Who is Afraid of Black Scholes
A Gentle Introduction to Quantitative
Finance
Day 2
July 12th 13th and 15th 2013
UNIVERSIDAD NACIONAL MAYOR DE SAN MARCOS
Ito Calculus
• Suppose the stock price evolved as
• Problem with this model is that the price can
become negative
Ito Calculus
• A better model is that the ‘relative price’ NOT the
price itself reacts to market fluctuations
• Q: What does this integral mean?
Constructing the Ito Integral
• We will try and construct the Ito Stochastic
Integral in analogy with the Riemann-Stieltjes
integral
• Note the function evaluation at the left end
point!!!
• Q: In what sense does it converge?
Stochastic Differential Equations
• Consider the following Ito Integral
• We use the shorthand notation to write this as
• This is a simple example of a stochastic
differential equation
Convergence of the Integral
• We have noted the integral converges in the ‘mean square
sense’
• To see what this means consider
• This means
Convergence of the Integral
So we have (in the mean square sense)
OR
How to Integrate?
• A detour into the world of Ito differential calculus
•
•
•
•
•
Q: What is the differential of a function of a stochastic variable?
e.g. If
what is
Is it true that
in the stochastic world as well?
We will see the answer is in the negative
We will construct the correct Taylor Rule for functions of
stochastic variables
• This will help us integrating such functions as well
Taylor Series & Ito’s Lemma
• Consider the Taylor expansion
• The change in F is given by
• We note that
behaves like a determinist quantity
is it’s expected value as
• i.e.
formally!!
that
Taylor Series & Ito’s Lemma
• We consider
when
• So the change involves a deterministic part
and a stochastic part
Ito’s Lemma
• We consider a function of a Weiner Process
and consider a change in both W and t
Ito’s Lemma
Ito’s Lemma
• Obtain an SDE for the process
• We observe that
• So by Ito’s Lemma
Integration
• Using Ito we can derive
• E.g. Show that
Example
• Evaluate
• Evaluate
Extension of Ito’s Lemma
• Consider a function of a process that itself
depends on a Weiner process
• What is the jump in V if
?
Extension of Ito
• So we have the result
Example
• If S evolves according to GBM find the SDE for
V
• Given
• Given
Stochastic Differential Equation
• We will now ‘solve’ some SDE
• Most SDE do NOT have a closed form solution
• We will consider some popular ones that do
Arithmetic Brownian Motion
• Consider dX=aXdt+bdW
• To ‘solve’ this we consider the process
• From extended Ito’s Lemma
Ito Isometry
• A shorthand rule when taking averages
• Lets find the conditional mean and variance
of ABM
Mean and Variance of ABM
• We have using Ito Isometry
Geometric Brownian Motion
• The process is given by
• To solve this SDE we consider
• Using extended form of Ito we have
Black Scholes World
• The value of an option depends on the price of the
underlying and time
• It also depends on the strike price and the time to
expiry
• The option price further depends on the
parameters of the asset price such as drift and
volatility and the risk free rate of interest
• To summarize
Assumptions
• The underlying follows a log normal process
(GBM)
• The risk free rate is known (it could be time
dependent)
• Volatility and drift are known constants
• There are no dividends
• Delta hedging is done continuously
• No transaction costs
• There are no arbitrage opportunities
A Simple One Step Discrete Case
The Payoff
Short Selling
Hedging with the Right Amount
And the value is…….
Drift and Volatility
Delta Hedging
• How did one know the quantity of stock to
short sell?
• Let’s re do the example:
– Start with one option
– And short on the stock
• The portfolio at the next time is worth
–
–
if the stock rises
if the stock falls
Delta Hedging
• We want these to give the same value
• In general we should go
The Stock Price Model
• Is out stock price model correct?
Derivation of Black Scholes Equation
• We assumed that the asset price follows
• Construct a portfolio with a long position in
the option and a short position in some
quantity of the underlying
•
• The value of this portfolio is
Derivation
• Q: How does the value of the portfolio
change?
• Two factors: change in underlying and change
in option value
• We hold delta fixed during this step
Derivation
• We use Ito’s lemma to find the change in the
value of the portfolio
• The change in the option price is
• Hence
Derivation
• Plugging in
• Collecting like terms
Derivation
• We see two type of movements, deterministic i.e.
those terms with dt and random i.e. those terms
with dW
• Q: Is there a way to do away with the risk?
• A: Yes, choose
in the right way
• Reducing risk is hedging, this is an example of
delta-hedging
Derivation
• We pick
• Now the change in portfolio value is riskless
and is given by
Derivation
• If we have a completely risk free change in
we must be able to replicate it by investing the
same amount in a risk free asset
• Equating the two we get
Black Scholes Equation
• We know what
should be
• This gives us the Black Scholes Equation
Black Scholes Equation
• This is a linear parabolic PDE
• Note that this does not contain the drift of the
underlying
• This is because we have exploited the perfect
correlation between movements in the
underlying and those in the option price.
Black Scholes Equation
• The different kinds of options valued by BS are
specified by the Initial (Final) and Boundary
Conditions
• For example for a European Call we have
• We will discuss BC’s later
Variations: Dividend Paying Stock
• If the underlying pays dividends the BS can be
modified easily
• We assume that the dividend is paid
continuously
• i.e. we receive
in time
• Going back to the change in the value of the
portfolio
Variations: Dividend Paying Stock
• The last terms represents the amount of
dividend
• Using the same delta hedging and replication
argument as before we have
Variations: Currency Options
• These can be handled as in the previous case
• Let be the rate of interest received on the
foreign currency, then
Variations: Options on Commodities
• Here the cost of carry must be adjusted
• To simplify matters we calculate the cost of
carrying a commodity in terms of the value of
the commodity itself
• Let q be the fraction that goes toward the cost
of carry, then
Solving the Black Scholes Equation
• We need to solve a BS PDE with Final Conditions
• We will convert it to a ‘Diffusion Equation IVP’ by
suitable change of variables
• Method of solution depends upon the PDE and BC
• Considering the BC in this case we will use the Fourier
Transform Methods to find a function that satisfies the
PDE and the BC
• Using different IC/FC will give the value for different
options
Transforming the BS Equation
• Consider the Black Scholes Equation given by
• As a first step towards solving this we will
transform it into a IVP for a Diffusion Equation on
the real line
Transforming the BS Equation
• We make the change of variables
• This transforms the equation into
• Where
Transforming the BS Equation
• Choosing
• Letting
• Choose
to simplify the expression
Transforming the BS Equation
• i.e. we take
• We get the following IVP
Introducing the Fourier Transform
• Our introduction will be very formal
• We will study only those properties that are
needed to solve the IBVP
• We will derive a solution called the fundamental
solution (Green’s function)
• This will allow us to find option pricing using BS
Fourier Transform
• We define the Fourier transform of ‘nice’
functions as
• The inverse transform is defined as
Fourier Transform: Properties
• We state some basic properties
– Linearity : If
then for
– Translation: for
– Modulation: If
and
Fourier Transform: Properties
• A very useful property for solving linear
constant coefficients differential equations is
• Pf: Integrate by parts in the definition
Fourier Transforms
Table of some common transforms
F(x)
F(k)
Solving an IBVP
• We will use the Fourier Transform Method to
solve the heat equation on an infinite domain
• PDE
• IC
• BC
as
Solving the Heat Equation
• Take the Fourier Transform of both sides and assuming
• We have using the properties of FT
The Heat Kernel
Solving the Heat Equation
• For a general initial condition
we note that the solution is given by
• The idea is that you ‘break’ your IC into tiny
bits and add them (integrate)
Back to the Black Scholes Equation
• We had transformed the B-S FVP to the IVP
• Where
Solving the Black Scholes for a
European Call
• Here the final condition is replaced by the IC
• Recalling the IVP and the fundamental
solution, we have
Solving the Black Scholes for a
European Call
• Going back to the original variables
• The call option has the payoff
Solving the Black Scholes for a
European Call
• Substituting into the solution we have
Solving the Black Scholes for a
European Call
• From which we get
Solving the Black Scholes for a
European Call
• These are integrals of the form
• By doing a little algebra (HW 4) we have
European Call Option
The Black Scholes PDE
• Consider the Black Scholes Equation given by
•
•
•
•
European Call
European Put
Binary Options
American Style Options
Solution of Black Scholes for a
European Call
• Solution for a European Call
Solution for a European Put
• We use the put call parity
• Using the solution for a European Call
• Noting
• We have
Calculating the Greeks
• To calculate
for a call we note
• The delta for a put is given by
Binary Options
• The payoff is of a binary call option given by
• The price of an European type option is given by
• So we have
American Options
• These can be exercised anytime prior to and at
expiry
• Issue with analyzing these is ascertaining
when to exercise
• This will lead to a free boundary problem
American Options
• Note that if the previous was the value of an
American Put then there would be an arbitrage
opportunity. i.e. If
an
arbitrage opportunity exists
• Pf: Buy the assent in the market for S and the
option for P, immediately selling the asset for K by
exercising the option. We make a risk free profit
of
• Hence for the American Option we have
Perpetual American Put
• This can be exercised at any point and there is
not expiry
• The payoff is
• We have already note that
• The option must satisfy the BS ODE
Perpetual American Put
• Solving the BS equation we have
• We also know that as
• Suppose we exercise when
so that the
payoff is
• Q: What is this ‘optimal’ exercise price?
Perpetual American Put
• This gives us the value of B
• Q: What
• Setting
maximizes V?
Perpetual American Put
• Notice the slope of the payoff function and
the option value are the same at
• This is called the smooth pasting condition
Perpetual American Options with
Dividends
• Let’s consider the perpetual American Call with
dividends
• This has solutions
• The perpetual put now has value
• It is optimal to exercise when
Perpetual American Call with
Dividends
• The solution for a perpetual call is
• Optimal to exercise when
• If there are no dividends then
NEVER optimal to exercise
and it is
American Options with Finite Time
Horizon
• Consider the case where the strike time is
finite
• We construct a portfolio with one option and
short Δ of the underlying
• Change in the value of the portfolio in access
of the risk free rate is
American Options with Finite Time
Horizon
• We delta hedge
to obtain
• We have three possibilities
American Options with Finite Time
Horizon
• We consider them each
•
– There is an arbitrage opportunity as one can buy the
option and short sell the asset and loaning out the
cash
•
– There is an arbitrage opportunity as one can sell the
option and buy the asset borrowing cash
• In the American Option case the second strategy
will not always lead to arbitrage as exercising the
option in no longer in the hands of the seller
American Options with Finite Time
Horizon
• So we have
(Payoff for early exercise)
is continuous
American Options with Finite Time
Horizon
• Q: How to go about ‘solving’ this?
• A: Closed form solutions not available
Will sue numerics (finite difference)
Recast as a Linear Complementarity
problem