I. INTRODUCTION
The analysis of European option pricing is by far the most complete in the sense that one can obtain useful no-arbitrage bounds without appealing to any underlying asset price evolution model and even an exact relationship known as put-call parity. Furthermore, exact pricing formulas when geometric Brownian motion is assumed for asset price evolution can be achieved.
II. NO ARBITRAGE BOUNDS
While it is possible to obtain exact pricing formulas for European options, it is instructive to first see what can be accomplished by consider no-arbitrage arguments. This is so for two principle reasons: one, the arguments do not require us to assume any particular asset price evolution model - the conclusions are valid in any complete market that does not permit arbitrage - and two, the arguments lead to easy to remember bounds on European option prices that are useful to keep in mind as a sanity check for when one is numerically computing prices; furthermore, when certain parameters approach extreme values, the bounds obtained tend to become sharp and approximately agree with each other, thus yielding the option's price (which is an even more legitimate price since, once again, no assumptions are made about how the underlying asset evolves in time).
Let us denote by $c(S,t)$ and $p(S,t)$ the values of European options at time $t$ where $S=S(t)$ is the underlying asset's price at time $t$. We are primarily interested in determining $c:=c(S_{0},0)$ and $p:=(S_{0},0)$, the premium paid by the buyer for the right to exercise an option at a future time. These quantities are of course non-trivial since options always have positive value at time $t=0$ (unless of course $S_{0}\equiv0$), and this is what makes pricing options significantly more difficult (and interesting) than the pricing of "zero-sum" derivatives like futures and swaps that have a value of $0$ and time $t=0$, and in any event are easily priced at all future times (we will not concern ourselves with the pricing of options at future times in this section, since as mentioned, it is much more important to know the values at the intial time $t=0$).
Suppose that we sell a put option for $p$ dollars and invest the proceeds into the risk-free money market account so that at expiry we have $pe^{rT}$ dollars. If $S_{T}\geq K$ then the option is worthless and we have $pe^{rT}$ dollars. If $S_{T}<K$, then the buyer will exercise the put option and we then have $pe^{rT}-K$ dollars. If $p>Ke^{-rT}$, then in both cases we have made a riskless profit of at least $pe^{rT}-K>0$ from a portfolio that cost nothing to setup. This arbitrage opportunity is only possible if $p>Ke^{-rT}$. This yields the no-arbitrage upper bound
$$p\leq Ke^{-rT}.$$
Similarly, if we sell a call option for $c$ dollars and cover our obligation to sell the underlying asset for $K$ dollars by buying the asset today for $S_{0}$ dollars, then we can make a riskless profit from a portfolio that cost nothing to setup if and only if $c>S_{0}$. Thus,
$$c\leq S_{0}.$$
(It is instructive to think about why the upper bound for $p$ involves $K$ and discounting while the upper bound for $c$ involves $S_{0}$ and no discounting, and that there is no alternative to this which is sharper.)
To obtain a lower bound, suppose that ...
III. PUT-CALL PARTIY
It is a rather remarkable fact that there exists a function $\Phi$ such that $\Phi(c,p)=0$. This fact is known as put-call parity and it allows on to price $p$ given $c$ and vice versa, with only the assumption of no-arbitrage.
To derive it, consider a portfolio $A$ consisting of ... and a portfolio $B$ consisting of ...
IV. EXACT PRICING FORMULAS
If we move from our realistic no-arbitrage world to that of Black-Scholes, we can obtain exact pricing formulas for $c$ and $p$ (for all times $0\leq t\leq T$ too!). The Black-Scholes model is discussed in this previous post ... The main quantitative assumption (in addition to the various qualitative market assumptions) is that the underlying asset follows the geometric Brownian motion process
$$S(t)=S(0)+\int_{0}^{t}\mu S(\tau)\;d\tau+\int_{0}^{t}\sigma S(\tau)\;dW(\tau),$$
where $\{W_{t}\}_{t\geq0}$ is a Brownian motion. Under this assumption, we derived the PDE
$$V_{t}+\frac{1}{2}\sigma^{2}S^{2}V_{SS}+rSV_{S}-rV=0,$$
for which any derivative depending on $S$ must satisfy (i.e. $c=c(S,t)$ and $p=p(S,t)$ must satisfy the PDE). By solving this PDE with boundary and terminal conditions characteristic of the particular derivative under consideration, we get a deterministic pricing function $V$ for the derivative, valid for all time $t\geq0$ and asset price paths $\{S_{t}\}_{t\geq0}$.
Basic financial considerations derived from the previous sections lead to the following BVP for $c$:
\[\left\{\begin{array}{rr}c_{t}+\frac{1}{2}\sigma^{2}S^{2}c_{SS}+rSc_{S}-rc=0,&(S,t)\in(0,\infty)\times(0,T)\\c(S,T)=\max(S-K,0)&\\c(0,t)=0&.\\\lim_{S\to\infty}c(S,t)=S&\end{array}\right.\]
(It should be understood that while $S=S(t)$ so that $c=c(S(t),t)=c(t)$, we generally consider $c$ to be a function two variables $(S,t)$, particularly when working directly with the PDE so that for instance the terminal condition is written as above instead of as $c(S(T),T)=\max(S(T)-K,0)$.)
Similarly, for $p$ we have the BVP:
\[\left\{\begin{array}{rr}p_{t}+\frac{1}{2}\sigma^{2}S^{2}p_{SS}+rSp_{S}-rp=0,&(S,t)\in(0,\infty)\times(0,T)\\p(S,T)=\max(K-S,0)&\\p(0,t)=Ke^{-r(T-t)}&.\\\lim_{S\to\infty}c(S,t)=0&\end{array}\right.\]
Let us solve the BVP for $c$. In this previous post on the analysis of the PDE, we discussed how to transform the Black-Scholes PDE into the heat equation by making a succession of variable changes in order to eliminate the various lower order terms. Review that post to see the transformation carried out in detail; here we just summarize the order of the transformations to be carried out and the resulting BVP:
