## European vs. American Options

All of the options that I’ve discussed so far on this blog have been European options. A European option gives us the right to buy or sell an asset at a fixed price, but only on a particular expiry date. In this post, I’m going to start looking at American options, which give the right to buy or sell at ANY date up until the expiry date.

Surprisingly for the case of vanilla options, despite the apparent extra utility of American options, it turns out that the price of American and European options is almost always the same! Why is this?

In general, American options are MUCH harder to price than European options, since they depend in detail on the path that the underlying takes on its way to the expiry date, unlike Europeans which just depend on the terminal value, and no closed form solution exists. One thing we can say is that an American option will never be LESS valuable than the corresponding European option, as it gives you extra optionality but doesn’t take anything away. So we can always take the European price to be a lower bound on American prices. Also note that Put-Call Parity no longer holds for Americans, and becomes instead an inequality.

How can we go any further? It is useful in this case to think about the value of an option as made up of two separate parts, an ‘intrinsic value’ and a ‘time value’, which sum to give the true option value. The ‘intrinsic value’ is the value that would be received if the exercise was today – in the case of a vanilla call, this is simply . The ‘time value’ is the ‘extra’ value due to time-to-expiry. This is the volatility-dependent part of the price, since we are shielded by the optionality from price swings in the wrong direction, but are still exposed to upside from swings in our favour. As time goes by, the value of the option must approach the ‘intrinsic value’, as the ‘time value’ decays towards expiry.

Consider the graph above, which shows the BS value of a simple European call under typical parameters. Time value is maximal at-the-money, since this is the point where the implicit insurance that the option provides is most useful to us (far in- or out-of-the-money, the option is only useful if there are large price swings, which are unlikely).

What is the extra value that we should assign to an American call relative to a European call due to the extra optionality it gives us? In the case of an American option, at any point before expiry we can exercise and take the intrinsic value there and then. But up until expiry, the value of a European call option is ALWAYS* more than the intrinsic value, as the time value is non-negative. This means that we can sell the option on the market for more than the price that would be received by exercising an American option before expiry – so a rational investor should never do this, and the price of a European and American vanilla call should be identical.

It seems initially as though the same should be true for put options, but actually this turns out not quite to be right. Consider the graph below, showing the same values for a European vanilla put option, under the same parameters.

Notice that here, unlike before, when the put is far in-the-money the option value becomes smaller than the intrinsic value – the time value of the option is negative! In this case, if we held an American rather than a European option it might well make sense to exercise at this point, since we would receive the intrinsic value, which is greater than the option value on the market [actually it’s slightly more complicated, because in this scenario the American option price would be higher than the European value shown below, so it would need to be a bit more in the money before it was worth exercising – you can see how this sort of recursive problem rapidly becomes hard to deal with!].

What is it that causes this effect for in-the-money puts? It turns out that it comes down to interest rates. Roughly what is happening is this – if we exercise an in-the-money American put to receive the intrinsic value, we receive cash straight away. But if we left the option until expiry, our expected payoff is roughly , where is the forward value

so we can see that leaving the underlying to evolve is likely to harm our option value [this is only true for options deep enough in the money for us to be able to roughly neglect the criterion]

We can put this on a slightly more rigourous footing by thinking about the GREEK for time-dependence, Theta. For vanilla options, this is given by

where is the forward price from to is the standard normal PDF of and  is its CDF, is a zero-coupon bond from to and the upper of the  refers to calls and the lower to puts.

The form for Theta shows exactly what I said in the last paragraph – for both calls and puts there is a negative component coming from the ‘optionality’, which is decreasing with time, and a term coming from the expected change in the spot at expiry due to interest rates which is negative for calls and positive for puts.

The plot below shows Theta for the two options shown in the graphs above, and sure enough where the time value of the European put goes negative, Theta becomes positive – the true option value is increasing with time instead of decreasing as usual, as the true value converges to the intrinsic value from below.

*Caveats – I’m assuming a few things here – there are no dividends, rates are positive (negative rates reverses the situation discussed above – so that American CALLS can be more valuable than Europeans), no transaction fees or storage costs, and the other sensibleness and simpleness criteria that we usually assume apply.

In between European and American options lie Bermudan options, a class of options that can be exercised early but only at one of a specific set of times. As I said, it is in general really tough to price more exotic options with early exercise features like these, I’ll look at some methods soon – but this introduction is enough for today!

## Digital Options

Today I’m going to talk about the valuation of another type of option, the digital (or binary) option. This can be seen as a bit of a case study, as I’ll present the option payoff and the analytical price and greeks under BS assumptions, and give add-ons to allow pricing with the MONTE CARLO pricer. I’ve also updated the ANALYTICAL pricer to calculate the price and greeks of these options.

Digital options are very straight-forward, they are written on an underlying S and expire at a particular date t, at which point digital calls pay $1 if S is greater than a certain strike K or$0 if it is below that, and digital puts pay the reverse – ie. the payoff is

$P_{\rm&space;dig\&space;call}&space;=&space;\Big\{&space;\&space;\begin{matrix}&space;\1&space;\quad&space;S&space;\geq&space;K\\&space;\0&space;\quad&space;S&space;<&space;K\end{matrix}$

We can calculate the price exactly in the BS approximation using the same method that I used to calculate vanilla option prices by risk-neutral valuation as follows

$C_{\rm&space;dig\&space;call}(0)&space;=&space;\delta(0,t)\&space;{\mathbb&space;E}[&space;C_{\rm&space;dig\&space;call}(t)&space;]$

where $\inline&space;C_{\rm&space;dig\&space;call}(t')$ is the price of the option at time $\inline&space;t'$, and we know that this must converge to the payoff as $\inline&space;t&space;\to&space;t'$, so $\inline&space;C_{\rm&space;dig\&space;call}(t)&space;=&space;P_{\rm&space;dig\&space;call}$

$=&space;\delta(0,t)\&space;\int^{\infty}_{-\infty}&space;P_{\rm&space;dig\&space;call}&space;\cdot&space;e^{-{1\over&space;2}x^2}&space;dx$

$\inline&space;P_{\rm&space;dig\&space;call}$ is zero if $\inline&space;S&space;=&space;S_0&space;e^{(r&space;-&space;{1\over&space;2}\sigma^2)t&space;+&space;\sigma&space;\sqrt{t}&space;x}&space;<&space;K$ which corresponds to $\inline&space;x&space;<&space;{\ln{K&space;\over&space;S_0}&space;-&space;(r-{1\over&space;2}\sigma^2)t&space;\over&space;\sigma&space;\sqrt{t}}&space;=&space;-d_2$

$=&space;\delta(0,t)\&space;\cdot&space;\1&space;\cdot&space;\int^{\infty}_{-d_2}&space;e^{-{1\over&space;2}x^2}&space;dx$

$=&space;\&space;\delta(0,t)\cdot&space;\Phi(d_2)$

where

$d_1&space;=&space;{\ln{S\over&space;K}+(r+{1\over&space;2}\sigma^2)t&space;\over&space;\sigma&space;\sqrt{t}}&space;\quad&space;;\quad&space;d_2&space;=&space;{\ln{S\over&space;K}+(r-{1\over&space;2}\sigma^2)t&space;\over&space;\sigma&space;\sqrt{t}}$

Since we have an analytical price, we can also calculate an expression for the GREEKS of this option by differentiating by the various parameters that appear in the price. Analytical expressions for a digital call’s greeks are:

$\Delta&space;=&space;{\partial&space;C&space;\over&space;\partial&space;S}&space;=&space;e^{-rt}\cdot&space;{\phi(d_2)\over&space;S\sigma&space;\sqrt{t}}$

$\nu&space;=&space;{\partial&space;C&space;\over&space;\partial&space;\sigma}&space;=&space;-e^{-rt}\cdot&space;{d_1&space;\&space;\phi(d_2)\over&space;\sigma}$

$\gamma&space;=&space;{\partial^2&space;C\over&space;\partial&space;S^2}&space;=&space;-e^{-rt}\cdot&space;{d_1&space;\&space;\phi(d_2)\over&space;S^2&space;\sigma^2&space;t}&space;=&space;-e^{-rt}\cdot&space;{d_1\&space;\phi(d_1)\over&space;S&space;K&space;\sigma^2&space;t}$

${\rm&space;Vanna}&space;=&space;{\partial^2&space;C&space;\over&space;\partial&space;S&space;\partial&space;\sigma}&space;=e^{-rt}\cdot&space;{\phi(d_2)\over&space;S&space;\sigma^2&space;\sqrt{t}}\Big(&space;d_1&space;d_2&space;-&space;1&space;\Big)$

${\rm&space;Volga}&space;=&space;{\partial^2&space;C&space;\over&space;\partial&space;\sigma^2}&space;=&space;e^{-rt}\cdot&space;{\phi(d_2)\over&space;\sigma^2}\cdot&space;\Big(&space;d_1&space;+&space;d_2&space;-&space;d_1^2&space;d_2&space;\Big)$

where

$\phi(d_1)&space;=&space;{1\over&space;\sqrt{2&space;\pi}}\&space;{e^{-{1\over&space;2}d_1^2}&space;}$

Holding a binary put and a binary call with the same strike is just the same as holding a zero-coupon bond, since we are guaranteed to receive \$1 wherever the spot ends up, so the price of a binary put must be

$e^{-rt}&space;=&space;e^{-rt}\cdot&space;\Phi(d_2)&space;+&space;C_{\rm&space;dig\&space;put}(t'=0)\quad&space;\quad&space;\quad&space;\quad&space;[1]$

$C_{\rm&space;dig\&space;put}(0)&space;=&space;e^{-rt}&space;\cdot&space;\Phi(-d_2)$

Moreover, differentiating equation [1] above shows that the greeks of a digital put are simply the negative of the greeks of a digital call with the same strike.

Graphs of these are shown for a typical binary option in the following graphs. Unlike vanilla options, these option prices aren’t monotonic in volatility: if they’re in-the-money, increasing vol will actually DECREASE the price since it makes them more likely to end out-of-the-money!

One final point on pricing, note that the payoff of a digital call is the negative of the derivative of a vanilla call payoff wrt. strike; and the payoff of a digital put is the positive of the derivative of a vanilla put payoff wrt. strike. This means that any binary greek can be calculated from the corresponding vanilla greek as follows

$\Omega_{\rm&space;dig\&space;call}&space;=&space;-{\partial&space;\Omega_{\rm&space;call}\over&space;\partial&space;K}$

$\Omega_{\rm&space;dig\&space;put}&space;=&space;{\partial&space;\Omega_{\rm&space;put}\over&space;\partial&space;K}$

where here $\inline&space;\Omega$ represents a general greek.

If you haven’t yet installed the MONTE CARLO pricer, you can find some instructions for doing so in a previous post. The following links give the header and source files for binary calls and puts which can be dropped in to the project in your C++ development environment

These will register the option types with the option factory and allow monte carlo pricing of the options (so far, all of the options in the factory also have analytical expressions, but I’ll soon present some options that can only be priced by Monte Carlo).