**Figure 1 : Term Structure of the Underlying Security. S is the spot, F the forward. **
(This idea came to me when I saw in London earlier this month a hotshot from UCLA who spent his life talking about "pricing" American options, but never tried to look outside the Gaussian "Black-Scholes" based models. As Yogi Berra says: "Don't tell them if they don't know").

** War Story 1 : The Currency Interest rate Flip**

I recall in the 1980s the German currency carried lower interest rates than the US. When rate 1 is lower than rate 2, then, on regular pricing systems, for vanilla currency options, the American Put is higher than the European Put, but American Call =European Call. At some point the rates started converging; they eventually flipped as the German rates rose a bit after the reunification of Deutschland. I recall the trade in which someone who understood model error (not a finance professor) trying to buy American Calls Selling European Calls and paying some trader who got an immediate marks-to-market P/L (from the mark-to-model). The systems gave an identical value to these -it looked like free money, until the trader blew up. Nobody could initially figure out why they were losing money after the flip --the systems were missing on the difference. There was no big liquidity but several billions went through. Eventually the payoff turned out to be big.

We repreated the game a few times around devaluations as interest rates would shoot up and there was always some sucker with a math degree willing to do the trade.

** War Story 2: The Stock Squeeze ** Spitz called me once in during the 2000 Bachelier conference to tell me that we were in trouble. We were long listed American calls on some Argentinian stock and short the delta in stock. The stock was some strange ADR that got delisted and we had to cover our short ASAP. Somehow we could not find the stock, and begging Bear Stearns failed to help. The solution turned out to be trivial: exercise the calls, enough of them to get the stock. We were lucky that our calls were American, not European, otherwise we would have been squeezed to tears. **Moral: **an American call has hidden optionality on model error.

These hiden optionalities on model errors are more numerous than the ones in the two examples I just gave. I kept discovering new ones.

** Misplaced Precision **

So many "rigorous" research papers have been involved in the “exact” pricing of American options, though within model when in fact their most interesting attribute is that they benefit from the breakdown of models. Indeed an interesting test to see if someone understand quantitative finance is to quiz him on American options. If he answers by providing a “pasting boundary” story but using a Black-Scholes type world, then you can safely make the conclusion that he represents an intellectual and financial danger. Furthermore, with faster computers, a faster pricing algorithm does not carry large advantages. The problem is in the hidden optionality...

**Major points to know**

An American option is always worth equally or more than the European option of the same nominal maturity. An American option has always a shorter or equal expected life than a European option. An American option’s value increases with the following factors: Higher volatility of interest rates. Higher volatility of volatility. Higher instability of the slope of the volatility curve. **DANGER:** A conventional pricing system will trick you into using the wrong parameter for the American option, as we will see.

**Rule:** The major difference between an American and European option is that the holder of the American option has the right to decide on whether the option is worth more dead or alive. In other words is it worth more held to expiration or immediately exercised ?

**War Story 3: American Option and The Squeeze**

I recall in the late 1990s seeing a strange situation: Long dated over-the-counter call options on a European Equity index were priced exceedingly below whatever measure of historical volatility one can think of. What happened was that traders were long the calls, short the future, and the market had been rallying slowly. They were losing on their future sales and had to pay for it –without collecting on their corresponding profits on the option side. The calls kept getting discounted; they were too long-dated and nobody wanted to toutch them. What does this mean?

Consider that a long term European option can trade below intrinsic value! I mean intrinsic value by the forward! You may not have the funds to arb it... The market can become suddenly inefficient and bankrupt you on the marks as your options can be severely discounted. I recall seing the cash-future discount reach 10% during the crash of 1987.

But with an American option you have a lower bound on how much you can be squeezed.

Let us look for cases of differential valuation.

**Case 1** (Simplest, the bang comes from the changes in the carry of the premium) Why do changes in interest rate carry always comparatively benefit the American option ? Take a 1 year European and American options on a forward trading at 100, i.e. with a spot at 100. The American option will be priced on the risk management system at exactly the same value as the European one. S=100, F=100, where S is the spot and F is the forward.

Assume that the market rallies and the spot goes to 140. Both options will go to parity, and be worth $40.

**Case 1 A. **Assume that interest rates are longer 0, that both rates go to 10%. F stays equal to S. Suddenly the European option will go from $40 to the present value of $40 in one year using 10%, i.e. $36.36. The American option will stay at $40, like a rock.

**Case 1 B. **Assume the domestic rate goes up to 10%, spot unchanged. F will be worth approximately of S. It will go from 140 to 126, but the P/L should be neutral if the option still has no gamma around 126 (i.e. the options trade at intrinsic value). The European option will still drop to the PV of 26, i.e. 23.636, while the American will be at 26.

We can thus see that the changes in carry always work to the advantage of the American option (assuming the trader is properly delta neutral in the forward).

We saw in these two cases the outperformance of the American option. We know the rule that ::

If in all scenarios option A is worth at least the same as option B and, in some scenarios can be worth more than option B, then it is not the greatest idea to sell option A and buy option B at the exact same price.

This tells us something but not too much: we know we need to pay more, but how much more?

**Case 2 :** Sensitivity (more serious) to changes in the Dividend/Foreign rate

Another early exercise test needs to be in place, now. Say that we start with S = 140 and F = 140 and that we have both rates equal to 0. Let us compare a European and an American option on cash. As before, they will initially bear the same price on the risk management system.

Assume that that the foreign rate goes to 20%. F goes to approximately S, roughly 1.16. The European call option will be worth roughly $16 (assuming no time value), while the American option will be worth $40. Why ? because the American option being a very smart option, chooses whatever fits it better, between the cash and the future, and positions itself there.

**Case 3:** More Complex: Sensitivity to the Slope of the Yield Curve.

Now let us assume that the yield curve has kinks it it, that it is not quite as linear as one would think. We often such niceties around year end events, when interest rates flip, etc.

As Figure 1 shows the final forward might not be the most relevant item. Any bubbling on the intermediate date would affect the value of the American option. Remember that only using the final F is a recipe for being picked-on by a shrewd operator. A risk management and pricing system that uses no full term structure would be considered greatly defective, as it would price both options at the exact same price when clearly the American put is worth more because one can lock-in the forward to the exact point in the middle -- where the synthetic underlying is worth the most. Thus using the final interest rate differential would be totally wrong.

To conclude from these examples, the American option is extremely sensitive to the interest rates and their volatility. The higher that volatility the higher the difference between the American and the European. Pricing Problems

It is not possible to price American options using a conventional Monte Carlo simulator. We can, however, try to price them using a more advanced version -or a combination between Monte Carlo and an analytical method. But the knowledge thus gained would be simply comparative.

Further results will follow. It would be great knowledge to quantify their difference, but we have nothing in the present time other than an ordinal relationship.

**The Stopping Time Problem**

Another non-trivial problem with American options lies in the fact that the forward hedge is unknown. It resembles the problem with a barrier option except that the conditions of termination are unknown and depend on many parameters (such as volatility, base interest rate, interest rate differential). The intuition of the stopping time problem is as follows: the smart option will position itself on the point on the curve that fits it the best.

Note that the forward maturity ladder in a pricing and risk management system that puts the forward delta in the terminal bucket is WRONG.

**Conclusion**

All we know is that an American option is worth more than a European. How much more is difficult to quantify.

Traders get get more bang for the complexity by trading barrier options. We suggest avoiding selling these instruments whenever possible and covering them back-to-back in the market. Their return on complexity is smaller than that of barriers, simply because they have more complexity and offer less edge.