Derivative pricing in discrete time
Definitions & notations
-
Derivative: financial product defined from another underlying asset
· S
: price of underlying stock
·
: price of derivative (call)
C
te [0 T] : time ,
payoff: b(Si) b(S 1c[o ])
-
or e+ =
b =
+
, ,+
We will look at 3 main approaches to determine the price Ct
Replication
I
Risk-neutral valuation
2
3
Deflator valuation
But we first exploit some useful theory
The binary one-period model
Br S+ (u)
erT P
Bank Bo
Stock So
this are the underlying assets
ert 1-
p
Br S + (d)
example Bo =
1
,
e =
1 .
1
,
So =
100 ,
U =
1 .
25
,
d =
0 .
&
,
p
= 0 .
8
1 .
1 125 =
Sou
X 1.1
XU
Bank ↑
Stock 100
xd
X 1 . 1
1 . 80 = God
For these underlying assets we can work backwards when we know the value at t = 1, using discount factor erT
e E (S ] ( p))
*T *
So = e (Sou +p +
=
+ Sod + -
N
How can we now price the derivative using these underlying assets, Co ?
f(5 )
We can first calculate the payoff of the derivative using the underlying asset C+ = +
C+ (u) = &(Sou) =
e
lo
e+ (d) =
b(Sod) =
red
example max[Sou-k o],
=
max
[125 -
100
,
03 =
25
Payoff European call option & (S ) +
= max[S +
-
k ,
03 &
take strike price K = 100 max [Sod-k 03
,
=
max 200 -
100
,
03 =
0
, As said before there are three methods to derive this , let's look at the first one Co
I
Replication: find a portfolio strategy investing in the stock and a risk-free asset that matches the derivative
price at each point
notation
3
Portfolio:
E 0 = (4 0 , .
- 1)
-1 derivative (sell one unit of derivative)
7
&
Price Po 8 =
4 Bo + $50 -
Co
M
shares in stocks
>
R
N
invested in bonds P (w) (w)
>
Payoffs +
.
0 =
4B + + 03 + -
er
Price vector Pa ( (a)
~
=
,
St ,
An arbitrage is a portfolio with either
i) (w)
A negative price and a non-negative payoff in both states : 0 .
Po o
,
0 .
P+ Lo
ii) (c) 20 PLA (n) o]
A non-positive price and a non-negative payoff, positive in at least one state : 0 .
PoEo ,
0 .
P
+
,
.
P
+
< > o
N
We rule out arbitrage opportunities and impose law of one price: a portfolio with payoff zero has price zero:
S
& S (u)
S (d)
yB
+B
&
+
+
e (u)
(d)
hence we can find 0 4
+ + =
and use these to solve
+
+
=
+
C+ So ↑Bo hence find
.
&o = + Co
note the risky position St hedges the payoff, so that Ve-0SA BE is risk-free again
- =
4
28at
=at
i
-
N
note u =
d =
=
e
We can rewrite this to explicit solutions:
E
en-ed
*
↓ Son 4 Bo
↑ Boe +
O =
(= hedge ratio
en
:
Son-Sod
edu-end
A
) Co =
050
en-ed
+
edu-end
"T rT
& Sod ↓Boe"
-
↓ Bo
+
+ =
ed =
e u -
d u -
d
(
u - eT
ed u - d
, ,
g1-q
This q is the risk-neutral probability,
this brings us to the second method
&
2
Risk-neutral valuation: construct a risk neutral probability measure Q under which the derivative price
equals the Q-expected discounted payoffs
e T(eu (1 g)) e z(e ]
Hence we find g
2 = + + ed + -
=
+
note we do not use the p probabilities as this is irrelevant for Co
We could also exploit this idea to a market with N assets and n states, the risk-neutral measure can be
uniquely determined if N = n
Complete market: any derivative with payoff depending on underlying assets can be replicated
>
Incomplete markets: no-arbitrage still may provide bounds on derivate prices, which price is realized
depends on market risk preferences
7
this happens when , hence more states then underlying assetsn2 N
, Binomial tree
The binary model is not rich enough in practice, we need more states and time periods, we introduce the
binomial tree: series of binary trees
T
example N = 2 T note stock prices are recombinant, Sc(nd) Sc(du), derivative price tree might be not
at = =
So un en (nu)
W W
Son & (u)
U U
d d
Stocks So So du (nd) (du)
U
Soud :
Call Co
d
U
en :
en
d
2
God C . (d)
At
d d
Sodd en(dd)
3
Using these binomial tree, we can calculate Co using backward pricing
step 1: calculate the payoffs at time N f (Sn- u) or &(Swd) :
,
step 2: using these payoffs and q, calculate en-1 e E [en /Sn ] et[ein /Si]
*
=
-1
In summary, li =
ere[en- /Sn-2]
r(N i) at
Ea[enISi]
-
step 3: repeat
-
&w - z =
or Ci = e
step 4: work backwards until Co
When we know all the derivative values, we also know all the hedging values
Miti (u) -
Citi (d) u(i +,
(d) -
dCi +
(u)
Di + 1
=
Sin-Sid Nit Bi ,
=
grat u -
d
This sequence (i + 1
,
Pi +
1) is a dynamic portfolio strategy with:
#
I
P
intermezzo: discrete-time martingales
definitions
·
probability space (r .
F ,
P)
>
probability measure I : gives probability to events in F ex. P(A) =
cp(i p)
-
collection of events A ex. F contains A End Y and A Sun da]
-field F : -l :
,
du =
,
sample space : set of all possible outcomes 7
ex. Enu dd] R wel 2 :
,
ud ,
du ,
random variable
X , assigns real numbers to outcomes
R : 1 >
R , random variable with extra dimension
stochastic variable : 2xT
~ >
example the variable Xt , takes 3 different values at t =
0 .
6
each corresponds to one sample path/trajectory W
Xe(w) is a collection of random variables, defined in one common probability space
, Y
the --field lists all events that might happen to X
F
7
we can define smaller O-fields Fr , collecting events that might have happened before n
>
filtration · [0 23 .
:
Fo F E .
. . .
[Fr ( : 5)
example 1 : Sunu ,
nud ,
udu ,
udd ,
dun ,
dud ,
du ,
Add 3
A: Eunu ,
und ,
uda ,
add 3 cr
F. : [0 ,
r ,
A , A ,
3]
In X" ((x3) [w X(w) Ye Fr( (B) (w X(w) BYE
·
measurability: = = = x
for continuous X = : = Fr
when this holds for all n, then Xr is adapted to the filtration Fr
&
I
ex. -fields Fr is the information set then the conditions 'X is G-measurable says 'the information set G I
S
contains X , and HEG is interpreted as 'all information in I is contained in G
important properties expected values
remember E(X) x(wi)p(wi) E(X1B) = X(wi) P(wilB) :
·
for finite field GEF E(XIG) (w) E(X/aw) An MStieg
· -
: = =
: we Ail
·
E(X1g) : X
if X is G-measurable
· E(X 150) : E(X)
·
E(XY(g) =
XE(y(g) if X
is G-measurable
·
EZE(Xig)] :
E(X)
5
E[E(X(g)(2] :
E(X (2) 288 = 5
(tower property)
>
ex. ELE(XIFn)IFn] E(XIFn) En E(X(fn) E(XIXo Xn)
if is generated by X, then
:
,
=
,
X , .
. .
.,
martingales
Xn
is a martingale with respect to In and I if: A stochastic process is said to have the
]
Xr
is adapted to In each Xn is measurable with respect to ,
= martingale property if, at any given time, the
expected value of the future values of the
-EP((Xn)) < process, conditional on the information
available up to the present time, is equal to
3E(Xn + 1 15n) : Xn
the current value.
example E(Xn + 1
15n) =
ELE(X1fn 1) (5n] +
:
E[X1Fn] :
Xn
martingale transform
when Xn is a (P Fn) -martingale, and .
on
is previsable ( On is Fn-measurable) . then In = 20 + "Pin (Xin -Xi) is also a
(P Fr) martingale
.
Definitions & notations
-
Derivative: financial product defined from another underlying asset
· S
: price of underlying stock
·
: price of derivative (call)
C
te [0 T] : time ,
payoff: b(Si) b(S 1c[o ])
-
or e+ =
b =
+
, ,+
We will look at 3 main approaches to determine the price Ct
Replication
I
Risk-neutral valuation
2
3
Deflator valuation
But we first exploit some useful theory
The binary one-period model
Br S+ (u)
erT P
Bank Bo
Stock So
this are the underlying assets
ert 1-
p
Br S + (d)
example Bo =
1
,
e =
1 .
1
,
So =
100 ,
U =
1 .
25
,
d =
0 .
&
,
p
= 0 .
8
1 .
1 125 =
Sou
X 1.1
XU
Bank ↑
Stock 100
xd
X 1 . 1
1 . 80 = God
For these underlying assets we can work backwards when we know the value at t = 1, using discount factor erT
e E (S ] ( p))
*T *
So = e (Sou +p +
=
+ Sod + -
N
How can we now price the derivative using these underlying assets, Co ?
f(5 )
We can first calculate the payoff of the derivative using the underlying asset C+ = +
C+ (u) = &(Sou) =
e
lo
e+ (d) =
b(Sod) =
red
example max[Sou-k o],
=
max
[125 -
100
,
03 =
25
Payoff European call option & (S ) +
= max[S +
-
k ,
03 &
take strike price K = 100 max [Sod-k 03
,
=
max 200 -
100
,
03 =
0
, As said before there are three methods to derive this , let's look at the first one Co
I
Replication: find a portfolio strategy investing in the stock and a risk-free asset that matches the derivative
price at each point
notation
3
Portfolio:
E 0 = (4 0 , .
- 1)
-1 derivative (sell one unit of derivative)
7
&
Price Po 8 =
4 Bo + $50 -
Co
M
shares in stocks
>
R
N
invested in bonds P (w) (w)
>
Payoffs +
.
0 =
4B + + 03 + -
er
Price vector Pa ( (a)
~
=
,
St ,
An arbitrage is a portfolio with either
i) (w)
A negative price and a non-negative payoff in both states : 0 .
Po o
,
0 .
P+ Lo
ii) (c) 20 PLA (n) o]
A non-positive price and a non-negative payoff, positive in at least one state : 0 .
PoEo ,
0 .
P
+
,
.
P
+
< > o
N
We rule out arbitrage opportunities and impose law of one price: a portfolio with payoff zero has price zero:
S
& S (u)
S (d)
yB
+B
&
+
+
e (u)
(d)
hence we can find 0 4
+ + =
and use these to solve
+
+
=
+
C+ So ↑Bo hence find
.
&o = + Co
note the risky position St hedges the payoff, so that Ve-0SA BE is risk-free again
- =
4
28at
=at
i
-
N
note u =
d =
=
e
We can rewrite this to explicit solutions:
E
en-ed
*
↓ Son 4 Bo
↑ Boe +
O =
(= hedge ratio
en
:
Son-Sod
edu-end
A
) Co =
050
en-ed
+
edu-end
"T rT
& Sod ↓Boe"
-
↓ Bo
+
+ =
ed =
e u -
d u -
d
(
u - eT
ed u - d
, ,
g1-q
This q is the risk-neutral probability,
this brings us to the second method
&
2
Risk-neutral valuation: construct a risk neutral probability measure Q under which the derivative price
equals the Q-expected discounted payoffs
e T(eu (1 g)) e z(e ]
Hence we find g
2 = + + ed + -
=
+
note we do not use the p probabilities as this is irrelevant for Co
We could also exploit this idea to a market with N assets and n states, the risk-neutral measure can be
uniquely determined if N = n
Complete market: any derivative with payoff depending on underlying assets can be replicated
>
Incomplete markets: no-arbitrage still may provide bounds on derivate prices, which price is realized
depends on market risk preferences
7
this happens when , hence more states then underlying assetsn2 N
, Binomial tree
The binary model is not rich enough in practice, we need more states and time periods, we introduce the
binomial tree: series of binary trees
T
example N = 2 T note stock prices are recombinant, Sc(nd) Sc(du), derivative price tree might be not
at = =
So un en (nu)
W W
Son & (u)
U U
d d
Stocks So So du (nd) (du)
U
Soud :
Call Co
d
U
en :
en
d
2
God C . (d)
At
d d
Sodd en(dd)
3
Using these binomial tree, we can calculate Co using backward pricing
step 1: calculate the payoffs at time N f (Sn- u) or &(Swd) :
,
step 2: using these payoffs and q, calculate en-1 e E [en /Sn ] et[ein /Si]
*
=
-1
In summary, li =
ere[en- /Sn-2]
r(N i) at
Ea[enISi]
-
step 3: repeat
-
&w - z =
or Ci = e
step 4: work backwards until Co
When we know all the derivative values, we also know all the hedging values
Miti (u) -
Citi (d) u(i +,
(d) -
dCi +
(u)
Di + 1
=
Sin-Sid Nit Bi ,
=
grat u -
d
This sequence (i + 1
,
Pi +
1) is a dynamic portfolio strategy with:
#
I
P
intermezzo: discrete-time martingales
definitions
·
probability space (r .
F ,
P)
>
probability measure I : gives probability to events in F ex. P(A) =
cp(i p)
-
collection of events A ex. F contains A End Y and A Sun da]
-field F : -l :
,
du =
,
sample space : set of all possible outcomes 7
ex. Enu dd] R wel 2 :
,
ud ,
du ,
random variable
X , assigns real numbers to outcomes
R : 1 >
R , random variable with extra dimension
stochastic variable : 2xT
~ >
example the variable Xt , takes 3 different values at t =
0 .
6
each corresponds to one sample path/trajectory W
Xe(w) is a collection of random variables, defined in one common probability space
, Y
the --field lists all events that might happen to X
F
7
we can define smaller O-fields Fr , collecting events that might have happened before n
>
filtration · [0 23 .
:
Fo F E .
. . .
[Fr ( : 5)
example 1 : Sunu ,
nud ,
udu ,
udd ,
dun ,
dud ,
du ,
Add 3
A: Eunu ,
und ,
uda ,
add 3 cr
F. : [0 ,
r ,
A , A ,
3]
In X" ((x3) [w X(w) Ye Fr( (B) (w X(w) BYE
·
measurability: = = = x
for continuous X = : = Fr
when this holds for all n, then Xr is adapted to the filtration Fr
&
I
ex. -fields Fr is the information set then the conditions 'X is G-measurable says 'the information set G I
S
contains X , and HEG is interpreted as 'all information in I is contained in G
important properties expected values
remember E(X) x(wi)p(wi) E(X1B) = X(wi) P(wilB) :
·
for finite field GEF E(XIG) (w) E(X/aw) An MStieg
· -
: = =
: we Ail
·
E(X1g) : X
if X is G-measurable
· E(X 150) : E(X)
·
E(XY(g) =
XE(y(g) if X
is G-measurable
·
EZE(Xig)] :
E(X)
5
E[E(X(g)(2] :
E(X (2) 288 = 5
(tower property)
>
ex. ELE(XIFn)IFn] E(XIFn) En E(X(fn) E(XIXo Xn)
if is generated by X, then
:
,
=
,
X , .
. .
.,
martingales
Xn
is a martingale with respect to In and I if: A stochastic process is said to have the
]
Xr
is adapted to In each Xn is measurable with respect to ,
= martingale property if, at any given time, the
expected value of the future values of the
-EP((Xn)) < process, conditional on the information
available up to the present time, is equal to
3E(Xn + 1 15n) : Xn
the current value.
example E(Xn + 1
15n) =
ELE(X1fn 1) (5n] +
:
E[X1Fn] :
Xn
martingale transform
when Xn is a (P Fn) -martingale, and .
on
is previsable ( On is Fn-measurable) . then In = 20 + "Pin (Xin -Xi) is also a
(P Fr) martingale
.
