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ORF 522

Linear Optimization

Lecture 21

More Applications

Options Pricing

1

Farkas’ Lemma for Systems in Equality Form

Recall Farkas’ Lemma:

Lemma. The system Ax ≤ b has no solutions if and only if there is a y such that A T^ y = 0 y ≥ 0 b T^ y < 0.

Today we need it in another form:

Lemma. The system Ax = b, x ≥ 0 has no solutions if and only if there is a y such that A T^ y ≥ 0 b T^ y < 0.

Proof is completely analogous to the one we had before. Hence, omitted.

Arbitrage

Big Assumption 1: We can hold positive or negative quantities of each asset—the return is the same.

Never satisfied in practice. If I give a bank 1 dollar to hold, they will return it after a year with 4% interest but if I give a bank −1 dollar to hold (i.e., I borrow a dollar), they will give −1 back to me with 10% interest.

It is, however, generally assumed to be true, at least for the big players.

Let x (^) j = number of units of asset j I hold.

Wealth at end of time period under scenario i: wi =

j

r (^) i j x (^) j.

In matrix notation: w = Rx.

Recall: the total current “price” for this portfolio is: p T^ x

An arbitrage is a portfolio x which is guaranteed (under every scenario) to have nonnegative value at the end of the time period but which has a negative price at the beginning: Rx ≥ 0 and p T^ x < 0.

Big Assumption 2: The scenarios considered cover all possibilities.

Efficient Market Assumption

Assumption: Prices will equilibrate so as to eliminate arbitrage.

Theorem. There is no arbitrage if and only if there is a vector y that satisfies: R T^ y = p y ≥ 0.

Proof. Immediate from Farkas’ Lemma (A = R T^ , b = p, and x and y interchanged).

Notes.

• If m = n and R is nonsingular, then the equality constraints uniquely determine y. Then, only need to check nonnegativity.
• If p is arbitrage-free, then any nonnegative constant times p is too. Therefore, at least one of the prices in p needs to be fixed arbitrarily (by, e.g., Alan Greenspan).

Options Pricing

Suppose that there are only two scenarios:

• The stock goes up by a factor u > 1, or
• down by a factor d < 1.

Under both scenarios, the bond goes up by a factor r > 1.

Suppose at the beginning that the stock price is S, the price of the bond is B, and of course the price of the option is to be determined. Let’s denote it by O.

The matrix R is then given by:

R =

Su Sd Br Br max( 0 , Su − K ) max( 0 , Sd − K )

and the vector p is given by:

p =

S

B

O

The no-arbitrage theorem says there must exist a vector y =

[

y (^) u y (^) d

]T

such that 



Su Sd Br Br max( 0 , Su − K ) max( 0 , Sd − K )

[

y (^) u y (^) d

]

S

B

O

[

y (^) u y (^) d

]

Black Scholes Formula

The first two equations can be solved for y (^) u and y (^) d : [ y (^) u y (^) d

]

[

Su Sd Br Br

]− 1 [

S

B

]

r(u − d)

[

r − d u − r

]

Then the last equation can be solved for O: O = y (^) u max( 0 , Su − K ) + y (^) d max( 0 , Sd − K )

This option pricing formula is the discrete analogue of the famous Black- Scholes formula.

Note:

• The nonnegativity requirement on y forces us to assume that d < r < u.