Option greeks

[Mezzala95]

Hi all,

Looking for a formula that depicts the relationship between all the option greeks; delta, gamma, vega, theta, and rho.

For example, the portfolio value *rf formula (below) is extremely useful for theta/delta/gamma/vega relationship (assuming non-delta neutral portfolio).

Such that I'm easily able to transpose to make any of them the subject, to derive a relationship.

Wanted to know if there was another formula that incorporates rho as well?

 

[Lu Shu Kai FRM]

Hi @Mezzala95 ,

Not sure if you've noticed, but you are actually quoting the famous Black-Scholes equation (not formula, the PDE) in your Theta-Delta-Gamma equation. I don't see a Vega in your equation and it isn't the variance, Vega refers to a certain quantity change in option price/premium per 1% change in the volatility. You can refer to section 5.4.4.1 Derivation of Black-Scholes equation from https://bookdown.org/maxime_debellefroid/MyBook/the-greeks.html where I read up on the explanation.

In summary, the value of a option can be replicated as a hedged portfolio that evolves at the risk-free rate by supplementing the option with delta times the underlying stock. This hedging process is also known as continuous delta hedging that was mentioned in The Pricing of Options and Corporate Liabilities in 1973. The equation given by bookdown is similar to yours:

Theta + 0.5*Gamma*S^2*sigma^2 = r*(V - Delta*S)

Where V is the value of the option. The right side of the equation r*(V - Delta*S) is the cost of constructing the hedged portfolio (interest over time). The left side just links this to the loss of the option value due to Theta and the profit due to Gamma (which is always positive for a call option and the stock component being squared) - this might require some thinking on your part because I can't explain it in a simpler way (sorry). The reason why the left and right sides are considered to be equal is because we are essentially fighting the loss of the option value from the Theta with the profit from Gamma and rebalancing with Delta, from section 5.4.4 Gamma and Theta are always flirting (bookdown explains this much better than me).

I don't think there are instances of Vega and Rho being included in the Black-Scholes PDE (but I could be wrong!). Perhaps a more intelligent Turtle may help out with this. Hope my two cents were helpful!