Market quotes and arbitrage argument

Question

DUPLICATE ON HOLD Use an arbitrage argument to construct a formula relating the price of the European to the price of an American.

Good day, I wanted to ask for help with a question from one of my exercise sheets.

For a share S the market quotes a given strike K in both european and american styles. Use an arbitrage argument to construct a formula relating the price of the european to the price of an american.

I do not understand what is meant by EU and US style Market quotes. From my research, it just seems that

• EU is quoted at K USD per share
• US is quoted at $\frac{1}{K}$ shares per 1 USD

But this does not tell me much about applying an arbitrage argument. I have no idea about the formula that relates the two styles.

Answer 1

I think that you can show this as follows:

Consider two portfolios, $A$ with one American call and $\frac{K}{(1+r)}$ of the riskless asset and $B$ with one American put and one stock.

Unlike European options, you can exercise American options at any moment of time and so your portfolio's value will depend on whether the options are exercised and when they are exercised.

So suppose you exercise your option at some moment $0\leq t \leq T$, then $A$ is worth:

$max(S_t,K)*[1+(r*(T-t))]$ at time T;

$B$ is worth

$(max(S_t,K)+K*t*r)* r*(T-t)$ at time $T$.

So you can see that $B$ will be more valuable than A at T:

$(max(S_t,K)+K*t*r) * r*(T-t) \geq max(S_t,K)*[1+(r*(T-t))]$

Note that at time $t = 0$, then

$American\ Call (0) + K(0) \geq American\ Put (0) + S(0)$

Now, consider $T=t$. Then, the above formulas become:

Portfolio $A$ = $max(S_t,0)$

Portfolio $B$ = $(max(S_t,K)+K*t*r) * r*(T-t) = 0.$

At time $T=t$, then

$American\ Put + S(0) \geq American\ Call + K^*$

You can combine the two cases as:

$S(0)-K \leq American\ Call - American\ Put \leq S(0)-K^*$

Now, by no arbitrage, the put-call parity for European options holds:

$S(0)+P(0)=C(0)+K^*$. Then if you rearrange $S(0)-K^*=C(0)-P(0)$.

Then you get a formula relating European and American options that is:

$S(0)-K \leq American\ Call - American\ Put \leq European\ Call - European\ Put$