Chapter 5 - Financial Forwards and futures Flashcards
what components of a stock purchase are we intrested in
1) The price
2) The time of money transfer
3) the time of asset transfer
what do we pay, when do we pay, when do we receive?
name the possible ways to buy a stock
1) Outright purchase. We buy and transfer money and receive shares at the same time (immediately)
2) Fully leveraged purchase. pay later, receive now
3) Pre paid forward contract. Pay now, receive later
4) forward contract. pay later, receive later
elaborate on what e^rT represent and why we bother with it
it represent the result of running a compounding formula towards continuous compounding by making the intervals in which the rate is divided smaller and smaller.
(1+r/payments)^(paymentsT). when we let payments grow large, this formula will approach e^(rT).
WHY DO WE USE IT?
As it turns out, when we consider the time value of money, it is sort of the reality. The intuition is that when we consider stock returns, there is a certain return we “require” from CAPM or something similar. this return is typically in annual format. but the question is what we as investors should expect for smaller time frames?
The key is arbitrage. in order for no arbitrage to be present, certain conditions must apply. If I enter the stock at a certain time, how much return should I expect until the end of the year? Say I invest at half year. If the stock is extremely cheap/undervalued, and I know this, I would short the forward contract and use the proceeds to go long in the asset. At the end of the year, I owe asset to the counter party of the forward contract, so I give him my assets. but I acquired these assets for cheap, and sold for forward price, making risk free profit. I’d make the difference between the forward price and the stock price.
but the key is what sort of return should I expect for a short interval? And how should the price curve look like? The answer is that is needs to be structured so that the return in any interval is given as annualReturn/periods. for instance, this means that every day should expect the same return in percentage. If this was not the case, arbitrage would be on the table. And because we can further say that every hour, every second etc needs to have the same percentage return expected, we end up with the formula for continuous compounding. Therefore, using e^rT is the appropriate term to use to model financial time value.
what is the price we should pay for the fully leveraged purchase?
We acquire assets NOW, but pay later. This gives the original asset holder a disadvantage, because we can use the cash to our services in the meantime. In other words, duirng the period, we hold everything.
So, since we still hold the cash, and the assets have a original price of S_0, we can invest in risk free bonds and earn S_0e^rT. Recall euler because this is how the time value of money works. This is how compounding works.
without going into any detail, name the 3 methods we can use to derive the price for the pre paid forward contract
1) analogy
2) Discounted present value
3) pricing by arbitrage
elaborate on dtermining the price of pre paid forward by use of analogy
pre paid forward is sort of the opposite of the fully leveraged purchase. We have a duration of time where we have neither hte cash nor the assets.
if there are no dividends involved, the time of transfer doesnt matter. We end up with the same value regardless. We have already payed for it.
we could buy stock without forward contract, buy at S_0 and not receive dividends.
we could also buy pre paid forwrd, and not receive any dividends.
Therefore, the paypffs are hte same. Therefore, the price must be the same. Therefore, the price must be S_0. (in the absence of dividends).
elaborate on pricing hte prepaid forward by discounted present valuw
The idea is that if we take the value of the stock at time T, we can discount it using some appropriate rate, and arrive at the present value, which will be our price.
First issue: we dont know what the stock price is at time T, so we use expected value.
Second issue: we need alpha, the appropriate discount rate. We use CAPM.
F^P_{0,T} = E[S_T] e^(-aT)
By the definition of expected return, we can say that:
E[S_T] = S_0 e^(aT).
So, we can insert this, and get that:
F^P_{0,T} = S_0
Still in the absence of dividends
elaborate on prepaid forward pricing with dividends
We have already established that in the absence of dividends, the price ofthe pre paid forward is equal to the spot price. With dividends, there is a benefit in holding the assets.
We need to account for the size of the dividend, the frequency, and how the dividends grow over time when they are assumed to be immediately re-invested in the stock.
The effect of dividends is captured in the discrete case as:
∑PV_i(D_i)
Note that we are discounting, not compounding.
elaborate on pricing of prepaid contracts when there are continuous dividends
The output of divifdends for certain indices bahves more liek a continuous dividend. In order to not oay all the time, the index will automatically re-invest the proceeds into the weighted portions similar to the index. Therefore, we need to model indices as continuous dividends.
If the annualized dividend yield is ∂, we get that the dividend per day would be: ∂/365 x S_0. Assuming no price change.
If we get this dividend each day, it becomes closer to:
(1+∂/365)^(365 T) = e^(∂T).
Recall that e^∂T represent a multiplier of compound. it is a percentage that we multiply our cash with to see how much our position is worth in some time. Therefore, this number is actually a measure of how many shares we can buy from the dividends. If we originally own 1 share, we end up at e^∂T shares, whether that is an amount between 1 or 2, or 10 and 100000 etc.
This means, if we long the prepaid forward, we loose this amount of new shares. So what is a fair price for us to pay for the forward, when we have to give away the opportunity to grow our shares by e^∂T?
If we buy it for S_0 e^(-∂T), the other dude receive this amount, and he’d long the index. He’d do this to acquire the shares he need to fulfill his deal, and he will make, at the end, S_0 amount.
IMPORTANT: We dont need to consider the price of the index, because here is the key: We pay a certain amount of money, and the question is whether or not the market maker can make risk free profit. Can he go long using the cash he receive, acquire the shares necessary, and still obtain an additional amount that can be invested in risk free bonds?
This is why the prepaid price is S_0e^(-∂T). He will go long with this amount and receive 1 share at the end. but this depends on the dividend yield rate being always the same. If the annual rate is 5%, the daily must be strictly 0.05/365 and so on. Otherwise, the formula mess up.
elaborate on LEPO’s
Low exervise price options.
Typically used to avoid taxes and transaction fees.
The idea is to have so low exercise prices that it will always be in the money. it is simply a way to trade the stock with less fees.
The payoff becomes “S_T - K” for a call option.
what do we need to compute the forward price, and why?
We need the pre paid forward price.
Recall that we can price pre paid forwards on analogy, discounted present value, or arbitrage. If we use discounted present value, we use the expected value of the stock at time T to say that we want to discount (using stock’s appropriate rate) the expected stock price. And the expected stock price is given by the Spot price multiplied by the time value of money using the same rate (capm rate).
Then terms will cancel out and we end up at the spot price.
but why do we only need the prepaid forward price to find the forward price?
Now the payment is deferred, and instead of the counter party having the advantage of the cash, we now have it.
So, we use the future value formula on the prepaid forward price to arrive at the forward price.
and this is then typically: F_{0,T} = S_0e^(rT), when F^P_{0,T} = S_0.
We generalize it:
F_{0,T} = FV(F^P_{0,T}), this holds for any dividend payment as well. But there is a special case with contniuous dividend, in which case we use the formula:
F_{0,T} = e^(rT)S_0e^(-∂T) = S_0 e^((r-∂)T)
define forward premium
forward premium is defined as the ratio between the forward price and the spot price
give the expressions for the two terms used in forward premium
F_{0,T} = FV(F^P_{0,T}) = typically S_0 e^((r-∂)T)
F^P_{0,T} = typically S_0
elaborate on annualized forward premium
the formula is 1/T ln(F_{0,T} / F^P_{0,T})
when we insert values, we get:
(1/T) ln(e^((r-∂)T))
which gives:
(1/T) (r-∂)T
which gives:
r-∂
In other words, the annualized forward premium is r-∂ in the case of no dividends.
is it possible to have scenarios where we observe the forward price but not the spot price?
The exchange for the index is not the same as the exchange for the futures on the index. Due to time zones etc there are differences.
does the forward price predict the future spot price?
no. we expect alpha, not r-∂.
define a tailed position
adjusting hte investment amount in order to offset the effect of incoming interest or dividends
how can a market maker hedge himself against a forward contract position where he has the short side (index forward)
as a market maker, we want to hedge hte payoff of
S_T - F_{0,T}
S_T - S_0 e^(r-∂)T
how can we replicate this payoff?
go long in the index with S_0e^(-∂T), which gives us one share at expiration.
to avoid paying anything, we borrow. Borrow at rate r, means we need to borrow S_0e^(-∂T) at time 0, but we have to pay S_0e^(r-∂)T at time T.
So, we acquire the asset, sell the asset for S_T.
This is mainly about how we can replicate the payoff of a long forward contract by using stock and borrowing.
how can we use a forward to create synthetic stocks?
Stocks:
Go long the forward and lend the present value of the forward price. We will get forward price at expiration, at which point in time we use it to purchase the share from the forward. At this point, we have received a payoff by the difference between the current stock price and the future price we payed.
The key is that at expiration, we end up with payoff S_T, while we had a net negative of S_0e^(-∂T) because this is what we lended out.
Time 0: nothing in regards to the forward, but negative S_0e^(-∂t) as the lending amount.
Time T: pay the forward price: S_0e^((r-∂)T). Then receive the lended payment + interest: S_0e^(r-∂)T. Then we sell the asset to get its value: S_T.
So, the payoff is: S_T - S_0e^(-∂T)
but why is this a stock?
we would have borrowed to get the stock, then sold the stock at time T, and payed back the loan. We’d make it so that we end up at one share at expiration.
Therefore, we borrow S_0e^(-∂T).
+S_T
-S_0e^(r-∂)T
-S_0e^(-∂T)
+S_0e^(-∂T)
Bottom two cancel out, and we get:
= S_T - S_0e^(r-∂)T
This is the same payoff as before, so we have the synthetic stock equal to an actual stock.
define a cash and carry
a transacation where you buy the underlying and short the offsetting forward contract
how can we create a synthetic bond?
I assume we have sold a bond, and we need to create a synthetic position to offset its losses.
Selling a bond imply receiving cash now, and having to repay later. So for the buyer of the bond, they receive FV at time T, and give away FVe^(-rT) at time 0.
If I buy a bond, I borrow first, invest in bond, repay loan after getting face value.
Time 0:
+ FVe^(-rT)
- FVe^(-rT)
Time 1:
+FV
-FV
borrowing to invest in risk free bond is does nothing.
Instead we buy stock and short a forward.
Time 0:
idfk, they keep changing the rules
what is reverse cash and carry?
short the index and enter a long forward position
what is the benefit of including borrowing and lending in the payoff tables?
Doing see automatically includes present value calculations, which are necessary to spot arbitrage.
what is the cost of carry
For a stock index forward, it is defined as r-∂. The difference between risk free rate and the dividend yield. it is called cost of carry because we say that if you fund a long position in a stock by borrwoing, you have to pay the difference r-∂ on an ongoing basis.
