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Option Leverage Measure: Lambda Vs. Delta

The delta of an option is the change in the value of that option for a given move in the price of the underlying asset. Because an option's delta is always less than one (in absolute value), it follows that for a given change in stock price, the option value will move accordingly for a smaller amount. This seems to contradict the general perception that options are leveraged instruments. Just as delta is the appropriate hedging ratio, however, another Greek letter lambda is the more appropriate leverage ratio for options, which incorporates both the delta factor and the gearing factor.

Option Gearing: Investment Gain/Loss Multiplier

For a start, let's look at some numerical examples. Suppose an investor has USD1,000 to invest and is bullish on ABC stock, which is trading at USD100. Obviously, the investor can buy 10 ABC shares directly. Alternatively, the investor may make the same investment via at-the-money call options (strike price at USD100) on ABC shares. Assuming a one-year investment horizon, a risk free rate of r = 1%, stock volatility *= 30%, and no dividend, the price for the at-the-money European call option is USD12.37, according to the Black-Scholes pricing formula.

At maturity the option holder will gain one-for-one for any price move above the original stock price USD100. The investor could invest in 10 at-the-money options for USD12.37 * 10 = USD123.7 and gain the same exposure for stock gains as if he invested USD1,000 in 10 shares. In other words, for the same USD1,000 capital the investor can invest in 1,000/12.37 = 80 options, giving eight times the exposure than the equivalent stock position.

An option position can give investors the same exposure in a stock position with less initial investment. This ratio of the underlying stock price over the corresponding option premium, often referred to as gearing (Stock Price/Option premium), provides investors with one measure of option leverage. For a given rate of return on an underlying security, an investor can increase those returns by applying the leverage of options.

The leveraging power of options, which can magnify profits, can also magnify losses if the underlying security moves in the opposite direction. This illustrates the classic financial trade-off between risk and reward. In the above example, if the stock price ends up at USD99 at maturity, the stock position will have a modest 1% loss, while the option position will suffer a 100% loss. This is because all the options will expire worthless.

The above example shows that options are highly leveraged. For the same initial capital investment, taking an option position will give an investor multiple times of exposures (both for gain and loss) as compared to a direct position in the underlying stock. In addition, this leverage can be even more pronounced for exotic derivatives that have additional embedded leverage, such as barrier options.

Lambda: Option Leverage Measure
In Derivatives Term

To describe the above leverage calculation in derivatives terms, we have Lambda (*), which is defined as the expected percent change in the value of an option for a given percentage change in the value of the underlying stock price. That is,

* = [(C1-C0)/C0] / [(S1-S0)/S0],

where C0 and S0 are the option price and stock price at starting point, and C1 is the new option price when the stock price moves from S0 to S1.

Applying the numbers in the early data, we have:

Recall that option delta (*) is determined as (C1-C0)/ (S1-S0), we can simplify the above as:

* = * S0 / C0

Option leverage can be viewed as the option delta muliplied by the gearing factor--the price ratio between underlying stock and option discussed above. Probably because its direct relationship with the delta, lambda is not a frequently used Greek letter for derivatives trading. But lambda is a useful and a more appropriate measure for options' leverage indication than delta.

The Black-Scholes pricing model shows why options always have more leverage than the underlying stock (i.e. * >1). Recall that for a European call option on non-dividend paying stock, the call option price can be expressed as:

C = S* N (d1) ­ PV (K)* N (d2)

Where S is the stock's spot price, PV (K) is the present value of the strike price and N(d1) and N(d2) are cumulative probability distribution functions. By definition, we immediately recognize N(d1) as the option delta (). For a detailed discussion of N(d1) and N(d2), please refer to my earlier Learning Curve article Option Delta Vs. Probability to Exercise (DW 04/20/03).

Because PV(K) * N(d2) is always positive, it follows that S* N(d1) > C, i.e., S* > C.

Therefore, * S0 / C0 >1, i.e., * >1.

Conceptually, the Black-Scholes formula above prices an option as a leveraged stock position. That is, an option can be thought of as being the equivalent to purchasing delta amount of stock and finance it with borrowing. For in the formula:

C = S* N (d1) ­ PV (K)* N (d2)

S*N(d1) represents the number of stocks that one must buy in a continuously rebalanced portfolio that replicates the payoff to the call and PV(K)* N(d2) represents the amount of borrowing for such a replicating portfolio. As one would expect, the borrowing amount is always less than or equal to the value of the long stock position so that the call option always has a positive value.


Options are leveraged financial instruments. The Greek letter * measures the option leverage, which is defined as the expected percent change in the value of an option for a given percentage change in the value of the underlying stock price. Although in absolute dollar terms, option prices move a smaller amount than a given stock price move, the option price will demonstrate a greater percentage change in value than the stock. That is because in addition to the delta relationship, investors must also take into account the gearing factor, which reflects the fact that options can achieve the same stock exposure with much a smaller initial investment.

This week's Learning Curve was written by Winston Wenyan Ma CFA, associate in the equity derivatives products and solutions
group at
JPMorgan Securities in New York.

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