Riemann surface
matlab

Suppose that a risky asset $S$ has spot price $S(0)=100$ and that the riskless return to $T=1$ year is $R=1.0223$. Assuming there are no arbitrages, compute the following:
(a) the current zero-coupon bond discount $Z(0, T)$,
(b) the Forward price for one share of $S$ at expiry $T$,
(c) the riskless annual interest rate (assuming continuous compounding),

Solution:
(a) From the formula on p. $3, Z(0, T)=\frac{1}{R}=0.9782$.
(b) By Eq.1.15, the fair price is $K=R S(0)=102.23$.
(c) By Eq.1.6, $r=(\log R) / T=0.022$, or $2.2 \%$.

Use the no arbitrage Axiom 1 to prove that Eq.1.7 holds.

Solution:
If the asset $A$ costs more, then short-sell $A$ for $A(0)$, buy the sequence of zero-coupon bonds, pocket the surplus, and use the proceeds from the bonds to pay the cash flow to the $A$ buyer over time, settling all liabilities.
Otherwise if the asset $A$ costs less, short-sell the bonds, use the money to buy $A$ for $A(0)$, pocket the surplus, and then pay the bond buyers back over time to settle all obligations. In either case there is an arbitrage.

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