. 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 an argument that Black Scholes (BS) option pricing model cannot
cover the case when volatility is a stochastic process.
Let us first recall the essence of the BS pricing. The BS option price is defined synthetically by constructing
perfectly hedged portfolio. This replicating portfolio is a long call option and a portion of short shares of
underlying asset. 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 < + ∞. 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 option and the  ( t ) = C
/
S ( t , S ( t ) ) short shares of the stocks.
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 with ‘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 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 definition of the call option payoff at maturity
C ( T , S ) = max { S – K , 0 }
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 ) = 
It was noted that volatility equation 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 more attention to the formula (6). As we can see the derivation of the (BSE) is quite formal
while equation (6) is written in a heuristic manner. Let us restore derivation of the equation (6) by following
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 }
Note that v ( t ) does not an asset. 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 ’ 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)
This is a heuristic equality. There is no formal evidence was presented in order to justify equality (7). If one
wishes 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 ) is replaced by its price counterpart. On the other
hand even this reduction does not make 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. The final conclusion is that there is no sense to consider
BS option pricing for assets with stochastic volatility.
Remark. Other curious fact is related to stochastic volatility the equation (5). The Feller’s square root
diffusion is known for a long time. It is nonstandard equation with peculiarity of the diffusion coefficient.
The diffusion coefficient does not satisfy Lipschitz condition at the point zero. The Lipschitz condition is a
sufficient condition for both existence and uniqueness strong solution of the stochastic differential equations.
In [2] the stochastic calculus proof of existence and uniqueness of the solution of the correspondent stochastic
differential equation was represented. There is a fundamental condition on coefficients of the Feller’s
diffusion (5) that guarantees existence solution on any finite time interval. This is inequality

4. 4
2 k  >  2
(8)
In our case we see that k = 2 ,  =
β2
δ 2
,  = 2. Thus latter condition does not true as the value
2 k  = 2 × 2 ×
β2
δ 2
= ( 2 ) 2
=  2
Equality sign in square root condition is not admitted and therefore we could not guarantee the existence of
the finite 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.