Option Pricing Form

Question

For option pricing, we have a formula for the price of an option like the following: $$\Phi(t,S_{t})$$ Where $S_{t}$ is the stock price.

Why not just write in a form of composite function: $$\Phi(S(t))$$ Clearly they are different. But both of them are stochastic processes w.r.t time t. Why should we use the first one? What is the mathematical reason behind that, or the physical meaning under that(I believe it has some).

Answer 1

In Merton's paper, "Theory of Rational Option Pricing," he developed properties of vanilla options (American and European) that relied on strict arbitrage laws and not on any presupposed model for price stochastics.

TIME DEPENDENCE

There are time dependency properties that are based only on these arbitrage laws. For example, Property $(4)$ of the paper states that for two call options with the same underlying asset and same strike, but different times to maturity, the value of the option with the greater time to maturity exceeds the value of the other.

Hence, the value of a call option increases with time to maturity $T$ or equivalently decreases with time $t$. So, there is explicit time dependence due to the "physical" nature of an option.

PRICE DEPENDENCE

There are also price dependent properties that are based on arbitrage only. Alongside these properties, Theorem $10$ of the paper proves that if the distribution of returns on the underlying price is independent of the price level, then the value of the option is a convex function of the underlying price.

---

Thus, an option depends on both time and price as independent variables.

Answer 2

The reason is that we decide that $S_t$ is a variable by itself, and that we may have a payoff that is a function of t, independently of $S_t$. The second expression is just saying that the payoff is only function of the underlying $S_t$, which is also a function of time. For example , $\phi(t,S_t)=S_t-t^2$, will not fit into your second definition but in the first one.

Answer 3

I see the following reasons to do that. First of all as dynamical system, its evolution function is explicitly dependent on $t$. Secondly It is explicitly dendent on $S$. For instance you can suppose $S$ to be something not dependent on time(like something trivial). Next , a convinient tool of stochastic analysis is Ito formula. And $S$ is almost always defined through some Stochastic differential equation, Ito formula can be used to derive the stochastic differential equation of $\Phi(t, S_t)$. And last. It is sometimes convenient to look at non stochastic PDE(or ODE)-s. There $S_{t}$ is almost always substituted with dummy variable $x$ In the form you have proposed, there is now dependence on t. Which merely means that if for some outcome $\omega$, we have $S_{t_{1}}= S_{t_{2}}$, than $\Phi(t_{1})=\Phi(t_{2})$, which is not always true.

Maybe I'm wrong.

Answer 4

The payoff may not depend on time, however, the option value, at time $t$, has a particular dependence on time $t$ through the discount factor, specifically, the discount factor is a function of time $t$, or defined by a term structure.