Synthetic Forward Contract & Caps

[777888]

1) "Synthetic Forward Contract can be represented by the following 'equation':
forward contract = stock purchase + loan
This can also be written as
forward contract = stock purchase - zero coupon bond
since '-zero coupon bond' means shorting the bond, or equivalently, taking out a loan."

Now I don't understand why shorting a zero coupon bond is the SAME as taking out a loan. When you short a zero coupon bond, you are simply selling YOUR asset (it is an asset that you own), you do NOT necessarily have to buy it back later on. You can just walk away with the money and keep the money, you don't have to pay it back.

But when you are taking out a loan, they lend you the money now, but later on, you MUST repay them.

So I don't understand how shorting a zero coupon bond is the same as taking out a loan. In my understanding, they are just completely different (as I described above).



2) "cap = short an asset + long (purchased) call
i.e. a cap is the combination of (i) a short position in the asset, along with (ii) a purchased call option in the asset."

I am having some troubles visualizing the "cap". In the term "cap", what is the meaning of "short an asset"? When we say we "short an asset", we sell the asset right now and take the money, but do we necessarily have to buy back the asset later on? I think the answer to this is important because it will affect the payoff and profit diagrams of a cap.



I have been deeply confused by questions like these since my study of derivatives (the "before" and "after" positions seem to be very vaguely defined), and I am never able to find the answers to these even after reading carefully the textbook Derivatives Markets.

Could someone please help me out? Any help is greatly appreciated!

[Rake]

Whenever the term 'short' is used, it is saying that you're taking a position with regard to a price that will benefit from a drop in that price. Thus being short is different than simply selling the underlying asset. If you just flat out sold the asset at time zero then you're completely indifferent to the future price..

[ActuarialHeroOfTime]

Shorting a Bond is the same as taking out a loan. When a company issues (shorts) a bond, they receive cash equal to the ZC bond price (which in this case is just the discounted redemption value). At bond maturity, the company pays the redemption amount to the owner of the bond.

This would run exactly with taking out a loan. Receive cash now, but in the future you must pay the loan back. Either way youre taking a short position on cash which must be settled later, with the time value of money taken into account.

A good way to visualize synthetic positions is to combine payoff graphs of the individual assets in your replicating portfolio. Your long asset position + a short cash position resembles (=) the payoff graph of a long forward contract.

Same thing with the cap. Picture the payoff graphs. And yes, when you short a stock now, it must be purchased later to settle the short position. Youre speculation is that stock price will drop, and you will be able to buy it back at a cheaper price than you shorted it at. When youre short the asset, youre exposed to upside price risk (it will be more expensive when you try to buy it back later). The long call option provides insurance against upside price risk, so its used to hedge youre short asset position, and put a cap on youre max losses.

[777888]

First of all, I would like to thank the helpful comments! Things make a lot make sense now, but I have some further questions:

When a company issues (shorts) a bond, they receive cash equal to the ZC bond price (which in this case is just the discounted redemption value). At bond maturity, the company pays the redemption amount to the owner of the bond.

I agree with this, but instead of a company, say when you buy a zero coupon bond, and YOU later then sell YOUR bond which is YOUR asset, why are you always obligated to buy it back? What I am thinking is that you can just walk away with the money and keep the money, no? Just think in reality, when you sell something, you don't necessarily have to buy it back later on. This is about where I got confused...

So "short" ALWAYS means "short sell"??

A good way to visualize synthetic positions is to combine payoff graphs of the individual assets in your replicating portfolio

In general, for synthetic positions, if we can show that the payoffs diagrams at expiration of the LHS is equal to that of the RHS, MUST they be the same? (i.e. is that enough?)
Don't we have to consider PROFIT diagrams which also take into account the net initial investment amount, and prove that the profit diagrams of LHS = RHS?

Your long asset position + a short cash position resembles (=) the payoff graph of a long forward contract.

payoff diagram (at expiration) of long asset is an upward sloping line starting from origin with slope 1
But what is the payoff diagram of a "short zero coupon bond"??


Thank you very much!

[Rake]

Originally Posted by 777888

So "short" ALWAYS means "short sell"??

From what I've seen, yes.

Originally Posted by 777888

In general, for synthetic positions, if we can show that the payoffs diagrams at expiration of the LHS is equal to that of the RHS, MUST they be the same? (i.e. is that enough?)

I think we're also demonstrating that the timing of the cash flows are the same as well. I believe that if the investments have the same net pay off and the same timing, then they are equivalent.

Originally Posted by 777888

But what is the payoff diagram of a "short zero coupon bond"??

Because the value at maturity of the bond is completely independent of the change of price of the asset, on the payoff diagram, it will simply be a horizontal line parallel to the X axis. McDonald has one graphed on page 29.

[777888]

Originally Posted by Rake

From what I've seen, yes.

So I guess "short" always means "short selling", and not simply "sell". (in short sale, we must buy it back and COVER the short sale)

Consider, for example, "short forward contracts". For a short position in a forward contract, we are describing the seller who is obligated to sell and deliver the asset at expiration date (time T). But this isn't short sale. He is simply selling HIS asset at time T, and never has to buy it back.
But nevertheless the DM textbook classifies a short forward contract as a "short" position. In this situtation the usage of "short" seems to be a little different from "short sale" (in which you always have to BUY BACK what you've sold earlier).
So is "short forward" a "short sale"?

I think we're also demonstrating that the timing of the cash flows are the same as well. I believe that if the investments have the same net pay off and the same timing, then they are equivalent.

So I believe to show that the LHS and the RHS are equivalent investments, we have to compare the PROFIT diagrams of LHS and RHS. If all we have shown is that the payoff diagrams at expiry are the same, this still does NOT prove that the LHS and RHS are equivalent investments because it's possible that the LHS and RHS have different INITIAL investment amounts.

Because the value at maturity of the bond is completely independent of the change of price of the asset, on the payoff diagram, it will simply be a horizontal line parallel to the X axis. McDonald has one graphed on page 29.

So for SHORT bond, I think the horizontal line would be BELOW the x-axis.
But what is the value of the payoff? (i.e. how far below the x-axis?)


Thanks!

[ActuarialHeroOfTime]

Okay, for the differences between uses of 'short' for different instruments.. Lets say the company writes (shorts) the bond. You, the investor, who purchases the bond, has a long position in the bond. You paid money to the company today, and they gave you a piece of paper (bond) promising to pay you the redemption amount at contract maturity.

Now the confusion youre having with 'selling your asset and walking away' is that when you do this, what youre selling is your long position on the bond. You sell to Investor B the right to the redemption amount to be paid by the company who wrote the bond. The company maintains its short position. You sold youre long position at its current value. You did sell the bond, but you didnt short, or write, the bond. The company did.

Difference btw short forward and short sale. In a forward contract, long and short simply refers to who does what at contract maturity. The buyer is long. The seller is short.
If you enter a short sale: as an investor you borrow a stock now, sell it at its current price, invest the cash, in the future you purchase (take a long position) on the asset, and then return it to the broker you bought it from, settling youre short position, and hoping to profit from a decrease in price.

In general, Profit = Payoff + AV( Initial Cash Flow to enter position )

Youre question about whether having identical payoff graphs meant same initial cost is YES they must have the same cost. In studying MFE its called the Law of one price. But from an FM standpoint and thinking about a no arbitrage argument, if you had 2 portfolios that were priced differently and that you knew would have the same payoff in the future, then you could earn risk-free profit by buying the cheap portfolio and selling the expensive one.

If the redemption amount is $X on a ZC Bond, then the payoff of a short ZC bond would be a line at -X
Profit would be -X + AV(bond price), but since bonds are priced w the risk free rate, this should be zero.

[Rake]

As for short positions... Just remember the textbook definition: "A position is short with respect to a price if the position profits from a decrease in that price." Short forwards profit from a decrease in price because they've sold the asset for some higher, pre-determined price.

Originally Posted by 777888

So I believe to show that the LHS and the RHS are equivalent investments, we have to compare the PROFIT diagrams of LHS and RHS. If all we have shown is that the payoff diagrams at expiry are the same, this still does NOT prove that the LHS and RHS are equivalent investments because it's possible that the LHS and RHS have different INITIAL investment amounts.

Pardon if I was unclear, but when I say that the timing of the cash flows are identical, to me this says that they have the exact same payments at the exact same times. The same initial investment, the same redemption at expiry, the same coupon payments, everything.

Originally Posted by 777888

So for SHORT bond, I think the horizontal line would be BELOW the x-axis. But what is the value of the payoff? (i.e. how far below the x-axis?)

Since a short position is opposite of a long, the payoff would be .

The addition of the bond is simply a method of taking the time-value of money into account within the payoff diagram.

[777888]

Originally Posted by ActuarialHeroOfTime

Difference btw short forward and short sale. In a forward contract, long and short simply refers to who does what at contract maturity. The buyer is long. The seller is short.

But still, for short forward, the payoff at time T is (F_0,T - S_T) as stated in my textbook, i.e. forward price - spot price at time T. This seems to assume that at time T he sells the asset for F_0,T and at the same time buys it back at S_T, right? Does he have to necessarily buy it back? The formula (F_0,T - S_T) seems to suggest that he ALWAYS do buy it back. It looks like that there are a lot of assumptions in the calculation of "payoff".

In general, Profit = Payoff + AV( Initial Cash Flow to enter position )
Youre question about whether having identical payoff graphs meant same initial cost is YES they must have the same cost. In studying MFE its called the Law of one price. But from an FM standpoint and thinking about a no arbitrage argument, if you had 2 portfolios that were priced differently and that you knew would have the same payoff in the future, then you could earn risk-free profit by buying the cheap portfolio and selling the expensive one.

So to prove that two investments are equivalent, all we have to do is to draw the payoff diagrams at expiration date (time T) and show that they are the same, and this automatically implies that the initial investments (at time 0) are the same as well, am I right?
i.e. payoff diagrams at time T identical => profit diagrams are identical ?

If the redemption amount is $X on a ZC Bond, then the payoff of a short ZC bond would be a line at -X
Profit would be -X + AV(bond price), but since bonds are priced w the risk free rate, this should be zero.

OK, so I have drawn the payoff diagrams of LHS and RHS and visualized the following equation in the case of no dividends.
long forward = long stock + short zero coupon bond

But what if the stock pays dividends? Is the above equation still true in this case? If so, how can we visualize it?
I know the payoff diagram for long stock is an upward sloping line starting from origin with slope 1, but what about the other two in the case of the stock paying dividends?


Thank you very much! You have cleared a lot of my doubts!

[777888]

Originally Posted by Rake

As for short positions... Just remember the textbook definition: "A position is short with respect to a price if the position profits from a decrease in that price." Short forwards profit from a decrease in price because they've sold the asset for some higher, pre-determined price.

Thanks!

Actually, I am having some troubles with that definition and the definition of "long" that a long would profit when there is an increase in price. I don't agree with it.

e.g. Let F_0,T = forward price
S_o = spot price at time 0
S_T = spot price at time T
F_0,T=1020, S_o=1000, S_T=1010

long forward payoff = long forward profit = S_T - F_0,T = 1010-1020 = -10
In this case, the price definitely went up (1010->1010), but the long still loses money. Same thing with "short", when the price falls, this still does not necessarily mean that the short will make money. So is that definition problematic?