The Black-Scholes formula can be used to create a hedge for an option. However, this model is derived in continuous time. What happens when we use it to hedge an option in discrete-time?

European options are financial securities which give their holder **the right** (but not the obligation) **to buy or sell an asset** (called the underlying asset) at a predetermined price (called the strike price) at a specified future date (but not before that). A European call option, for example, gives the holder the right to buy the underlying asset at the specified strike price on the expiration date.

In this post, we will carry out an experiment using European call options on the S&P 500 index. The options are first traded on the 22nd of January 2020 and expire on the 17th of April 2020, so they have a life of 60 trading days. Two strike prices will be considered: $2500 (the call ends in the money and is exercised) and $3000 (the call is not exercised). The goal of this experiment is to test the accuracy of a Black-Scholes theoretical hedge on the option in discrete time.

**Calculating the option prices series**

The theoretical no-arbitrage price for a call option can be calculated through the Black-Scholes formula, which earned its finders the Nobel prize in economic sciences in 1997. The formula is derived from **the assumption that there is no arbitrage in the market** and therefore **any riskless portfolio must earn the risk-free rate of the market**. You can read more about how the formula is derived in the provided link. The resulting European call price for a non-dividend paying stock can be calculated from the following 5 inputs:

- The current price of the underlying asset.
- The volatility that the underlying is projected to have until the expiration of the option (this is the only non-observable variable in the formula).
- The risk-free rate in the market.
- The time remaining until the expiration date.
- The strike price.

You can also calculate Black-Scholes option prices using a Monte Carlo method, which allows to avoid the complex formulae. See this post to find out how to implement this method in Python.

Using the Black-Scholes formula and assuming the market’s expectations for volatility remain constant during the life of the option and equal to the actual volatility of the underlying during this time period (this is quite an assumption but it will be valid for the point of this post), the call option price series during its lifetime can be calculated from the S&P 500 series (the underlying). The riskless asset is assumed to be perfect and earn an annual rate of return of 1%:

Note that the price of the call options follows a similar trend to that of the index, especially for the first call. Generally speaking, **when the index rises so does the call price**, which is intuitive from the definition of the option. The falls in the index also comes with falls in the price of the call options.

**How to build a hedge for an option**

Now, imagine that the holder of the option wants to limit their exposure to the variations in the index. This can be done via a strategy called delta hedging, which in our case simply means taking a certain short position on the index. In this manner, the movements in the call price will be compensated by the movements in the short position on the index, regardless of the direction of the changes in the underlying asset price. This strategy reduces the investors’ return if the index goes up but reduces their loss if it goes down as well. In other words, **the strategy reduces the investors’ risk at the cost of less expected return**. A delta hedge can be used by the call holder to secure profits if the underlying asset is predicted to fall for some time.

Option prices also change due to the passing of time. The value of a call option, for example, is reduced as time goes by, even if the underlying asset price remains the same. The holder of the option can also cover their position on the call against this effect by investing in the risk-free asset. National treasury bonds can make an approximation for the concept of a riskless asset.

From all the above, we can see that a complete hedge (for both effects) for a call option can be achieved by using a portfolio invested in the underlying asset and the risk-free asset only. It is useful to realise that, at the end of the day, **finding a perfect hedge for the option is equivalent to finding a replicating portfolio for the option**, on which the investor can take a short position to achieve their hedge.

From the Black-Scholes formula, the theoretical replicating portfolio P for a call option can be achieved by buying Δ shares in the underlying asset S and taking a short position b on the riskless asset B:

Where S is the underlying price, sigma is the implied volatility, r is the risk-free rate and delta, gamma and theta are three of the greeks (partial derivatives of the call price C):

Note that these weights are functions of time and S (see “The Greeks”), and therefore the weights change over time. The more frequently the weights are updated, the more accurate the replicating portfolio will be. Such a replicating portfolio can be shown to be self-financing too.

In order to evaluate the accuracy of this method in discrete time, a replicating portfolio for the considered call is constructed in this way. The portfolio weights are updated only once every trading day, using closing prices. The returns on this portfolio are calculated from these weights and the returns on the index and the riskless asset:

As you can see, the replicating portfolio is a reasonable approximation for the call option but the errors can be significant after a certain amount of time. The end value in the replicating portfolio for the first call has an error of -12.5%, for example.

## Conclusion

Theoretically, the replicating portfolio of our experiment should provide a perfect hedge for the option (under some assumptions and if updated continuously in time). However, hedging an option position in real life is much more complicated.

As shown in the previous experiment, **even under the easiest case assumptions** that implied volatility remains constant during the life of the option and that a perfect riskless asset exists, **the Black-Scholes theoretical hedge may be quite inaccurate in discrete time**. A more frequent update in the portfolio weights together with more advanced hedging strategies are used for a more precise hedge on an option. These strategies cover against changes in implied volatility and include second-order approximations (see vega and gamma hedging). Such hedging strategies may also use puts and other derivatives.

Finally, transaction costs and options premiums should also be considered as they can spoil the accuracy of a hedge when considering the net return on the portfolio. Transaction costs create an interesting trade-off between the accuracy of a hedging strategy and its costs.

As always, a model can be extremely useful but one must be aware of its limitations for a real-life application.