**INTRODUCTION**

**What are they used for?**

Let’s imagine a financial institution who sells options or other derivatives on an OTC market, they could buy the same options on the regulated markets if they are available, otherwise hedging these positions could be difficult, so we need to introduce some strategies that traders can use in every context. Well, the institution could even keep a naked position, just by selling the option and not trying to cover it. The alternative is to create a covered position, by buying the underlying asset. However, both these strategies could produce some large losses and so this is not the answer we are looking for.

**STOP LOSS strategy**

To better explain this strategy we first introduce an example: a financial institution has written a call option (european), with strike price K. The stop loss strategy consists of buying the underlying asset when the price rises over K and selling it when it falls below K. If the call option will expire in the

money the institution has the asset available bought at K and if the option expires out of the money it doesn’t have anything. The cost of this strategy will be equal to

Q=max(S0-k,0)

This is just a theoretical situation in which there are no transaction costs and buying and selling the underlying assets happens at the price K. However, we know that this value is not correct, since every transaction will take place at a price different from K, given that the hedger cannot know if when the price is K it will rise or fall. The price will probably be something in the form of K+epsilon or K-epsilon, so that every transaction costs epsilon. To try and reduce epsilon the hedger could increase the frequency of its transactions.

**DELTA HEDGING strategy**

The stop loss strategy is not the most efficient one and traders know that, so they usually compute some more advanced statistics for their portfolios, which are called Greek letters or Greeks. These are used to assess different faces of the risk in a portfolio or in an option position. To calculate the Greeks the models used by traders are the Black-Scholes-Merton for european options and binomial trees for american ones. Another assumption we need to make when talking about Greeks is that the volatility is equal to the implied volatility of the option, so that the price of the option is an exact function of price, volatility, interest rate and dividend yield.

The delta can be expressed as the derivative of the option price over the price of the underlying asset, so it represents the change in price of the option for a certain change in the price of the underlying asset. In symbols:

With c that is the price of a call option and S is the price of the underlying asset. The target for the institution is to keep a delta neutral position for the portfolio. This means that if it has a short position on N call option contracts, each for 100 shares of a certain stock, to hedge this position the institution will need to buy deltaxNx100 shares. This is also the value of the delta of the option position, since when the stock price rise of deltaS the position on the options losses deltaSxdeltaxNx100, and the delta for the stock position will be the reciprocal, so that if we sum up the two positions we end up with zero (the delta for each share is 1). It is seems easy, however, the value of the delta changes as shown below:

So the hedger has to rebalance his position continuously, to keep a delta neutral position. For example if the stock price rises, also the delta of the call option rises and the hedger will need to buy (deltat+1-deltat)xNx100 shares. This strategy of dynamic hedging is the most popular, but the institution can even set a delta at the beginning of the period and never rebalance, this is called static hedging.

But at this point the question is how can we compute the delta?

For european call options on stocks that do not pay any dividend the formula can be derived from the Black-Scholes model, since their idea was to price options by creating a delta neutral portfolio with the same expected value.

In this case the delta is negative, this means that to hedge a long position on a put you need to have a long position on the stock.

Here it is an example of a delta hedging strategy with its cost: imagine a situation in which a financial institution sold for $300,000 a european call with strike price $50 (current price is $49) and expiring in 20 weeks, so that deltaT=0.3846, on 100,000 shares of a stock that do not pay any dividend; the risk-free rate is at 5%, the expected return is 13% and we assume that the volatility is constant at 20%. Applying the Black-Scholes-Merton formula we end up with a value of $239,466 and a delta of 0.521 at t=0.

This is what our input looks like. We can then simulate the path of the underlying stock, assuming it follows a geometric brownian motion, described by the following equation:

Where epsilon is a random value from a normal distribution. The antithetic case, where the final value of the stock is lower than the strike price, has been created by symmetry:

And this is what our results look like, assuming also a constant volatility and no transaction costs:

The final value is the cumulative cost computed as (final cumulative cost - (final Delta*final stock price*number of shares)). The Delta has been computed by applying the Black-Scholes-Merton formula and the same table has been developed for the antithetic case. This case is valid when the rebalancing happens once a week, however, more often the delta hedging strategy is rebalanced on a daily basis and we made a simulation of both scenarios just to see if the results are significantly different; this is what we got:

The first thing we should notice is how both the base and the antithetic case have similar results, which are not far from the value of the option we got with the BSM formula, well this is because performing such a strategy can be intended has creating a synthetic option, opposite to the one originally sold by the financial institution. The results of the simulation include both the cumulative cost we explained before and the payoff from the shares whose value grew. The more we rebalance our position, the more we should approach the value of the option we computed. From the example above we can also see how the delta behaves to the reduction of the option’s life, since if the option is in the money it tends to 1 as the life tends to zero, while if it is out of the money it tends to zero.

**Synthetic options and index futures**

Not always the option needed to hedge the position is available on the market and the market could be not liquid enough to absorb the trade; one way to overcome these problems is to use synthetic options. It consists of a position on the underlying asset, or on futures on the underlying asset that has the same delta of the option. It needs to be the opposite of the position that would be necessary to hedge it. Recalling what delta is

It means that at any time a proportion of

Will be sold and the money will be invested in some risk-free assets. When the value of the portfolio decreases the delta becomes more negative and the proportion of the portfolio that needs to be sold is reduced. When the value of the portfolio increases a portion of the risk-free asset is sold. The cost of this procedure derives from the fact that a portion of the portfolio is sold after it registered some losses.

Often the costs of buying directly the underlying asset could have high transaction costs, for this reason usually traders use index futures. Considering that there is no perfect correspondence between the number of contracts of the futures contracts and the value of the portfolio we need to adjust it:

Where the portfolio is worth A1 times the index and each index future contract is written on A2 times the index. This equation gives us the number of contracts that needs to be shorted. However, this is a specific case in which the portfolio has the same composition of the index, otherwise we will need to adjust for the beta, find the position of the options on the index and the position in index futures.

**Implications with portfolio insurance crisis (1987)**

Let’s imagine what would happen if every portfolio manager adopted a delta hedging strategy and they could invest in just 2 stocks. If stock 1 happened to lose for example 1% of its value all the trading algorithms would sell enough of the portfolio to keep a delta neutral position. If everyone thought like this it would generate a vicious circle of selling orders that would easily wipe out the value of this stock. Well, this is an extreme example that resumes what happened in 1987 with the portfolio insurance crisis, where many synthetic put options were created by buying and selling index futures. This shows how this trading strategy could easily increase the volatility in the market if it is not liquid enough.

**Transaction costs and hedging optimization**

The situation we analyzed until now does not take into account transaction costs. Since usually the price of stocks is described by a continuous motion this implies that there are infinite price changes and transaction costs tend to be infinite, so this strategy is not applicable. There have been proposed numerous approaches that develop in discrete time at various intervals. What is considered the optimal hedging strategy should achieve the best tradeoff between risk and transaction costs (Zakamouline). What is fundamental in this case is the risk-aversion of the investor, here we recall the utility function, this theory was first introduced by Hodges and Neuberger (1989). Having a finite time horizon [t,T], and assuming that at time t the investor has xt cash and yt shares:

J0(t, xt, yt,St) = max Et[U(xt+ytSt)]

Where U(x) is the utility function of the investor. Now if we include the option liability it becomes:

Jw(t, xt, yt,St) = max Et[U(xt+ytSt-(St-K)+]

The price of the option should make the following equation true

Jw(t, xt+p, yt,St) = J0(t, xt, yt,St)

However these equations have no closed solution so we need to think of a different approach and we introduce a negative exponential utility function so that:

gamma represents the risk-aversion, and as we can see the utility does not depend on the initial cash. Actually this utility function works pretty well since as shown in Andersen and Damgaard (1999), an option price is approximately invariant to the specific form of the hedger’s utility function, and mainly only the level of absolute risk aversion plays an important role. Recalling the utility based approach of the option hedging strategy we can define it as the difference

If the risk-aversion is moderate we cannot derive much from this equation, however if the risk-aversion is quite high we can assume that y0(t)=0. The hedging strategy will develop between two boundaries which are deltau and deltal where u stands for upper and l for lower, and it represents the region where no hedging happens.

This model, however, requires many numerical computations, so it has been simplified with the “delta tolerance strategy”, which gives a range in which delta can move before needing to rebalance.

H reflects the risk-aversion of the investor, if it is low the investor is more risk-averse.

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