It is easier to understand Taleb's distinction between 'soft' and 'hard' American options if we understand from the beginning that he is talking about FX options or behaving similarly options on another assets.
Under Garman-Kohlhagen model for valuation, the price of European FX call is:
$$ C_{e}(S(\tau), K, \tau, r, r^f, \sigma ) = e^{-r^{f}\tau}S(\tau)N(d_+) - e^{-r\tau}KN(d_{\_}) \\
d_{+} = \frac{1}{\sigma \sqrt{\tau}}\Big[\log{\frac{S(\tau)}{K}}+(r-r^f+\frac{1}{2}\sigma^2)\tau\Big] \\
d_{\_} = d_{+} - \sigma \sqrt{\tau}$$
where:
- $S(\tau)$ is an exchange rate in units of domestic currency($\$$) per $N_f$ units of foreign currency
- $N_f$ is a foreign notional amount, its actual value is not important
- $K = \$80$ is an option strike price in units of domestic currency per $N_f$ units of foreign currency
- $\sigma = 0.157$ is the volatility of the exchange rate $S(\tau)$
- $\tau$ is time to expiration
- $r = 0.06$ is a domestic currency risk-free rate
American FX call price $C_a$ is higher or equal than $C_e$ for any type of options. But for FX options this boundary can be augmented further because under Garman-Kohlhagen model European call may have negative time value or cost less than its intrinsic value. Obviously, American option can't cost less than intrinsic value due to early exercise feature, so:
$$ C_{a}(S, K, \tau, r, r^f, \sigma ) \geq \max (C_{e}(S, K, \tau, r, r^f, \sigma ), S-K)$$
It is instructive to see how the low boundary of the price of the American option depends on $r^f$ assuming the other parameters fixed as in Taleb's 'soft' American option example, i.e. $\tau= \tau_0 = \frac{90}{360}$, $S(\tau_0) = \$100$:
When foreign risk-free rate rate $r^f$ is less than $\approx 5\%$, time-value of both American and European option is greater than zero. This value is lost when the American option is exercised early and generally speaking we would have to take this into consideration, but we will not since Taleb most likely assumes that this is not the case (see below).
Instead, if foreign interest rate is greater than $5\%$, European FX call costs less than its intrinsic value while cost of American one is equal to its intrinsic value. That's what Taleb probably refer to when he emphasizes that the option is an American one:
Assume also that the 3-month 80 call is worth $20, at least if it is American.
Assume (as Taleb does) that the price of put available on the market is negligible and $r^f$ is greater than $5\%$. Compare final payoffs in domestic currency for two scenarious proposed by Taleb:
- We keep the call till expiration and receive payoff $V_{call}$
- We exercise the call borrowing $K$ and receiving $N_f$ units of foreign currency, buy put with strike $K$ and keep this position till expiration of put and then exchange $N_f$ back to domestic currency. The received payoff is $ V_{N_f + \text{put}}$
Thus:
$$V_{call} = \max(S(0) - K,0) $$
Here $S(0)$ is the exchange rate at the time of expiration ($\tau = 0$).
Then:
$$ V_{N_f + \text{put}} = \underbrace{N_f\frac{\max(S(0), K)}{N_f} - K}_{N_f + \text{put} = \max(S(0) - K,0) = V_\text{call}} + \underbrace{N_f (e^{r^f \tau_0}-1)\frac{S(0)}{N_f} - K(e^{r\tau_0}-1)}_\text{the interest earned} \tag{1} \label{one}$$
Here $N_f (e^{r^f \tau_0}-1)$ is the interest earned in units of foreign currency and $\frac{S(0)}{N_f}$ is an exchange rate in units of domestic currency per 1 (not $N_f$!) unit of foreign currency
It is "the interest earned" part of $ V_{N_f + \text{put}}$ which is used by Taleb to gauge 'hardness' of the FX option. In turn, we can further split it into:
$$ N_f(e^{r^f \tau_0}-1)\frac{S(0)}{N_f} - K(e^{r\tau_0}-1) = S(0)(e^{r^f \tau_0}-1) - K(e^{r\tau_0}-1) \\ \approx \tau_0 S(0)r^f - \tau_0 Kr = \underbrace{\tau_0 (S(0) - K)r}_\text{financing of intrinsic value} + \underbrace{\tau_0 S(0)(r^f-r)}_\text{carry cost of the underlying } \tag{2} \label{two} $$
So if "financing of intrinsic value" is much greater than "carry costs of the underlying": $$\tau_0 (S(0) - K)r \gg \tau_0 S(0)(r^f-r)$$ then American option is 'soft'. Otherwise it is 'hard'.
Now everything is ready to answer your questions.
Q1. Taleb's distinction between 'soft' and 'hard' options is easier to understand assuming that he is talking about FX options. An FX option is a 'call' option in one currency and 'put' in the other simultaneously, so you can assume that 'Soft American Option' may be 'put' as well as 'call'.
It is relationship between foreign and domestic interest rates, not the growth or change of the exchange rate, that defines 'softness' of the American option as described above.
It is interesting that Hull's appreciate this fact:
In general, call options on high-interest currencies and put options on low-interest currencies are the most likely to be exercised prior to maturity.
but then gives what appears to be an incorrect explanation to this fact:
The reason is that a high-interest currency is expected to depreciate and a
low-interest currency is expected to appreciate.
Q2. Taleb assumes that
- foreign and domestic interest rates are the same ($r^f \approx r =
0.06$)
- final exchange rate $S(0)$ is equal \$100 per $N_f$ units of foreign currency.
Then he calculates "financing of intrinsic value" as in formula $\eqref{two}$ above. "carry cost of the underlying" is zero in this case. Forgoing early exercise would lost this interest.
Q3. Replication might have better costs if "the interest earned" component in $\eqref{one}$ is positive. It is likely the case when the call is very deep in-the-money, volatility is low and $r^f \geq r$. If $r^f < r$ "the intelligent operator" will need to take into consideration the loss of time value due to early exercise (i.e. difference between intrinsic value and the market price of the exercised option).
