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I just started my financial maths master and was introduced to binomial option pricing for European options.

I am slightly confused by the derivation as I saw a different version. Some straightly get into "we want to replicate an option with a riskless and a risky asset" while others use a covered call to introduce the derivation.

  1. I am wondering if different hedging strategies (I am not entirely sure it is hedging) affect the valuation of the option.

  2. When replicating a portfolio for an option, why do we need a risky and a riskless asset at the same time? can we just use either one?

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They are both doing the same thing. 1 derivation is constructing a portfolio containing shorted calls and longed stocks (or the reverse), the other is setting up a replicating portfolio. They both achieve the same result.

There are different types of hedging strategies. The one you are currently learning is to be "delta-neutral", which is hedging the price of the stock. So if S goes from 20 to 10, you would have not lost money because you profited from the call options you sold. There are other types of hedging like Vega hedging, which is hedging the volatility of a stock.

To your second question, we have a risk-free account because the discounted call option is a martingale. More intuitively, the call option requires money and you are looking at the difference of your bankroll going up and down versus how much the call option is.

If you look at the derivation of the Heston PDE, it's similar to what you would have done in class with the black-scholes, but we have a new account to hedge the volatility, so the portfolio has the value:

$$\Pi = V + \Delta S + \phi U$$

https://www.frouah.com/finance%20notes/The%20Heston%20model%20short%20version.pdf

If you want a more technical explanation of why we require a risk-free account, try to prove that the call option option (without discounting) is a martingale (it won't be possible).

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  • $\begingroup$ Thank you so much for your answer. One quick question, do you think it would be better for me if I just accept some of the derivation as a 'given' at the moment? I sense that I need to further my learning and reflect back so I will understand things better rather than try to understand everything at once $\endgroup$
    – TJT
    Oct 15, 2023 at 1:32
  • $\begingroup$ Probably. I didn't fully understand it until I did my upper level stochastic calculus class and interest rate theory class. Everything will loop back around and fit together when you learn the 1st fundamental theorem of asset pricing. It might make more sense when you look at SDEs and when they're martingales $\endgroup$ Oct 15, 2023 at 3:14

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