金融工程讲义(上海财经大学)_1

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Consider a European call option; which confers on its owner the right to buy but not the obligation to buy one share of the asset at time one for the strike price K. Assumption 4: S1(T) (H) K if S S 0 K
(b)*The option will give
(c)*The money market give 1 (d)Money market return ((1+r)(1.20- V0)>0 <=(net profit)
(a*) and (b*)+(c*) offset each other with net profit (1+r)(1.20- V0)>0 at time one.
This cash position (or payoff of the proposed portfolio) is identical to buy a contract of a call option at strike price K=5. (S1(H)-5))+= 3 0
Assumption 3: r is also the borrow rate.
Definition 1: Arbitrage is a trading strategy that begins with no money, has zero probability of losing money and a positive probability of making money. Result 1:
Notation: (S1-K)+ =
denotes the profit of owning this option. The question an investor wants an answer is the price for owing such option at time zero.
The other direction is left as exercise.
< hint >: Let X0 be the wealth at time 0 for an investor. This investor can buy ∆ share of the asset at price S0 and invest the rest money for interest return. Let X1 be the wealth at time one following this strategy, then X1 = ∆ S1 + (1+r) (X0 - ∆ S0), where S1 = or X1 = ∆ H ∆ S T ∆ u ∆ d 1 1 1 1 r r X r X ∆ ∆ ∆ S when H occurs ∆ S when T occurs when H occurs when T occurs S 1
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Therefore, we have demonstrate that in the absence of arbitrage the option price has to be equal to V0=1.20
In the above operation, the implicit assumptions are: (1) Share of asset can be subdivided for sale (2) Interest rate are the same for borrowing and investing (3) The selling and buying stock are at the same price (NO bid-ask spread) (4) The asset price at time one can only take two possible values.
The initial wealth X0=1.20 is required to acquire the proposed portfolio. If the option price is higher than 1.20, an investor can sell this option and use $1.20 of it to construct the portfolio and the keep the difference as profit. If the option price V0 is lower than $1.20 (V0<$1.20), then (a) Short sell ∆0 share of asset to receive ∆0S0 = 2 (b) Use V0 buy the option (c) Deposit 0.8 in the money market (d) Deposit (1.20-∆0 ) into money market At time one: (a)*Need to buy -∆0 share of asset to return which can cost 3 0 ∆ ∆ 8 2 4 1
Consider a numerical example with S0= 4 S1(H)=8, 2 S1(T)=2 d= r=
Suppose the strike price of this European call is K=5.
Assumption 5: Initial wealth X0=1.20 and buy ∆0= shares of asset at time zero, i.e. requires to borrow $0.80 to do this. i.e. At time zero, the cash position is X0-∆0S0= -0.80 at time one, the cash position of this investment is (1+r)( X0-∆0S0)=-1 with the above specification. The stock value at time one can be either S1(H)=4 or S1(T)=1.
Question: How to determine the price V0?
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Strategy using existing instruments (money market + the asset market)
With initial wealth X0 and buy ∆0 shares of asset, cash position is: X0-∆ S Stock position owns ∆0 share of asset A. Portfolio value at time one is X1 = ∆ S + (1+r) (X0- ∆ S ) The approach is to choose X0 and ∆ to make X1(H) = V1 (H) and X1(T) = V1 (T) then X0 would be the price required for the considered derivative security. Time one value of the portfolio is X1 = ∆ S + (1+r) (X0-∆ S ) = (1+r) X0 + ∆ (S - (1+r) S ) = (1+r) [X0+ ∆ (
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i.e. the portfolio ( buy ∆0 share of asset and invest the rest money market) value at time one is S H S T 1 1 r X r X ∆ S ∆ S 3 when H occurs 0 when T occurs
S
and
S
We further assume that
.
Assumption 2: There is an interest rate r such that one dollar invested at time 0 will yield (1 + r) dollars at time one. Here we assume the (1+ r) 0.
Now, let’s treat the derivative in a general form. We define a derivative security be a security that pays V1 (H) when a head is realized and V1 (T) when a tail is realized offer a coin tossing on day one. Call option V1 (H) = (S1-K)+= S1-K V1 (T) = 0 or V1 = (S1-K)+ Put Option Forward contract V1= S1-K V H V T 0 V1 = (K- S1) + K S
Chapter 1: A Binomial Model for Asset price. Assumption 1: Let S0 be the price of an Asset at time 0. At time one, the price per share for this asset will be one of the two positive values, S1(H) and S1(T) where H and T denoting for head and tail respectively. We further assume that the probability the S1(H) appears is p and hence the probability to have S1(T) is 1-p. Define: u
1
Assuming the absence of arbitrage, iff (if and only if) 0 : (a) If (b) If 1 1 , then borrow money to buy stock.
1
.
, then short sell stock to save money for interest rate payment.
S
-S )]
To make this portfolio identical to the derivative, it requires: X 0 + ∆0 (
S H
-S ) =
V1 (H)
X 0 + ∆0 (
X S
=
If X0
X S
0, when initial wealth equals zero, then ∆ u ∆ d 1 1 r when H occurs r when T occurs 1
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=
Since the two situations cannot be positive simultaneously, therefore we have u and d 1 => d 1