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Short Selling

As was pointed out in footnote 2, the delta of a vanilla European-style call option lies in the interval [0,1]. Furthermore, as discussed in footnote 3, to manage a riskless portfolio associated with the sale of the call option, the hedger needs to acquire ΔCall shares, in the process making the net portfolio delta neutral 0 – all of which is illustrated in Tables 7.5 to 7.7.

By the same token, if the hedger has instead bought a call option or sold a put option, the sensitivity of this position to movements in the underlying stock price results in a positive delta. As a consequence, for the hedger to have a riskless portfolio, the hedger has to short delta shares, which would not be possible if short selling was not allowed – something that was touched on in footnote 5.

Given the above backdrop, it is easy to conclude that if short selling is not allowed (which has happened in certain markets and during financial crisis), some of the positions like going short a call or long a put cannot be initiated and delta hedged – forcing risk-managers to take unhedged positions. As a consequence, market makers during such times will either not transact in any positions resulting in them needing to short the markets or simply price in a hefty premium to take into account the inability to short the market while trying to capitalize on the fact that other market makers are no longer making markets on such products or do a mirror trade with another counterparty.

Transactions Costs, Continuous Trading, and Divisibility

In illustrating the implementation of a delta-hedging program outlined in Tables 7.5 to 7.8,1 assumed that there are no transaction costs associated with rebalancing the portfolio. As a consequence, it is easy to rebalance the portfolio delta as frequently as one desires. All the delta-hedging dynamics fall apart in the presence of transaction costs simply because the hedging P&L associated with a continuously rebalanced delta-hedging program far outweigh the theoretical premium charged for the option in the absence of transactions costs – pointing to the fact that the generated premium is inadequate.

Before discussing the impact of transactions cost on a delta-hedging program, it is important for the reader to first understand that in practice there are two types of transaction costs that a trader has to deal with. These are:

1. Bid-offer spread.[1]

2. Commissions associated with each transaction.[2]

Table 7.9 illustrates the impact of transaction costs on the hedging P&L along a sample path associated with the sale of a European-style call option. In fact, Table 7.9 represents the rerun of Table 7.7 in the presence of transaction costs.

As can be seen from Table 7.9, running the delta-hedging program in the presence of transaction costs results in a lower hedge P&L compared to Table 7.7. Thus, the more frequent the rebalancing of the deltas, the lower the hedge P&L of this scheme – which, as a consequence, implies that the option premium generated from the sale of this option has to be higher than the theoretical one without any transaction costs.[3]

The disadvantage of hedging less frequently is that there will be a deviation from the theoretically expected result of 0 (due to continuous rebalancing). In fact, there have been several publications discussing the impact of hedging frequency on the standard deviation of the hedge P&L. See, for example, Leland (1985).[4]

In addition to the above, there are two more issues I would like to touch on before moving to the assumption relating to dividends. The first relates to the fact that it is not possible to transact in a fractional number of shares (e.g., 10.25 shares, 13.57 shares). This problem can easily be rectified in the simulations by ensuring that one is only transacting in an integer number of shares (or number of lots – where each lot would contain 100 shares). The second issue relates to the fact that financial markets are only open during certain times of the day. While this may be a small concern for most options, trying to model a delta-hedging program on an hourly basis involving a very short-term option means stopping all hedging at market close and starting again at market open – something that can be easily incorporated in a simulation-based methodology.

TABLE 7.9 Settlement Associated with a Delta-Hedging Program for Vanilla European-Style Call Option on a Dividend-Paying Stock (In-The-Money Expiry) With Transaction Costs

  • [1] The price paid to buy a stock is different from the price obtained by selling the same stock. The more illiquid the stock, the wider the bid-offer spread.
  • [2] An example of this would be paying $7 for each transaction (regardless of the number of shares bought or sold in that transaction) or $7 plus $1.25/option contract (regardless of the number of option contracts bought or sold).
  • [3] In practice, this is done through implied volatilities. In fact when trading, practitioners use implied volatility as a currency to compensate them for the cost associated with manufacturing the option they are selling (by taking into consideration the transactions cost, risks exposed, and the cost of capital needed to execute the transaction).
  • [4] Most of these papers were published before the availability of cheap, high-powered computer hardware and hence were focused on finding the bounds of errors associated with running a delta-hedging program in the presence of transaction costs. Given the current environment of cheap, high-powered computers, this problem has morphed into a simulation-based one in which hedging strategies are simulated in practice so as to be able to numerically identify bounds for various error or loss functions.
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