16 Mathematics for Finance
Proof
Suppose thatC(0) + 114 A(0)>^12 S(0).If this is the case, then at time 0:
- Issue and sell 1 option forC(0) dollars.
- Borrow 114 ×100 =^40011 dollars in cash (or take a short positiony=− 114 in
bonds by selling them). - Purchasex=^12 shares of stock forxS(0) =^12 ×100 = 50 dollars.
The cash balance of these transactions is positive,C(0) + 114 A(0)−^12 S(0)>0.
Invest this amount risk-free. The resulting portfolio consisting of shares, risk-
free investments and a call option has initial valueV(0) = 0. Subsequently, at
time 1:
- If stock goes up, then settle the option by paying the difference of $20
between the market price of one share and the strike price. You will pay
nothing if stock goes down. The cost to you will beC(1), which covers both
possibilities. - Repay the loan with interest (or close your short positiony=− 114 in bonds).
This will cost you 114 ×110 = 40 dollars. - Sell the stock for^12 S(1) obtaining either^12 ×120 = 60 dollars if the price
goes up, or^12 ×80 = 40 dollars if it goes down.
The cash balance of these transactions will be zero,−C(1)+^12 S(1)− 114 A(1) = 0,
regardless of whether stock goes up or down. But you will be left with the initial
risk-free investment ofC(0) + 114 A(0)−^12 S(0) plus interest, thus realising an
arbitrage opportunity.
On the other hand, ifC(0) + 114 A(0)<^12 S(0), then, at time 0:
- Buy 1 option forC(0) dollars.
- Buy 114 bonds for 114 ×100 =^40011 dollars.
- Sell shortx=^12 shares of stock for^12 ×100 = 50 dollars.
The cash balance of these transactions is positive,−C(0)− 114 A(0)+^12 S(0)> 0 ,
and can be invested risk-free. In this way you will have constructed a portfolio
with initial valueV(0) = 0. Subsequently, at time 1:
- If stock goes up, then exercise the option, receiving the difference of $20
between the market price of one share and the strike price. You will receive
nothing if stock goes down. Your income will beC(1), which covers both
possibilities. - Sell the bonds for 114 A(1) = 114 ×110 = 40 dollars.
- Close the short position in stock, paying^12 S(1),that is,^12 ×120 = 60 dollars
if the price goes up, or^12 ×80 = 40 dollars if it goes down.
The cash balance of these transactions will be zero,C(1) + 114 A(1)−^12 S(1) = 0,
regardless of whether stock goes up or down. But you will be left with an