In this short notice, we present a critical arguments that Black Scholes (BS) option pricing model cannot cover the case of stochastic volatility.

1. 1
CRITICAL POINT ON STOCHASTIC VOLATILITY OPTION PRICING.
Ilya I. Gikhman
6077 Ivy Woods Court Mason,
OH 45040, USA
Ph. 513-573-9348
Email: ilya_gikhman@yahoo.com
JEL : G12, G13.
Keywords . Option pricing, stochastic volatility, square root diffusion.
Abstract. In this short notice, we present a critical arguments that Black Scholes (BS) option pricing model
cannot cover the case of stochastic volatility.
I. Let us recall the essence of the BS pricing. The BS option price is defined synthetically by constructing a
perfectly hedged portfolio. This replicating portfolio is constructed by a long call option and a portion of
shares of underlying asset in short position. Assume that underlying asset price S ( t ), t ≥ 0 follows a
Geometric Brownian Motion equation
dS ( t ) = µ ( t ) S ( t ) dt + σ ( t ) S ( t ) dw ( t ) (1)
where coefficients µ ( t ), σ ( t ) are known deterministic functions and S ( 0 ) > 0. Let 0 ≤ t < T < + ∞. For
any fixed moment t the value of the hedged portfolio Π ( u , t ) at a moment u ≥ t is defined by the formula
Π ( u , t ) = C ( u , S ( u ) ) – C
/
S ( t , S ( t ) ) S ( u ) (2)
It represents the value of the one long call option and the number of  ( t ) = C
/
S ( t , S ( t ) ) short shares
of the stock. The change in the value of the portfolio is risk free during the future infinitesimal period
[ t , t + dt ). Indeed, applying Ito formula one can see that
dΠ ( u , t ) |u = t = dC ( u , S ( u ) ) – C
/
S ( t , S ( t ) ) dS ( u ) |u = t =
= [ C /
t ( t , S ( t ) ) +
2
1
C
//
SS ( t , S ( t ) ) σ 2
( t ) S 2
( t ) ] dt

2. 2
As far as dΠ ( u , t ) |u = t does not hold risky term ‘dw’ the rate of return on Π ( u , t ) at t is the risk free rate r,
i.e.
d u Π ( u , t ) |u = t = r Π ( u , t ) du |u = t (3)
Therefore
C /
t ( t , S ( t ) ) +
2
1
C
//
SS ( t , S ( t ) ) σ 2
( t ) S 2
( t ) = r [ C ( t , S ( t ) ) – C
/
S ( t , S ( t ) ) S ( t ) ]
The latter equality is true if the option price is a solution of the Black Scholes equation
C /
t ( t , S ) + C
/
S ( t , S ) ) S +
2
1
C
//
SS ( t , S ) σ 2
( t ) S 2
– r [ C ( t , S ( t ) ) ] = 0 ( BSE)
Boundary condition comes from the definition of the call option payoff at maturity T
C ( T , S ) = max { S – K , 0 }
II. In [1] it was introduced the option pricing model for a stochastic volatility asset. It was assumed that
stock price follows
dS ( t ) = µ S ( t ) dt + v ( t ) S ( t ) dw 1 ( t )
d )t(v = –  )t(v dt +  dw 2 ( t ) (4)
E w 1 ( t ) w 2 ( t ) = 
The volatility equation in (4) can be rewritten in the form
dv ( t ) = [  2
– 2  v ( t ) ] dt + 2  )t(v dw 2 ( t ) (5)
which can be represented as a square root process
dv ( t ) = k [  – v ( t ) ] dt +  )t(v dw 2 ( t ) (5)
It was stated that “standard arbitrage arguments demonstrate that the value of any asset U ( S , v , t ) must
satisfy the partial differential equation (PDE)
2
1
v S 2
2
2
S
U


+   v S
vS
U2


+
2
1
v  2
2
2
v
U


+ r S
S
U


+ { k [  – v ( t ) ] –
(6)
– λ ( S , v , t ) }
v
U


– r U +
t
U


= 0
The unspecified term λ ( S , v , t ) represents the price of volatility risk, and must be independent of the
particular asset ”.

3. 3
We should pay attention to the formula (6). As we can see the derivation of the (BSE) is quite formal while
equation (6) is declared heuristically. Let us restore derivation of the equation (6) by following the standard
BS scheme represented above. Equation (6) implies the choice of the BS’s hedged portfolio Π U ( u , t ) in the
form
Π U ( u , t ) = U ( S ( u ) , v ( u ) , u ) – U
/
S ( S ( t ) , v ( t ) , t ) S ( u ) – U
/
v ( S ( t ) , v ( t ) , t ) v ( u )
Applying Ito formula we note that the change in the value of the hedged portfolio is risk free and therefore
dΠ U ( u , t ) |u = t =
t
U


+
2
1
v ( t ) S 2
( t ) 2
2
S
U


+   v ( t ) S
vS
U2


+
2
1
v ( t )  2
2
2
v
U


=
= r [ U – U
/
S S ( t ) – U
/
v v ( t ) ] = r Π U ( u , t ) |u = t dt
where U = U ( S ( t ) , v ( t ) , t ). This equality is true if U ( S , v , t ) is chosen as a solution of the problem
t
U


+ r S U
/
S + r v U
/
v +
2
1
v S 2
2
2
S
U


+   v ( t ) S
vS
U2


+
+
2
1
v  2
2
2
v
U


– r U = 0 (BSE U)
U ( S , v , T ) = max { S – K , 0 }
The critical point is that stochastic process v ( t ) does not represent a traded asset price. It does not traded in
the market along with asset S ( t ). Therefore it does not make sense to consider a portfolio with short ‘shares’
of v ( t ). In order to correct derivation of the equation (6) the term price of volatility was introduced. In such
case the BSE term ‘ r  v U
/
v ’ in (BSE U) is assumed to be equal to the corresponding term in the equation
(6), i.e. one should assume that
r v = k (  – v ) – λ ( S , v , t ) (7)
Equality (7) is a heuristic equality. There is no formal evidence was presented in order to justify equality (7).
In order to get a correct derivation of the equation (6) it is necessary to consider stochastic price of the
underlying asset in the form (4) where the volatility v ( t ) of S ( t ) is replaced by its pricing counterpart. On
the other hand even this reduction does not make the price of the risk to be a traded asset and therefore there
is no sense to talk about buy-sell or long-short of volatility price. This remark suggests that there is no sense
to consider BS option pricing for assets with stochastic volatility.
Remark. Other curious fact that looks missed in [1] relates to stochastic volatility represented by the
equation (5). The Feller’s square root diffusion is known for a long time. It is nonstandard stochastic
equation with a peculiarity of the diffusion coefficient. The diffusion coefficient does not satisfy Lipschitz
condition at the point S = 0. The Lipschitz condition is a sufficient condition for both existence and
uniqueness of the strong solution of the stochastic differential equations. In [2] the stochastic calculus proof
of existence and uniqueness of the solution of the square root stochastic differential equation was presented.

4. 4
The fundamental condition on coefficients of the Feller’s diffusion (5) that guarantees existence solution on
any finite time interval is
2 k  >  2
(8)
In Heston’s case we have k = 2 ,  =
β2
δ 2
,  = 2. Then the condition (8) does not true. Indeed,
2 k  = 2 × 2 ×
β2
δ 2
= ( 2 ) 2
=  2
Equality sign in square root condition is not admitted. Therefore we could not guarantee the existence of the
volatility of the asset on a finite interval [ 0 , T ].

5. 5
References.
1. Heston, S., Closed-Form Solution for Option with Stochastic Volatility with Applications to Bond
and Currency Options. The Review of Financial Studies, v. 6, issue 2 (1993), 327-343.
2. Gikhman, I., A short remark on Feller’s square root condition.