This is a continuation of the Black and Scholes post. After we have computed the fair price with the B&S formula, we can compute more parameters that are related to the formula.
Consider the buyer of stocks. What is he worried about? Obviously, a market decline. Anyone who only buys the underlying has one real concern, which is the direction of the underlying. In contrast, the option trader is exposed to many parameters besides the move of the underlying. This post will be concerned with the effects of what is known as the delta.
The Delta
The delta, simply stated, is the sensitivity of the option price to the underlying price. The delta of a Call option is between 0 and 1, and it is usually written as a number between 0 - 100. This means that a delta of 0.5 is written as 50.
A call option with a delta of 25 can be expected to change it’s value at 25% of the rate of the underlying. If the underlying rises by 1 point, the option would rise by .25 and vice versa. If the underlying rises by .60, the option would rise by 0.15.
An important note: For deep in-the-money calls, the delta approaches 100, which means that for each point, the option gains a point. The intuition is simple. Deep in-the money options have high probability of expiring in the money, and hence each point gained in the underlying almost surely guarantees a point of profit for the option. Hence, as the option gets closer to the at-the-money strike, it’s delta rises, as seen below in the graph (a call option with strike of 60)
Notice that the delta of the at-the-money price is 0.5. This also agrees with our intuition that when the price of the stock is at-the-money, it has 50% chance of either going up or down, hence an expected return of 50%.
How about puts? they behave the same as Call options just in the opposite direction. The Put option gains value when the underlying falls and vice versa. For this reason, Put options have negative deltas, again from 0 to 1. Deep in-the-money options have a delta of -100 and deep out-of-the money options have a delta of 0.
Hedging Ratio
If we wish to hedge (eliminate risk) an option position against the underlying, the delta tells us the proper ratio of options we need to buy. An underlying always has a delta of 100. This means that if we want to hedge our risk when we decide to buy the underlying, and we have an at-the-money option with a delta of 50, we’d have to sell 2 options. The deltas sum up: . This is the concept of being delta neutral.
With a delta neutral position, no matter where the underlying will move, we would be covered. If the underlying moves up 1 point, we profit it but lose 1 point because we sold 2 options, and vice versa. Consider a call with a delta of 40. This means that to be delta neutral, we’d have to sell two underlying contracts for each five options purchased, or vice versa.
In here we only discussed a combination of an option and an underlying, but we can make many more combinations. Suppose we buy four calls with a delta of 50 each, and ten puts with a delta of -20 each. The position is delta neutral since .
Final note
In here we only discussed the delta. Remember though that there are more parameters that affect the price, which will be discussed in the next post! Also remember that deltas can change over time and you need to adjust. This is known as dynamic hedging. This will also be discussed in the next posts. Stay tuned, and please e-mail me if you have questions
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I just discovered your blog. You have one of the best stock-market related blogs I have seen.
I have a question regarding how to handle a potential arbitrage position. Let’s say that I think that the S&P 500 (or some other index) will outperform long-term U.S. bonds over the next year (or longer).
To profit off an outperformance of the S&P 500 versus bonds, I could sell short an ETF (e.g., TLT) that tracks long-term bonds and establish a long position in an ETF that tracks the S&P 500 (e.g., SPY). However, my understanding is that the money one “receives” from a short position cannot be used to purchase another position. Accordingly, one would either have to already have sufficient additional money in a brokerage account to purchase the long position or would have to borrow on the margin.
I actually did what I just described in early 2006 and made some money. However, the margin fees basically destroyed most of the potential benefits of this long-short hedge.
Are you aware of a way I could capture a potential outperformance by using options instead of purchasing simultaneous long and short positions to establish a hedge?
Thanks.
Thanks for the compliment Jim!
As for your question, what you describe is actually more suitable to be considered speculation than arbitrage unless you have found an anomaly in the price and in that case good for you :)
As for doing the exact trade with option is a little bit tricky because options are subject to more variables that affect their price (time, interest rates, volatility and etc..)
I think there are options on bonds, so in theory, you could buy options on the stock index and sell bond options. The problem would still be margin requirements by selling the option so it would be back to square one sadly.