some questions about pricing an asset or nothing put option with a strike price equal to St

Question

I am working on a homework exercise where the aim is to price an asset or nothing put with K = St, offcourse the normal formula could be used St * N(-d1), but I was wondering if pricing the asset by making a replicating portfolio with the same cumulative greeks was also possible. In this context, I have a few questions.

• If two portfolios have the same greeks does this mean the price should be the same by no-arbitrage?

• Why use the greeks to price an exotic option and not the 'normal' formula, why would the results be better (as both are derived under the black Scholes framework)? Are for instance fewer assumptions needed?

• Are my formulas for the greeks correct? I derived them and checked them multiple times(see attachment), but I am not sure they are correct.

• When searching for a portfolio with cumulative greeks equal to the asset or nothing put how far can you realistically lever up before the flaws in the black Scholes framework make the answer unreliable. For instance, I expect a long position in 300 calls with strike 2600 and a short position in 300 calls with a strike of 2605 would be problematic?

I offcourse tried googling, but the answers I find seem contradictionary to me.

Answer 1

You can find the answers to most of your questions in the Taylor's series and/or approximation theory articles, but I will add a bit more detail below (in order):

A simplistic example would be $y=a+bx$ vs $z=bx$, so greeks being equal does not necessarily mean that the prices will be equal. But you can use hedging/replicating argument, though it needs more assumptions - e.g., financing, frictionless trading etc.

Analytical formula will be preferred if it is available and feasible. Assuming Greeks are derived based on the same analytical formula (which I think is the case here), or under similar assumptions, then Greeks based approach can at best be an approximation, which is not ideal. But if you don't have the analytic formula, or it is slow to evaluate, then you can employ some approximation (e.g., polynomial fitting, simpler models), though these will require more inputs than just the Greeks (some of the best ones don't require greeks at all). The Greeks based approaches (e.g., Taylor's series) are usually used to approximate how much the price will change by when the input changes. These could be to evaluate the need for/assess the effectiveness of hedging, or to simulate P&L etc.

To check the formulae, please compare your formulae against the results/derivations here, and let us know if you see any differences that you cannot explain. For put, please set $\phi=-1$, and you can ignore the $r_f$, which represents the foreign currency interest rate(in FX) or the dividend yield (stock).

If you are planning to use Taylor's series, then it would be the truncation error - if you have accounted for greeks of all orders, including the cross greeks (e.g., $\frac{\partial}{\partial S \partial \sigma}$), and the market moves are small, then the estimate would be more accurate, but then accuracy depends on the use!

Hope this helps!