Hedge Accordingly
Vol #82: May 20th, 2022
Not all change is equal.
There are very linear changes that we can expect and predict their impact. There are also convex shifts that can suddenly and disproportionately impact your position.
With options, we define change as delta. According to Fischer Black and Myron Scholes it is the change in the price of an option relative to the change in price of the underlying asset. How much does your PnL change when stock moves?
The underlying logic of the Black Scholes options pricing model is hedge replication. They arrive at the price of an option based on the cost to replicate its payoff function through a continuously implemented hedge. If you can define your hedge, you have defined your delta, and vice versa.
Financial engineering and arbitrage are all about creating synthetic payoffs, and buying the cheap cash flows to sell the expensive ones in as riskless a manner as possible. From a lifestyle guru perspective, change is good. From a financial perspective change is exposure.
Managing a book of risk means keeping your deltas in line, and staying robust to change. Understanding and hedging exposures is the key to good risk management and allows one to find and capitalize on opportunities.
This replication formula is intuitive for the options like exposure we see in real life. While a fatal car accident is a one in ten thousand chance, the premium that every driver pays each month helps cover those unfortunate incidents. In an efficient market, the aggregate paid by all drivers - minus the administration costs and some profit margin - should be close to the amount paid out in benefits. The option on protection is worth approximately the likelihood of an accident times the cost of that accident.
The probability of an accident is relatively small, but changing certain assumptions might shift that chance dramatically. If all of the sudden an insurance company took on a disproportionately large number of younger drivers, or residents of stormy climates, the actuarial arithmetic would change, and the company would need to adjust premium (option) prices as their benefits (hedges) change.
Delta is not as simple as a single number, and we see that hedge ratios will change and shift as time, implied volatility, and price movement happen. Gamma is the mathematical way we define changes in delta, but more broadly you can think about deltas being relative degrees of hard and soft.
A hard delta is fixed, and won’t change very much as other factors move. A soft delta is more squishy and susceptible to change if the inputs move. At any given snapshot their values can net against each other, but their forward looking exposure is dramatically different.
The less an option is susceptible to its price changing from anything other than the stock movement, the harder those deltas. Hedge accordingly.
The insurance example for a customer is a relatively soft delta, but it’s extremely important. As far out of the money as that option is, we don’t want to be “short” by not having insurance and drowning in medical bills.
Hard deltas in real life tend to look like liabilities. If you lease an apartment in Paris, you’re short Euros - you owe some fixed amount in the future that could go up or down in your native currency. The amount you pay in dollars is a direct function of that exchange rate, and little else will change that.
Perhaps the hardest delta we’re all exposed to is tax liability. Paycheck deductions are a bi-weekly hedge against your IRS exposure. For every dollar you earn, there’s a direct relationship towards how much you owe Uncle Sam. (Excusing the 2,600 pages of deductions, credits, and loopholes in the tax code.)
A slightly softer delta would be a fixed interest rate loan. While the value of that loan will stay the same in dollars, interest rate changes can reduce its value in purchasing power terms. As we see inflation numbers spike around us, our dollars go less far but our debts are also relatively less too.
Hedging financial exposure has an even softer delta. The cost of protection varies significantly based on the overall environment. If you buy puts during a calm market, all of the sudden their delta might balloon when implied volatility spikes on rising fear and uncertainty. You’ve made money, but your exposure profile now looks rather different.
The different risks we have all have different degrees of hedgeability. There are robust markets for health and car insurance, much less so career risk. Financial markets provide every possible mechanism for hedging economic risk, but it’s usually significantly more expensive than the pricing for very predictable things like term life insurance.
As a rule of thumb, hard deltas are worth hedging completely. April 15th is going to come knocking one way or another, and it’s impossible to magically change how much money you’ve made.
The softer the delta, the more it’s worth weighing the costs of that out of the money option. A small premium in car insurance helps us sleep well at night, but as comforting as puts seem, hedging away all market uncertainty is a big drag on returns.