Built-In Leverage

Question

A stock trades at 50; its at-the-money call costs 2.50 with delta 0.52. The stock rises 1% today. Roughly what percent return does the call holder make, and what is the general formula for an option's effective leverage? Why isn't that leverage free?

Accepted Answer

The arithmetic. Stock up 1% = up 0.50. The option picks up Delta times 0.50 = 0.52 times 0.50 = 0.26, on a 2.50 premium, a 10.4% return on capital for a 1% stock move. The general object is the option's elasticity (often written lambda or "lambda"): lambda = frac%,Delta C%,Delta S = Delta cdot fracSC = 0.52 times frac502.50 = 10.4. An ATM option is roughly a 10x levered position in this example; further OTM options have smaller deltas but far smaller premiums, so lambda rises with strike for calls, cheap lottery tickets are the most levered instruments on the board. Why it isn't free: - Theta. The premium decays every day; the levered exposure has a daily rent. Hold the call through two quiet weeks and the 10x leverage applies to a position that bled while you waited. - It cuts both ways. A 1% down move costs 10.4% of capital. Leverage is symmetric locally, only the bounded loss (premium) is the asymmetry, and you paid for it. - Lambda is unstable. Delta and premium both change with spot and time, so the leverage you bought is not the leverage you'll hold tomorrow. Desk relevance: elasticity is why small option books can carry huge effective exposure, why risk views position in del

Built-In Leverage

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