Deriving Delta for European Options
The delta of an option describes how the price of the option \(V\) changes with respect to changes in the underlying asset \(S\).
\[\Delta = \frac{\partial V}{\partial S}\]This article explains how the delta for a European option which does not pay dividends can be derived by evaluating the partial derivative of the of the value with respect to the underlying price. The same methodology can be used to derive the other option greeks, however this article is only intended to provide an example derivation for the delta.
Delta of a European Call: \(\Delta_c\)
The value of a call option for a non-dividend-paying underlying stock is
\[C_t(S_t,t) = S_t N(d_1) - Xe^{-r(T-t)}N(d_2) \label{eq1}\tag{1}\]where
\[N(d) = \frac{1}{\sqrt{2\pi}}\int_{\infty}^d e^{-\frac{s^2}{2}}ds, \label{eq2}\tag{2}\]is the cumulative distribution of a standard normal distribution,
\[\begin{align} d_1 &= \frac{ \ln{ \left(\frac{S_t}{X}\right) } + \left( r + \frac{\sigma^2}{2}\right)(T-t)}{\sigma\sqrt{(T-t)}} \label{eq3}\tag{3}\\ d_2 &= \frac{ \ln{ \left(\frac{S_t}{X}\right) } + \left( r - \frac{\sigma^2}{2}\right)(T-t)}{\sigma\sqrt{(T-t)}} \label{eq4}\tag{4} \end{align}\]- \((T - t)\) - is the time to maturity
- \(S_t\) - is the spot price of the underlying asset at time \(t\)
- \(X\) - is the strike price
- \(r\) - is the risk free interest rate
- \(\sigma\) - is the volatility of the underlying asset returns
The delta is defined as the change in value of the European call option \(C\) with respect to the price of the underlying asset \(S\).
\[\Delta_c = \frac{\partial C_t}{\partial S_t} = \frac{\partial [S_t N(d_1)]}{\partial S_t} - Xe^{-r(T-t)} \frac{\partial N(d_2)}{\partial S_t} \label{eq5}\tag{5}\]To evaluate the first term of (\ref{eq5}) we can use a combination of the product rule and the chain rule as follows
\[\begin{align} \frac{\partial S_tN(d_1)}{\partial S_t} &= \frac{\partial S_t}{\partial S_t} N(d_1) + S_t \frac{\partial N(d_1)}{\partial S_t} \label{eq6}\tag{6} \\ &= N(d1) + S_t \frac{\partial N(d_1)}{\partial d_1} \frac{\partial d_1}{\partial S_t} \label{eq7}\tag{7} \end{align}\]We can evaluate the first partial derivative in (\ref{eq7}) as
\[\frac{\partial N(d_i)}{\partial d_i} = \frac{1}{\sqrt{2\pi}} \frac{\partial}{\partial d_i} \int_{-\infty}^{d_i} e^{-\frac{s^2}{2}}ds = \frac{e^{-\frac{d_i^2}{2}}}{\sqrt{2\pi}} \label{eq8}\tag{8}\]We can also rewrite \(d_{1,2}\) conveniently to evaluate the second partial derivative in (\ref{eq7}) as
\[d_{1,2} = \frac{\ln{\left(\frac{S_t}{X}\right)}}{\sigma\sqrt{T-t}} + \underbrace{\frac{\left( r \pm \frac{\sigma^2}{2} \right)(T-t)}{\sigma\sqrt{T-t}}}_{\text{Constant w.r.t } S_t} \label{eq9}\tag{9}\]and therefore
\[\frac{\partial d_1}{\partial S_t} = \frac{\partial d_2}{\partial S_t} = \frac{1}{S_t \sigma\sqrt{T-t}} \label{eq10}\tag{10}\]By substituting (\ref{eq7}), (\ref{eq8}) and (\ref{eq10}) into (\ref{eq5}), we get the expression for the delta of a European call option
\[\begin{align} \Delta_c &= N(d_1) + \frac{e^{-\frac{d_1^2}{2}}}{\sigma\sqrt{2\pi (T-t)}} \\ & \hspace{2.2cm} - Xe^{-r(T-t)} \frac{e^{-\frac{d_2^2}{2}}}{S_t\sigma\sqrt{2\pi (T-t)}} \\ &= N(d_1) + \frac{Ee^{-\frac{d_1^2}{2}}}{S_t\sigma\sqrt{2\pi(T-t)}} \left( \frac{S}{E} - e^{-r(T-t)-\frac{d_2^2}{2}+\frac{d_1^2}{2}} \right) \label{eq11}\tag{11} \end{align}\]Equation (\ref{eq11}) can be reduced further by noting that
\[\frac{S_t}{E} = e^{-r(T-t)-\frac{d_2^2}{2}+\frac{d_1^2}{2}}\]and hence
\[\Delta_c = N(d_1)\]The Python code to calculate the delta for a European Call option on a non-dividend paying stock would be
from scipy.stats import norm
from math import log, sqrt
def call_delta(S, v, X, T, r):
'''
S: Stock Price
v: Volatility
X: Strike Price
T: Time to maturity
r: Risk free interest rate
'''
d1 = (log(S / X) + (r + 0.5 * v**2) * T) / (v * sqrt(T))
return norm.cdf(d1)European Call Option Delta
Delta of a European Put: \(\Delta_p\)
The price of a European Put option can be derived through put-call parity under the assumption that there is no arbitrage
\[C(S_t,t) - P(S_t,t) = S_t - Xe^{-r(T-t)} \label{eq12}\tag{12}\]By rearranging and substituting the price of the European call, we can write the price of a put option as
\[P(S_t,t) = Xe^{-r(T-t)}N(-d_2) - S_tN(-d_1) \label{eq13}\tag{13}\]The delta of a European put option can be derived in a similar fashion to the call option
\[\Delta_p = \frac{\partial P_t}{\partial S_t} = Xe^{-r(T-t)}\frac{\partial N(-d_2)}{\partial S_t} - \frac{\partial [S_tN(-d_1)]}{\partial S_t} \label{eq14}\tag{14}\]By evaluating (\ref{eq14}), much like we did for (\ref{eq5}), we arrive at the delta of a European put
\[\Delta_p = N(d_1) - 1 \label{eq15}\tag{15}\]The Python code to calculate the delta for a European put option on a non-dividend paying stock would be
from scipy.stats import norm
from math import log, sqrt
def put_delta(S, v, X, T, r):
'''
S: Stock Price
v: Volatility
X: Strike Price
T: Time to maturity
r: Risk free interest rate
'''
d1 = (log(S / X) + (r + 0.5 * v**2) * T) / (v * sqrt(T))
return norm.cdf(d1) - 1European Put Option Delta
Summary
This article demonstrated how we can derive the delta of a European call and put option from their respective price equation.