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On Page 24-25 of N. Taleb's "Dynamic Hedging" the author talks about "Soft American Options"

A soft American option (also called a pseudo-European option) is only subjected to early exercise from the standpoint of the financing of the intrinsic value.

An extension of this definition is that only one interest rate, that affecting the the financing of the premium for the operator, impacts the decision to early exercise.

For risk management and trading purposes, soft American options will be largely similar to the European options, except when interest rates become very high relative to volatility. The reason they are often called pseudo-European options is that they behave in general like European options, except when very deep in the money. The test of early exercise is whether the total option value is less than the time value of the money between the time of consideration and expiration.

Example: Assume that an asset trades at \$100, with interest rates at 6% (annualized) and volatility at 15.7%. Assume also that the 3-month 80 call is worth \$20, at least if it is American. Forgoing early exercise would create an opportunity cost of 20 x 90/360 x .06 = .30 cents, the financing of \$20 premium for 3 months. The time value of the equivalent put is close to zero (by put-call parity), so the intelligent operator can swap the call into the underlying asset and buy the put to replicate the same initial structure at a better cost. He would end up long the put and long the underlying asset.

I have three questions here

Q1. My understanding is, "soft american options" are in general call options, that the option's value is due to the increasing of the stock value. Since stock value grows faster than interest rate, it's not advisable to early execute, unless, the interest rate is too high. The other case is put options, as time goes by, stock price increases and option's intrinsic value decreases, it's more advisable to early execute. Is my understanding correct?

Q2. How did Taleb come to the conclusion that "Forgoing early exercise would create an opportunity cost of 20 x 90/360 x .06 = .30 cents, the financing of \$20 premium for 3 months. "? I couldn't figure out why the financing cost of an American options, is the difference between an American and an European option's price.

Q3. Taleb says "so the intelligent operator can swap the call into the underlying asset and buy the put to replicate the same initial structure at a better cost", I do agree that call = asset + put, i.e. a call option can be replicated by an asset and a put, but why such replication has a better cost?

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  • $\begingroup$ I think Q2 & Q3 are more or less answered here quant.stackexchange.com/q/33019/15154 $\endgroup$ – zer0hedge Mar 20 '17 at 18:24
  • $\begingroup$ @zer0hedge I saw your comment before post this question -- actually I don't understand : even if "under the assumptions the spot price is unlikely to go below 80", why "an intelligent operator should swap"? Anything cheaper? $\endgroup$ – athos Mar 20 '17 at 23:43
  • $\begingroup$ Also posted here: forum.wilmott.com/viewtopic.php?f=8&t=100620. $\endgroup$ – LocalVolatility Mar 26 '17 at 13:00
  • $\begingroup$ yes. ppl said Wilmott forum is one of the best quant forum, so i also seek help there. however, it attracts no attention, maybe it's too primary.. $\endgroup$ – athos Mar 26 '17 at 13:02
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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. $\tau= \tau_0 = \frac{90}{360}$, $S(\tau_0) = \$100$: enter image description here

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:

  1. We keep the call till expiration and receive payoff $V_{call}$
  2. 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

  1. foreign and domestic interest rates are the same ($r^f \approx r = 0.06$)
  2. 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).

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  • $\begingroup$ thx for the explanation, i agree with the "soft" part, but for the "hard" one, won't the option values and future rates change, if eoru interest is 14%? $\endgroup$ – athos Mar 25 '17 at 9:44
  • $\begingroup$ @athos I got your point, thinking .... it will take some time it seems ... $\endgroup$ – zer0hedge Mar 27 '17 at 8:23
  • $\begingroup$ thx... i guess it's something traders do from a business point of view.. $\endgroup$ – athos Mar 27 '17 at 8:25

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