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Let's say you want to find the fair price of a call option. One way is to use the black scholes formula. This assumes you can delta-hedge the underlying asset and the option to eliminate risk, and hence come up with a fair price such that there are theoretically no arbitrage opportunities.

However, what if you were NOT allowed to delta hedge? For example, what if the call option was bought over the counter (off exchange). Does the rationale of the black scholes option price still hold?

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3 Answers 3

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it's the liquidity of the underlying that matters not the market in which the option was bought.

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    $\begingroup$ Please ellaborate or post your remark as a comment. $\endgroup$
    – lehalle
    Commented Nov 12, 2014 at 22:23
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The Black Scholes formula still holds without any hedging (as long as the underlying follows a GBM) by the first fundamental theorem of asset pricing. The original paper by Black and Scholes actually made an equilibrium argument as well as a no-arbitrage argument to show that the formula is valid.

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    $\begingroup$ Without delta hedging how do you intend to replicate the payoff? It is the replication that creates no arbitrage in the BS argument. Are you saying you can statically hedge? $\endgroup$
    – Drew
    Commented Dec 14, 2014 at 21:14
  • $\begingroup$ You don't replicate the payoff...you have to rely on equilibrium arguments. $\endgroup$
    – user9403
    Commented Dec 14, 2014 at 21:30
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    $\begingroup$ Can you explain the difference? It seems to be the same thing. Define an equilibrium argument and then define the BS equilibrium argument. If you cant delta hedge or statically hedge, how does one remove the effect of the drift? Remember BS' equilibrium arguments are not the correct ones, those belong the Merton (1973), and Ross shows even those are too strong. BS require delta hedging for the modification of $\mu$ in Sprenkle's formula to r. $\endgroup$
    – Drew
    Commented Dec 14, 2014 at 22:24
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    $\begingroup$ Please see pages 642-645 of BS. They invoke a delta hedging argument there. $\endgroup$
    – Drew
    Commented Dec 14, 2014 at 22:26
  • $\begingroup$ They use a no arbitrage argument as their main argument, but they offer an alternate argument using equilibrium (CAPM) considerations in pages 645-646. $\endgroup$
    – user9403
    Commented Dec 15, 2014 at 1:01
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I use the Black-Scholes formula here http://www.seleno.us/options.php often when I buy and sell options. Sometimes the Black-Scholes price equals the price of the option on the exchange. A lot of times when companies grant stocks options they use the Black-Scholes to report the expense of the options or determine how much the stock price might be diluted by issuing new shares and it is useful in this regard. I think if you use Black-Scholes and the price of the option is really high in comparison to Black-Scholes you shouldn't buy the option. Maybe you should sell it. I've often noticed on exchanges where the price isn't the same. Probably goes the option is getting bid up. Maybe this is an arbitrate opportunity.

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  • $\begingroup$ Could you clarify your answer? It seems to be entirely wrong. Namely, that deviations from BS prices indicate arbitrage opportunities or even statistical arb opportunities. Thats not true, as it goes against the whole idea of the skew. If you mean, deviations from the local volatility indicate statistical arbitrage opportunities, I will agree. Otherwise, this answer doesnt seem to add much value. $\endgroup$
    – Drew
    Commented Dec 14, 2014 at 21:11
  • $\begingroup$ Come on. Black-Scholes is a theoretical price, which is sometimes representative of the price of an option, but not always. If someone off the exchange wanted to sell me an option really bad I could ignoring BS negotiate a price. Think about it. As for arbitrage opportunities, arbitrageers are a big part of the market. Deviations from BS prices could indicate arbitrage opportunities. What decided the price of option pre Black-Scholes? $\endgroup$
    – JTHouseCat
    Commented Dec 15, 2014 at 22:43
  • $\begingroup$ Your post doesn't really answer the question and it seems your affiliated with the website selono.us. Please disclose affiliations and clarify your answer otherwise this seems like spam and that merits deletion. $\endgroup$
    – Bob Jansen ♦
    Commented Dec 20, 2014 at 12:34
  • $\begingroup$ "You're" affiliated. Not "your" bob. The normal dist of d1 in black scholes is the call delta. I don't think you guys get it. Maybe take a finance class $\endgroup$
    – JTHouseCat
    Commented Dec 21, 2014 at 6:36
  • $\begingroup$ If you are going to reprimand me and call me a spammer (because I know how to write code) please at least spell correctly. $\endgroup$
    – JTHouseCat
    Commented Dec 22, 2014 at 3:24

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