Critical point of view on Local Volatility concept

1. Remarks on Local Volatility.
Ilya Gikhman
6077 Ivy Woods Ct
Mason OH 45040
Ph. (513)-573-9348
Email: iljogik@yahoo.com
Abstract.
In this short notice some comments on local volatility are provided.
The Black–Scholes (BS) model of the options pricing has advised a ‘fair’ price
that interpreted as the PV of the ‘neutralized’ pay off value at maturity. In BS equation
(BSE) the real stock return µ is replaced by the risk-free rate of return r. These
parameters are assumed to be constants though the setting admits that µ, r, and volatility
σ could be known deterministic functions on time t.
If the market option prices are given one could calculate volatility of the
underlying. This calculation leads to a notion known as implied volatility. If the model
and calculation are perfect, this implied volatility value would be the same for the options
with different strike prices and maturities. In reality the calculations show that this is not
the case.
Implied Black–Scholes volatilities highlight a strong dependence on maturity T and the
strike price K. The dependence of the implied volatility on K and T is known as “smile”
effect. The way how to solve the smile problem was presented in [1-3]. We briefly recall
following [1] the main steps of this derivation.
The risk- neutral stock price in the BS model is assumed to be written in the form
d Sr ( t )
= r ( t ) dt + σ ( t ) dw ( t ) (*)
Sr ( t )
Here w ( t ) is Wiener process on original probability space { Ω , F , P }. Denote
C ( T , K ) the BS European call option price at the date t given
S r ( t ) = x. Parameters T and K are maturity date and strike price correspondingly. In
order to develop a solution of the smile problem it was assumed that
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2. d Sr ( t )
= r ( t ) dt + σ ( t , S r ( t )) dw ( t )
Sr ( t )
Let φ T ( x ) denote the probability density of the solution of the equation (1). We put for
simplicity r = 0. Then
d Sr ( t )
= σ ( t , S r ( t )) dw ( t ) (1)
Sr ( t )
Then
C ( T , K ) = ∫ ( x - K ) + φ T ( x ) dx (2)
Twice differentiation in (2) respect to K leads to the equality
∂ 2 C( T , K )
φT(K) = (3)
∂K 2
We pay attention only to martingale diffusions x ( t )
d x = b( x, t ) d W (4)
Here diffusion coefficient b ( t , x ) ≥ δ 2 > 0 is unknown function. The Fokker-Planck
( Kolmogorov’s first ) equation for the density f ( t , x ) is
∂2 ∂f
2
(b2 f ) = (5)
∂x ∂t
∂2C
As f = we obtain, after changing the order of derivatives:
∂x2
∂2 ∂ 2 ∂C
(b2 f ) =
∂ x2 ∂ x2 ∂ t
∂C
Assuming that lim = 0 and integrating twice in x leads to the equation
x→+∞ ∂t
1 2 ∂2C ∂C
b 2
= (6)
2 ∂x ∂t
Solving this equation for b we have
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3. ∂ C ( x,t )
2
∂t
b ( x,t ) = 2
(7)
∂ C ( x,t )
∂ x2
Compare (1) and (4) we see that
xσ(x,t) = b(x,t) (8)
Reminding that ( x , t ) is actually ( K , T ) we rewrite (5) as
1 2 ∂2C ∂C
b 2
= (9)
2 ∂K ∂T
This equation has the same ‘flavor’ as the classical BSE but distinct from it by
independent variables involved.
Comments.
I. First we remind that the underlying density is a function of the several
variables. That is φ T ( x ) = φ ( t , y ; T , x ) where the variables y, x ∈ ( 0 , + ∞ ) are
associated with the coordinate points for security price S (*) at dates t and T . The BS’s
call option price is introduced above and is also the multivariable function C ( T , K ) =
= C ( t , y ; T , K ). The equality (3) we present in the form
∂ 2 C ( t , y ;T , K )
φ(t,y;T,K) =
∂K 2
Here in the left hand side of the density φ its value is calculated at the point in which the
pricing variable x = K though variables x > 0 and K> 0 have different meaning.
The Fokker-Planck equation for the x-diffusion defined in (4) we rewrite in
following form
∂2 ∂ f ( t , y; T , x )
2
[ b 2 ( T , x ) f ( t , y ; T , x )] = (5′)
∂x ∂t
If we wish to apply in (5′) the substitution
∂ 2 C( t , y;T, K )
f(t,y;T,K) = (10)
∂K 2
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4. in which parameter K is considered as a parameter taking values from ( 0 , + ∞ ) we need
to exchange the independent variable x on K that is the strike price coordinate. This
transformation leads us to equality
∂2 ∂ 2 C( t , y; T , K ) ∂ ∂ 2 C( t , y; T , K )
[ b2(T , K ) ]= [ ] (5′′)
∂ x2 ∂K 2 ∂t ∂K 2
This equation shows that the substitution (10) does not replace x-diffusion by K-diffusion
having the same finite distributions as it was done above. From general point of view it is
difficult to expect that one could translate the diffusion (1) taking place in the ( x, t )-
coordinate space into ( K, T )-diffusion implied by (9) with the help of (2).
On the other hand if we define the ( K, T )-diffusion implied by (9) explicitly then there
is no path wise connection this diffusion with ( x, t )-diffusion.
∂2C
The substitution f = in (5) is correct only if f = φ T and the second derivatives in
∂x2
x here is actually the second derivatives in K.
II. The assumption (1) could not be applied to the stock market. Indeed, in this
case equation (1) is nonlinear and rate of return on 1, 2 or more stocks does not equal.
Indeed sum of two equations (1) does not lead to the same type of equation with doubled
initial condition. That means if an investor hold 2 the same stocks and one stock bring a
dollar profit over a day then 2 stocks do not bring 2 dollars. The linearity of the stock
prices is an axiom of the stock market and could not be violated at least in standard
situations.
Very Last remark. It also might be make sense to highlight BS pricing approach itself.
The BS pricing is a theoretical approach that using mathematics attempts to construct a
logical concept of the option price. If we logically assume that a stock has an expected
return µ and volatility σ then BS price for a particular initial stock value x for given K
and T will be the same regardless of µ. Thus for the same BS-premium one can get at
maturity the right to buy a junk or a good stocks. This interpretation of the notion ‘price’
applied for the option contradicts the common sense.
Note that another definition introduced in [4] does not present such confusion.
Here we add a new remark to Local Volatility Concept.
The solution C ( T , K ) = C ( t , x ; T , K ) , T > t of the problem (9) and satisfies
initial condition
C ( t , y ; t , K ) = max ( y – K , 0 ) (11)
Thus, C ( t , x ; T , K ) is a solution of the Cauchy problem (9), (11) and it has “ the same
‘flavor’ as the classical Black–Scholes Equation. One distinction of the ( T , K )-Cauchy
problem from its ( t , x ) counterpart is that the independent variables are different. We
can change the direction of time by introducing new time variable. Denote fixed initial
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5. moment as t = t 0 and let T ∈ [ t 0 , T* ]. Introduce new variable t = T* – T + t 0 .
∂ ∂
Bearing in mind that = − the equation (9) can be rewritten as
∂T ∂t
~ ~
∂C 1 2 ∂2C
+ b (t,K) = 0 (9′)
∂t 2 ∂K 2
~ ~
Here C ( t , K ) = C ( T* , y ; t , K ) = C ( t – T* + T , y ; T* – t + t 0 , K ). The
initial condition (11) given at t = t 0 corresponds to the terminal condition at t = T*
for the solution (9′)
~
C ( T* , y ; T* , K ) = max ( y – K , 0 ) (11′)
Cauchy problem (9′) , (11′) is similar to classical Black–Scholes Equation and the only
difference is the diffusion coefficient in (9′).
Recall that the risk neutral stock price in the BS model is assumed to be written in
the form
dS r ( t )
= r(t)dt + σ(t)dw(t) (*)
S r (t)
where w ( t ) is a Wiener process on original probability space { , F , P }. In smile
problem we arrive at the equation (1) and let us replace the variable K by S. Recall that
equations (*) and (1) are interpreted in the integral form and they have the same initial
value S ( t 0 ) = x. Assume that initial value of In formula (4) it was used a Wiener
process W ( t ) that can be differed from the Wiener process w ( t ) used in (*). Note that
as far as we deal with strong solutions, i.e. the coefficients of the SDEs are assumed to be
Lipchitz continuous functions the Wiener process in (1) , (4) could be chosen the same.
For technical simplicity we put r = 0.
Statement 1. Equalities (*) and (1) take place when and only when if
σ ( t ) = b ( t , S ( t )) and therefore σ ( t , S ( t )) actually does not a random function and
it is independ on S.
Indeed, subtraction (1) from (1) leads to the equality
t
∫
0
[ σ ( t ) – b ( t , S ( t )) ] S ( t ) d w ( t ) = 0
Raising into second power and taking expectation leads us to equality
t
∫
0
E [ σ ( t ) – b ( t , S ( t )) ] 2 S 2 ( t ) d t = 0
Bearing in mind that S 2 ( t ) > 0 it follows that b ( t , S ( t )) = σ ( t ), where σ ( t ) is
a nonrandom function . The case r ≠ 0 can be studied analogously.
Hence, the smile underlying which corresponds to the (9′), (11′) either equal to the
original security (*) or is a different unobservable diffusion process.
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6. References.
1. Dupire, B., “Pricing and Hedging with Smiles” , Research paper, 1993, p.9.
2. Dupire, B., “Pricing with a Smile”. Risk Magazine, 7, 1994, p.8-20.
3. Derman, E., and Kani,I., Riding on a smile, Risk, 7 (1994), p.32—33.
4. Gikhman, I., 2003, On Black-Scholes Equation ICFAI J.of Applied Finance, v.10,
No.4, April 2004, p.47-74 ( http://papers.ssrn.com/sol3/papers.cfm?abstract_id=500303).
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