
I wrote that 1st proof wrong, that was sloppy, should be:
Value [forward] = Spot - Discounted Delivery
So the proof itself is still okay; i.e., V'[forward] = dV/dS = 1.0. And, you will rightly ask, how is the future different. And we are down the rabbit hole because (i) the forward/future are treated the same in these Hull readings (yet technically there is a tiny theoretical difference btwn them, which Hull full knows, and is ignoring for purposes of cost of carry] but yet (ii) he acknowledges the difference in regard to delta (a 1st derivative). Put another way, he could have given either 1.0 or EXP[rT] for a single delta and he'd have been "correct" and consistent with the simplifying assumption (i.e., to treat them identically). But he acknowledges their subtle difference at delta, not having acknowledged it before that.
I think the first point is simply to see that a future settles daily so the +$1 is a future change (FV change, a TVM term not an instrument term!) of +$1*exp[rT] while the forward does not settle daily, so it is a FV change of +$1. If you see that fundamental difference, you probably are okay to entertain they have tiny price/delta differences that, in the short run, don't impact cost of carry.
David
Ened,
No, proof cannot be wrong, because i just intergrated from his answer of 1.0. Must be = Spot + constant. So, i just used his answer to infer his premise. Seriously, this is all in John Hull's lap, i was trying to help you interpret him.
delta is dP/dS or dV/dS. Wilmott uses dV/dS. I would not get to literal about delta - it is just the first term in truncated Taylor series. delta can be applied to other financial instruments. (note: that's what we are talking about here-yes? Delta for futures/forwards where otherwise delta connnotes options). To illustrate, duration could have been called delta, as they are both 1st term in taylor series, but it would have been confusing. So, yes, by all means delta is dP/dS.
and, again i'll say the key point here is daily settlement. It occurs to me this may be why Hull introduced the forward/future distinction at delta rather than before, merely to make the daily settlement point.
David
No, proof cannot be wrong, because i just intergrated from his answer of 1.0. Must be = Spot + constant. So, i just used his answer to infer his premise. Seriously, this is all in John Hull's lap, i was trying to help you interpret him.
delta is dP/dS or dV/dS. Wilmott uses dV/dS. I would not get to literal about delta - it is just the first term in truncated Taylor series. delta can be applied to other financial instruments. (note: that's what we are talking about here-yes? Delta for futures/forwards where otherwise delta connnotes options). To illustrate, duration could have been called delta, as they are both 1st term in taylor series, but it would have been confusing. So, yes, by all means delta is dP/dS.
and, again i'll say the key point here is daily settlement. It occurs to me this may be why Hull introduced the forward/future distinction at delta rather than before, merely to make the daily settlement point.
David
Hi, David,
After reading all your posts, I have two question:
1. the delta of the future contract can be greater than one?
2. I want to try to describe how future and forward contract works as time passes:
Suppose we setup the future and forward contract at time 0. Now we are at time t and the maturity date is T.
With all condition beting the same, a forward and a future have the same price K at time 0.
After time 0, suppose we at time t. If we want to buy the forward contract we must pay the amount of (F0-K)*exp(-rT), which is the value of the forward and we also need to pay K at time T to get the underlying asset.
For the future, if we want to buy it, we only need to pay the initial margin at time t, maintain it , and pay St*exp(r*(T-t)) at time T(or still pay K?). In other words, will the delivery price of future contract change.
Thanks for your help!
After reading all your posts, I have two question:
1. the delta of the future contract can be greater than one?
2. I want to try to describe how future and forward contract works as time passes:
Suppose we setup the future and forward contract at time 0. Now we are at time t and the maturity date is T.
With all condition beting the same, a forward and a future have the same price K at time 0.
After time 0, suppose we at time t. If we want to buy the forward contract we must pay the amount of (F0-K)*exp(-rT), which is the value of the forward and we also need to pay K at time T to get the underlying asset.
For the future, if we want to buy it, we only need to pay the initial margin at time t, maintain it , and pay St*exp(r*(T-t)) at time T(or still pay K?). In other words, will the delivery price of future contract change.
Thanks for your help!
Hi jcjc0602,
1. Yes, as the futures price = F0 = S*EXP[rt], the delta = dF/dS = EXP[rT] (i.e., F=aS so dF/dS = a) so the delta is slightly greater than 1.0. (Hull says the delta of a forward is 1.0 where the key difference is daily settlement: if the spot is +$1, assuming a flush margin account, only the futures contract gives you the $1 today)
2. Starting with your final sentence, the delivery price is like the vanilla strike price on a stock option, it does not change: K is constant for the forward/futures.
At time 0:
delivery = K
assuming a fair deal, futures price F0 = K, such that
per the fair deal value (f) = 0
forward to time t - x days:
delivery = K (unchanged)
but futures price like changed: F(t-x) = ?
and now value is non-zero: [F(t-x) - K]*EXP(-rT)
at settle, assuming future price converges to spot:
delivery = K (unchanged)
F(t) ~= S(t)
value is "intrinsic value": [F(t) - K]*EXP(-rT) but expiring so T=0 , so value = F(t) - K = S(t) - K
i.e., the long is always promising to pay the fixed K to recieve the unknown value of S(t) and the F(x) is changing over time
Hope that helps, David
1. Yes, as the futures price = F0 = S*EXP[rt], the delta = dF/dS = EXP[rT] (i.e., F=aS so dF/dS = a) so the delta is slightly greater than 1.0. (Hull says the delta of a forward is 1.0 where the key difference is daily settlement: if the spot is +$1, assuming a flush margin account, only the futures contract gives you the $1 today)
2. Starting with your final sentence, the delivery price is like the vanilla strike price on a stock option, it does not change: K is constant for the forward/futures.
At time 0:
delivery = K
assuming a fair deal, futures price F0 = K, such that
per the fair deal value (f) = 0
forward to time t - x days:
delivery = K (unchanged)
but futures price like changed: F(t-x) = ?
and now value is non-zero: [F(t-x) - K]*EXP(-rT)
at settle, assuming future price converges to spot:
delivery = K (unchanged)
F(t) ~= S(t)
value is "intrinsic value": [F(t) - K]*EXP(-rT) but expiring so T=0 , so value = F(t) - K = S(t) - K
i.e., the long is always promising to pay the fixed K to recieve the unknown value of S(t) and the F(x) is changing over time
Hope that helps, David
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