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In reality, companies hardly change their valuations on a day-to-day basis, but their stock prices and valuations change nearly every second. This difficulty in reaching a consensus about correct pricing for any tradable asset leads to short-lived arbitrage opportunities. But a lot of successful investing boils down to a simple question of present-day valuation— what is the right current price today for an expected future payoff?

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Here, riskless means that the portfolio has a known value at the end of the period, no matter what happens to the stock price. If the portfolio is riskless, then we know its current value; it is simply the future value discounted by the risk-free interest rate. Then, since we know the portfolio value and the stock price, we can determine the option price. Why are we doing this?

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The idea is that both a derivative and the underlying asset depend on the same random diffusion term, since Ito lemma shows that both asset processes share the same random term. By adjusting the weights assigned to the derivative and the underlying asset, one can form a portfolio which is instantaneously risk-free. Such portfolio earns the risk-free rate. The starting point is the process of the underlying asset and the process of the derivative is obtained from Ito lemma.

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By riskless portfolio, he means a portfolio with totally predictable payoff. He then uses this argument to give the correct current price of the option which makes arbitrage impossible. Now I understand why the risk-neutral price of the call option is the only arbitrage-free price. If the call were overpriced an arbitrageur would long a replicating portfolio which easily exists in this model by some elementary linear algebra and short the call and if it were underpriced he or she would do the opposite.

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