Deriving Black–Scholes Call Theta with Rates and Dividends

# Derivation of Call Theta from Black Scholes Model


# Derivation of Call Theta from Black Scholes Model












How is call theta mathematically derived from Black Scholes Model (without approximation) ? Please help me understand each step mathematically.

## Answer by siou0107 (score 2)

https://quant.stackexchange.com/a/73955

In the original Black-Scholes-Merton model, with the interest rate $r$ and the dividend yield $q$ constant, you have $$ c = S e^{-q \left(T - t\right)} \Phi \left(d_1\right) - Ke^{-r \left(T - t\right)} \Phi \left(d_2\right); \quad \begin{cases} d_1 = \frac{1}{\sigma \sqrt{T - t}} \ln \left(\frac{S e^{-q \left(T - t\right)}}{K e^{-r \left(T - t\right)}}\right) + \frac{1}{2} \sigma \sqrt{T - t} \\ d_2 = d_1 - \sigma \sqrt{T - t} \end{cases} $$ In this formula, $t$ only appears in the difference $T - t$, so you could change variables $\tau := T - t$ to simplify it, and just use the chain rule $\frac{\partial c}{\partial t} = \frac{\partial c}{\partial \tau} \cdot \frac{\partial \tau}{\partial t} = - \frac{\partial c}{\partial \tau}$. This yields $$ \frac{\partial c}{\partial t} = q S e^{-q \left(T - t\right)} \Phi \left(d_1\right) - r Ke^{-r \left(T - t\right)} \Phi \left(d_2\right) - S e^{-q \left(T - t\right)} \phi \left(d_1\right) \frac{\sigma}{2 \sqrt{T - t}} $$

Put-call parity $p = c + K e^{-r \tau} - S e^{-q \tau}$ would then yield $\frac{\partial p}{\partial \tau} = \frac{\partial c}{\partial \tau} - r K e^{-r \tau} + q S e^{-q \tau}$.

In the more general case of non-constant, but deterministic, rates and dividends, the $q \tau$ / $r \tau$ terms would be replaced by $\int_t^T{q_s \mathrm{d}s}$ / $\int_t^T{r_s \mathrm{d}s}$, whose derivative with respect to $t$ would be $-q_t$ / $-r_t$, and the extension would be straightforward.

Note: in the constant parameter model, $\frac{\partial c}{\partial T} = - \frac{\partial p}{\partial t}$, so the price of an infinitesimal calendar spread is equal to the opposite of theta. In a non-constant but deterministic parameter model, this is no longer the case : the former depends on the long-end of the yield/dividend curve, while the latter depends on the short-end.

## Answer by LvM_ (score 1)

https://quant.stackexchange.com/a/73954

You have a great website that show the derivation, step by step. It involves both the chain rule and product rule.

https://quantpie.co.uk/bsm_formula/bs_theta.php

If there is a step you don't understand, please let me know.

Best, Jules