This document provides an overview of Lesson 2 which covers Itô's lemma and the Black-Scholes model. It first discusses the log-normal property of stock prices and how Itô's lemma can be used to derive the geometric Brownian motion process. It then introduces the Black-Scholes model, its assumptions, and the partial differential equation it yields. Finally, it presents the Black-Scholes formulas for European call and put options and defines the Greeks (Delta, Gamma, Vega, Theta, Rho) as sensitivities in the Black-Scholes model.

1. LESSON 2
ITÔ’S LEMMA AND BLACK-SCHOLES MODEL
Derivative Pricing - Module 4

2. Outline
I Log-normal property of stock prices
I Itô’s Lemma and the solution to GBM SDE
I Black-Scholes model
2

3. Log-Normal Property of Stock Prices
In order to analytically solve the SDE in the previous lesson, we are going to make use of two things:
(i) Log-normal property of stock prices
(ii) Itö’s Lemma to solve the SDE
Let’s explore the first of these ⇒ Is it realistic to assume stock prices follow a log-normal distribution?
Go to the first part of the Jupyter Notebook accompanying this lesson to see this:
3

4. Itô calculus for GBM
Conveniently for us, a log-normal transform of a log-normally distributed variable will follow a normal
distribution.
I By taking St = Ln(St ) and applying Itô’s Lemma, we can get a closed-form solution for St from our SDE:
Ln St
S0
=
µ − σ2
2
t + σWt ,
LnSt ∼ N
LnS0 +
µ − σ2
2
t , σ2t
which yields:
St = S0 e
µ− σ2
2
t+σWt
where,
E(St ) = S0eµt
Var(St ) = S2
0 e2µt
eσ2
t − 1
4

5. Black-Scholes-Merton Model
The last SDE gave rise to one of the most important option pricing models ever: Black-Scholes.
The model departs from some important assumptions (some will be relaxed in the future):
(i) Prices follow a GBM with µ and σ constant.
(ii) Perfect capital markets: No taxes, transaction costs, no dividends, absence of arbitrage, etc.
(iii) Risk-free rate is constant and equal for all maturities.
I Next, apply the same intuition as in binomial setting: No-arbitrage and riskless portfolio replica!
I This yields the following Black-Scholes-Merton SDE (see additional references for proof):
∂f
∂t
+ rS
∂f
∂S
+
1
2
σ2
S2 ∂2f
∂S2
= rf
for f being the claim associated with any derivative instrument.
I Importantly, note how the previous equation only depends on r → Risk-neutral valuation
5

6. Black-Scholes Formulas for Option Pricing
Using the Black-Scholes-Merton SDE, we can get closed-form solutions for call and put option prices.
I The formulas are the following:
Call option → c = S0N(d1) − Ke−rT
N(d2)
Put option → p = Ke−rT
N(−d2) − S0N(−d1)
where N() is the cumulative density function (CDF) of a standard normal and:
d1 =
Ln(S0/K) + (r + σ2/2)T
σ
√
T
d2 =
Ln(S0/K) + (r − σ2/2)T
σ
√
T
= d1 − σ
√
T
I Again, only discounting on r → Risk-neutral valuation
6

7. Greeks in Black-Scholes
In the final part of the lesson, we are going to deal with the Greeks in a Black-Scholes model.
We can interpret Greeks as the sensitivity of option price to different variables → remember Delta?
I Sensitivity to underlying stock price (Delta):
Call Option ∆ =
∂C
∂S
= N(d1); Put Option ∆ =
∂P
∂S
= N(d1) − 1
I Sensitivity to changes in underlying stock price (Gamma):
Γ =
∂2V
∂S2
=
N(d1)
Sσ
√
T − t
I Sensitivity to volatility (Vega):
ν =
∂2V
∂σ
= SN(d1)
√
T − t
7

8. Greeks in Black-Scholes (con.)
I Sensitivity to time (Theta):
Call Option Θ =
∂C
∂t
= −
SN(d1)σ
2
√
T − t
− rKe−r(T−t)
N(d2)
Put Option Θ =
∂P
∂t
= −
SN(d1)σ
2
√
T − t
+ rKe−r(T−t)
N(−d2)
I Sensitivity to risk-free rate (Rho):
Call Option ρ =
∂C
∂r
= K(T − t)e−r(T−t)
N(d2)
Put Option ρ =
∂P
∂r
= −K(T − t)e−r(T−t)
N(−d2)
⇒ You can consult additional references to see more on the derivation of each of these.
8

9. Summary of Lesson 2
In Lesson 2, we have looked at:
I Log-normality of stock prices
I Derivation of GBM process for stock prices using Itô’s Lemma
I Black-Scholes-Merton SDE and option pricing model
I Greeks (sensitivities) in Black-Scholes model
⇒ TO-DO NEXT: Now, please go to the associated Jupyter Notebook for this lesson to learn how to
implement the Black-Scholes option pricing model in Python. Also, you have a working example of the different
Greeks there. Lastly, please make sure that you check the additional material in this lesson to ensure that you
have a comprehensive understanding of how to derive the previous formulas (Itô’s Lemma, GBM,
Black-Scholes, etc.)
⇒ In the next lesson, we will see the application of a previously introduced method, Monte Carlo, for pricing
options under the Black-Scholes framework.
9