Options Puzzles #1 - Vega Hedging in your Head!

Level 1 - Zero to Options Hero

Hello aspiring Options Heroes! In honor of the 50th anniversary of the seminal paper on Option Pricing by Fischer Black and Myron Scholes, we are launching a series of option brainteasers which will help sharpen up your mental math skills when it comes to trading live, impressing Wall Street interviewers, and perhaps serve as a good parlor trick at your next dinner party. Let’s go!

Scenario: You are on your first week at the job on a Wall Street trading floor, and the head of sales’ favorite client lifts your offer for $100mio notional of a 2-year, at-the-money call option on the S&P 500 (traded on an over-the-counter basis). It is 5 minutes to the close of trading on the NYSE and CBOE (Chicago Board Options Exchange), so you cannot wait for the salesperson to enter all the trade details into your risk system (we know boomers type slowly!) before doing some vega hedging of this trade.

Question: How much of 1-year ATM call options on S&P 500 should you buy to have a vega-hedged portfolio? Let’s assume that 1-year is the longest maturity options that have reasonable exchange liquidity.

Recall that Black Scholes vega is defined as:

\( \text{Vega} = \frac{\partial C}{\partial \sigma} = S e^{-r \cdot T} \sqrt{T} \cdot \phi(d_1)\)

where:

\(d_1 = \frac{\ln(F/K) + (r + (\sigma^2)/2)T}{\sigma \sqrt{T}}\)

r is the risk-free rate of the numeraire

T is time to maturity in years

S is the spot price of underlying.

For an at-the-money option,

\(F = K\)

and in particular,

\(\phi(d_1) = 0\)

for both the 1-year and 2-year ATM options, and for small levels of r,

\(e^{-r \cdot 1} \approx e^{-r \cdot 2}\)

therefore, the Vega of a 2y ATM call / Vega of a 1y ATM call is equal to:

\(\sqrt 2 / \sqrt 1 \approx 1.4\)

So, you need to buy 1.4*100 = 140mio USD of 1-year at-the-money call options before the end of the day to vega hedge the portfolio before market close. Don’t mess it up, and don’t forget the salesperson’s sales credits!

On a more serious note, the vega multiplier as a square root of the time to maturity is something that every option trader knows by heart, and we highly recommend committing some of these common multipliers to memory as it will help you to quickly calculate the vega risk of your options portfolio if it contains options with different maturities. See the chart below!

Hope this was fun and hope you’re excited about the next edition of this series! Until next time,

The OptionsNerds

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