The document summarizes the Black-Scholes equation, which provides a theoretical estimate for the price of European-style options. It describes the assumptions of the model, including constant riskless interest rates and stock price behavior following geometric Brownian motion. The derivation shows how forming a hedged portfolio eliminates uncertainty and leads to the Black-Scholes PDE. Transforming this into a heat equation allows the value of a European call option to be expressed in terms of the standard normal distribution.

1. Black Scholes Equation

2. Few financial terms
› Options
Options are financial contracts that offer the buyer the
opportunity to buy or sell- depending on the type of contract
they hold-the underlying asset. The holder is not required to
buy or sell the asset if they choose not to.
European option is a kind of a option which comes with
expiration date and can’t be exercised before that date.

4. Difference between stock and strike price
STOCK PRICE
› Current price of the
underlying asset
STRIKE PRICE
› The fixed price at which buyer of the
option can exercise his option. It
always remain constant throughout the
life of contract period.
Example of European Option-XYZ Limited
› Option price: ₹1
› Strike price: ₹30
› Stock (spot) price: ₹27.59
› Option price: ₹1
› Strike price: ₹30
› Stock (spot) price: ₹35

5. › Stochastic process: It is process involving many random
variables depending on a single variable (time).
› Brownian motion: It is a simple continuous stochastic
process that is used widely in finance used for modelling
random behavior that evolves over time. For example
fluctuation in stock prices.
› Portfolio: It refers to combination of financial assets such
as bonds, stocks and cash. Can be held individuals or
managed by financial professionals.
› Itô’s Lemma: It is an identity used in Itô’s Calculus to find
the differential of time dependent functions stochastic
processes.
› Itô’s Calculus: Methods used to solve Brownian motion.

6. Definition
› Black–Scholes model is a mathematical model for the
dynamics of a financial market containing financial
contracts or derivatives. From the partial differential
equation in the model, known as the Black–Scholes
equation, one can deduce the Black–Scholes formula,
which gives a theoretical estimate of the price of European-
style options and shows that the option has a unique price
regardless of the risk of the security and its expected
return.

7. Derivation
› Assumptions:
• Constant riskless interest rate,
• No transaction costs,
• Any no. of stocks can be bought,
• Option is of European style.

8. Notations
› S = stock price
› t = time
› V = (S,t)……….option price
› Portfolio: 1 option,
𝝏𝑽
𝝏𝑺
no. of shares
• Value of portfolio = 𝑷 = −𝑽 +
𝝏𝑽
𝝏𝑺
𝑺
• Change in value of portfolio: 𝚫𝑷 = −𝚫𝑽 +
𝝏𝑽
𝝏𝑺
𝚫𝑺

9. › Per the model assumptions above, the price of the
underlying asset (typically a stock) follows a geometric
Brownian motion. That is
𝒅𝑺
𝑺
= µ 𝒅𝒕 + 𝝈 𝒅𝑾
where W is a stochastic variable (Brownian motion). Note
that W, and consequently its minute increment dW,
represents the only source of uncertainty in the price history
of the stock.
› The payoff of an option 𝑽(𝑺, 𝒕) at maturity is known. To find
its value at an earlier time we need to know how 𝑽 evolves
as a function of 𝑺 and 𝒕. By Itô's lemma for two variables
we have

10. 𝒅𝑽 =
𝝏𝒗
𝝏𝒕
+ 𝝁𝑺
𝝏𝑽
𝝏𝑺
+
𝟏
𝟐
𝝈𝟐𝑺𝟐
𝝏𝟐𝑽
𝝏𝑺𝟐
𝒅𝒕 + 𝝈𝑺
𝝏𝑽
𝝏𝑺
𝒅𝑾
Now consider a certain portfolio, consisting one option and
𝝏𝑽
𝝏𝑺
shares at time 𝒕. The value of these holdings is
𝑷 = −𝑽 +
𝝏𝑽
𝝏𝑺
𝑺
Over the time period 𝒕, 𝒕 + 𝜟𝒕 , the total profit or loss from
changes in the values of the holdings is
𝚫𝑷 = −𝚫𝑽 +
𝝏𝑽
𝝏𝑺
𝚫𝑺

11. Now discretize the equations for dS/S and dV by replacing
differentials with deltas:
𝚫𝑺 = µ𝑺𝚫𝒕 + 𝝈𝑺𝚫𝑾
𝚫𝑽 =
𝝏𝑽
𝝏𝒕
+ 𝝁𝑺
𝝏𝑽
𝝏𝑺
+
𝟏
𝟐
𝝈𝟐
𝑺𝟐
𝝏𝟐𝑽
𝝏𝑺𝟐
𝚫𝒕 + 𝝈𝑺
𝝏𝑽
𝝏𝑺
𝚫𝑾
and appropriately substitute them into the expression
𝚫𝑷 = −𝚫𝑽 +
𝝏𝑽
𝝏𝑺
𝚫𝑺
We get,
𝚫𝑷 = −
𝝏𝑽
𝝏𝒕
+ 𝝁𝑺
𝝏𝑽
𝝏𝑺
+
𝟏
𝟐
𝝈𝟐𝑺𝟐 𝝏𝟐𝑽
𝝏𝑺𝟐 𝚫𝒕 − 𝝈𝑺
𝝏𝑽
𝝏𝑺
𝚫𝑾 +
𝝏𝑽
𝝏𝑺
(µ𝑺𝚫𝒕 +

12. 𝚫𝑷 = −
𝝏𝑽
𝝏𝒕
+
𝟏
𝟐
𝝈𝟐
𝑺𝟐
𝝏𝟐𝑽
𝝏𝑺𝟐
𝚫𝒕
Notice that the term 𝚫𝑾 has vanished. Thus uncertainty has been
eliminated and the portfolio is effectively riskless. The rate of return on
this portfolio must be equal to the rate of return on any other riskless
instrument. Now assuming the risk-free rate of return is 𝒓 we must
have over the time period 𝒕, 𝒕 + 𝜟𝒕
𝚫𝑷 = 𝒓𝑷𝚫𝒕
−
𝝏𝑽
𝝏𝒕
+
𝟏
𝟐
𝝈𝟐𝑺𝟐
𝝏𝟐
𝑽
𝝏𝑺𝟐
𝚫𝒕 = 𝒓 −𝑽 +
𝝏𝑽
𝝏𝑺
𝑺 𝚫𝒕
Simplifying, we arrive at the celebrated Black–Scholes partial
differential equation:
𝝏𝑽
𝝏𝒕
+
𝟏
𝟐
𝝈𝟐𝑺𝟐
𝝏𝟐𝑽
𝝏𝑺𝟐
+ 𝒓
𝝏𝑽
𝝏𝑺
𝑺 − 𝒓𝑽 = 𝟎

13. Transformation Into Heat Equation
› General form of the heat equation:
𝝏𝒖
𝝏𝒕
= α
𝝏𝟐
𝒖
𝝏𝒙𝟐
+
𝝏𝟐
𝒖
𝝏𝒚𝟐
+
𝝏𝟐
𝒖
𝝏𝒛𝟐
α is constant of Diffusivity

14. Changes To Make In Variables
› To get the time running in the right direction, you can define a new
variable 𝜏 = 𝑇 − 𝑡. Then 𝑡 = 𝑇 will correspond to 𝜏=0.
› Since it was
𝑑𝑆
𝑆
= 𝑑(𝑙𝑜𝑔𝑆) that satisfied the standard Brownian motion
that leads to the usual heat equation, it makes sense to define a new
variable 𝑥 = log 𝑆 (natural logarithm). This should get rid of the
appearances of the independent variable 𝑆 or 𝑥 multiplying the various
derivatives.
› A substitution of the form 𝑢 = 𝑒𝛼𝑥+𝛽𝜏𝑉 can be used to get rid of unwanted
constants and first order derivatives.

15. Derivation

21. › Now with previous result we will prove that value of European call
option with strike price 𝐸 and expiry time 𝑇 is given by:
𝑉 𝑆, 𝑡 = 𝑆𝐹 𝐴+ − 𝐸𝑒−𝑟 𝑇−𝑡 𝐹 𝐴−
Where 𝐹 is function of standard normal distribution
and the constants 𝐴± are given by

22. =

25. Finally, we get
𝑉 𝑆, 𝑡 = 𝑆𝐹 𝐴+ − 𝐸𝑒−𝑟 𝑇−𝑡 𝐹 𝐴−

26. Question:

28. STARRING YAZUR GARG
PRODUCER YAZUR GARG
DIRECTOR YAZUR GARG
SUBJECT PRANJUL GARG
BACKGROUND SCORE AVENGERS THEME
Special Thanks
MRS. KALIKA SRIVASTAVA