Most people with even the slightest familiarity with option pricing will know of time decay, or the tendency of the option value to decrease through time independent of any interest rate effect. The chart above shows the price of call option through 40 years, struck at 100 with the underlying price also constantly at 100, and with interest rate set to zero to remove the effect of interest carry. I.e. the only change to the model is the time to expiry. The effect of time decay or theta is apparent.
Now I have discussed in several posts, such as here, how a real option can be replicated by means of a synthetic option – a series of linear positions in the underlying market adjusted frequently to match the delta or sensitivity of the real option.
But it is not so well understood that the time decay of a real option has a counterpart in the synthetic option, indeed this is the key to understanding Black Scholes, and also to understanding why the model is relatively independent of distribution, contrary to what many actuaries, including the Institute of Actuaries itself, believe.
The mathematical treatment is somewhat complex, and I shall discuss this in a subsequent post, but the intuition is to consider what happens when we try to construct the synthetic with the underlying price trading around 100, but varying around it with a volatility equal to the input volatility to the standard B-S formula.
We are synthesising a long option position, so the delta goes up as the underlying price rises, and falls as the underlying price falls. Thus if we have matched the delta perfectly at 100, and the delta goes up as the price goes to 101, we will have to buy a little to replicate the delta again. So we buy at 101. But then the price falls back to 100, the delta falls back again, and we have to sell. So we take a small loss. Then it goes back to 101, we buy a second time, and then sell a second time as the price reverts to 100, producing another small loss. This continues until the option expires at the money. Throughout this process, assuming constant mean reversion to 100, we will bleed money on the synthetic position.
This is an intuitive demonstration that a synthetic option also has a form of time decay.
The trick is to show mathematically that the B-S time decay matches the expected time decay of the synthetic, on the single assumption that the ex post volatility of the underlying price matches the ex ante volatility input to the B-S formula.
That is a (slightly) more tricky question that I shall return to shortly.