Option Pricing with Stochastic Volatility
Chu, Yung Chi
Department of Finance
National Taiwan University
Contents
1 Introduction 1.1 Introduction
1.2 Structures of the Thesis 2 Bivariate Binomial Model
2.1 Stochastic Volatility Models 2.2 Constructing the Lattice
2.3 Binomial Jumps and Probabilities 3 Numerical Results
3.1 Bivariate Binomial Option Pricing 3.2 Evaluating European Put Options 3.3 Evaluating American Put Options
4 Conclusions Appendix
Bibliography
Chapter 1 Introduction 1.1 Introduction
The volatility smile is frequently observed in options prices. But in the pure Black-Scholes world, there should not be any smile as the volatility should be constant across the strike price and time. Although the Black-Scholes formula has been successfully used in pricing stock option prices, it does have known biases. This is not surprising since the Black-Scholes model makes the strong assumption that stock returns are normally distributed with known variance, but the constant variance assumption is somewhat simplisitc.
Pricing models with stochastic volatility have been addressed in the literature by many authors (see Scott [1987,1991], Hull and White [1987], and Wiggins [1987));
they generalize the Black-Scholes model to allow stochastic volatility. However, these models all assume zero correlation between volatility and price. Heston [1993]
provides a closed-form solution for European options when the stochastic volatility is correlated with the spot asset.
The bivariate binomial framework presented by Hilliard and Schwartz [1996] not only allows non-zero correlation between the volatility and the underlying process but can also be used to value American options. It uses a simple recombining binomial tree with a set of four joint, but possibly non-independent, probabilities. The node probabilities,Pij, can be determined by simple calculations.
Although the underlying process has stochastic volatility, it is still driven by the usual Brownian motion. The stochastic volatility process is driven by Brownian motion except that the drift term can be quite general; for example, it can be mean-reverting.
Hilliard and Schwartz [1996] present an efficient method to calculate option prices on the bivariate binomial model. But, strangely, they do not use the tree method to derive the values in their paper; instead, they only use simulation to obtain the numbers. In other words, the tree does not seem to have been implemented. This thesis fills that gap by implementing the bivariate binomial tree method to price options. It then uses Monte-Carlo simulation to compare the accuracy of the tree method.
1.2 Structures of the Thesis
There are four chapters in this thesis. The first chapter introduces stochastic volatility model. The second chapter covers the bivariate binomial model which is developed by Hilliard and Schwartz [1996]. The third chapter presents the numerical results using the bivariate binomial model. The fourth chapter concludes this thesis.
Chapter 2
Mathematical Models
2.1 Stochastic-Volatility Model
We consider continues-time risk-neutral diffusion process of the form ( ) ( )
S
V V
dS m dt f S h V dZ
dV m dt bVdZ
= + S
= + (2.1)
where S is the spot asset, V is the stochastic volatility, and are Wiener processes with correlation
dZS dZV
Corr(dZS, dZV) =ρSV , and f S h V( ) ( ) is typically of the form S Vθ α.
2.2 Constructing the Lattice
We follow the tree method of Hilliard and Schwartz [1996]. Consider first the volatility transformation. The transformation is
ln( ) V Y = b ,
which yields a process with unit volatility (see Appendix A for the proof of Eq. (2.2)):
( )
2
V
V
y V
m b
dY dt dZ
bV m dt dZ
= − +
= +
(2.2)
where my is the drift term of Y.
Since the coefficient of is a constant, the lattice in Y recombines as required.
But the transformation of S to constant volatility is not straightforward because the volatility of S includes both random variables S and V.
dZV
We use a two-step transformation. First, we consider a transformation H of the form
( , ) 1( ) ( )
S dS H H S V h V
f S
= = −
∫
,and the diffusion process is
2 2
1[ 2
2
( ) ( )
S V SS SV VV
S S V V h
dH H dS H dV H dS H dSdV H dV
H f S h V dZ H bVdZ m dt
= + + + +
= + +
]
V
)
where mh is the drift term of H and depends on mS, m , and second-order partials.
The second transformation of H to Q is
( ) ln(1 SV h Q = αb − αbH − ρ + σ where
2 2 2
h 1 2 bH SV b H
σ = − α ρ +α , and the diffusion is of the form
q h
dQ = m dt + dZ . The diffusion for Q has unit volatility now, as required.
By Ito’s theorem, we can get the drift terms of H and Q as follows (see Appendix B and C for the proofs of Eqs. (2.3) and (2.4)):
1 1 2
(1 )
( ) 2 2
S V
h S
m m H
m f V H
f S V V
α
α α α α b αb SV
= − − + + − ρ (2.3)
and
( 2 2
1 2
h SV
q
h h
m b b
m α ρ α
σ σ
= + − H)
Y
(2.4) Since both Y and Q have unit volatility, the bivariate binomial grid can be easily constructed on the Y×Q space.
The values of V and S variables are given by the inverse transformation (see Appendix D for the proof of Eq. (2.5)):
exp( ) V = b
2 (1 2) exp( ) exp( )
2
SV SV
r r abQ abQ
H ab
− − × − +
= (2.5)
1
[ (1 ) ] 1 , 1
exp( ( ) ), 1
a q
V q H q
S
h V H q
⎧⎪ − − ≠
= ⎨⎪⎩ =
Under these transformations, the increments dZV and dZh have correlation
Corr( V, ) h ( SV
h
dZ dZ ρ αbH) σ
= −
and Corr(dY dQ, ) Corr(= dZV, dZh).
2.3 Binomial Jumps and Probabilities
As in the standard univariate model with unit volatility, the binomial jumps for the transformation process Y and Q are given by:
1 0
Y± = Y ± Δ t and
1 0
Q± = Q ± Δ t
where is the size of time step. The associated probabilities for upward jumps for Y and Q are, respectively:
Δt
0.5(1 y )
p = + m Δt
and
0.5(1 q )
q = + m Δt
Joint probabilities are defined by:
11 1 1
12 1 1
21 1 1
22 1 1
prob( , ) prob( , ) prob( , ) prob( , )
P Q
P Q
P Q
P Q
Y Y Y Y
+ −
+ +
− −
− +
=
=
=
=
When and are independent, joint probabilities are easily derived by multiplication. For example,
dZV dZh
11 (1 ), , 12 21 (1 )(1 ), 22 (1 ). P = q − p P = pq P = −q −p P = p −q When they are dependent, the joint probabilities are given by:
11 12 21 22
(1 ) Corr( , ) Corr( , )
(1 )(1 ) Corr( , ) (1 ) Corr( , )
P q p dY dQ
P pq dY dQ
P q p dY d
P p q dY dQ
Q κ κ
κ κ
= − −
= +
= − − +
= − −
where (1κ = p −p q) (1−q).
Chapter 3
Numerical Results
3.1 Bivariate Binomial Option Pricing
We consider a special diffusion processes examined by Hull and White [1987]:
S V
dS rSdt S V dZ dV bVdZ
= +
=
that is ms = Sr , f S h V( ) ( ) = S V , and mv = in Eq. (2.1). The resulting 0 transformations from Q and Y back to the original variables V and S are given by equations illustrated in section 2.2 with θ = and 1 α=0.5.
We first calculate the values of Q and Y at each node by a bivariate binomial tree.
We can also calculate the values of V and S at the expiration day. Then we can get the value of option price by backward induction.
We notice that the probabilities of upward jumps p and q are determined by my of Eq. (B) and of Eq. (D), respectively. These two values should be calculated by the initial values of each variable involved in the equations in section 2.2.
mq
3.2 Evaluating European Put Options
The following is an example of the effect of stochastic volatility on the prices of European put options when the volatility is uncorrelated with the underlying asset price.
Exhibit 1 shows the value of a European put with stochastic volatility parameter b
= 25% and zero correlation between volatility and price. The bivariate binomial values with 270 time steps are compared with the values generated by Hilliard and Schwartz [1996], the Hull-White stochastic volatility model [1987], and the standard Black-Sholes model [1973] which the volatility is fixed and equal to the initial
volatility.
The simulation values are from Hilliard and Schwartz [1996]. Observe that the simulation values are almost indistinguishable from the Black-Scholes and Hull-White values. Recall that Hilliard and Schwartz do not use the bivariate binomial tree method to obtain the values of European puts. This thesis uses the bivariate binomial tree method to get the values of European options, in contrast.
Observe that our values are close to the Black-Scholes, Hull-White, and Hilliard and Schwartz’s values. In fact, the difference of bivariate binomial tree method from the other three methods is less than 0.1. We also use Monte Carlo simulation to get the values of European puts with 100,000, 1,000,000, and 10,000,000 sample paths. The values by Monte Carlo simulation also match the above values. In fact, the difference of Monte Carlo simulation from Black-Scholes, Hull-White, and Hilliard and Schwartz’s simulation values is less than 0.01.
Exhibit 2 shows the convergence of put prices as the number of time steps varies.
The exercise prices are 80, 100, and 120. The parameters are identical to those in Exhibit 1 except the numbers of time steps. We first set the number of time steps to be 6 and calculate the put values. After this, we set the number of time steps to be 7 and recalculate again. And we keep increase the number of time steps and calculate the put values until the number of time steps reaches 358.
In Exhibit 2, we can see that the prices change a lot when the number of time steps is less than 25 and they converge to a level when the number of time steps exceeds 200.
Exhibit 1
Effect of Stochastic Volatility on European Put Prices S/X
X=100
Black- Scholes
Hull-White Simulation values (H&S)
Bivariate Binomial
Monte Carlo N=100,000
0.80 17.643 17.645 17.646 17.6781 17.6432
0.84 13.876 13.878 13.878 13.9317 13.8900
0.88 10.398 10.397 10.397 10.4597 10.3868
0.92 7.365 7.362 7.361 7.4094 7.3654
0.96 4.903 4.898 4.898 4.9042 4.8937
1.00 3.058 3.053 3.054 3.0155 3.0529
1.04 1.784 1.782 1.782 1.7408 1.7698
1.08 0.975 0.975 0.975 0.9501 0.9753
1.12 0.500 0.501 0.501 0.4772 0.5008
1.16 0.241 0.244 0.243 0.2291 0.2485
1.20 0.110 0.112 0.112 0.1091 0.1182
S/X X=100
Monte Carlo N=1,000,000
Monte Carlo N=10,000,000
0.80 17.6476 17.6479
0.84 13.8799 13.8786
0.88 10.4004 10.3993
0.92 7.3524 7.3570
0.96 4.8864 4.8898
1.00 3.0409 3.0401
1.04 1.7754 1.7746
1.08 0.9726 0.9743
1.12 0.5044 0.5038
1.16 0.2484 0.2488
1.20 0.1172 0.1168
Bivariate binomial is a stochastic volatility model with 270 time steps. The volatility parameter of the volatility diffusion (dV) is b = 25%. There is zero correlation between price and volatility. European puts are priced with the parameters: risk-free rate = 5%, time to maturity = 0.5 year, stock volatility = 15%, and exercise price =
$100. For stochastic volatility models, the initial volatility, V , is equal to 15%, the 0 stock volatility. There are no dividends. N is the number of sample paths.
Exhibit 2
The Convergence of Put Prices
S0=80 Price=17.676
16.500 17.000 17.500 18.000 18.500 19.000 19.500 20.000
6 30 54 78 102 126 150 174 198 222 246 270 294 318 342
time step
put value
S0=100 Price=3.0155
0.000 0.500 1.000 1.500 2.000 2.500 3.000 3.500
6 29 52 75 98 121 144 167 190 213 236 259 282 305 328 351 time step
put value
S0=120 Price=1091
0.000 0.250 0.500 0.750 1.000 1.250 1.500 1.750 2.000
6 29 52 75 98 121 144 167 190 213 236 259 282 305 328 351
time step
put value
Exhibit 3 is the distribution of stock at maturity and the initial stock price is 100 with all parameters equal to Exhibit 1 except time steps are 100. There would be 101*101 stock prices after we use bivariate binomial tree method. We sort these values from small to large and then add their probabilities every 100 stock prices, so we can get 102 stock prices with their probabilities. The horizontal axis is the logarithm stock price at maturity. The vertical axis is the probability of the stock price.
Some logarithm stock prices less than 1.7 and more than 2.3 with almost zero probability are cut.
In Exhibit 3, we can find that the distribution of the logarithm stock prices at maturity is like a normal distribution.
Exhibit 3
The Distribution of the Stock Price at Maturity S0=100
0.00 0.10 0.20 0.30 0.40
1.7 1.8 1.9 2 2.1 2.2 2.3
log ST
Probability
The volatility smile is a widely accepted phenomenon. The smile describes the convex shape of the implied volatility with respect to moneyness (S/X) computed via
the Black-Scholes formula. Hull and White [1998] use Monte Carlo simulation to evaluate the effects of stochastic volatility and correlations on option prices. Their results are consistent with the smile when implied volatilities are computed by matching Black-Scholes prices to theoretically correct values.
Similarly, the bivariate binomial model also reveals significant convexities. Exhibit 4 graphs the implied volatility when the initial stock volatility is 15%, the risk-free rate is 5%, the time to maturity is 0.5 year, and the volatility drift rate is zero. There is indeed a smile across the initial stock prices. All points below the implied volatility of 15% represent overpricing by Black-Scholes.
Exhibit 4
Black-Scholes Implied Volatility Smile S/X Implied Volatility
0.80 15.736432 0.84 15.563522 0.88 15.376780 0.92 15.192946 0.96 15.002478 1.00 14.842511 1.04 14.814724 1.08 14.863096 1.12 14.818848 1.16 14.845893 1.20 14.984344
Volatiliy Smile
14 15 16
0.80 0.84 0.88 0.92 0.96 1.00 1.04 1.08 1.12 1.16 1.20 S/X
Implied Sigma
European puts are priced by the bivariate binomial model with 270 time steps. The exercise price is $100, the risk-free rate is 5%, the time to maturity is 0.5 year, the volatility drift is zero, and the stock volatility is 15%. The volatility parameter of the diffusion process (dV) is b = 25%. The initial volatility, V , is equal to15%, the 0 stock volatility.
Exhibit 5 is another illustration of the effects of stochastic volatility on European put options with a long maturity. We let the volatility be correlated with the underlying asset price.
Exhibit 5 shows the value of a European put with stochastic volatility parameter b
= 1.00 and the correlation between volatility and price being −0.5, −0.25, 0, 0.25, and 0.5, respectively. The bivariate binomial model with stochastic volatility uses 1000 time steps.
Some values calculated by the tree method are close to the simulation values, but others are different from the simulation values. We can see the convergence of the put prices when the number of time steps varies in Exhibit 6. The convergence pattern is not clear even when the number of time steps exceeds 300. So the tree method for pricing long-maturity options is not without problems, and we should let the time steps be large to achieve convergence.
Compared with the values calculated by Black-Scholes which the volatility is constant and equal to the initial volatility, we find that the Black-Scholes model overprices in-the-money puts when there is negative correlation as well as out-of-money when there is positive correlation.
Exhibit 5
Effect of Stochastic Volatility on European Put Prices (Long Maturities)
ρSV = -0.50 ρSV = -0.25 S/X
X=100 Black-Scholes Simulation
(H&S) Bivariate Simulation
(H&S) Bivariate
0.80 15.716 14.267 15.7476 14.735 15.9531
0.85 12.848 11.465 12.5868 11.816 13.1262
0.90 10.393 9.217 9.9550 9.424 10.4335
0.95 8.325 7.451 7.9368 7.512 8.0610
1.00 6.611 6.073 6.3011 6.014 6.4325
1.05 5.208 4.999 4.9053 4.849 5.1444
1.10 4.074 4.159 3.8004 3.945 3.9978
1.15 3.167 3.495 2.9870 3.240 2.9660
1.20 2.449 2.965 2.3760 2.689 2.2606
ρSV = 0 ρSV = 0.25 ρSV = 0.50 S/X
X=100 Simulation
(H&S) Bivariate Simulation
(H&S) Bivariate Simulation
(H&S) Bivariate 0.80 15.145 15.8232 15.499 15.8975 15.796 15.7668 0.85 12.110 13.0572 12.351 12.7745 12.538 12.7522 0.90 9.581 10.6938 9.683 10.4086 9.721 10.0662
0.95 7.518 8.3190 7.464 8.2786 7.338 7.8827
1.00 5.890 6.0560 5.694 6.2737 5.409 6.1161
1.05 4.626 5.0036 4.319 4.6571 3.905 4.5511
1.10 3.655 4.0464 3.274 3.7122 2.775 3.2651
1.15 2.910 3.1141 2.492 2.9329 1.961 2.3614
1.20 2.340 2.1989 1.912 2.2015 1.385 1.8019
Bivariate binomial model with stochastic volatility uses 1000 time steps. The volatility parameter of the volatility diffusion (dV) is b = 1.00. European puts are priced with the parameters: risk-free rate = 5%, time to maturity = 2.0 years, stock volatility = 20%, and exercise price = $100. For stochastic volatility models, the initial volatility, V , is equal to 20%, the stock volatility. There are no dividends. 0
Exhibit 6
The Convergence of Put Prices
S0=80
16 16.5 17 17.5 18
1 28 55 82 109 136 163 190 217 244 271 298 325 352 time step
put value
S0=100
0 1 2 3 4 5
1 28 55 82 109 136 163 190 217 244 271 298 325 352 time step
put value
S0=120
0 0.5 1 1.5 2 2.5 3
1 28 55 82 109 136 163 190 217 244 271 298 325 352 time step
put value
3.3 Evaluating American Put Options
The bivariate binomial model is more general than the other methods because it can evaluate American options with stochastic volatility. The following is an example for pricing American put options when the volatility is correlated with the underlying asset price.
Because the Black-Scholes formula can not evaluate American options, and Monte Carlo simulation is not appropriate for the evaluation of the early exercise premium found in American options, we use the CRR binomial tree to serve as a benchmark for the bivariate binomial model. At each node, a comparison of the early exercise value and the continuation value of the option is made. The maximum of these two values is then retained.
Exhibit 7 shows the value of an American put with stochastic volatility parameter b
= 1 and correlation between volatility and price being -0.5, −0.25, 0, 0.25, and 0.5.
The bivariate binomial values are estimated with 500 time steps.
In Exhibit 7, we find that the values calculated by the bivaviate binomial tree method are close to the values given by Hilliard and Schwartz [1996]. Comparing with the values calculated by CRR binomial tree, we find that the CRR binomial tree (fixed volatility) overprices in-the-money puts when there is negative correlation as well as out-of-money when there is positive correlation.
Exhibit 7
Effect of Correlation and Moneyness on Univariate and Bivariate American Put Prices
ρSV = -0.50 ρSV = -0.25 S/X
X=100
CRR Binomial
Tree
Simulation
(H&S) Bivariate Simulation
(H&S) Bivariate
0.80 20.000 20.000 20.0000 20.000 20.0000
0.85 15.020 15.000 15.0000 15.000 15.0000
0.90 10.668 10.429 10.3857 10.550 10.4973
0.95 7.226 6.992 6.8253 7.081 6.8686
1.00 4.649 4.566 4.3683 4.570 4.2006
1.05 2.864 2.935 2.7325 2.865 2.6736
1.10 1.675 1.872 1.6576 1.763 1.6107
1.15 0.927 1.193 0.9961 1.076 0.8882
1.20 0.500 0.765 0.6010 0.656 0.5251
ρSV = 0 ρSV = 0.25 ρSV = 0.50 S/X
X=100 Simulation
(H&S) Bivariate Simulation
(H&S) Bivariate Simulation
(H&S) Bivariate 0.80 20.000 20.0000 20.000 20.0000 20.000 20.0000 0.85 15.037 15.0000 15.092 15.0000 15.158 15.0241 0.90 10.663 10.4441 10.767 10.4747 10.863 10.6131
0.95 7.160 7.1005 7.228 7.0672 7.285 7.0577
1.00 4.565 3.8554 4.549 4.1369 4.523 4.3217
1.05 2.785 2.6226 2.689 2.4743 2.581 2.4127
1.10 1.642 1.6898 1.504 1.5935 1.352 1.3547
1.15 0.946 0.7821 0.806 0.8500 0.651 0.7843
1.20 0.539 0.4703 0.419 0.4233 0.293 0.4088
American put prices computed by the univariate and bivariate binomial models. Both models use 500 time steps. The volatility parameter of the volatility diffusion (dV) is b = 1.00. European puts are priced with the parameters: risk-free rate = 5%, time to maturity = 0.5 year, stock volatility = 20%, and exercise price = $100. For stochastic volatility models, the initial volatility, V , is equal to 20%, the stock volatility. There 0 are no dividends.
Chapter 4 Conclusions
In the paper by Hilliard and Schwartz [1996], they develop a stochastic volatility model that is simple and accurate. The basis of the model is a lattice formed from a possibly correlated volatility process and an underlying price process. These processes are then transformed to form a recombining bivariate binomial tree with attractive convergence properties.
However, they do not seem to implement the method they develop. Instead, they only use simulation to get the prices. In this thesis, we use the model they develop and get the numbers by bivariate binomial tree method. The numbers show that the values given by the bivariate binomial tree method are almost identical to those in Hull and White [1987] and Hilliard and Schwartz [1996]. Unlike the Hull-White model, the methods in this thesis are also appropriate for non-zero correlations. In addition, they are effective for value American options.
Appendix A The Derivation of the Process of Y The transformation of Y is defined by:
ln( )V Y = b
1 2
V 2 VV
dY Y dV Y dV
∴ = +
From Eq. (2.3):
V V
dV =m dt+bVdZ , so we can get
2 2 2
dV =b V dt
2 2 2
1 1 1
( ) ( )(
2 ( )
2
V V
V
V
dY m dt bVdZ b V dt
bV bV
m b
dt dZ bV
∴ = + + − )
= − +
Appendix B The Derivation of mh
The transformation H =H S V( , ) is defined by:
( , ) 1( ) ( )
S dS H H S V h V
f S
= = −
∫
With ( )h V =Vα, the derivatives are given by:
2
2 2
1 , ( )
( ) ( ) (1 )
,
( )
S
S SS
V VV
SV
H H f
f S V f V
H V H
H V H
H H
V V
H f S V V
α α
α
α α α
V
α α α
α
= = −
− − − − +
= − = =
= −
2 2
2 2 2 2
1[ 2 ]
2
( ( ) ( ) ) ( )
1[ ( ) 2 ( ) ]
2 ( )
S V SS SV VV
S S S V V v
SS SV SV VV
S S V v h
dH H dS H dV H dS H dSdV H dV
H m dt f S h V dZ H m dt bVdZ
H f S V dt H bf S V V dt H b V dt H f S V dZ H bVdZ m dt
α α
α
ρ
∴ = + + + +
= + + +
+ + +
= + +
(B.1)
2 2 2 2
2 2 2
2 2
2
1 1
( ) ( )
2 2
( )
1 1
[ ( ) ]
( ) 2 ( ) 2
1 1
(1 )
( ) 2 2
h S S V V SS SV SV VV
S V S SV
S V
S SV
m H m H m H f S V H bf S V V H b V
m m H f bf S V V H(1 ) 2
f S V b V
f S V V f V f S V V V
m m H
f V H b b
f S V V
α α
α α
α α α
α α
ρ
α α ρ α α
α α α α ρ
∴ = + + + +
= − + − − + +
= − − + + −
Appendix C The Derivation of mq
The transformation Q=Q H( ) is defined by:
( ) ( ) ln(1 SV h) Q=Q H = αb − αbH − ρ + σ ,
where σh = 1 2− αbHρSV +α2 2b H2 . The derivatives are given by:
2 2 2 2
2 2 2
2 2
1 1
2 2
( ) 2 1 2
( ) 1
( ) ( )
1
( ) ( 1
SV SV
h h
SV h
SV SV
h h h h
h
SV h SV h SV h
h SV
h
SV h h
hh h h
b b H b b H
bH b H
b b H b
b b
Q b b
bH bH bH
bH
bH
Q Q
α ρ α α ρ α
σ α ρ α σ
α ρ α ρ α
α σ α σ σ
α α
H
α ρ σ α ρ σ α ρ σ
σ ρ α
σ
α ρ σ σ
σ
− −
− + − +
= =
− +
− + − +
+ +
= + = =
− + − + − +
− +
= =
− +
= =
2 2
2 2
2 2 3
) ( )
SV
h h h SV
h
h h h h
b b H
b b H
α ρ α
σ σ α ρ α
σ σ σ
− +
− − −
= = =
From Eq. (B.1):
dH =m dth +H f S V dZS ( ) α S +H bVdZV v.
2 2
σh
= So we can get
2 [ S ( ) S V v]2 [ S V]2 [1 2 SV 2 2 ]
dH = H f S V dZα +H bVdZ = dZ −αbHdZ = − αbHρ +α b H dt dt
2 2
2 2
3
2 2 2 3
2 2 2 3
1 1 1
( ( ) )
2 2
1 1 1
( ) [ ]
2
1 1
( ) 1 2
SV
h hh h S S V v h
h h
h SV
S S V v h
h h h h
S S V v q
h h
h SV h
q h
h h
b b H
dQ Q dH Q dH m dt H f S V dZ H bVdZ dt
m b b H
H f S V dZ H bVdZ dt
H f S V dZ H bVdZ m dt
m b b H m
m
α
α
α
α ρ α σ
σ σ
α ρ α σ
σ σ σ σ
σ σ
α ρ α σ
σ σ
∴ = + = + + + −
= + + + −
= + +
∴ = + − = 1 2 2
2
SV
h h
b b H
α ρ α
σ σ
+ −
Appendix D The Derivation of H from Q (Proof of Eq. (2.5)) The transformation of Q=Q H( ) is defined by:
1 1
( ) ln( SV h) ( ) ln( SV 1 2 SV
Q= αb − αbH−ρ +σ = αb − αbH−ρ + − αbHρ +α2 2b H2)
2 2 2 2 2 2 2 2 2
2 2 2 2
2 2
ln( 1 2 )
exp( ) 1 2
exp( ) 1 2
[exp( ) ] 1 2
[exp( ) ] 2[exp( ) ] 1 2
[exp( ) ]
SV SV
SV SV
SV SV
SV SV
SV SV SV
SV
bQ bH bH b H
bQ bH bH b H
bQ bH bH b H
bQ bH bH b H
bQ bQ bH bH
bQ
α α ρ α ρ α
α α ρ α ρ α
α ρ α α ρ α
α ρ α α ρ α
α ρ α ρ α α
α ρ
ρ
∴ = − + − +
∴ = − + − +
∴ + − = − +
∴ + − = − +
∴ + − + = −
∴ + −
2
1 2 exp( )
2 (1 ) exp( ) exp( )
2
SV
bH bQ
bQ bQ
H b
α α
ρ ρ α α
α
=
− − − +
∴ =
Bibliography
S. Heston, A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options, The Review of Financial Studies, Vol. 6, No. 2 (1993), pp. 327-343.
J. Hilliard and A. Schwartz, Binomial Option Pricing Under Stochastic Volatility and Correlated State Variables, Journal of Derivatives, Fall 1996, pp. 23-39.
J. Hull and A.White, The Pricing of Options on Assets with Stochastic Volatility, Journal of finance, Vol. 42, No. 2 (June 1987), pp. 281-300.
H. Johnson and D. Shanno, Option Pricing when the Variance is Changing, The Journal of Financial and Quantitative Analysis, Vol. 22, No. 2 (June 1987), pp.
143-151.
