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I have just started studying finance and stochastic calculus so apologies if this question is too naive.

I was first introduced to stocks and bonds as risk and riskless investment assets. Then a new idea is introduced, that of a call option. At first sight, this seems like a new independent investment product, and that you wouldn't be able to create such a portfolio using stocks and bonds only.

But now to calculate the price of this call option, we assume no-arbitrage in our model (Binomial and Black Scholes are the two I have studied so far) and hedge using the stocks and bonds. In other words, just using stocks and bonds we create a portfolio that is the same as a call option. This was surprising to me in a nice way.

But at the same time, it has left me confused. If I could achieve such a portfolio using stocks/bonds then why are call options studied separately?

In fact, it seems to me like introducing call options as an additional investment asset class adds to your "risk dimension". In the sense that, when someone sells a call option, they need to hedge against this risk. So why should someone sell a call option, instead of shorting some stocks or bonds? I agree that options risk will still exist implicitly, but introducing them as a separate asset class just adds to the risk dimension that I would need to hedge.

Also, are there any other merits of options that stocks/bonds alone cannot provide, perhaps in the real world?

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2 Answers 2

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"But at the same time, it has left me confused. If I could achieve such a portfolio using stocks/bonds then why are call options studied separately?"

Good question! The short answer: because it turns out options cannot be synthesized using stocks and bonds only, except in the highly idealized case of Black-Scholes which assumes constant volatility, constant interest rates, infinite liquidity, no transaction costs to name but a few sources of replication "slippage".

Hence options should really be regarded as a new source of risk, even perhaps a different asset class.

The uses of options are speculation and/or taking a view without the need to buy the underlying security, but for a large part also hedging the risk of other (more complex) options such as options embedded in structured notes and life insurance guarantees, and to create bespoke investment vehicles that address investors needs for particular risk profiles.

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  • $\begingroup$ So if I understand correctly, options cannot be synthesized using stocks/bonds in the real world. But to price them, we create a model where we can synthesize them and this gives us an "approximation" for its price which works out well in the real world? $\endgroup$
    – kishlaya
    Oct 7, 2021 at 13:38
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    $\begingroup$ Yes you need a model to price them. Blck Scholes is the 'simplest', and you will encounter others such as stochastic vol models etc. All parametric models, no matter how sophisticated, are approximations. Hence, if possible, you'd like model free results. But that's another topic. $\endgroup$ Oct 7, 2021 at 13:42
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    $\begingroup$ For vanilla options the market price is what is observed in the market. For exotic options it's more difficult to answer. You will see that under the Black Scholes model there is no way you can match market price of vanillas. However if you "abuse" the BS call/put price formula by assigning a different volatility (called implied volatility) to each option strike you can match the market price of vanillas. Feel free of course to post another question or search existing questions and answers when you arrive at the wonderful world of implied vol. $\endgroup$ Oct 7, 2021 at 13:50
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    $\begingroup$ "because it turns out options cannot be synthesized using stocks and bonds only", I think this can be misinterpreted. It can be, but this synthetization does not work so well in the real world as it does in theory because of the things you mentioned $-$ I would particularly highlight transaction costs and non-constant volatility. $\endgroup$ Oct 7, 2021 at 14:01
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    $\begingroup$ Indeed an interesting question that is related to the Hakansson's paradox. $\endgroup$
    – AKdemy
    Oct 7, 2021 at 14:20
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Yes, conceptually, you can replicate options with stocks and bonds, but why would you when options are readily available?

Put another way, you can replicate a Coke with the right amount of sugar, water, flavorings, etc., but why would you do that when you can just buy a bottle. How easy are those ingredients to get individually (and in the minute quantities that you need)?

It's impractical to replicate an option exactly with the right about of stocks and bonds. That replication is great for creating pricing models, but is not practical to do in reality.

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  • $\begingroup$ Because someone has to start selling coke bottles... someone has to start selling options. Why would they sell an option when they can create an equivalent portfolio for themself using stocks/bonds. $\endgroup$
    – kishlaya
    Oct 7, 2021 at 13:42
  • $\begingroup$ Again, it's highly impractical to replicate options (whether buying or selling) with stocks and bonds. Replication is much more a theoretical exercise than a practical one. $\endgroup$
    – D Stanley
    Oct 7, 2021 at 13:58

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