金融工程是一个多学科交叉的领域涵盖从财经，数学，统计，工程和数值计算方法。FE&RM Part 1 这门课的重点会放在是用简单随机模型来给包括股票，固定收益工具，信用以及抵押支持证券在内的各种资产类别衍生品证券定价。我们也将思考其中证券在金融危机中的角色。这门课程一个引人注目的特别栏目是对Emanuel Derman （著名“宽客”，畅销书“宽客人生”的作者）的一段采访。我们希望完成这门课的学生将能够开始理解金融工程背后的高深原理。但更重要的是，我们希望你们也能理解这一理论在实践上的局限性以及明白为什么总是应该用批判的眼光看待金融模型。接下来的FE&RM Part 2将会进一步深入学习衍生品定价模型但是它也会关注资产配置和资产组合优化以及其他金融工程的应用实例，像实物期权，商品衍生品，能源衍生品和程序化交易。
Forward Contracts
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來自 Columbia University 的課程
金融工程与风险管理，第 1 部分
1367 個評分
從本節課中
Introduction to Basic Fixed Income Securities
Review of interest and basic fixed income securities; introduction to arbitrage pricing.
與講師見面
Martin HaughCo-Director, Center for Financial Engineering
Industrial Engineering & Operations Research
Garud IyengarProfessor
Industrial Engineering and Operations Research Department
In this module, we are going to introduce you to the forward contract
and also how to use the no arbitrage principal to price forward contracts.
A forward contract gives the buyer the right and also
the obligation to purchase a specified amount of an asset.
At a specified time capital T, at a specified price capital
F, called the forward price, which is set at time T equal
to 0. Just to get the mechanics right,
here's time T equal to 0.
Here's time, the expiration time, capital T.
At time T equal to 0, you're specified at price capital F.
Which allows you to buy a specified quantity of an
asset at that price F, but at time capital T.
So you fix the price at time T equal to 0
for a purchase that happens at a later time, capital T.
They are many examples of forward contracts.
Forward contracts on delivery of a stock with a maturity of six months.
Forward contract for sale of gold with a maturity of one year.
Forward contract to buy ten million dollars
of Euros with a maturity of three months.
Forward contract for a delivery of a nine
month T bill with a maturity of three months.
In all of these cases, you are being quoted a price today to purchase
the quantity, which has been specified in the contract at a later time, capital T.
For example, in the forward contract for a delivery of a stock with
maturity six months, you've been quoted a price today for buying a stock
in six months.
Our goal in this set of slides is to set
the forward price capital F for a forward contract at time
T equal to 0 for one unit of an asset with
asset price ST at time T and maturity of capital T.
Let little fd denote the price or the value at
time little t for a long position of forward contract.
The value of the forward contract at time capital T is known.
F little t at time capital T is going to be
the price of the asset at time capital T as sub T.
This is the price of the asset
minus the forward price. This is the forward price.
And why is this exactly the value of the forward contract at time capital T?
This is because a long position in the forward,
you must purchase the asset at price capital F.
You can immediately sell that in the market at a
spot price of S capital T, so you make the difference.
If the forward price happens to be greater than S capital T, you make a loss.
If the forward price happens to be less than S capital T, you make a profit.
The forward price capital F is set in such a manner, that at time t equal
to 0, the value of the price of the forward contract f sub 0 is exactly equal
to 0.
F0, this is the price at time t equal to 0, and that must be set equal to 0.
Which means that you're indifferent between taking either a long
position or buying the contract or taking a short position or selling the contract.
The buyers and sellers are indifferent about taking any position.
Just to remind you, little f0 is going to
relate to the price of the value of the contract.
Capital F would relate to the forward price.
This is the full price that is set at time t equal to 0.
But the purchase is going to happen at time t equal to capital T.
We're going to use the no arbitrage principle to set the forward price
F.
In order to construct this price, I have
to get into a concept called short selling.
Short selling is the selling of shares in a stock that the seller doesn't own.
What happens is that the seller borrows the shares from a broker.
The shares come from the brokerage's own inventory.
The shares are sold and the proceeds are credited to the seller's account.
However, sooner or later, the seller must close the short
by buying back the shares. This is also called covering.
So you sell the shares in the market by borrowing it from the broker.
And then later on, you buy back from the market and give it back to the broker.
Why would one be interested in short selling?
Because one feels that the price of the asset is going to drop.
If the price of the asset drops, then you'd make a profit by short selling.
If the price of the asset goes up then a short sale results in a loss.
The short positions can be very risky and have to be handled very carefully.
The price can only drop to zero, so the potential
profit that you can get from a short position is bounded.
But the price can increase to arbitrary large values, and
therefore the potential loss from a short position is unbounded.
So if you're taking short positions, one has to be very careful about
how you manage those short positions and
the risks associated with those short positions.
We are going to use short selling as a technique for pricing formulas.
Suppose the asset has no intermediate cash flows.
For example, dividends, or storage costs, and so on.
Construct the following portfolio.
You buy the forward contract, short sell the
underlying, and lend S0 up to time capital T.
And let's look at one of the cash flows associated with this.
If you buy the contract, you can get into the
contract by paying nothing because f0 is equal to 0.
This is how it's constructed at
time capital T.
What are the cash flows associated with it?
It's equal to the price or the value of the
contacted time capital T, which is S capital T minus F.
You short sell the asset and buy back at time capital T.
What happens when you short sell the asset?
You get S0 into your account because you sold it.
You immediately, the funds get credited to your account.
You buy it back at capital T and
therefore you have a cash flow minus S capital T happening at time capital T.
Notice as zero is a price today, you know exactly what it's going to be.
S capital T is a random quantity.
It's some quantity that is going to be there in the future.
Why did I construct this particular object in my portfolio?
I put that because now this ST and that minus ST cancel each other.
Remember what we did
for the floating rate bonds.
We removed the randomness and that's we are doing here too.
We are removing the randomness.
You lend S0 up to time capital T, what happens now?
Minus S0 is the cash flow at time T equal to 0,
S0 divided by d(0, T) is the cash flow at time capital T.
This is just a fancy way of writing, S0 times 1 plus
r to the power capital T.
But because now the interest rate could
have a term structure associated with it, instead
of writing it as 1 plus r to the capital T, I'm just using the discount.
And that allows me to give you a more
general result, which doesn't rely on flat interest rates.
What is the net cash flow?
I paid nothing for this portfolio. What is the cash flow at time capital T?
It's S0 divided by d(0, T) minus F. The portfolio
has a deterministic cash flow at time capital T and the cost is
equal to 0, therefore, discounted, you should get price V equal to 0.
So S0 divided by D0 T minus F times D0 T.
Discounted back you end up getting equal to 0 and
therefore F is nothing but S0 divided by D0 T.
Now why is F strictly greater than the stock price?
It's because of cost of carry.
You are, you're not committing your cash for up to time capital T.
The opportunity cost associated with the, that cash is essentially 1 over
d(0,T), and that's why the F, which is the forward
price, which is the price at which you can purchase an
asset sometime in the future, is greater than the spot price.
Here's an example.
A forward contract on a non-dividend paying stock that matures in six months.
The current stock price is $50.
The six-month interest rate is 4% per annum.
Assuming semi-annual compounding. Why semi-annual?
Because the maturity is six months, so I'm taking, sort
of, the largest period that is associated with the problem.
What is the discount rate?
It's going to be 1 plus the annual interest rate divided by 2.
The discount rate turns out to be 0.9804 and therefore, F, which is the forward
price is going to be the spot price, 50 divided by 0.9804 and turns out to be $51.
Now what we want to do is construct the value
of the forward at some time T strictly greater than 0.
Remember this is the value in the price of a forward contract.
We already know that at time T equal to 0, F sub 0 is equal to 0.
At time T equal to capital T, F sub t is going to be s capital T minus F.
The spot price
at the expiration, capital D minus the forward price.
Now we want to construct F sub little t, which
for time t which is not 0 or capital T.
And we could do this using two different forward contracts.
So, let f 0 denote the forward price at time 0 for delivery at time capital T.
So, here's what is going to happen.
Here's my time t.
Here's time 0. Here time little t.
Here time capital T.
So F0 is a forward strike price that
is negotiated here for delivery at time capital T.
Ft is a forward price negotiation at time
little t also for delivery or expiration capital T.
Now we are going to again construct a portfolio.
We are going to short this contract.
I'm going to short the Ft contract.
What are the cash flows associated with that?
At time t equal to little t, because this contract has been struck at time
t equal to little t, I don't have to pay anything to take the short position.
At time T equal to capital T, I get F
sub little t minus St, notice, this is the
worst. Because
I take a short position.
I long, go long on a contract F0.
If I had struck this contract at time 0, I would have paid nothing for it.
Because the expiration at time capital T, but the
contract was struck at time T equal to 0.
However, now I'm buying this contract at time little t.
Its value, or its fair price is F sub little t.
And therefore, I have to pay that amount in order to get it.
So with the cash flow at time little t
is going to be minus Ft associated with this contract.
The cash flow at time capital T is going to be s capital T minus F0.
F0 is the forward price associated with the contract that was written at time 0
for expiration at time capital T. If you take the difference, ST cancels.
F little t minus f0 is a deterministic quantity
minus Ft is a deterministic quantity, and therefore you end
up getting that the price, ft, must be just
a discounted value of the cash flow in the future.
F capital T minus F0 times the discount
rate over the window little t to capital T.
Let me review what happened here.
We're constructing a portfolio using two different forward contracts.
The forward contract that I'm call, calling f sub 0, is a
contract that was written at time 0, for delivery at time capital T.
If I had gotten into this contract time T
equal to 0, I would have not paid anything.
However, in my portfolio I'm getting into this contract at time little t.
And therefore, I have
to pay the value or price of the contract at little t which is F sub little t.
I have another forward contract which was constructed
at time little t for expiration of capital T.
And for this contract, I know that the value is equal to 0 so I
can get into this contract or I could sell this contract at price equal to 0.
The payoff from this contract is going
to be something different.
It's going to depend on the forward price constructed at time little t.
I take both of these into my portfolio and I do
this in order to cancel out the random quantity S capital T.
The spot price of the underlying asset on which these forward contracts are written.
When you do the net cash flow, you end up
getting a deterministic quantity, and I know how to price that.
And that's how I end up computing the value of a forward contract at some
time T in the future, which is not the time at which it was struck.
And it's not the time at which it's expired.
