# Why hold options when you can dynamically replicate their payoff?

When holding vanilla options, you can cancel out, theoretically, all risk with dynamic (delta) hedging. Then you earn the "risk free rate of return".

Why would you make such a portfolio when you can simply buy a bond that earns the "risk free rate of return"?

• Hi Andrei, welcome to quant.SE and thanks for posting your question. Sep 7 '11 at 14:50
• sometimes, the imputed interest rate works out to be much much higher than the risk free theoretical
– user3232
Feb 15 '13 at 5:46

Short Version : Two main uses

1. I'm doing an arbitrage/statarb strategy (volatility for instance) which should not be dependant on the Delta (I'm an arbitragist).

2. I HAVE to keep a product in my portfolio, but I don't want to be EXPOSED to it (I'm a market maker).

Long Version :

The goal of Dynamic Hedging is not down the line to earn risk free rate of return. You are probably talking about a Delta Hedge, Delta is not the only Greek you can hedge, you could hedge over Parameters, but I assume you're talking about Delta.

If I'm an option trader, I can basically Buy or Sell Volatility, I will hedge my Delta at the start of the day (Usually before the close of the day before). But I will gain money on the day after if realized Volatility is Higher than Implied Volatility if I'm a Volatility Buyer (and vice versa). Pay off of the strategy below :

So I need dynamic hedging, which if done every minute will provide me risk free interest rate minus fees (i.e. probably negative return), if I do it once a day I will realize profit that is not related to the direction of the change of the underlying but rather it's intensity.

Other businesses in Finance need to Hedge Frequently as they are not supposed to have a directionnal bias on the market, most notably Market Makers and Liquidity Suppliers. Sometimes you also need to keep some position you can't sell, like an OTC swap, in your portfolio, you probably want a very good hedge on this.

• In Black Scholes equations, dynamic hedging will cancel out the dZ factor. The portfolio value becomes dependant just on dt. I'm talking exactly about "coutinous time Dynamic Hedging" equations.
– Andr
Sep 7 '11 at 14:23
• Then it's second part : you're a market maker and have open positions or you have to keep an OTC Deal which you don't want to be exposed to. I bolded it. Sep 7 '11 at 14:52

I think there is an error implicit in your question. Dynamic delta hedging, even assuming the underlying process is a continuous martingale and trading entails zero transaction costs, only eliminates the directional risk. A number of residual risks remain, most notably volatility risk, embodied in both the gamma and vega. A dynamically hedged portfolio of stock and option will only yield the risk-free rate if the realized volatility equals the implied volatility at which the option was purchased. Actual P&L from the portfolio will differ from the risk-free rate if realized differs from implied (gamma risk) or if the implied volatility in the market for similar options changes (vega risk). In addition, some interest rate risk (rho) will also remain.

Note that the Black-Scholes derivation assumes that changes in volatility are only functions of time and underlying (spot) price. Thus, even in the theoretical limit of continuous hedging, one would still want to trade options to take a view on unexpected changes in volatility (or on differences between their expectation and the market's expectation of volatility).

• Great answers, guys! Not much difference between you and the mainstream textbooks. This was just an obvious question that one may get after going through the proof and expecting some surprise. Anyway, you're right.
– Andr
Sep 7 '11 at 17:25
• Do I understand your point correctly here: in BS model the only risk is the directional one, and it can be completely eliminated by $\Delta$-hedging (in theory). Now, if we relax our assumptions a bit and still allow for continuous and cost-free $\Delta$-hedging, we are still exposed to volatility and interest-rate related risks (that are not assumed to be present in the BS model though) - right?
– Ilya
Sep 29 '13 at 11:45

You have to differentiate here between the risk-taking and the market-making side. As a risk-taker, like e.g. a hedge-fund, you are right, you could just buy the bond!

But as a market-maker you sell these options but don't want to bear the risk, so you have to counterbalance it. You could of course counterbalance it with another option which would be the best case when you find another customer as the counterparty (you would earn double here). Another market-maker as your counterparty would not be a good idea since it is too expensive (he wants to earn something too!).

So you have a completely different business model here: As a market-maker you live on the spread but don't want to bear risks. The risk-taker lives on some kind of model or edge but has to assume adequate risks as a consequence.

The first issue here is that the argument underlying your comment is not always valid. It is true that in a Black-Merton-Scholes economy, you can build a perfect hedge, but that is a pretty narrow sliver of all possible price processes. If you make just a tiny step outside of that world and allow for jumps, the whole thing breaks down to pieces. This is an extremely serious problem because:

1. There is ample evidence of excess kurtosis in returns on equity;
2. Without immediate conditionally non-gaussian features present in your model of underlying stocks and related indexes, you simply will not be able to match the volatiltiy surfaces at shorter maturities.

It doesn't matter how fast you trade here: if there are jumps, your delta hedge will be useless. So, even before you look at issues of transaction costs, the whole idea falls apart.

On top of the many other good answers here, you also have to be careful about distinguishing between the "payoff" and the risk exposure.

Let's assume there are no jumps; that zero-cost instant frictionless trading were possible; and none of the classic shortcomings of the Black-Scholes assumptions apply.

Pick any option you like: long or short, call or put, any instrument, any strike, any date. How do you replicate the payoff in the scenario that the market simply doesn't move? The option will obviously return either (0-premium) or (spot-strike-premium), depending on the strike. Your substitute hedge in the underlying will always return 0.

Imagine instead you buy a 100 ATMF call with a ~16% implied vol (ie ~1% a day), and the price goes 100, 101, 100, 101, 100... expiring at 100. Your hedge will "buy high and sell low" every day, losing you a lot more than the premium on the equivalent option you're trying to hedge.

The hedge guarantees the same risk exposure as the option at any point in time. This obviously prevents arbitrage between the option and the underlying. However, this does not then in turn guarantee the same payoff from the hedge as from the option.

Put differently, Black-Scholes is an unbiased and asymptotically consistent estimate of the value of an option... but the hedge that delivers these attractive properties is no guarantee in finite samples... ;-)