Consider the usual one-period binomial model.

The delta-hedging formula, following Shreve's convention, is:


Shreve states:

"The agent has hedged a short position in the derivative security...Although we have determined the no-arbitrage price of a derivative security by setting up a hedge for a short position in the security, one could just as well consider the hedge for a long position. An agent with a long position owns an asset having a certain value, and the agent may wish to set up a hedge to protect against loss of that value. This is how practitioners think about hedging. The number of shares of the underlying stock held by a long positon hedge is the negative of the number determined by the above expression, $\Delta_0$."

$\textbf{My question:}$ What does he mean by this? What is the short position of the European call option he is considering in the text? I think my confusion stems from the fact that I view European or American call option as a hedge against long position. A trader wants to lock-in a price and hedge against its steep depreciation. As for the put option, a trader is preparing for the worst possible outcome, so wouldn't the trader be hedging against the short position of the put option? I am primarily confused when Shreve uses the term short or long position of the derivative, and in his case, it is the European call option.

Shreve, Steven E. Stochastic Calculus for Finance I : The Binomial Asset Pricing Model. New York ; London :Springer, 2005.


The word "hedge" can be ambiguous because it is not always clear what the risk is that we are trying to eliminate.

The "business model" that Shreve has in mind here (which is very common) is that an investment bank sells a derivative to a customer and now is short that derivative. They are exposed to changes in the market value of that derivative which they do not particularly wish to be exposed to. (The idea of buying the derivative was the customer's and the bank merely accomodated that demand). The bank needs a "hedge" that is a position or strategy that neutralizes the unwanted risk: something that goes up one dollar if the derivative goes down one dollar and vice versa. The hedge works both ways, neutralizing all the risk (you are protected against profits as well as losses, so the bank will only earn by charging a small fee for its role).

This is very similar to hedging in the futures market (are you familiar with agricultural futures?) A maize farmer who has 1000 bushels of maize growing on his field can hedge the price risk by taking a position in maize futures whose cash flows exactly offset the market price fluctuations of his outright holdings of physical maize. Once this is done the farmer will not suffer from price declines nor benefit from increases in the crop price.

Bookmakers who accept gambles from the public also work similarly. The bets for Manchester United to win the football match (roughly) offset the bets it will lose. The bookmaker is hedged.

The other sense of hedging is "to insure against one specific bad outcome that one is particularly worried about" without completely eliminating the exposure. If we ask the customer why he has bought a call option and not just the stock itself, he might reply that "I am hedging against a short term decline in case there is some bad news when the company announces results next week". This is a different kind of "hedging", an asymmetric hedging where you are willing to pay a price (the option premium) to selectively eliminate one contingency which you believe has a higher chance of occurring than other people allow for.

so in one case "hedging" means complete elimination of risk by taking on an exactly offsetting risk (complete hedging), in the other it means paying for insurance against a specific future event (asymmetric hedging). Shreve (and an investment bank that makes markets in options) is using the word in the former sense, people who trade options directionally in the latter.

  • $\begingroup$ "What is the short position of the European call option he is considering in the text?" It is a position which a customer bought from us and that we now carry as a short position in our books. And it represents a risk to our firm that we need to get rid of. (The analogy with the bookmaker should be clear, we need to start taking bets in the opposite direction so we are in a balanced position). $\endgroup$
    – Alex C
    Jun 28 '19 at 17:50

That means that to determine the price of a security by non arbitrage you can either find the strategy that will hedge your short position ( when you sold an option) or you can find the strategy that will hedge your long position (when you buy an option). Both need to lead to a profit of 0.
With the short position of an European call, he is considering here is the case of someone who sold the option to a third person. So to hedge himself, he needs to buy the quantity derived from the delta hedging formula. Indeed, it will earn money when the price of the stock decrease (as the option is less likely to be exercised) and lose money in the other case. So, to hedge, he needs to find a way to earn money when the price of the stock increases and to lose money when the security price decreases. Intuitively, it will buy a certain quantity of the security itself and be assured at the end of the day that the profit or loss of one position (the call) is compensated by the other position (the hedge through the underlying).

A call option price increases when the price of the underlying increases and conversely. So, if you are long a position, in order to hedge ( ie have no risk and thus no profit at the end) you will short a certain amount of call option. Hedging means just erasing the risks ( ie the uncertainty). Thus, to hedge, you basically take an opposite bet relative to your current position.


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