anyone understood equation 36-5, pg 400, CFAI 4. Fo = Eo(St) e ^(r-alpha)T The equation for forward uses current spot prices. but this one uses expected spot price at time T. on pg 398, they say Fo = So e ^(r-alpha)T and obviously So is not equal to St… so eq 36-5 is wrong but the way they have derived it makes complete sense… confusing.
No, So is the Current Spot Price today. EoSt is the Expected Spot Price at Time t. So in other words say Gold is around $920 today. I Expect the price of Gold to be $875 1 week from today. What would be the 1M future 1 week from now based on my expected spot price. OR…you can look at it as if I have a Futures price I can back into it to see what the Expected Spot Price would be based on the No-Arb formula.
Those equations aren’t right (alpha is risk premium, no?). Are you sure that’s what the book says? So the equations are really talking about very different things. Start with the second. That’s the no-arbitrage formula F=S*exp(r*t). That says that if I want St i can either by it now or invest in risk free deposits and buy Ft. Since those give me the same things at t, they must be equivalent now. The first formula should be Fo = Eo(St) e ^(-r-alpha) t. This one is tougher and it’s not even necessarily true. E0(St) is the expectation at time 0 of the stock at time t. This is based on some probability distribution you will never know, can’t observe even in the rearview mirror, and people argue about all the time. So if you have the expected value of St presumably a fair price is to discount that by the risk-free rate + risk premium that compensates you for taking on risk. That’s a common finance concept but it’s not useful for pricing the forward contract when the no-arbitrage formula works (because you can make a risk-free hedge). This formula though is an important part of talking about normal backwardation.
in cfai book, the fomula it gave is indeed the face value of forward price at time T (not 0). it goes like: Fo = cost of cash-and-carry = F(face value of forward price) * exp(-rT) (1) on the other hand, forward price should equal to “uncertain” expected spot price at T discounted by alpha to compenstate the uncertainty. Fo = E(S) * exp(-alpha*T) (2) since (1) and (2) should be the same you get F(face value of forward price) = E(S) * exp(r - alpha)T (3) ==== (3) is the fomula bips got ==== but, if we can remove the uncertainty, then alpha becomes r (risk-free), and combine (1) and (3), we got a formula we all familiar, Fo = E(S) * exp(-rT) (4) (note: schweser follows a similar logic on this subject book4 p56-57)
rand0m Wrote: ------------------------------------------------------- > in cfai book, the fomula it gave is indeed the > face value of forward price at time T (not 0). > > it goes like: > > Fo = cost of cash-and-carry = F(face value of > forward price) * exp(-rT) (1) > Yes my equation 1 above > on the other hand, forward price should equal to > “uncertain” expected spot price at T discounted by > alpha to compenstate the uncertainty. > > Fo = E(S) * exp(-alpha*T) (2) Right in this case you have alpha as some # greater than r. Above we have alphais risk premium (on a technical thing - T should be ‘t’ or you are usually taking about a random time and then we have more work to do). > > since (1) and (2) should be the same > > you get > > F(face value of forward price) = E(S) * exp(r - > alpha)T (3) > > ==== (3) is the fomula bips got ==== > > but, if we can remove the uncertainty, then alpha > becomes r (risk-free), and combine (1) > and (3), we got a formula we all familiar, > > Fo = E(S) * exp(-rT) (4) > > (note: schweser follows a similar logic on this > subject book4 p56-57) Right, but that “if we can remove uncertainty” is just weird. If we remove uncertainty, then the distribution of St is one point. Then the formula becomes even simpler because if the distribution is at one pt a.s. then E(St) = St. which gives us back the no-arbitrage formula.
agree, i dont think cfai wants to take us all the way to risk-neutral (brownian motion, etc), but still likes to show us there are a lot more yet to explore beyond L3.
sorry, still not very clear. can anyone please elaborate it a little more. I get confused when expected spot price comes into question.
The spot price t days from now is random. It has some distribution which, alas, I don’t know. This distribution is well enough behaved that it has an expectation. The expected spot price is just the expectation of that distribution (or to be more accurate the expectation of a rv with that dist’n).
its starting to make sense… can you please explain the equation using some numbers Fo = Eo(St) e ^(r-alpha)T
You can’t directly observe either E(St) or alpha so it’s a littel hard to put numbers in there.
ok… thank you… i’m understanding it now, still not proficient enough to explain it to someone else but its definitely clearer… thanks.
bips Wrote: ------------------------------------------------------- > its starting to make sense… can you please > explain the equation using some numbers > > Fo = Eo(St) e ^(r-alpha)T black-scholes-merton assumed the St follows a stationary log-normal process although it may not be true at all. but, at least, we can imagine log of St has a bell curve distribution and be happy. alpha would be very hard mathematically (at least to me) . it requires understanding of martingale, risk-neutral, etc. but, dont worry they are not for cfa l3. (may be l5)
Whoa - This is much more general than that. BSM says something like “if St has log-normal distn (or something that leads to that) then [blah, blah]”. The above equation is true for any distribution of St as long as it has an expectation.
as i said, that’s one explanation just in case someone likes to picture what distribution of St may look like. given the book used exp(-alpha*T), i assume the author might have that framework in mind too. but, it’s purely my speculation. in addition, i am not an expert on this at all, to be honest. if you happen to be one, i would like to learn from you. it will be a lot of fun! the only thing about distribution of St i have seen so far is something like (i could be wrong cause it’s out of memory) dSt = r*St*dt + sig * St * dWt; where Wt is standard brownian motion … as to if there are many other ways to explain movement of St, i really have no knowledge of (zero) …
But this is just one model for stock (or other security) prices which gives you this beautiful calculus and allows you to value everything in the planet. But it also says stuff like there are no jumps and all moments of St exist. Above all we require for that statement to be true is that E(S(t)) exists which is, of course, much lighter. If you want to learn all that stuff about martingales and stuff and you have a decent background in calc and probability, the books by Shreve are really great (#2 is much more useful IMHO than #1).
if i can pass l3, if i still have energy left afterward, if my gf still lets me , if … but, for sure, i will get hold of the book … it seems someone can download the book for free … anyway, thx
I get the equation below as the basis for the chapter. Fo = So e ^(r-alpha)T From the equation above you can just add costs (storage) or subtract benefits (convenience or leasing). And I understand the math in the beginning of the chapter they take the PV of the forward and rearragne the equatoins to get the Fo = Eo(St) e ^(r-alpha)T BUT IT IS KILLING HOW IN SCHWESER IT SAYS, ON PAGE 57 BOOK 4, "IF THERE IS NO UNCERTTAINTY (RISK) ASSCOISTED WITH St, THE EXPRESSOIN REVERTS BACK TO OUR ORIGINAL EXPRESSION: Fo = E(S) * exp(-rT) How can you disocunt by a negative interest rate? I understand that is the expected value of the spot so it really is the forward. I hope this is a moot point and not on exam. How much of this commodity forward infor are you guys banking on seeing on the exam?
you are not discounting by negative interest rate E(S) * exp(-rT) = E(S) / exp(rt)