Using Black–Scholes Identities to Simplify the Theta Derivative

Summary

The note explains an algebraic step in deriving Black–Scholes option theta. It combines two terms involving the time derivatives of d1 and d2 by using the identity that the discounted strike-density term equals the spot-density term in the Black–Scholes model. This makes the expression proportional to the derivative of d1 minus d2.

The simplification then uses the relationship between d1 and d2, whose difference is volatility times the square root of time to maturity. The explanation identifies the identity and the substitution needed for the step, but does not provide a full proof of the identity or a complete derivation of theta. Its applicability is within the standard Black–Scholes framework and notation used in the derivation.

Key ideas

• The theta derivation combines terms by applying a Black–Scholes identity relating the spot and discounted-strike density terms.
• After that identity is used, the expression depends on the time derivative of d1 minus d2.
• In the stated notation, d1 minus d2 equals volatility times the square root of time to maturity.
• The explanation addresses one algebraic step rather than deriving the full theta formula.

Tags

# Proof Black Scholes Theta


# Proof Black Scholes Theta












I saw the following proof of theta in a paper I read, and I thought it looked pretty neat. Unfortunately I don't understand the step that they do. This is what they do:

Now, I don't get how they go from $S_0 n(d_1)\frac{\partial d_1}{\partial t} - Xe^{-rt}n(d_2) \frac{\partial d_2}{\partial t}$ to $S_0 n(d1) \frac{\partial (d_1-d_2)}{\partial t}$. Could anyone explain to me why this is true?

## Answer by Alex C (score 5, accepted)

https://quant.stackexchange.com/a/42759

There is a well known identity for the Black Scholes model: $S_0 n(d_1)-X e^{-rT} n(d_2) = 0$ (proof).

Using this allows you to combine these two terms:

$$S_0 n(d_1)\frac{\partial d_1}{\partial t} - Xe^{-rT}n(d_2) \frac{\partial d_2}{\partial t}$$

into

$$S_0 n(d1) (\frac{\partial d_1}{\partial t}-\frac{\partial d_2}{\partial t})$$

or

$$S_0 n(d1) \frac{\partial (d_1-d_2)}{\partial t}$$

Then we use the fact that $d_1-d_2=\sigma\sqrt{t}$

## Answer by Gotham (score 0)

https://quant.stackexchange.com/a/42367

Since Black Scholes Theta is for the Black–Scholes option pricing formula, the above step holds true.

For more info, refer page 3 and 4 of this pdf. http://moya.bus.miami.edu/~tsu/jef2008.pdf

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.