2. Outline for the presentation
My Background and Motivation for the Project
Aim of the project
Short Introduction to underlying concepts
Background of Black-Scholes and its importance
Derivation of the Black-Scholes-Merton Equation
Solution of the Black-Scholes-Merton Equation
Next Steps
3. Motivation
Past work experience
Commerzbank AG - Back Office Qaunt
LLoyds Banking Group - Data/Risk Analyst
Raiffeisen Fund Management - Investment Analyst
Project related extra curricular learning
Asset Pricing - University of Chicago
Computational Investing - Georgia Tech
Mathematical Methods for Quantitative Finance - University of
Washington
Introduction to Computational Finance and Financial
Econometrics - University of Washington
R programming
Python course
Machine Learning and Data Visualization at Stanford
University
4. Aims of Project
Learn about the well established financial models
Get an idea about how the newer financial models work
Compare the different pricing techniques and see how they are
related
5. Options
An option is a contract between two parties, where the buyer has
the right but not the obligation to buy or sell an asset. The
contract buyer is called the option holder and the seller is called
the option writer.
Options can be of many types:
American
European
Asian
Barrier
Digital
Knock-in/Knock-out
6. Options cont.
Call Option - gives the holder the right but not the obligation
to buy the underlying asset for a certain price by a certain
time in the future (or at a certain time for European Calls)
Put Option - gives the holder the right but not the obligation
to sell the underlying asset for a certain price by a certain
time in the future (or at a certain time for European Puts)
7. Options cont.
The option is a function of the following variables:
S0 is the initial value of the underlying asset
E is the exercise/strike price
r is the risk-free rate
τ is the time to expiration
σ2 is the volatility of the underlying asset
Changing one of the variable in the below table give what effect it
will have on the value of the option given caeteris paribus(all other
things are held constant):
Table 1: Option Value
S0 E r τ σ2
Ac + - + + +
Ap - + - + +
Ec + - + ? +
Ep - + - ? +
8. Efficient Market Hypothesis
The participants have access to all applicable information in
the market.
With the collective effort of the participants all relevant
information is entirely and instantly reflected in the the
market price of the asset.
Market prices represent accurate estimates of assets’ intrinsic
value.
Assets are priced so that they can be expected to deliver
risk-adjusted returns.
9. Model for Stock Prices
dS = µSdt + σSdz
Geometric Brownian Motion is a continuous-time stochastic
process in which the logarithm of the randomly varying quantity
follows a Brownian motion (also called a Wiener process) with
drift.
µ is the expected return of the stock
σ is the volatility of the stock
Note: z follows a Wiener process with and dz =
√
dt and
∼ φ(0, 1). For any non overlapping periods of time dz-s are
independent.
The mean of dz is 0 and the variance is dt.
10. Ito’s Lemma
If we know the stochastic process followed by x, Ito’s lemma
tells us the stochastic process followed by some function
G(x,t)
Since a derivative is a function of the price of the underlying
and time, Ito’s lemma plays an important part in the analysis
of derivatives securities
In an Ito process the drift rate and the variance rate are
function of time
dx = a(x, t)dt + b(x, t)dz
The lemma says that
dG =
∂G
∂x
a +
∂G
∂t
+
1
2
∂2G
∂x2
b2
dt +
∂G
∂x
bdz
11. Concepts of BS
Options have been around since ancient Greece, but no-one
had a proper way of valuing them
There are many methods which use numerical methods to
approximate the value of options
Binomial Tree Model
Monte Carlo Methods
Monte Carlo Markov Chains
The Black-Scholes-Merton method is the only one with a
closed end solution for pricing options
12. Assumptions of Black-Scholes
The underlying asset and the derivative, have the same source
of underlying uncertainty
Constructing a portfolio this uncertainty can be eliminated.
The portfolio is risk-less. (A risk-less portfolio then must earn
the risk-free rate.)
No transaction costs
No taxes
Risk-free rate is constant across all maturities
The stock price is characterised by a Geometric Brownian
Motion
The stock is non-dividend paying
Trading takes place in continuous time
Short selling is allowed
No risk-less arbitrage opportunities
13. Putting things together
1. The stock process:
dS
S = µdt + σdz
After rearranging the variables:
dS = µSdt + σSdz
2. Ito’s lemma gives the following result:
df = ∂f
∂S µS + ∂f
∂t + 1
2
∂2
f
∂S2 σ2
S2
dt + ∂f
∂x σSdz
3. Constructing a portfolio consisting of ∂f
∂S shares and −1
derivative (option):
The value of the portfolio is:
Π = −f + ∂f
∂S S
The change in the value of the portfolio is then:
dΠ = −df + ∂f
∂S dS
4. The return on the portfolio must be the risk-free rate, hence:
dΠ
Π = rdt
After rearranging the variables:
dΠ = Πrdt
14. Putting things together cont
Using 3. and 4. from the above list and substituting in for Π, dS
and df :
dΠ = −df +
∂f
∂S
dS
= −
∂f
∂t
+
1
2
∂2f
∂S2
σ2
S2
dt
(1)
dΠ = Πrdt
= −rf +
∂f
∂S
rS dt
(2)
15. Derivation of the BS equation
Equation (1) and (2) gives:
∂f
∂t
+
∂f
∂S
rS +
1
2
∂2f
∂S2
σ2
S2
= rf (3)
The above equation is the Black-Scholes Differential Equation.
This is a PDE and by the special portfolio construction we
managed to remove the stochastic components.
16. BS for European Call Options
To derive the BS formula for European call options we will need to
use the following: f is C(S, t) with C(0, t) = 0,
C(S, T) = max(0, S − E) and C(S, t) = S as S → ∞.
∂C
∂t
+
∂C
∂S
rS +
1
2
∂2C
∂S2
σ2
S2
= rC (4)
17. Solution of the BS Equation
∂C
∂t
+
∂C
∂S
rS +
1
2
∂2C
∂S2
σ2
S2
= rC
Setting:
S = Eex
, t = T −
2τ
σ2
, C(S, t) = Ev(x, τ)
x = ln
S
E
, τ =
1
2
σ2
(T − t), v (x(S), τ(t)) =
C(S, t)
E
This transformation is a good way of simplifying the problem.
18. Solution of the BS Equation
The change of variables leads to the following changes in the
partial derivatives:
∂
∂t = ∂τ
∂t
∂
∂τ = −1
2σ2 ∂
∂τ
∂
∂S = ∂x
∂S
∂
∂z = 1
S
∂
∂x
∂2
∂S2 = ∂
∂S
∂
∂S = 1
S
∂
∂x
1
S
∂
∂x = 1
S − 1
S
∂
∂x + 1
S
∂2
∂x2 =
− 1
S2
∂
∂x + 1
S2
∂2
∂x2
19. Solution of the BS Equation
After rearrangements and simplifications we get:
∂v
∂τ
=
∂2v
∂x2
+ (k − 1)
∂v
∂τ
− kv
where k = r
1
2
σ2
The initial condition changes to:
v(x, τ) =
C(S, t)
E
v(x, 0) = max(ex
− 1, 0)
(5)
20. Solution of the BS Equation
Trying to get a solution of the above equation with a general form:
v(x, τ) = eαx+βτ
u(z, τ)
where α and β are constants.
After some algebra and rearrangements we get that choosing
2α + (k − 1) = 0 and β = α2 + (k − 1)α − k we can eliminate the
u and ∂u
∂x this gives α = −1
2(k − 1) and β = −1
4(k + 1)2. Yielding
v(x, τ) = e−1
2
(k−1)x−1
4
(k−1)2τ
u(x, τ)
21. Solution of the BS Equation
∂ (u(x, τ))
∂τ
=
∂2 (u(x, τ))
∂x2
for − ∞ < x < ∞, τ > 0
u(x, τ) = v(x, τ)e
1
2
(k−1)x+1
4
(k+1)2τ
, so the initial condition
becomes:
u(x, τ) = v(x, τ)e
1
2
(k−1)x+1
4
(k+1)2τ
u(x, 0) = max(e
1
2
(k+1)x
− e
1
2
(k−1)x
, 0)
(6)
22. Solution of the BS Equation
It can been shown that A√
τ
e
−x2
4τ is a solution to the heat equation.
If taking the constant A = 1
2
√
π
the solution becomes:
1
2
√
πτ
e
−x2
4τ
∞
−∞
1
2
√
πτ
e
−x2
4τ dx = 1
23. Solution of the BS Equation
The fundamental solution if the diffusion equation can be used to
derive an explicit solution to the initial value problem, in which one
has to solve the diffusion equation for −∞ < x < ∞ and τ > 0,
with arbitrary initial data u(x, 0) = u0(x) and suitable growth
conditions at x = ±∞.
This yields:
u(x, τ) =
1
2
√
πτ
∞
−∞
u0(s)e−
(x−s)2
4τ ds
which has initial data
u(x, 0) =
1
2
√
πτ
∞
−∞
u0(s)δ(s − x)ds = u0(x)
24. Solution of the BS Equation
u(x, τ) =
1
2
√
πτ
∞
−∞
u0(s)e−
(x−s)2
4τ ds
with initial condition
u0(x) = max(e
1
2
(k+1)x
− e
1
2
(k−1)x
, 0)
The following transformation will translate the integral into
something more tractable:
x = (s−x)
√
2τ
→ s =
√
2τ + x, dx
ds = 1√
2τ
→ ds =
√
2τdx and the
limits are unchanged.
25. Solution of the BS Equation
u(x, τ) =
1
√
2π
∞
−∞
u0(x
√
2τ + x)e−x 2
2 dx (7)
When substituting in u0 we need to only consider the region where
u0(δ) ≥ 0, so when e
1
2
(k+1)δ
= e
1
2
(k−1)δ
or δ = 0. Therefore when
substituting in the lower bound of the integral can be changed to
− x√
2τ
which is found by solving x
√
2τ + x = 0.
28. Solution of the BS Equation
After completing the square in the exponent we get:
I1 =
e
1
2
(k+1)x
√
2π
∞
− x√
2τ
e
−1
2
x −
(k+1)
√
2τ
2
2
+
(k+1)2τ
4
dx
=
e
1
2
(k+1)x+
(k+1)2τ
4
√
2π
∞
− x√
2τ
e
−1
2
x −
(k+1)
√
2τ
2
2
dx
(10)
29. Solution of the BS Equation
I1 =
e
1
2
(k+1)x+
(k+1)2τ
4
√
2π
∞
− x√
2τ
−
(k+1)
√
2τ
2
e−1
2
ρ2
dρ
=
e
1
2
(k+1)x+
(k+1)2τ
4
√
2π
N
x
√
2τ
+
(k + 1)
√
2τ
2
(11)
I2 can be solved similarly.
Note :
Φ(x) =
1
√
2π
x
−∞
e
t2
2 dt
Φ(x) =
1
√
2π
∞
−x
e
t2
2 dt
30. Solution of the BS Equation
Combining the results to get the solution for u(x, τ) and using the
definition of v(x, τ) :
u(x, τ) = ex
N(d1) − e−kτ
N(d2) (12)
31. Solution of the BS Equation
Now substituting back
C = Ev(x, τ), x = ln(S
E ), τ = 1
2σ2(T − t):
d1 =
ln S
E + (r + 1
2σ2)
√
T − t
σ (T − t)
(13)
d2 =
ln S
E + (r − 1
2σ2)
√
T − t
σ (T − t)
(14)
C(S, t) =E e
ln S
E
N(d1) − e
− r
1
2 σ2
1
2
σ2(T−t)
N(d2)
= SN(d1) − Ee−r(T−t)
N(d2)
(15)
32. Next Steps/Further Work
Solve the Binomial Model in a programming language and
confirm its convergence to the Balck-Scholes solution (The
background has been established for this)
Create Monte Carlo simulations of asset paths and price
financial derivatives using it (The code has been created
already)
Learn about the ”Greek” parameters to see how the
derivative’s price is affected by changes in the underlying
price, time to maturity, risk-free rate etc.
33. Next Steps/Further Work cont.
The long term goals, which might not be finished due to time
limitations of this project:
Learn about Markov Chain Monte Carlo and see how it could
be used to price financial derivatives
Lear Artificial Intelligence and Machine Learning techniques to
predict financial markets and price financial derivatives
I have been admitted to Oxford for a Master Degree course in
Mathematical and Computational Finance. This project builds on
many modules taught at that course and I feel if I do not have
enough time to finish everything I wanted I will be working on it
during my masters.
34. Thank you
Thank you for your attention. I am happy to take any additional
questions or suggestions regarding the project.
