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A partial
review:
BlackScholes
and beyond
MN50429 Financial
Engineering
MN50429  Financial Engineering
Precursor:
It¯o ’s lemma
Deriving
BlackScholes
Extension 1:
Stochastic
volatility
Extension 2:
Jumpdi¤usion
processes
Lecture 1
A partial review: BlackScholes and beyond
Dr Andreas Krause and Dr XiaoHua Chen
The BlackScholes option pricing model: Main
assumptions and formula
A partial
review:
BlackScholes
and beyond
MN50429 Financial
Engineering
Precursor:
It¯o ’s lemma
Deriving
BlackScholes
Extension 1:
Stochastic
volatility
Extension 2:
Jumpdi¤usion
processes
Main assumptions:
The stock price follows Brownian motion (i.e. Wiener
process). Thus, the distribution of stock prices is
lognormal.
The volatility of stock return is constant.rre
There are no riskless arbitrage opportunities.
The option is European.
Security trading is continuous.
The short selling of securities with full use of proceeds is
permitted.
...
The BlackScholes option pricing formula
A partial
review:
BlackScholes
and beyond
MN50429 Financial
Engineering
Precursor:
It¯o ’s lemma
Deriving
BlackScholes
Extension 1:
Stochastic
volatility
Extension 2:
Jumpdi¤usion
processes
c = S0 N (d1 ) − K e−rT N (d2 )
p = K e−rT N (−d 2 ) − S0 N (−d1 )
ln( S0 / K ) + (r + σ2 / 2)T
where d1 =
σ T
ln( S0 / K ) + (r − σ2 / 2)T
d2 =
= d1 − σ T
σ T
The need to go beyond BlackScholes
A partial
review:
BlackScholes
and beyond
MN50429 Financial
Engineering
Precursor:
It¯o ’s lemma
The market consists of a wide variety of derivatives
Deriving
BlackScholes
Payo¤ structures are di¤erent
Extension 1:
Stochastic
volatility
Extension 2:
Jumpdi¤usion
processes
Underlying assets’stochastic processes are di¤erent
Need for accurate pricing
A partial
review:
BlackScholes
and beyond
MN50429 Financial
Engineering
Precursor:
It¯o ’s lemma
Deriving
BlackScholes
Extension 1:
Stochastic
volatility
Extension 2:
Jumpdi¤usion
processes
Issuers need to be able to price derivatives in order to sell
them appropriately
Hedgers need to be able to price derivatives in order to
obtain coste¢ cient solutions to their risk exposures
Traders need to be able to price derivatives in order to
ensure an e¢ cient market
A partial
review:
BlackScholes
and beyond
MN50429 Financial
Engineering
Precursor:
It¯o ’s lemma
1
Precursor: It¯o’s lemma
2
Deriving BlackScholes
3
Extension 1: Stochastic volatility
4
Extension 2: Jumpdi¤usion processes
Deriving
BlackScholes
Extension 1:
Stochastic
volatility
Extension 2:
Jumpdi¤usion
processes
Brownian motion
A partial
review:
BlackScholes
and beyond
MN50429 Financial
Engineering
Precursor:
It¯o ’s lemma
Deriving
BlackScholes
Extension 1:
Stochastic
volatility
Extension 2:
Jumpdi¤usion
processes
A common assumption is that stock returns are normally
distributed
St St
St
For
t
t
ln St
ln St
t
N
t;
2
t
t ! 0 we can write the dynamics as
dSt
= dt + dzt
St
(1)
is the drift (expected return)
is the volatility
p
dZt = "t dt is a Wiener process, " N(0; 1). Wiener
process is a partricular type of Markov stochastic process
with a mean change of 0 and a variance rate of 1.
A partial
review:
BlackScholes
and beyond
MN50429 Financial
Engineering
Precursor:
It¯o ’s lemma
Deriving
BlackScholes
Extension 1:
Stochastic
volatility
Extension 2:
Jumpdi¤usion
processes
Source: Hull, J, Options, futures, and other derivatives, 7th ed., p.264.
Functions of stochastic processes
A partial
review:
BlackScholes
and beyond
MN50429 Financial
Engineering
Precursor:
It¯o ’s lemma
Deriving
BlackScholes
Extension 1:
Stochastic
volatility
Extension 2:
Jumpdi¤usion
processes
The value of a derivative will be a function of the value of
the underlying asset and time
Second order Taylor series approximation:
C (St ; t)
C (St
t; t
t) +
@C
(St
@S
@C
(t (t
t))
@t
1 @2C
+
(St St t )2
2 @S 2
1 @2C
+
(t (t
t))2
2 @t 2
@2C
+
(St St t ) (t (t
@S@t
St
t)
+
t)) (2)
A partial
review:
BlackScholes
and beyond
MN50429 Financial
Engineering
Precursor:
It¯o ’s lemma
Deriving
BlackScholes
Extension 1:
Stochastic
volatility
Extension 2:
Jumpdi¤usion
processes
Note: second order Taylor series approximation formula:
f (x) = f (x0 + h) = f (x0 ) + f 0 (x0 )h + 12 f 00 (x0 )h2
It¯o’s lemma
A partial
review:
BlackScholes
and beyond
MN50429 Financial
Engineering
Precursor:
It¯o ’s lemma
Deriving
BlackScholes
Extension 1:
Stochastic
volatility
Extension 2:
Jumpdi¤usion
processes
In Eq.(2), If we move C (St
t ! 0, this becomes
dC
=
t; t
t) to LHS and, let
@C
1 @2C
@C
dS +
dt +
(dS)2
@S
@t
2 @S 2
@2C
1 @2C
2
(dt)
+
dSdt
+
2 @t 2
@S@t
@C
@C
1 @2C
dS +
dt +
(dS)2
@S
@t
2 @S 2
This relationship is knows as It¯o’s lemma
(3)
Application to Brownian motion
A partial
review:
BlackScholes
and beyond
MN50429 Financial
Engineering
Precursor:
It¯o ’s lemma
Deriving
BlackScholes
From Eq.(1), we have
p
dS = Sdt + Sdz = Sdt + S dt"
As dt ! 0 and E ["2 ] = 1, we have
(dS)2 =
2 2
S (dt)2 + 2
=
2 2
) dC =
+ S
2 2
S dt"2
S dt
Extension 1:
Stochastic
volatility
Extension 2:
Jumpdi¤usion
processes
3
S 2 (dt) 2 " +
S
@C
dz
@S
(4)
@C
1
+
@S
2
2 2@
S
2C
@S 2
+
@C
@t
dt
(5)
This is the basis from which to derive the BlackScholes
formula
A partial
review:
BlackScholes
and beyond
MN50429 Financial
Engineering
Precursor:
It¯o ’s lemma
1
Precursor: It¯o’s lemma
2
Deriving BlackScholes
3
Extension 1: Stochastic volatility
4
Extension 2: Jumpdi¤usion processes
Deriving
BlackScholes
Extension 1:
Stochastic
volatility
Extension 2:
Jumpdi¤usion
processes
Hedge portfolio
A partial
review:
BlackScholes
and beyond
MN50429 Financial
Engineering
Precursor:
It¯o ’s lemma
Deriving
BlackScholes
Extension 1:
Stochastic
volatility
Extension 2:
Jumpdi¤usion
processes
De…ne a portfolio V of the option C and a short position
of units of S, i.e. the underlying asset
V =C
S
) dV = dC
(6)
dS
(7)
Eliminating risk
Using Eqs. (1) and (5), the above portfolio of Eq.(7)
becomes
1 2 2 @2C
@C
@C
+
S
+
S dt
dV =
S
2
@S
2
@S
@t
@C
+ S
dz
(8)
@S
Eliminate risk by setting
) dV =
=
@C
@S
@C
1
+
@t
2
2 2@
S
2C
@S 2
dt
(9)
Riskless portfolio
A partial
review:
BlackScholes
and beyond
MN50429 Financial
Engineering
This portfolio is riskless, hence dV = rVdt, where r is the
riskfree asset rate of return.
Inserting Eq.(9) in the above and substituting V we have
Precursor:
It¯o ’s lemma
Deriving
BlackScholes
Extension 1:
Stochastic
volatility
Extension 2:
Jumpdi¤usion
processes
rv =
@C
1
+
@t
2
2 2@
S
2C
(10)
@S 2
Using Eq.(6), delta hedge = @C
@S and Eq.(10) we get the
BlackScholes partial di¤erential equation:
@C
1
+
@t
2
2 2@
S
2C
@S 2
+ rS
@C
@S
rC = 0
(11)
BlackScholes partial di¤erential equation
A partial
review:
BlackScholes
and beyond
MN50429 Financial
Engineering
Precursor:
It¯o ’s lemma
Deriving
BlackScholes
Extension 1:
Stochastic
volatility
Extension 2:
Jumpdi¤usion
processes
This equation holds for all derivatives where the
underlying follows a Brownian motion
Solution for a derivative depends on boundary conditions
Because a solution of a PDE is generally not unique,
additional conditions must generally be speci…ed on the
boundary of the region where the solution is de…ned.
Boundary conditions are typically values at expiry, values
at low or high prices of the underlying, etc.
Partial di¤erential equations can in general not be solved
analytically
Many can be solved numerically
European call option
A partial
review:
BlackScholes
and beyond
MN50429 Financial
Engineering
Precursor:
It¯o ’s lemma
Deriving
BlackScholes
Extension 1:
Stochastic
volatility
Extension 2:
Jumpdi¤usion
processes
For a European call option the boundary conditions are:
C (0; t) = 0
limS !1 C (S; t) = S
C (S; T ) = maxfS
E ; 0g
Solving for the BlackScholes formula
A partial
review:
BlackScholes
and beyond
MN50429 Financial
Engineering
Precursor:
It¯o ’s lemma
Deriving
BlackScholes
Extension 1:
Stochastic
volatility
Extension 2:
Jumpdi¤usion
processes
With these boundary conditions exceptionally an analytical
solution exists
Derivation via variable substitution and Heat equation
using Fourier transforms (See Hull, J, Options, futures,
and other derivatives, 7th ed., p.307)
In future lectures we will look at numerical procedures to
solve such equations as well as other approaches to
determine the value of derivatives
A partial
review:
BlackScholes
and beyond
MN50429 Financial
Engineering
Precursor:
It¯o ’s lemma
1
Precursor: It¯o’s lemma
2
Deriving BlackScholes
3
Extension 1: Stochastic volatility
4
Extension 2: Jumpdi¤usion processes
Deriving
BlackScholes
Extension 1:
Stochastic
volatility
Extension 2:
Jumpdi¤usion
processes
Changing volatility
A partial
review:
BlackScholes
and beyond
MN50429 Financial
Engineering
Precursor:
It¯o ’s lemma
Deriving
BlackScholes
Extension 1:
Stochastic
volatility
Extension 2:
Jumpdi¤usion
processes
Volatility is the main drive of the option value in
BlackScholes
Volatility is not constant over time in real markets
More realistic to assume that it changes by some
mechanism, e.g. stochastic di¤erential equations (SDEs):
dS = Sdt +
d
2
= dt +
t SdzS
(12)
dz
(13)
dzS dz = dt
(14)
where is the volatility of , is a function of (S; 2 ; t).
is the correlation between the two stochastic processes.
Multivariate It¯o’s lemma
A partial
review:
BlackScholes
and beyond
MN50429 Financial
Engineering
The multivariate version of It¯o’s lemma is
Precursor:
It¯o ’s lemma
Deriving
BlackScholes
Extension 1:
Stochastic
volatility
Extension 2:
Jumpdi¤usion
processes
dC
=
@C
@C
@C
dS +
d 2+
dt
2
@S
@
@t
1 @2C
1 @2C
2
+
(dS)
+
d
2 @S 2
2@ 4
(15)
2 2
+
@C
dSd
@S@ 2
2
Preliminaries
A partial
review:
BlackScholes
and beyond
MN50429 Financial
Engineering
Precursor:
It¯o ’s lemma
Deriving
BlackScholes
Extension 1:
Stochastic
volatility
Extension 2:
Jumpdi¤usion
processes
In Eq.(4) we derive
(dS)2 =
2 2
S dt
p
From Eq.(13), let dt ! 0, dz = "t dt, we have
d
2 2
=
2
=
2 2
t dt
dSd
2
3
2
2
t t (dt)
(dt)2 + 2
=
=
2 2 2
t t dt
(16)
3
S (dt)2 +
+
+
tS
t
S (dt) 2
(dt) 2 +
2
t Sdt
2
t Sdt
3
(17)
A partial
review:
BlackScholes
and beyond
MN50429 Financial
Engineering
Precursor:
It¯o ’s lemma
Deriving
BlackScholes
Extension 1:
Stochastic
volatility
Extension 2:
Jumpdi¤usion
processes
Plug Eqs.(4,16,17) in Eq.(15), we have
dC
@C
@t
=
+
+
1
2
2 S 2 @2 C + 1 2 2 @2 C
@S 2 2 2
@ 4
2S @ C
+
@S @ 2
@C
dS
@S{z }
Uncertainty
+
1
@C
d 2
2
@
 {z }
Uncertainty
!
dt
(18)
2
Hedge portfolio
A partial
review:
BlackScholes
and beyond
MN50429 Financial
Engineering
Precursor:
It¯o ’s lemma
Deriving
BlackScholes
Extension 1:
Stochastic
volatility
Extension 2:
Jumpdi¤usion
processes
We have two uncertainties, so we need two instruments to
hedge
One instrument is the underlying asset S, the other is
b on the same asset S.
another option C
V =C
) dV = dC
b follows the same rules as C
C
S
dS
bC
b
b dC
b
(19)
(20)
Riskless portfolio
A partial
review:
BlackScholes
and beyond
MN50429 Financial
Engineering
Precursor:
It¯o ’s lemma
Deriving
BlackScholes
Extension 1:
Stochastic
volatility
Extension 2:
Jumpdi¤usion
processes
Plug Eq.(18) in Eq.(20), we have
dV
=
@C
@t
b
@C
+
@S

!
2 S 2 @2 C + 1 2 2 @2 C
@S 2 2 2
@ 4
dt
2S @ C
+
2
@S @
!
b
b
b
@C
1 2 2 @2 C
1 2 2 @2 C
+
+
S
@t
2
2
@S 2
@ 4
dt
b
2 S @2 C
+
@S @ 2
+
1
2
b
b @C
@S
{z
Uncertainty
!
1
dS +
}

@C
@ 2
b
b @C
@ 2
{z
Uncertainty
(21)
!
2
d
2
}
A partial
review:
BlackScholes
and beyond
MN50429 Financial
Engineering
To elimiate dS terms, Delta hedge:
@C
@S
Precursor:
It¯o ’s lemma
Deriving
BlackScholes
Extension 1:
Stochastic
volatility
Extension 2:
Jumpdi¤usion
processes
To elimiate d
2
b
b @C
@S
=0
(22)
terms, Vega hedge:
@C
@ 2
b
b @C = 0
@ 2
(23)
First step towards a solution
A partial
review:
BlackScholes
and beyond
MN50429 Financial
Engineering
Precursor:
It¯o ’s lemma
Inserting the hedges and noting that dV = rVdt, plug
Eq.(19, 22, 23) in Eq.(21), we have:
@C
@t
+
=
b
@C
@t
Deriving
BlackScholes
Extension 1:
Stochastic
volatility
Extension 2:
Jumpdi¤usion
processes
1
2
+
2 S 2 @2 C
@S 2
1
2
+
b
2 S 2 @2 C
@S 2
+
2 S @ 2 C + 1 2 2 @ 2 C + rS @C
rC
2
@S
@S @ 2
@ 4
@C
@ 2
2b
b
b
2 S @2 C
+ 12 2 2 @@ C4 + rS @@SC
@S @ 2
b
@C
@ 2
(24)
To hold Eq.(24), both side have to be equal to certain
function. We deduce the following function (ignore the
reasoning here):
b
rC
Linear structure
A partial
review:
BlackScholes
and beyond
MN50429 Financial
Engineering
Precursor:
It¯o ’s lemma
Deriving
BlackScholes
Extension 1:
Stochastic
volatility
Extension 2:
Jumpdi¤usion
processes
f (S;
where
2
and
; t) =
(S;
2
; t) +
(S;
2
; t)
(25)
are de…nded in Eq.(13)
and will in general be nonlinear. Since they depend
on other variables so it’s not necessarily linear.
Combining the RHS and this function gives us a solution
to the di¤erential equation
Interpretation of
A partial
review:
BlackScholes
and beyond
Assume we have a delta hedged portfolio:
MN50429 Financial
Engineering
Precursor:
It¯o ’s lemma
Deriving
BlackScholes
Extension 1:
Stochastic
volatility
Extension 2:
Jumpdi¤usion
processes
V =C
@C
S
@S
(26)
and applying It¯o’s Lemma in Eq.(18), we have:
dV
@C
@t
=
+
+
@C
d
@ 2
1
2
2 S 2 @2 C +
@S 2
2
+ 21 2 2 @@ C4
2
This portfolio still has volatility risk!
2 S @2 C
@S @ 2
!
dt
(27)
A partial
review:
BlackScholes
and beyond
To elimiate the volatility risk, we plug Eq.(26) in the
following expression:
MN50429 Financial
Engineering
Precursor:
It¯o ’s lemma
dV
rVdt = dV
(28)
Plug Eqs.(13, 25, 27) in the above Eq.(28), we have
Deriving
BlackScholes
Extension 1:
Stochastic
volatility
@C
S)dt
@S
r (C
dV
rVdt = (
+
Extension 2:
Jumpdi¤usion
processes
=
+
)
@C
dt
@ 2
@C
( dt +
@ 2 0
@C @
(S;
@ 2
dz )
2
; t)dt +
dz
{z}
Uncertainty
1
A
(29)
Market price of volatility risk
A partial
review:
BlackScholes
and beyond
MN50429 Financial
Engineering
Precursor:
It¯o ’s lemma
This expression dV
risk free rate
Deriving
BlackScholes
Taking expectations, the last term dz vanishes
Extension 1:
Stochastic
volatility
The excess return (premium) increases in
Extension 2:
Jumpdi¤usion
processes
rVdt is the excess return over the
is the market price of volatility risk
at a rate of
A partial
review:
BlackScholes
and beyond
MN50429 Financial
Engineering
Precursor:
It¯o ’s lemma
1
Precursor: It¯o’s lemma
2
Deriving BlackScholes
3
Extension 1: Stochastic volatility
4
Extension 2: Jumpdi¤usion processes
Deriving
BlackScholes
Extension 1:
Stochastic
volatility
Extension 2:
Jumpdi¤usion
processes
Jumps in stock prices
A partial
review:
BlackScholes
and beyond
MN50429 Financial
Engineering
Precursor:
It¯o ’s lemma
Deriving
BlackScholes
Extension 1:
Stochastic
volatility
Extension 2:
Jumpdi¤usion
processes
Sometimes stock prices move by discrete amounts
Reasons can be crashes but also news arrivals, e.g
unexpected changes to earning, merger announcements,
etc.
Such discrete movements are called jumps
A partial
review:
BlackScholes
and beyond
MN50429 Financial
Engineering
Precursor:
It¯o ’s lemma
Deriving
BlackScholes
Extension 1:
Stochastic
volatility
Extension 2:
Jumpdi¤usion
processes
Representing jumps
A partial
review:
BlackScholes
and beyond
MN50429 Financial
Engineering
Precursor:
It¯o ’s lemma
Deriving
BlackScholes
Extension 1:
Stochastic
volatility
Extension 2:
Jumpdi¤usion
processes
dS
S
=(
m) dt + JdP( ) + dz
is the intensity, i.e. probability of observing a jump in
one time period
J is the size of the jump, which has some distribution with
m = E [J]
P( ) is a Poisson process
f (x) =
x
e
x!
;
> 0; E (x) = var (x) =
Other representations of jumpdi¤usion processes exist
Known jump size
A partial
review:
BlackScholes
and beyond
MN50429 Financial
Engineering
Precursor:
It¯o ’s lemma
Deriving
BlackScholes
Extension 1:
Stochastic
volatility
Extension 2:
Jumpdi¤usion
processes
If we know the jump size we have two sources of risk: dz
and dP( )
As with stochastic volatility we proceed using a hedge
portfolio with two assets
Di¤erent di¤erential equation as stochastic process is
di¤erent
Unknown jump size
A partial
review:
BlackScholes
and beyond
MN50429 Financial
Engineering
Precursor:
It¯o ’s lemma
Deriving
BlackScholes
Extension 1:
Stochastic
volatility
Extension 2:
Jumpdi¤usion
processes
This is not a third uncertainty
Uncertainty is about jump of size J1 , J2 , J3 ,....
For real valued jumps an in…nite number of uncertainties
exists
Market is then not complete
Derivative would then be a bene…t to the market, not a
mere replication of existing assets
It could not be perfectly priced, only price bounds be
determined
A partial
review:
BlackScholes
and beyond
MN50429 Financial
Engineering
Precursor:
It¯o ’s lemma
Deriving
BlackScholes
Extension 1:
Stochastic
volatility
Extension 2:
Jumpdi¤usion
processes
Summary: The general steps to price derivative C
Use It¯o’s Lemma to derive dC
Construct riskless hedge portfolio V
Derive dV in the terms of stochastic process
Derive PDE by elimiating the risk in dV expression
through hedging
Solve the PDE either analytically or numerically. If
solution is numerical, set the boundary conditions for the
PDE and then solve.
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