What Is the Price Return?

The state price refers to the current price of an asset with a return of 1 when a specific state occurs, otherwise the return is 0.

State price pricing

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The state price refers to the current price of an asset with a return of 1 when a specific state occurs, otherwise the return is 0.

If there are N states in the future at a time, and we know the prices of these N states, then as long as we know the return status of an asset in various future states, we can price the asset, which is the state price pricing. technology.

A is a risky security. Its current price is PA. After one year, its price will either rise to uPA or fall to dPA. These are the two states of the market: rising state (probability is q) and falling state (probability is 1-q).

Basic securities 1 have a value of 1 when the securities market rises, and a value of 0 when they fall; basic securities 2 have the opposite, with a value of 0 when the market rises and a value of 1 when they fall. The current market price of basic security 1 is & pi; u, and the price of basic security 2 is & pi; d.

Buying uPA Basic Securities 1 and dPA Basic Securities 2 makes up a hypothetical portfolio. This combination can generate the same cash flow as securities A no matter what happens at time T

PA = & pi; uuPA + & pi; ddPA or 1 = & pi; uu + & pi; dd

At any time, the combination of the unit's basic securities returns 1 yuan. This is a risk-free portfolio, and its rate of return should be a risk-free rate of return r

& pi; u + & pi; d = e & minus; r (T & minus; t)

Suppose a stock meets the two market states we mentioned above. The beginning value is S0 and the ending value is S1. Here S1 can only take two values: one is S1 = Su = uS0, u> 1, and the other is S1 = Sd = dS0, d <1. What we want to determine now is the value of the call option attached to the stock?

We construct such an investment portfolio so that it has exactly the same value characteristics as the call option: borrow a portion of the capital B at the risk-free interest rate r (equivalent to shorting a risk-free bond), and at the same time purchase N-share underlying stocks on the stock market. The cost of the combination is N S0 & minus; B. At the end of the period, the value of the combination V is N S1 & minus; RB, and R is the interest rate factor. Corresponding to the two possibilities of S1, V has two values: if S1 = Su, then V = Vu = N Su & minus; RB; if S1 = Sd, then V = Vd = NSd & minus; RB.

Vu = NSu & minus; er (T & minus; t) B = cu

Vd = NSd & minus; er (T & minus; t) B = cd

N = (cu & minus; cd) / (Su & minus; Sd) = ((cu & minus; cd) / [(u & minus; d) S0]

B = (Sdcu & minus; Sucd) / [(Su & minus; Sd) er (T & minus; t)] = (NSd & minus; cd) e & minus; r (T & minus; t) = (dcu & minus; ucd) er ( T & minus; t) / (u & minus; d)

Since the portfolio at the beginning of the period should be equal to the value of the call option, that is, N S0 & minus; B = c0, substituting N and B into this formula to obtain the value formula of the call option

c0 = [pcu + (1 & minus; p) cd] e & minus; r (T & minus; t)

Where p = (er (T & minus; t) S0 & minus; Sd) / (Su & minus; Sd) = (er (T & minus; t) & minus; d) / (u & minus; d) ^[1]