Replace the T-bill with the credit you receive upfront (when selling puts or calls)... In other words: the credit is your T-bill...
Looking at the example I mentioned prior, your return will be the credit minus the net of "buy high sell low" hedging and I suspect the outcome will be the risk free interest rate (T-bill).
This according to the paper you quoted: i.e., your return, when you hedge continuously, is the risk free rate, not the premium you collected. By the way, thank you for the link. I finally understand hedging, as he also explained how one could also hedge gamma and vega together with delta. But the outcome is there is no free lunch, when you are risk free, you don't get anything extra other than the risk free rate.
If you buy or sell a fairly valued option and delta hedge at the correct actual volatility continuously, then you will earn the risk-free rate guaranteed. The point is not to try and earn the risk-free rate. You can't hedge continuously, so you are not risk-free! The point is to buy cheap contracts and sell expensive ones in order to earn more than that. Moreover, unless the fair value of the option is 0, and you can sell it for more than that, your expected profit is not simply the premium you collected. More generally: Expected P&L (option buyer) = Black-Scholes Value (actual volatility) - Black-Scholes Value (implied volatility). or equivalently, E [P&L] = (Actual Vol - Implied Vol) * Vega This expectation is there regardless of whether you hedge or not. Hedging is about P&L variance reduction, not increasing the expectation of trade profitability. Furthermore, you gamma hedge to reduce the necessary frequency of delta hedging in a world of transaction costs. You vega hedge because you don't know what actual vol is for sure, so you hedge with other options to help reduce the potential damage to your P&L from being wrong. Vega represents the integral of gamma profits over the life of the option -- i.e. the expected sum of how much you'll make on trading the gamma. If vega is +500, you hedge at 15% but actual vol comes in at 14%, then you expect your P&L to take a negative hit of $500 on average if you are long the option. This is equivalent to the market bringing the implied vol of the option from 15% down to 14%. The option holder loses $500. Hence the normal definition of vega -- the $ change in option price for a % point increase in implied vol. Same idea.
imo, hedging financial trading/investment risk is not simple nor easy for carrying it out correctly, without faults. Already caused many court cases against banks for hedging contracts, as a result of losing great money. LTCM (including Scholes) is a good example of losing great money while performing a perfect dynamic hedging that they wanted/designed! ( http://www.investopedia.com/terms/l/longtermcapital.asp Long-term capital management (LTCM) was a large hedge fund led by Nobel Prize-winning economists and renowned Wall Street traders that nearly collapsed the global financial system in 1998 as a result of high-risk arbitrage trading strategies.) Let's don't forget the potentially expensive risk/cost of carrying out hedging! On top the practicality of any theoretically feasible approaches/models! Here is another book of interest to practitioners as it is written by more than 25 contributors/practitioners (including Scholes), perhaps. Portfolio Insurance: A Guide to Dynamic Hedging There have been several new books in German recently. (But I don't read German.)
Longthewings, I really appreciate your comments. Thank you for your comments in language and terms that I can understand. In general, for a call option buyer, when you hold an option to expiration, P&L = final stock price - strike price - option price (the reverse is true for option seller). If I understand you correctly, if you hedge, the outcome is different. For your hedge to return greater/less than the risk free rate the options you purchased/sold must be mispriced? i.e., if you hold the contract to expiration and continuously delta hedge, assuming Black Scholes, then, Expected P&L (option buyer) = Black-Scholes Value (actual volatility) - Black-Scholes Value (implied volatility)? Is the Expected P&L over and above the risk free rate? Otherwise when actual volatility = implied volatility the return will be zero instead of the risk free rate. I am ignoring transaction costs of course. Another question, if you hold the contract to expiration, is the actual volatility the same as the historical volatility for the period? I appreciate any comments from anyone here so we all can learn. Regards,
Hi longthewings, but what does this practically mean nowadays when the interest rate is zero? Selling options is attractive when one needs immediate cash, or a continous cash flow, so to say. It is useful for those who sit on relatively big funds like OPM managers, and/or have the stock of the underlying in their portfolio (covered selling). What strategy/method would you recommend for profiting from selling options for example for a moderate value of say 300k stock value? Would hedging always be an integral part of it? I guess yes. But then how can one earn any money with options selling nowadays when the interest rate is zero? I rather think that from a logical POV it makes more sense if one sees it so: interest earned = credit collected Ie. the risk-free interest rate here is already integrated into the premium, together with the yield of the underlying company's stock. Is that a wrong assumption? So, what would you recommend for profiting from options selling nowadays with ZIRP, and even NIRP? [OPM = Other People's Money, ZIRP=Zero Interest Rate Policy, NIRP=Negative Interest Rate Policy]
LTCM was a Black Swan event (Asian Crisis, Russia's default etc.). It is IMO also unclear whether they had used any hedging, or were doing casino-gambling, or maybe a pre-planned fraud?... I think most here are more interested in the more general case, ie. under normal market conditions. Hmm, you better shouldn't make recommendations for books/papers you haven't already read yourself, as it otherwise sounds more like an advertisement...
imo, I think we cannot easily frame some proper questions in options trading to learn options trading correctly. Of course, some would not have this same view as mine. Many options traders on ET in the past suggested some very good books for options that could be useful, and it's worth to do a simple search about these books. So long!
ironchef, as usual a very good observation and question formulation. Thx being so precise in your postings, it's a pleasure to read such good postings. I think you should read about this magical term called "discounted price" in the BSM options pricing model.