Volatility Trading 101. An intuitive guide to delta hedging | by Roman Paolucci | Towards Data Science

A large client wants to purchase 100,000 AAPL call options with the following parameters:
Strike Price: 350
Spot Price: 320
Implied Volatility: 30%
Risk-free rate: 8%
Time until expiration: 1 year
You decide to consult your team.
Your team has research suggesting that realized volatility will be less than the volatility implied by the option, your intention is to profit off of this spread. However, they maintain competing perspectives about the direction AAPL is heading. You are tasked with constructing a portfolio to take advantage of the potential volatility spread while mitigating directional risk.
Let’s take a look at how we can accomplish this...
In 1973 Fischer Black, Myron Scholes, and Robert Merton developed the notorious Black-Scholes option pricing model in the paper The Pricing of Options and Corporate Liabilities published to the Journal of Political Economy. If you wish to understand the entire derivation you can see Deriving the Black-Scholes Model. This equation effectively prices an option given a strike price, expiration date, volatility, and an appropriate risk-free rate.
N : Cumulative normal distribution function
K : Strike price
T : Time until expiration
r : Risk-free rate
S : Spot price of the underlying asset
σ : Volatility
In essence this equation is a multi-variable function where the inputs K, T, r, S, and σ yield the theoretical option price. Consequently, it can be partially differentiated with respect to each variable to find the effect a change in each variable has on the option’s price holding all others constant.
Delta is the result of partially differentiating with respect to the underlying asset price. The following is a derivation of the function describing delta...
If the option is a put then delta is negative or zero: [-1, 0], and if the option is a call delta is positive or zero: [0, 1]. Though the function of delta is in terms of the cumulative normal distribution function there is a straightforward interpretation.
When the spot price of the underlying asset increases/decreases by $1, holding all other variables constant, the option’s value will increase/decrease by delta. It’s important to note that the settlement for options is the same as stock, these profits/losses generated by increases/decreases in the underlying asset are unrealized until the position is liquidated.
Since delta tells us how an option’s value changes as the underlying asset changes in value we can take offsetting positions to create a temporary risk-neutral or delta-neutral portfolio. To construct this portfolio we take a position relative to our option position based on the quantity of our position and delta.
Let’s consider an example:
Position: Long 100,000 call options, delta = .7
To construct a delta-neutral portfolio go short (100,000)*(.7) = 70,000 shares of the underlying asset.
Underlying asset increases by $1
Option position decreases in value by (100,000)*(.7) = $70,000
Underlying asset position decreases in value by (70,000)*(-1) = -$70,000
Net change in portfolio value = $0
Underlying asset decreases by $1
Option position decreases by (100,000)*(.7) = -$70,000
Stock position increases in value by (70,000)*(1) = $70,000
Net change in portfolio value = $0
While a portfolio remains delta-neutral, directional movements in the underlying asset do not change the overall portfolio value. This allows us to capitalize on our research suggesting implied volatility will be greater than the realized volatility without concern of where the underlying asset price will end up.
In the context of the example above, we have all the parameters required to find the delta of the option. I have taken the liberty of building classes for call and put options in Python to assist with pricing and computing delta...
After creating an instance of the EuropeanCall class with the given parameters from the example we get a delta of...
.5469501144776616
To construct a risk-neutral portfolio...
Position: short 100,000 1 year 350 AAPL call options, delta = .54695
To construct a delta-neutral portfolio go long (100,000)*(.54695) = 54,695 shares of the underlying asset.
AAPL price 320 →321
Option position decreases in value by (100,000)*(.54695) = -$54,695
Stock position increases in value by (54,695)*(321–320) = $54,695
Net change in portfolio value = $0
AAPL price 320 →319
Option position increases by (100,000)*(.54695) = $54,695
Stock position decreases in value by (54,695)*(319–320) = -$54,695
Net change in portfolio value = $0
We have effectively constructed a portfolio to take advantage of the potentially overstated volatility while mitigating direction risk.
Herein lies a major problem with the idea of a risk-neutral portfolio, and why I specified temporary above. The offsetting position in the underlying asset is based on a linear approximation of a non-linear function. This means as the price of the underlying asset deviates further away from our initial hedge the error between the current delta and the initial delta creates directional exposure. To continue to maintain a risk-neutral portfolio the stock position will have to be periodically updated by buying or selling shares to ensure the correct amount of stock for the current delta remains in the portfolio for a risk-neutral position. Keep in mind that this process isn’t free, when purchasing/selling stock you will be subject to transaction costs while simultaneous (potentially) realizing an open P/L.
I made a video explaining volatility trading in the context of this example:
As always all of the code for this project can be found on my GitHub.