1. Delta or sometimes called spot delta is the first derivative of option market value (MV) with respect to spot price.
If P is an option price and S is a spot price, then

This is the most basic definition from a theoretical point of view. Delta is a sensitivity measure.

2. Mathematically, delta is the slope of the tangent line to the graph of the option price.

As an example, consider a simple 3-months plain vanilla call option with strike 100.

The idea is that a small movement along with the tangent line is roughly the same as a small movement along the market value of an option.
Therefore, if you don’t know how to visualize delta, you can always use charts as the above and draw tangent lines at different points.

Example. Let’s look at the small interval (90-91) and zoom it. This is simply the straight line which is the tangent line.

Tan(X) = MV Change / Stock price change = Delta or slope.

Tan(X) = (1.55 -1.32) / (91-90) = 0.23 which is equal Tan(13°).

Therefore, our delta is 0.23 and the slope is 13°. If we continue moving forward, the slope will increase and thus delta will increase.

Deep out of the money option (OTM): delta and the slope are close to zero; the tangent line is almost flat.

Deep in the money option (ITM): delta becomes 1 because the slope is 45°, Tan(45°) is 1.

At the money option (ATM): delta is 0.5 and the slope is 27°; Tan(27°) is 0.5.

Therefore, the steeper the slope of the tangent, the greater the delta. Finally, it is easy to plot our delta values if we peak many different points.

3. Delta is the option hedge ratio and is a percent of notional amount (or a cash amount in the notional currency).

With other words, delta is the equivalent position in the underlying product.

Example, a plain vanilla call option on equity:
• Underlying: AXP US Equity (American Express)
• Spot price: $57.83
• Strike: $62.5
• One option: 100 000 shares
• Maturity: 45 days

The option is an OTM option with delta around 13.44% (or 0.1344) which is equal:

100 000 shares * 0.1344 = 13 440 shares at the moment or simply 13 440 shares multiplied by $57.83=$777 235.

Delta aggregates option risk on a given underlying and expresses it in terms of the underlying.

4. Delta is a measure of the probability of the option expiring ITM.

This definition is considered the trader’s definition and mathematically imprecise. An option with a 0.5 delta has 50% chance of being ITM at expiration and of cause 50% chance of being OTM under this definition.

Of cause, this probability is not the risk neutral probability and as for me, this probability is highly theoretical and can be applied only to plain vanilla options with delta from 0 to 1 (look at the picture above).

Example, an Up-and-In barrier call option:
• Underlying: stock
• Strike: $100
• Maturity: 3 months
• Barrier: 120
• Risk free rate: 8%
• Volatility 40%

The charts below illustrate the market value at different spot values and delta.
The Up-and-In barrier call option is cheaper than plain vanilla call option but gives potential upside if the stock is going up too much.
The barrier is above the strike price and option holder gets usual payout if the stock price hits the barrier during the life of the option.

By using market Value chart we can draw tangent lines at different spot points and build our Delta profile.
For example:

• Spot price 95: delta becomes is near zero because the slope is near zero
• Spot price 114: delta becomes more than 1 because the slope is around 60°, Tan(60°) is 1.73
• Spot price 120: delta is near 1 because the slope is 45°, Tan(45°) is 1

We see that delta is not monotonous and thus the hedging of a barrier option is more involved than for vanilla options, as the delta near the barrier can be significantly more than 1.

The absolute level of the delta can be very high, from the delta chart we see it can reach even value of 2. So what do we say about probability in this case? Is it 200% to be ITM?