Basis Risk
- Thread starter liordp
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Hi Convexity,
You are right about basis risk within the model. As basis variance = S^2 + F^2 -2*vol(S)*vo(F), under those assumptions (where vol(S) = vol (F), then basis risk = S^2 + S^2 - 2*S^2 = 2*S^2 - 2*S^2 = 0
... note memorization is not required if we remember Gujarati's variance: VAR(A-B) = VAR(A) + VAR(B) - 2*COV(A,B)
... and, btw, this is equivalent to saying the beta = 1.0, yes? beta = correlation * vol(F)/vol(S). In this case, beta = 1*1
The storage cost is very interesting ... but I would say: no, it makes no difference (within the theoretical model) because it does not itself alter the correlation of 1.0. As F = S*EXP((r+u)T), the delta of this futures contract (dF/dS) is EXP((r+u)T) such that storage implies a slightly higher delta (1.x) but this merely changes the slope of a line (F vs S) which remains a perfectly straight line and will therefore still have correlation of 1.0.
That's just within the static model. It is a huge difference from saying "storage does not impact basis risk;" e.g., this model has constant storage as %. Immediately after the "perfect" hedge, time enters, storages costs change, impacting both future/spot and the model's assumption will almost certainly be violated. In oil futures (eg), varying storage costs (in practice) impact basis, if for no other reason than they time-vary
David
You are right about basis risk within the model. As basis variance = S^2 + F^2 -2*vol(S)*vo(F), under those assumptions (where vol(S) = vol (F), then basis risk = S^2 + S^2 - 2*S^2 = 2*S^2 - 2*S^2 = 0
... note memorization is not required if we remember Gujarati's variance: VAR(A-B) = VAR(A) + VAR(B) - 2*COV(A,B)
... and, btw, this is equivalent to saying the beta = 1.0, yes? beta = correlation * vol(F)/vol(S). In this case, beta = 1*1
The storage cost is very interesting ... but I would say: no, it makes no difference (within the theoretical model) because it does not itself alter the correlation of 1.0. As F = S*EXP((r+u)T), the delta of this futures contract (dF/dS) is EXP((r+u)T) such that storage implies a slightly higher delta (1.x) but this merely changes the slope of a line (F vs S) which remains a perfectly straight line and will therefore still have correlation of 1.0.
That's just within the static model. It is a huge difference from saying "storage does not impact basis risk;" e.g., this model has constant storage as %. Immediately after the "perfect" hedge, time enters, storages costs change, impacting both future/spot and the model's assumption will almost certainly be violated. In oil futures (eg), varying storage costs (in practice) impact basis, if for no other reason than they time-vary
David
.... sorry to append, but I woke up thinking about this (dang you convexity!). Not to geek out but I think maybe i neglected a technical finesse. As the cost of carry says forward, with storage (u), is given by:
F0 = S0*EXP[(r+u)T], then assuming EXP[(r+u)T) is a constant = a where 1 must be greater than 1 b/c r > 0 and u > 0
variance(F0) = a^2*Variance(S0)
... this implies that variance(F) cannot quite equal variance (S) at least ex ante (it can still ex post)
... or put another way, and I am less than < 100% confident, it seems to me that just under the model, the incremental impact of + u (+storage) is to multiply the futures variance by EXP(u*2*t);
i.e., before storage, variance(F)
= variance(S)*[EXP(rT)]^2
after storage, variance(F)
= variance(S) * [EXP((r+u)T]^2
difference = EXP(2rT)/EXP(2*T*(r+u)) = EXP(2ut)
... so now i am thinking that additional storage cost, just in the model, while it ought to have no impact on correlation, ought to transmit a bit more volatility to the futures price and thusly introduce a bit of non-zero basis risk.
F0 = S0*EXP[(r+u)T], then assuming EXP[(r+u)T) is a constant = a where 1 must be greater than 1 b/c r > 0 and u > 0
variance(F0) = a^2*Variance(S0)
... this implies that variance(F) cannot quite equal variance (S) at least ex ante (it can still ex post)
... or put another way, and I am less than < 100% confident, it seems to me that just under the model, the incremental impact of + u (+storage) is to multiply the futures variance by EXP(u*2*t);
i.e., before storage, variance(F)
= variance(S)*[EXP(rT)]^2
after storage, variance(F)
= variance(S) * [EXP((r+u)T]^2
difference = EXP(2rT)/EXP(2*T*(r+u)) = EXP(2ut)
... so now i am thinking that additional storage cost, just in the model, while it ought to have no impact on correlation, ought to transmit a bit more volatility to the futures price and thusly introduce a bit of non-zero basis risk.
Hi David,
Very interesting - it's what I thought .. like you always say the basis risk is the mother of all risks and so Theoretically speaking i can say - Even if cofficient correlation between asset and contract equal one and the variance identical - Hypothetical there is some BASIS RISK
nice
Very interesting - it's what I thought .. like you always say the basis risk is the mother of all risks and so Theoretically speaking i can say - Even if cofficient correlation between asset and contract equal one and the variance identical - Hypothetical there is some BASIS RISK
nice
sucheta_isi
New Member
David,
Is there anything called "Variance of Basis"? If yes then how is it defined?
Thanks,
Is there anything called "Variance of Basis"? If yes then how is it defined?
Thanks,
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