Think Value

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Trading options requires risk hedging.
Suppose an investment fund uses the Black-Scholes-Merton model to calculate the value of a call option and then sells it in the market at a higher price. In substance, it has made a profit, but this profit is not yet realized because the option has not expired.

For example, let’s assume we have the following initial data:

Based on these data, if we calculate the option price using the Black-Scholes-Merton formula, the price will be 2.4 USD per share, and the unrealized profit will therefore be 60,000 USD.

The financial institution may do nothing until expiration and simply wait to see what the final profit is (an open/naked position). However, if the stock price rises before expiration, the option holder will exercise, which can become very expensive for the option writer. In our example, if the stock price increases to 60 USD, the financial institution will have to buy 100,000 shares at 60 USD and deliver them to the option holder at 50 USD. Taking into account the 300,000 USD already received, the final result will be a loss of 700,000 USD.

The second option is for the fund to immediately buy 100,000 shares (covered position). In this case, it is protected against increases in the stock price, but if the price falls, losses are unavoidable. Overall, for open and covered positions at different stock prices, we get the following picture:

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STOP-LOSS STRATEGY

A stop-loss strategy is a theoretical hedging approach that in practice (and really even in theory) does not work well. In our example the suggested approach would be: buy the stock as soon as the price crosses the strike upward, and sell it immediately when the falling price crosses the strike downward. The assumption was that in this case losses would be significantly reduced.

The strategy does not work because first, it is practically impossible to determine which direction the price will move after hitting the crossing point. And even if you could determine it, buying and selling at exactly the strike price is impossible because of the bid-ask spread. This means every transaction will produce some loss.

Moreover, crossings are likely to occur frequently because of volatility and because of fair pricing around intrinsic value. To reduce the loss from transactions, you must buy and sell as close to the strike price as possible. But the closer you make that range, the more frequently the trades will occur (in the limit, infinitely often 🙂).

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Hedge Performance

Hedge performance is a measure that shows how the standard deviation of hedging costs compares to the Black-Scholes-Merton option price. In the ideal case, this measure should be equal to zero. Here, the standard deviation shows the range of the problem because, in an efficient market, the average hedging cost and the option price should be equal.

Below are values calculated via Monte-Carlo simulation for our example. You can see that, even though the re-hedging frequency increases, the measure does not change much (and it cannot fall below 70%, no matter how small Δt becomes).

Time between re-hedging	Performance
5 weeks	1.02
4 weeks	0.93
2 weeks	0.82
0.5 weeks	0.77
0.25 weeks	about 0.76

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Δ — Delta Hedging

Δ (delta) shows the sensitivity of the option price to changes in the stock price — that is, how much the option price changes when the stock price changes by one unit. It is the slope of the curve describing the relationship between the stock price and option price.

For example, take delta = 0.6; stock price = 100 USD; option price = 10 USD.

If we sell call options on 2,000 shares, the result can be hedged by buying:

2,000 × 0.6 = 1,200 shares

If the stock price changes, the option price changes as well, and the changes offset each other. This is called a delta-neutral position.

The important point is that hedging works over short intervals of price and time, because the relationship between stock and option prices changes. Therefore, rebalancing is needed from time to time (dynamic hedging).

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Delta of European Stock Options

The delta of American options is calculated using binomial trees or other complex methods. For European options, using the Black-Scholes-Merton model, for a non-dividend-paying stock, delta equals the cumulative normal distribution of stock price movements.

Similarly, for put options we derive the equivalent expression.

Graphically, it looks like this:

Delta also varies with time to expiration. Below are typical paths of call option deltas.

Returning to our starting example,
d₁ = 0.0542 and N(d₁) = 0.5216,
which means that if the stock price changes by ΔS, the option price changes by 0.5216 ΔS.

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Excel – Delta Hedging

Finally, the table shows the hedge-performance measure (standard deviation of hedging costs relative to the Black-Scholes price) from 1,000 iterations. As you can see, delta-hedging is much more effective.

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P.S.
In this note, the discussion is developed from the perspective of hedging a written option using the underlying stock, but delta-hedging strategies can also be run in reverse or applied to other assets. The key idea is to make the portfolio delta equal to zero. However, the technique is especially important for option writers, because the risk here is much higher than simply buying or even shorting a stock.

Source: Options, Futures & Other Derivatives, John C. Hull

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