1. Derivatives Pricing Basics and
Randomization
Ilya Gikhman
13th
International Research Conference on
Finance, Accounting, Risk and Treasury
Management, Cambridge UK
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2. Pricing
There are two primary asset classes in stochastic Finance; deterministic
bond B ( t , T ) and stochastic stock S ( t , ), t ≥ 0. A market scenario
is associated with a particular path, i.e. = S ( t , ). The difference
between the returns on stochastic stock and bond can be interpreted as a
market risk of the investment in stock. It is risk associated with the set of
scenarios
risk
= { : rate of return on stock < the rate of return on bond}
This definition of the market risk remains valid when bond price is a
stochastic process. We can develop statistical characteristics of
the market risk. If the bond price is stochastic then unified pricing model
of the bond and stock should take into account the market risk of the
stock and bond simultaneously. Rates of return as well as market risk are
parameters that bind spot prices of the stock and bond.
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3. Financial equality. Over an interval of time, two investments are equal
if they have equal instantaneous rates of return at any moment
during this interval. Applying this definition to European call option on
[ t , T ], we arrive at
S( T )
x
{S(T)>K} =
C ( T , S( T ) ; T , K )
C( t, x ; T, K )
(1)
The solution C = C ( t , x ; , ) of the equation (1) is a random function
that promises the same rate of return as its real underlying (1) for any
meaningful scenario . The price which depends on market scenario ,
we call the market price.
The spot price c ( t , x) is interpreted as the settlement price between
sellers and buyers of the option. The buyers - sellers market risk of the
call option is measured by the probabilities
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4. BR = P { c ( t , x ) > C ( t , x ) } , SR = P { c ( t , x ) < C ( t , x ) }
Consider calculations of the implied stock values for t = 0, T = 1.
Let S j , p j = P ( S (1) = S j ) , j = { S (1) = S j } , j = 1, ... , n.
For each scenario ω j we put
Sj
x
Sj
K
if S j > K
C (t, x ; ω j )
and
C ( t , x ; ω j ) = 0 , if S j ≤ K .
Market price can be approximated by expression
n
C(t,x;ωj ) =
j
1
x
(Sj - K)
Sj
( ω j)
(2)
Value of the stock at t = 1 is unknown. Buyer of the call will loose the
premium c ( t , x ) if S j ≤ K. Therefore, c ( t , x ) should have a higher
return than it is implied by the market price C ( t , x ; ω ).
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5. For example, the spot option price c can be assumed to equal
c = E
or
x
[S(T;t,x) - K] χ{S(T;t,x)>K}
S( T; t , x )
EC(t,x;ω)χ{C(t,x;ω)>c} =
EC(t,x;ω)
for
( 0 , 1 ).
Let c ( t ) = c ( t , S ( t ; 0 , x )) denote spot option price at t = 0.
One can assume that
c (t) =
y
= E
[S(T;t,y)-K]χ{S(T;t,y)>K}|
S( T; t , y )
y S( t ;0, x )
Dynamics of the c ( t ) can be studied analytically. Nevertheless, there
is no evidence how the option‘s spot price is formed.
Note. We have not made any assumption on distribution S ( t ).
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6. B&S pricing is defined by assumptions:
Let d S ( t ) = S d t + S d w (t).
*) Black and Scholes defined hedging portfolio using the formula
C ( t , S( t ) )
Π ( t , S ( t )) = − C ( t , S ( t )) + S ( t )
S
**) change in value of the portfolio Π is given by the formula
C ( t , S( t ) )
d Π ( t , S ( t )) = − d C ( t , S ( t )) +
dS(t)
S
Right hand side (4) does not contain the ‘white’ noise. Hence
d Π ( t , S ( t )) =
(3)
(4)
r Π ( t , S ( t )) d t
(5)
Eq. (5) leads to the Black Scholes equation with solution
C ( t , x ) = E exp – (T – t) max{ S r (T; t, x) – K , 0 }
(6)
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7. Final Remarks to B&S pricing:
One can easily verify that if Π is given by (3) then d Π can be
calculated and it does not satisfy (4), (5). If dΠ is given by (4) then for
arbitrary Π ( 0 ) value of Π ( t ) is uniquely defined and it does not
equal to (3).
1) B&S option price is based on perfect (dynamic) hedging concept.
This price represents spot price. The concept described is formally
incorrect.
2) There is no perfect hedging strategy in B&S option pricing theory.
3) BS price can be considered as a model for the spot price, however, it
is not a complete formal definition. Technical methods, parameters,
and developed price characteristics in B&S model can be used in the
expanded theory introduced in earlier slides.
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8. Calibration in BS pricing can be interpreted as an attempt to diminish
deviation between theoretical price and real world data.
In our expanded model, BS parameters are random variables for the
market pricing and are deterministic for the spot pricing.
Interest Rate Swap, IRS
Let L ( t , T ), 0 ≤ t ≤ T < + ∞ be a risk free floating rate;
t 1 < … < t n ; T 1 < … < T m , max { t n , T m } = T , denote fixed
and floating dates of payments correspondingly ; t = 0 denote
initiation date of the IRS. Let t j - t j – 1 = Δ t and T k - T k – 1 = Δ T
do not depend on j , k. Cash flow from fixed rate payer
m
n
CF = N [
k
1
L( T k–1 , T k )ΔT χ ( t = T k ) –
j
1
c Δ t χ ( t = t j )]
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9. IRS
Spread value c is a solution of the equation
PV ( 0 ) CF = Σ L ( T k – 1 , T k ) L ( 0 , T k ) – Σ c L ( 0 , t j ) = 0
N = 1. Rates of the type L ( 0 , ) are known while the real future
rates L ( T k – 1 , T k ) , k > 1 are unknown at t = 0.
Denote l ( T k – 1 , T k ; 0 ) date-0 forward rate over [ T k – 1 , T k ].
Then it is easy to see that
Σ L( 0 , T k ) l ( T k–1 , T k ; 0 )ΔT = 1 – L( 0 , T m )
It follows that date-0 implied value of the spread is equal to
1
c =
n
L ( 0, T m )
L ( 0, t j ) Δ t
j 1
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10. Suppose in theory that distributions L ( T k – 1 , T k , ω ) are
known. Then
m
L(T k
,T k , ω) L ( 0,T k )
1
k 1
C(ω) =
n
L ( 0, t j )
j 1
c =
1
n
L ( 0,T m )
L ( 0, t j ) Δ t
j 1
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11. δ C ( ω ) = C ( ω ) – c defines market risk of the swap. The set
Ω A = { ω : δ C ( ω ) > 0 } represents counterparty A profit. A paid
less than it is implied by a scenario. For any scenario
Ω B = { ω : δ C ( ω ) < 0 } counterparty A pays more than it is
implied by the scenario ω. The market risk of the swap payer A is
P ( Ω B ). Numeric characteristics of the market risk for party A at
t = 0 are defined by the random variables
[C(ω)–c] L(0,tj)
These variables represent PV of the deviations C ( ω ) and the
market estimate c. One can define primary statistical
characteristics of the Interest Rate Swap market risk by making
reasonable assumptions regarding distributions of the rates
L( T k–1 , T k , ω ) .
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12. Local Volatility (mathematical aspect).
BS theoretical value of an option is a function of volatility and assume
that there exists inverse function, i.e. C = f ( * , ), = f – 1 ( * , C ).
The value of volatility im implied by the option prices is called
Implied volatility. It is noted that volatility estimates based on historical
Data show dependence im = im ( * , K ).
In BS model parameters T , K are fixed and variable t changes from 0
to T. For simplicity, lets say r = 0. For fixed ( t , S ( t ) = x ) consider
function
C ( T , K ) = E max{ S 0 ( T; t, x ) - K , 0 }
where
dS 0 ( t ) =
S 0 ( t ) dw ( t )
It was shown that in domain ( T , K )
[t,+
)
(0,+
)
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13. C
T
1 2 2C
b
2
K2
(7)
2
b(T,K)
C(T, K )
T
2
C(T, K )
K2
Function C = C ( t , x ; T , K ) of the variables T > t , K ≥ 0
C ( t , x ; t , K ) = max { x - K , 0 }
(7 )
The solution to the problem (7,7 ) admits probabilistic representation. Let
T
b ( k ( s ; T, K ) , s ) d w ( s )
k ( t ; T, K ) = K +
t
be a backward Ito eq. in which variable t changes from T to 0.
The primary result of local volatility concept is the representation
C ( t , x ; T , K ) = E max { x – k ( t ; T , K ) , 0 }
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14. Conclusions.
1) Coefficient b ( K , T ) is not an adjustment for .
2) BSE presents ( t , x )-dynamics of the C when ( T , K ) are fixed.
3) LV presents ( T , K )-dynamics of the C when ( t , x ) are fixed.
Function C = C ( t , x ; T , K ):
For a fixed ( T , K ) function, C is a solution of BSE and estimations
of the diffusion coefficient based on C-data would lead us to the
estimate.
For a fixed ( t , x ), function C ( t , x ; T , K ) is a solution eq. (7) and
estimations of the diffusion coefficient leads us to the b ( K , T )
estimate.
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15. Remark. There are a few questions can be raised.
*) Assume that estimates of unknown drawn from stock prices and
option prices are different. LV theory cannot be applied as far as it
deals with option’s implied volatility and does not deal with the
stock volatility.
**) Assume that for some 2 strikes K j , j = 1,2 and fixed T the
estimates of the implied volatility lead to j , j = 1,2. In the BS
framework it is difficult to find an answer. LV concept can be used
when we observe a number of option prices with different K and T
for the fixed ( t , x ).
One explanation is that the date-t pointwise volatility estimate of the
price for both stock and options do not match to time series technique
of the estimate volatility.
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