The Upper Bound of an American Put Option

Question

I have just read the following paragraph (in bold) and have a question on the upper bound of an american put option:

http://www.sharemarketschool.com/option-valuation-upper-and-lower-bounds-part-iii/

"Upper bounds of American puts:

The american put cannot have a value that’s greater than the strike price.So, if the underlying stock is at Rs 70 with strike price at Rs 75, the value of the put cannot be greater than 75. That’s simple to understand. The existence or non existane of dividends in the underlying stock desn’t make any difference in this case. The upper bound will always be the strike price in the case of american puts."

My question is:

The lower bound of an american and european put option can be optained by the put-call parity:

$P(S, \tau; X) \geq p(S, \tau; X) \geq \text{max}(XB(\tau)+D-S, 0)$

where
$S$ : spot price of the stock,
$\tau$ : time to maturity
$X$ : strike price
$P$ : price of an american put option
$p$ : price of an european put option
$B(\tau)$ : zero-coupon discount bond price
$D$ : present value of multiple deterministic dividend payments

Thus, with $D$ being sufficiently high and $S$ being sufficiently low, it is possible that

$\text{max}(XB(\tau)+D-S, 0) \gt X$,

which is a contradiction to the statement that the upper bound of an american put option is the strike.

How to resolve this contradiction? Thanks for helping!

Answer 1

Dividends do not matter for the determination of the upper bound. Indeed, the maximum profit which the holder of a put option can make (be it through a European or an American exercise feature) is exactly equal to the strike price $X$. This can be seen by simply looking at the payout function: the maximum profit is finite and located on the downside when the underlying is worth 0. Consequently, there is no way that the price of a put option with time to maturity $\tau$ can be greater than $X$, the maximum achievable gain (assuming positive discount rates).

That being said, the lower bound for the price of a European put is actually easily derived using Jensen's inequality (i.e. if $f$ is a convex function then $E[f(X)] \geq f(E[x])$): \begin{align} P(S_0;\tau,X) &= B(\tau)E_0[ \max(X-S_\tau, 0) ] \\ &\geq B(\tau)\max(E_0[X-S_\tau], 0) = \max(B(\tau)X-B(\tau)F(0,\tau), 0) \end{align} where $F(0,\tau)$ figures the forward price.

From the above expression, you see that the first argument of the $\max(.,.)$ function is always smaller than $B(\tau)X$ because $F(0,\tau)$ is always positive in the equities world. There is therefore no conflict between the above result and the theoretical upper bound $X$ (assuming positive discount rates).

Note that your equation gives exactly the same result since in the absence of arbitrage opportunity: $$B (\tau)F (0,\tau) = S_0 -D$$ Which can be shown by a simple cash and carry argument. Thus, when I say that $F (0,\tau)$ is positive, it is equivalent to saying $S_0 \geq D $ in your equation. In other words, the current stock price reflects future expected dividend payments: you cannot reasonably have $S_0$ sufficiently small and $D $ sufficiently large at the same time.

Also there is no such thing as call-put parity for American options (because of the early exercise feature). Well, there is, but it becomes an inequality rather than the well-known equality observed for European options.

Answer 2

Assuming that $max(KB(\tau) + D - S, 0) > K$, and assuming that the strike $K$ is positive, we know that $max(KB(\tau) + D - S, 0) = KB(\tau) + D - S$, since $0$ is less than $K$.

Given that, we get:

$KB(\tau) + D - S > K$

$D - S > K (1 - B(\tau))$

Assuming that $B(\tau) \leq 1$ (non-negative interest rate), we get:

$D - S > K (1 - B(\tau)) \geq 0$

$D > S$

Or, in other words, the asset is selling for less than the present value of the dividends, which is an opportunity for arbitrage.