Delta as a hedge factor - charts, formula

By roma, 14 August, 2020

Δ = δVS

V - option price
S - price of the underlying asset

To find a fair price for put and call options, an investor can use the Black-Scholes model (Read Key assumptions of the Black-Scholes model). The model also allows to determine the sensitivity of an option to changes other model parameters, for example, the price of the underlying asset, volatility, interest rate, time. Delta, gamma, theta, vega and rho are the units of measurement of the sensitivity of the option price to all parameters of the option model.

In practice, Greeks of options, as they are called by traders and academics, are extremely crucial in the risk management process of derivatives portfolio. A special place among Greeks is given to the Delta.

 

Delta

The Delta measures the change in the option price with respect to a small movement in the price of the underlying asset, assuming that the other variables from the Black-Scholes model remain unchanged. Very often the delta is measured as a percentage. For example, if the purchased call has a delta of 0.5 or 50%, it means that if the price of the underlying asset rises/(falls) by $1, the option value will increase/(falls) by $0.50.

For option traders, the delta sign is very important, since a negative delta and a positive delta sign are very different. The delta sign gives an idea of the direction.

Long Call: Positive Delta. The value of an option position increases as the price of the underlying asset increases and decreases as the price of the asset falls.

Short Call: Negative Delta. The position will turn out to be unprofitable if the price of the asset increases, because the price of the asset increases the value of the call. If the price of the asset drops, the position will be profitable because the option can be bought back at a lower price. Therefore, a short call option has the characteristics of a short position on the underlying asset, hence the negative delta.

Long Put: Negative Delta. The value of an option increases as the price of the underlying decreases, and falls as the price of the asset rises.

Short Put: Positive Delta. The value of the position increases as the price of the underlying increases (it becomes cheaper to buy back to close a short position). A fall in the price of the asset causes the option to rise in value, which has a negative effect on the short put position. Hence, a short put option has the characteristics of a long position on the underlying asset, hence the positive delta.

 

Delta

Long call

+

Short call

Long put

Short put

+


Option Delta

The delta sign indicates a bullish or bearish position. A positive delta means that the position will profit when the price of the underlying asset rises. A negative delta in the option portfolio will be profitable if the price of the underlying asset falls.

For example, let us assume that the trader purchased a put option on Walmart stock with a delta of -0.5 or -50%. Thus:
If the price of Walmart shares decreases by $1, the value of the long put will increase by $0.50.
If the price of Walmart shares increases by $1, the trader will incur a loss of $0.50.

 

Delta behavior

The Delta is the first derivative of the option price with respect to the price of the underlying asset.

Fig. 1. Call price

Fig. 2. Call delta

Call price

Call price

For a standard call option, the delta value is in the range from zero to 1 (or 0% to 100%). The Delta of deep out of the money call is almost equal to zero: the option reacts poorly to small changes in the price of the underlying asset. As it gets closer to at-the-money, the Delta tends to 0.50 or 50%, where the option price changes by half the price of the underlying asset.  As the call option moves in-the-money, the Delta tends to an upper limit of 1 or 100%. When the delta reaches one, the option price changes one to one relative to the value of the underlying asset. Therefore, a deep ITM call is like a long position in the underlying asset, while deep ITM put is like a short position on the underlying asset.

 

Delta as tangent slope coefficient

Fig. 3. Call price
Call price

Figure 3 shows the price line for a call option with a strike of $50 relative to different price levels of the underlying asset. The chart also shows the tangent line at the point where the option is at-the-money. The slope coefficient of this tangent line, equal to 0.50, is the option delta. Note that the slope coefficient of this tangent line tends to zero when the call is deeply out of the money, i.e. the option price has a low sensitivity to changes in the asset price. On the other hand, the slope coefficient is close to 1 when the option is deep in-the-money, i.e. the call option price and the underlying asset move one to one.

 

Delta as a hedge factor

The delta is not only useful as a unit of sensitivity. Options traders use it to reduce their derivatives portfolio risk. For example, suppose that a trader sold a call option with a delta of 0.50 on 1000 Macy’s shares. Thus, the trader has a negative delta because he sold a call option. If the price of Macy’s shares rises significantly, the value of the call options will increase, which will lead to significant losses for the trader: since the trader will spend more than he received from the sale of the same call options to buy them.

Let us consider the delta of the trader's position. The trader has sold options on 1000 shares of Macy’s. If the share price moves, the value of options will change by half of the share price change. Consequently:
Delta position: - 1000 x 0.50 = - 500 shares

This means that (with small changes in the price of the asset) the trader's risk is comparable to that of a short position in 500 shares. If the Macy’s share price increases by 1%, the losses (from a short position in 1000 call options) will equal to:
0.01 x 500 x stock price

To fully hedge the risk associated with Macy’s stock price fluctuations, a trader can buy 500 shares immediately after selling 1000 calls. The delta for each share purchased is 1 or 100%. Therefore, the total delta of the position (from 1000 short calls and 500 shares purchased) will be 0. All losses on the short call option position will be compensated by profits from the long position in Macy’s shares.

For example, let us assume that the share price increased by $1.
Losses from the option position = -1000 x 0.5 x 1 = - $500
Profit from shares = 500 x 1 = $500

Option portfolio that is hedged by selling/buying shares to reduce the delta to zero is called delta-neutral.

Read Hedging oil contracts through options - a detailed example.

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