Arbitrage-Free Option Pricing and Dynamic Market Completion

# What is the principle of determining an arbitrary option price


# What is the principle of determining an arbitrary option price












First I want to talk about one of my wrong ways of pricing an European call option. When I consider the simplest case of European call option, the first idea of determining the price is to calculate the probabilistic average of the output at the end time point, $\displaystyle\int_{\omega\in\Omega}(S(\omega,T)-K)^{+}p(\omega)d\omega$, and pull it back to the current time point by multiplying the mean value by $e^{-r(T-t)}$, where $r$, by assumption, is the stable and constant interest rate of a money market. However, as I continue to learn this subject, I found the logical flaw in this definition. Although there is no doubt that $\displaystyle\int_{\omega\in\Omega}(S(\omega,T)-K)^{+}p(\omega)d\omega$ is the correct average price of the call option in terms of the money at time $T$, there is no natural reason to believe that $e^{-r(T-t)}$ is the correct link between the money at time $T$ and the money at time $t$.

In the above example the output $(S(\omega, T)-K)^{+}$ only depends on the final price of the stock. And there are options that the output actually relies on more information of $\omega$, e.g. the exotic option. So what I want to know is that, what is the eventual/most fundamental of determining the price of a call option? I guess you may immediately refer to "arbitrage-free", or some technical tools like "risk-neutral measure". But "arbitrage-free" looks more like a philosophical principle. What I really want to investigate, is the idea behind "risk-neutral measure", or the original idea of how to describe "arbitrage-free" in terms of mathematical language. In other words, how should one start to consider, by him/herself, the mathematical problem of option pricing, before any of those ideas/tools were developed?

## Answer by LocalVolatility (score 2)

https://quant.stackexchange.com/a/30798

You make the argument that the option risky and thus its expected payoff should be discounted at a risk-adjusted rate by a risk averse agent. I would even go one step back. You say that

\begin{equation} \int_{\omega \in \Omega} (S(\omega, T) - K)^+ p(\omega)\mathrm{d}\omega \end{equation}

is the "correct" average price of the call option. What growth rate are you using for the underlying asset which is itself also risky?

These were exactly the open questions in option pricing pre Black-Scholes. The concept of log-normally distributed stock prices was not new and structurally similar equations for European plain vanilla call options were obtained before. A very good account can be found here. The major problem pre Black-Scholes was that both the appropriate growth rate for the underlying asset and the discounting rate used for the option were unobservable and utility dependent.

The concept of state prices existed already earlier but the problem was that $n$ terminal states required $n$ linearly independent assets to complete the market and obtain arbitrage prices for contingent claims. Black and Scholes' insight was that a market could be dynamically completed through continuous trading. I.e. although there is an infinite number of terminal states, two assets are sufficient when the risky asset is driven only by a diffusion.

Both state pricing and Black Scholes heavily rely on the concept of arbitrage or the absence thereof. I will not take the bait and elaborate why this is not a "philosophy" as you claim. I do however suggest you read up on the topic to improve your understanding. Here are a few good references:

- Pennachi (2008) is a good general introduction to asset pricing theory. See in particular Chapter 4 "Consumption-Savings and State Pricing" and Chapter 9 "Dynamic Hedging and PDE Valuation".

- Musiela and Rutkowski (2005) is an excellent overview of derivatives pricing in a diffusion context. See in particular Chapter 3 "Benchmark Models in Continuous Time".

References

Musiela, Mark and Marek Rutkowski (2005) "Martingale Methods in Financial Modelling", Springer, 2nd Edition

Pennachi, George G. (2008) "Theory of Asset Pricing", Pearson