Consider a market with a risk-free asset such that A(0) = 100, A(1) = 110, A(2) = 121 dollars and a risky asset, the price of which can follow three possible scenarios,

Scenario S(0) S(1) S(2)

$ω_{1}$ 100 120 144

$ω_2$ 100 120 96

$ω_3$ 100 90 96

Is there an arbitrage opportunity if a) there are no restrictions on short selling, and b) no short selling of the risky asset is allowed?

Question

Consider a market with a risk-free asset such that A(0) = 100, A(1) = 110, A(2) = 121 dollars and a risky asset, the price of which can follow three possible scenarios,

Scenario	S(0)	S(1)	S(2)
[latex]ω_{1}[/latex]	100	120	144
[latex]ω_2[/latex]	100	120	96
[latex]ω_3[/latex]	100	90	96

Is there an arbitrage opportunity if a) there are no restrictions on short selling, and b) no short selling of the risky asset is allowed?

Accepted Answer

a) If there are no short-selling restrictions, then the following strategy will realise an arbitrage opportunity:
• At time 0 do not invest at all.
• At time 1 check the price S(1). If S(1) = 120 dollars, then once again do
not invest at all. But if S(1) = 90 dollars, then sell short one share of the
risky asset and invest the proceeds risk-free. The time 0 and time 1 value of this admissible strategy is 0. The value at time 2 will be

[latex] \begin{cases} 0 &in \ scenarios \ ω_1 and \ ω_2,\3 & in \ scenario \ ω_3, \end{cases} [/latex]

which means that arbitrage can be achieved.
b) In a) above arbitrage has been achieved by utilising the behavior of stock prices at the second step in scenario ω2: The return on the risky asset is lower than the risk-free return. Thus, shorting the risky asset and investing the proceeds risk-free creates arbitrage. However, when short selling of risky assets is disallowed, then this arbitrage opportunity will be beyond the reach of investors.

Question 4.5: Consider a market with a risk-free asset such that A(0) = 10...

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