Here we don't see any advantage of a put towards a call. This is in reality because b=0, meaning the stock does not depreciate/appreciate.
Hull American Price
fabien @ 31416.org
American options can be exercised anytime until expiration date, European only at expiration date.
At first when I read in Hull “Options, Futures, and other Derivatives” that it was optimal to exercise an in the money american option put immediately and exercise an in american option call at expiry, I was a bit surprised. It seemed the american feature of the option was not so interesting after all.
When I later looked at the price of american options compared to european ones, I was surprised how in many conditions the difference was small between their prices.
One key element here is that the option payout is only paid at expiry if it is american or european. This is what makes the put/call lower bound (the main Hull argument) very important:
c >= S0 – K*exp(-r*t)
p >= K*exp(-r*t) – S0
and for americans as you can exercise immediately, for a put this translates to:
p >= K – S0
while for a call we see it is obviously not optimal to exercise immediately because
S0-K <= S0-K*exp(-r*t)
If we compare american with european, it seems that the American call price would always be very close to the European call price (exercise optimal at maturity) but the American put price would be quite a bit higher (exercise optimal instantly). To find out how much could be the difference, I plotted the graph of (americanPrice – europeanPrice)/europeanPrice for different strikes and for put/call.
The first graph is obtained using parameters in Haug book namely
TTE=6 months, S=100, vol=0.15, r = 0.1, b = 0 [TTE= time to expiry, S=initial spot, vol=volatility, r=discount rate, b=cost of carry rate (r-q), q=dividend rate]
The prices are obtained using two binomial methods (CRR and RendlemanBartter). I used those parameters to validate with the prices in Haug book for analytic models of american/european.
Here
we don't see any advantage of a put towards a call. This is in
reality because b=0, meaning the stock does not
depreciate/appreciate.
In more real conditions:
r = 0.04, vol=0.20, q = 0.02
Here
we clearly see the put is more interesting especially in the money.
The american call is almost the same as the european call.
The pink curve is the minimum american price as described by Hull relative to the european price. The minimum american price is K-S, the pink curve is =max(K-S, european_put_price). This approximation is quickly exact (when more than 125% in the money). Therefore the complex formula american put price in only interesting when not in the money or just in the money.
Still the price difference between an american and a european is only at most 5%.
“Extreme” conditions:
r = 0.10, vol=0.15, q=0
Same
shape as previously, but more intense. This is all due to the
interest rate.
What if the market goes down?
I used a simple trick to simulate a down market, in the RB binary tree algorithm, I change the probability of up move from 0.5 to 0.49. Therefore the stock price will globally move down. The other parameters are: r = 0.04, vol=0.20, q = 0.02 (like the 2nd graph). The result is quite shocking given the slight probability change:
The RB call price (of course here only RB algo was changed) is much higher (as expected, but not by traditional models). So arguments about american call being same as european call have to be taken very carefully, it only happens if the market is “perfect”, it does not seem very stable, as changing probability to go up from 0.5 to 0.49 shows already a big difference.
Average S(expiry) = 1000 under “perfect” market assumption
Average S(expiry) = 94.81 under 0.49 up probability. It is not a completely foolish assumption.
Under more extreme assumption (p=0.4) Average S(expiry)=53.64 (not too unreasonable these days) and the max relative american-european call price difference is 6500000 percent!
Another replication for down market
There is another way to “simulate” a down market, it is to change the cost of carry rate to a negative value. A final average spot in 6 months of 94.81, starting from 100 means an average cost of carry rate of -10.38%.
r = 0.04, vol=0.20, b = -0.1038
This is the exact same graph as the one with probabily in the RB tree of 0.49. We can guess now with the graphs with b>0, b=0, b<0 that the issue wether the call should be exercised first or the put should be exercised first is directly correlated to b (cost of carry) being negative or positive. What Hull essentially says it is that in a up market, it is better to exercise put early (an evidence). But in a down market it is better to exercise the call early (another evidence).
After all these graphs trying to understand what Hull was really saying, it turns out there was nothing to look for but common sense.