Convexity of option prices in the Heston model

Convexity of option prices in the Heston model

Jian Wang

U.U.D.M. Project Report 2007:3

Examensarbete i matematik, 20 poäng Handledare och examinator: Johan Tysk

Januari 2007

Abstract

Acknowledgment

· My supervisor: Professor Johan Tysk

· Every teacher who ever taught me at Uppsala University

Contents

1 Introduction………....1

2 Volatility model ……….…….…3

2.1 Implied volatility and local volatility……….…….…….3

2.2 Stochastic volatility……….……….4

2.3 The Heston Model……….…………..5

2.3.1 Motivation……….………….5

2.3.2 Model and parameters………6

3 Pricing method……….7

3.1 The Heston PDE………...…7

3.2 Closed form solution……….……….12

4 Convexity………..………14

4.1 Introduction of convexity……….……….14

4.2 Theorem for convexity...………16

5 Numerical method………..……21

5.1 Introduction……….………..…….21

5.1.1 Main idea………..21

5.1.2 Detail……….………...22

5.2 Implementation……….………..……24

5.2.1 Closed form solution………....24

5.2.2 Finite difference method………..24

5.3 Results ……….…….…….29

5.3.1 Results from the closed form solution………...……..29

5.3.2 Results from the finite difference method………32

5.3.3 Comparison between the results of the previous sections……..35

6 Conclusion……….38

Chapter 1 Introduction

In 1973, there was a landmark paper about option pricing published by F. Black and M. Scholes [1]. The option market grew significantly after this, which is expressed not only by the amount of deals rising, but also by the kinds of option increasing due to the requirement from investors and risk hedging.

However there is a major problem for the original Black-Scholes namely the choice of volatility. There is a general phenomenon of volatility varying by strike which is referred as volatility skew or volatility smile. After this, the term local volatility became another choice of volatility. In 1994, Dupire [2] proposed a local volatility function which can be calculated with different strike price and maturity time. Because of the statistical difficulties for finding local volatility function, people needed more appropriate models.

In this paper, I decide to use the Heston model [5], which is one of the most widely used stochastic volatility models. First, the convexity for European call option in the Heston model will be shown. Then, the convexity for the general case will be discussed using an approximation argument.

Chapter 2 Volatility model

2.1 Implied volatility and local volatility

In [1], a stochastic differential equation is given to present the behavior of an underlying asset. It is given as follow:

dS =rSdt+σ ωSd (1) This σ represents the volatility. In [1], the volatility σ is assumed to be constant. However, most derivative markets indicate that the volatility varies by the strike price. One could easily get the implied volatility by using the Black-Scholes pricing formula to calculate backward if the strike price and the corresponding option price are given. As described in [6], “Implied volatility is the wrong number to put into wrong formula

to obtain the correct price”. The implied volatility is always calculable. The

phenomenon which is referred as volatility skew or volatility smile is illustrated in Figure 1.

Even though the volatility smile or the volatility skew exists, the Black-Scholes pricing formula is still used widely in practice.

A formula for calculating local volatility was proposed by Dupire [2] in 1994. Dupire assumed that asset price acts differently comparing with Black-Scholes model, the difference is that one could use a volatility function instead of constant volatility. Such models have the following form:

( , )

dS=rSdt+σ t S dω. (2)

The volatility function can for instance have following form:

( , )t S S α

σ ₌ −

(3) where α is a real number.

The main idea of his formula is that one can obtain the local volatility if the option price for all strike price and maturity time is given. Theoretically the investor can obtain the local volatility using Dupire’s formula. However the choices of volatility function are extensive.

2.2 Stochastic volatility

Hull-White model s s v s dS Sdt vSdW dv vdt vdW dW dW dt σ σ μ μ ξ ρ = + = + = i Scott model ( ) y s y s y dS Sdt e SdW dy y dt dW dW dW dt μ κ θ α ρ = + = − + = i Heston model ( ) s v s v dS Sdt vSdW dv v dt vdW dW dW dt μ κ θ α ρ = + = − + = i Table 1 stochastic volatility

In those models we note that the Brownian motions are correlated.

2.3 The Heston Model

2.3.1 Motivation

2.3.2 Model and Parameters

Standard stochastic differential equations (SDE) for the Heston model are given as follows: 1 2 1 2 ( ) t t t t t t t dS S dt v S dW dv v dt v dW dW dW dt μ κ θ σ ρ = + = − + = i . (4)

where and represent the price and the volatility of underlying asset

respectively, and are two Brownian motions with correlation

t

S v_t

1

W W2 ρ . In the

process of volatility, a mean reversion process is chosen. κ is the rate of reversion,

θ is the long-term mean, and σ is the volatility of volatility.

The correlation also represents the relation between the volatility and the underlying asset. The process of is called mean reversion process which is proposed by Cox, Ingersoll and Ross in [7]. If we set

t

v

κ and θ to positive, the drift of volatility will decrease as the volatility increases. This property makes sure that the volatility does not increase without a limitation. Furthermore, the process of volatility never reaches zero if 1 2

0 2

κθ− σ ≤ is fulfilled. A short proof for this property is given as follow [8]:

For a n-dimensional Bessel process B , the stochastic differential equation is denoted _t

The Bessel process represents the Euclidean distance from origin to n-dimensional Wiener process. It is well known that the Bessel process B never reaches origin if _t

. One could denote that 2 n≥ 2 2 4 t

V =σ B , where _t σ is a real number. By using Ito’s

formula, one can obtain stochastic differential equation as follow:

2

4

t

dV =σ ndt+σ V dW_{t t}. (6)

Compare (6) with the volatility process in the Heston Model.

2

( )

t t t

dv =κ θ−v dt+σ v dW . (7) So far, we can see the similarity from their forms. Furthermore, the process

performances similar to our volatility process near zero only if

t V 2 4 nσ κθ = . Since we already know that the Bessel process B never reaches origin if _t , thus it is

Chapter 3 Pricing method

Once we have the stochastic volatility model, there are many ways to price the option. However, it is important to begin with the partial differential equation (PDE).

3.1 The Heston PDE

First, it is very important to notice that the Heston model is incomplete. There are different methods to hedge for deriving the Heston PDE, the same method also could be used to other stochastic volatility models. In this paper, the idea used for hedging is that the value of hedging portfolio is at least same as the payoff of the option at time maturity T. Since it is an incomplete market, the value of option therefore depends on the hedging strategy which is different from the case of complete markets.

We denote contingent claim as c S v t( , , )_{t t} , bond as B , and our portfolio consists of _t

the underlying asset, the bond and the contingent claim. We assume that the underlying pays no dividend. In details, those assets have dynamics as follows:

Then we choose a certain proportion ( , , )α β γ for our assets, the value of our portfolio becomes:

( , , )

t t t t t

F =αB +βS +γc S v t . (11) Under the requirement of self financing, it becomes:

( , , )

t t t t t

dF =αdB +βdS +γdc S v t . (12) Because of the non arbitrage assumption, the option price must be same

as the value of the portfolio:

( , , )_{t t} u S v t ( , , )_{t t t} u S v t = . (13) F From (13) we have: ( , , )_{t t t} du S v t =dF. (14)

Using Ito’s formula we obtain:

Since , the expressions in front of , and should be same. This leads to following equations:

( , , )_{t t t} du S v t =dF dW₁ dW₂ dt t t t t t t t t u c S v S v S v s s u u v v v v γ β σ γσ ∂ ₌ ∂ ₊ ∂ ∂ ∂ ₌ ∂ ∂ ∂ . (17)

From these equations, we can obtain the hedging proportion:

u c s s u v c v β γ γ ∂ ∂ = − ∂ ∂ ∂ ∂ = ∂ ∂ . (18)

Now we substitute following expression into the term at front of dt in dFt

( , , ) ( , , )

t t t t t t t t

B F S c S v t u S c S v t

α = −β −γ = −β −γ . (19)

We set those terms at front of dt in du S v t( , , )_{t t} and dF_t equal to each other

Plug in β and γ , we can obtain following equation after appropriate adjustment: 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 1 1 ( ( ) ( ) 2 2 1 1 1 ( ( ) ( ) 2 2 t t t t t t t t t t t t t t t t u u u u u u u S v S v v S v ru r S u t s v s v s v s v c c c c c c c S v S v v S v rc r S c t s v s v s v s v μ κ θ σ σρ μ μ κ θ σ σρ μ ∂ ∂ ∂ ∂ ∂ ∂ ∂ + + − + + + − − − ∂ _{∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂} ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ + + − + + + − − − ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ) ) = (21)

Since we can derive such equation with any kind of contingent claim ,

hence the left hand side of the equation above only depends on , and t.

Furthermore, we can denote a function

( , , )_{t t}

c S v t

t

S v_t

( , , )S v t_{t t}

λ , and set ( , , )λ S v t_{t t} equal to left hand side of (21). 2 2 2 2 2 2 2 1 1 ( ( ) ( 2 2 ( , , ) (22) t t t t t t t t t u u u u u u S v S v v S v ru r S t s v s v s v u S v t v μ κ θ σ σρ μ λ ∂ ₊ ∂ _{+ −} ∂ ₊ ∂ ₊ ∂ ₊ ∂ _{− − −} ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ = ∂ ) _t u) s ∂ After appropriate re-arrangements, we obtain:

2 2 2 2 2 2 2 1 1 ( ( ) ) 0 2 2 t t t t t t t u u u u u u rS v S v v S v ru t s κ θ λ v s σ v σρ s v ∂ ∂ ∂ ∂ ∂ ∂ + + − − + + + − ∂ ∂ ∂ ∂ ∂ ∂ ∂ = . (23)

The above equation is the Heston PDE with market price of volatility riskλ, in the case of no dividend. According to the proposal in [5], function ( , , )λ S v t_{t t} should have following form:

( , , )S v t_{t t} v_t

3.2 Closed form solution

It is always nice to have an explicit formula for pricing the option. Fortunately, Heston proposed a particular method for pricing European call option in stochastic volatility models. In this thesis, the derivation of the closed form solution for European call option in the Heston model will be briefly given, and more details could be found in [5].

First, we can guess a solution based on the Black-Scholes formula:

( ) 1

( , , , )_{t t t} r T t

C S v t T =S P−Ke− − P₂. (25) where P_j could be interpreted as “adjusted” or “risk-neutralized” probability [9].

This probability also could be explained as follow:

( , , , ) [ ( ) ln( ) ( ) , ( ) ]

j t

P x v T K = probability x T ≥ K x t =x v t = (26) v

where ln( )x= S_t ; j = 1, 2.

Next, we can plug our proposed solution into the Heston PDE. Then the following equation must be satisfied:

2 2 2 2 2 2 1 1 ( ) ( ) 2 2 j j j j j j j P P P P P v v v r u v a b v Pj 0 x ρσ x v σ v x v t ∂ ∂ ∂ ∂ ∂ ∂ + + + + + − + ∂ ∂ ∂ ∂ ∂ ∂ ∂ = (27) where u₁ =0.5,u₂ = −0.5,a=κθ,b₁= + −κ λ ρσ,b₂ = +κ λ.

twice-differentiable function f x v t_j( , , ) which represents a conditional expectation of . By Ito’s Lemma, we obtain following expression:

( ( ), ( )) g x T v T 2 2 2 2 2 2 1 2 1 1 ( ( ) ( 2 2 ( ) ( ) j j j j j j j j j j j f f f f f df v v v r u v a b v_j ) fj)dt x x v v x v t f f r u v dW a b v dW x v ρσ σ ∂ ∂ ∂ ∂ ∂ ∂ = + + + + + − + ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ + + + − ∂ ∂ . (28)

Since f x v tj( , , ) must be a martingale, we should have no term in above equation: dt 2 2 2 2 2 2 1 1 ( ) ( ) 2 2 j j j j j j j f f f f f f v v v r u v a b v j 0 x ρσ x v σ v x v t ∂ ∂ ∂ ∂ ∂ ∂ + + + + + − + ∂ ∂ ∂ ∂ ∂ ∂ ∂ = . (29)

If g x T v T( ( ), ( ))=1_{{ ( ) ln(x T ≥ K)}} , the solution for above PDE is the conditional probability at time t that x T( )≥ln( )K . If g x T v T( ( ), ( ))=exp(i xφ ), the solution is the characteristic function. Furthermore, we guess that our solution has the following form:

( , , ) exp ( ) ( )

j t t j j t

f x v t = ⎡_⎣C T− +t D T −t v +i xφ ⎤_⎦. (30) where C Tj( − )t and D Tj( −t) are unknown functions.

Plug in our guess into the PDE, we can obtain the ODE as follows:

By solving above ODE, we can get the solution as follows: 2 2 2 2 2 1 2 ( , , , ) exp ( , ) ( , ) 1 ( , ) [( ) 2 ln( 1 1 ( , ) ( ) 1 ( ) (2 ) 0.5, 0.5, j j j j t j j t d j j j d j j j d j j j j j j j j f x v T C T t D T t v i x a g C T t r ir b i d g b i d e D T t ge b i d g b i d d i b u i u u a τ τ τ φ φ φ φ τ φ φ ρσφ τ σ ρσφ τ φ σ ρσφ ρσφ ρσφ σ φ φ ⎡ ⎤ = _⎣ − + − + _⎦ − = − = + − + − − − + − = − = − − + = − − = − − − = = − =κθ,b₁= + −κ λ ρσ,b₂ = +κ λ,x= _t )] e ln( ).S (32)

As long as we have the characteristic functions, we can invert them to corresponding probabilities: ln( ) 0 ( , , , ) 1 1 ( , , , ) Re( ) 2 i K j t j t e f x v T P x v T K d i φ _φ φ π φ − ∞ = +

∫

In all, the closed form solution for a non dividend European call option is:

Chapter 4 Convexity

4.1 Introduction of Convexity

Definition of convexity:

Suppose that we have a continuous function f defined on an interval I. The function

f is convex on I if and only if:

( (1 ) ) ( ) (1 ) ( )

f θa+ −θ b ≤θ f a + −θ f b (35) for any θ∈[0,1], and any a b, ∈I.If f is twice differentiable, then it is convex when f''≥0

One example of convex function can be illustrated by the following figure which is the price of European call option in the Black-Scholes model.

From this figure, we can easily see that the price of European call option is a convex function. Also the price of European call option increases as the stock price increases. However, the option price is not likely to change linearly as the stock changes; instead it behaves as some nonlinear function of the stock price. The convexity is the measurement of how the option price changes as the stock price changes.

One might ask why convexity is an important property for the option price. The reasons come from different aspects. One of them is that the convexity is useful for comparing different options. We can view convexity as a measure of option risk. The option with less convexity is less influenced by the variation of underlying asset price than one with greater convexity. Also convexity is useful for risk management; if the combined convexity is low, one would lose less even though fairly big price variation happens.

4.2 Theorem for convexity

Recall the Heston PDE, equation (23) in Section 3.1. First, we should take a look at Black-Scholes formula and corresponding Greeks. One can easily obtain the Greeks and based on the Black-Scholes formula because of the homogeneity properties of financial markets [11]. The expressions are shown as follows:

Based on the Heston model, the closed form solution of a non dividend European call option is already given in Section 3.2. The solution which is similar to the Black-Scholes formula is:

( )

1 2

( , , , )t t t r T t

C S V t T =S P −Ke− − P .

By taking the appropriate derivatives or by exploiting homogeneity properties of financial markets [11, 12], we can easily get:

1 2 1 2 ln( ) 1 1 0 ( , , , ) ( , , , ) ( , , , ) 1 1 ( , , , ) Re( ) 2 ln( ). t t t t i K t t t C S V t T P S C S V t T P S S e f x V T where P x V T K d i x S φ _φ φ π φ ∞ − ∂ Δ = = ∂ ∂ ∂ Γ = = ∂ ∂ = + =

∫

The functions may be interpreted as adjusted or risk-neutralized

probability [9]. Since is the cumulative distribution function (in the variable of

1( , , ,t )

P x V T K

1

P

ln( )K ) of the log-spot price after time T-t, staring at x for some drift μ , hence the first order derivative of with respect to spot price should be the corresponding density [12]. It has the following form:

1 P ln( ) 1 1 1 0 1 ( , , , ) Re( ( , , , )) ln( ). i K t t t t P p x V T K e f x V T d S S where x S φ _{φ φ} π ∞ − ∂ Γ = = = ∂ =

∫

Theorem 4.1:

For the Heston model, if the contract function of an option is convex, then the option price is convex in the underlying asset for all fix time before maturity.

Proof:

Denote the convex contract function of an option Φ( )S , where and

. It is well known that any convex function could be approximated by a

convex piecewise-linear function. This approximation could be written as follow: ( )S 0 Φ ≥ max [0, ] S∈ s 1 ( )S a S_i( b_i) when S [ ,s s_{i i+} ] Φ = − ∈ (39)

It also could be illustrated by following figure:

Figure 3 Piecewise-linear approximation of a convex function

It is important to notice that b_i is positive since Φ( )S ≥0 and S∈[0,s_max]. Further, could be positive or negative. When is positive, we could consider in

the interval

i

a a_i Φ( )S

1

[ ,s si i+ ] as a European call option with strike price and weight .

When is negative, we could consider

i

b ai

i

European put option with strike price bi and weight − . Hence, we could consider ai

as a non negative weight combination of European call and put options with

corresponding strike price. The price of ( )S

Φ

( )S

Φ for any fix time before maturity is the

non negative weight combination of corresponding price of European calls and puts.

From expression (38), we already verified that the price of European call option for any fix time before maturity in the Heston model is convex. From Lemma 4.1, it is also true that if the price of European call option is convex, the price of European put option is convex. It is also well known that the non negative weight combination of convex is also convex.

Finally, we can conclude that the option price is convex in the underlying asset for all fix time before maturity if the contract function is convex.

Lemma 4.1:

If the price of European call option is convex, the price of identical1 European put optionis convex.

Proof:

It is well known that there is a relation called call-put parity which is described as follow:

( )

( , ) r T t ( , )

C S t + ∗K e− − =P S t + (40) S

where and are the price of European call and put option with identical strike price and maturity time,

( , )

C S t P S t( , )

K is strike price and is the value of the underlying asset.

S

From (40), we have:

( )

( , ) ( , ) r T t

P S t =C S t + ∗K e− − − (41) S

Since C S t( , ) and −S are convex, K e∗ −r T t( −) is constant. Thus, the sum of these three terms is convex.

Chapter 5 Numerical methods

5.1 Introduction

The numerical analysis method is the most common method for dealing with partial differential equation. There are many different numerical analysis methods, such as: finite difference method, finite element method, finite volume method and so on. In this paper, I decide to use the finite difference method.

5.1.1 Main idea

The main idea for the finite difference method is that one could apply discretisation of partial differential equation on a grid in the finite domain. This method was fully developed in the 1960s. It became popular because it is relatively easy to program; and it provides considerably accuracy which depends on the choice of grid density and time step. Furthermore, discretisation can be performed as uniform grid, non-uniform grid and random grid. The uniform grid which is used in this thesis is the common choice. The following graph shows main idea of the method.

The value of each small mesh point depends on several others around it. All the values inside the domain are unknown, and the values on the boundary are deterministic.

5.1.2 Detail

One could use different forms, such as forward difference, backward difference and central difference, to substitute the derivatives in PDE, and obtain a numbers of equations which have the same number of unknown variables.

forward difference F F x( x) F x( ) x x ∂ ₌ + Δ − ∂ Δ backward difference F F x( ) F x( x) x x ∂ − − Δ = ∂ Δ central difference ( ) ( 2 ) F F x x F x x x x ∂ ₌ + Δ − − Δ ∂ Δ

Table 2 Forms of derivative substitution

where represents the equations after the difference substitution in domain. Since the order of accuracy is 2, Crank-Nicholson method is recommended. Respectively Euler backward has order of accuracy 1, Euler forward depends on strictly stable condition. Finally one could iteratively compute the numerical solution from one side of time to another; the value of starting time should be deterministic.

t

AF

Boundary condition is also one important aspect for the finite difference method. It represents the value or the function lying on the boundary of the domain. Concretely, there are three kinds of boundary condition:

Dirichlet boundary condition _{F(∂Ω =) u}

Neumann boundary condition F( ) _u

n

∂ ∂Ω ₌ ∂

Cauchy boundary condition _{aF( ) b} F( ) _u

n

∂ ∂Ω

∂Ω + =

∂ Table 4 Boundary conditions

∂Ω represents the boundary of domain, is a deterministic function or value, and are some certain numbers.

u a

b

In the Heston PDE, there are terms of second order derivative and cross derivative. They could be approximated by following expressions:

5.2 Implementation

We will use Matlab in the implementation part of this thesis. Both the finite difference method and the close form solution are implemented. More details will be discussed in the following sections.

5.2.1 Closed form solution

The formula (34) looks complicated, but the only problem is that the integer cannot be calculated directly. Practically, it is fairly easy to calculate with Matlab. Using certain command “quadl” which involves recursive adaptive Lobatto quadrature, the integer could be approximated within an acceptable error.

5.2.2 Finite difference method

Boundary conditions

In the Heston PDE, since we have derivatives with respect to time, stock price and volatility, we need the boundary conditions on each direction. Based on the inequality which is obtained in Section 2.3.2, we should notice that if 1 2 0

2

κθ− σ ≤ is fulfilled, volatility process will never reach zero. However, it is very important to have all the boundary conditions for the finite difference method. We will use some “artificial” boundary conditions which are reasonable in our case. The consideration of boundary conditions is also based on [1, 5, 10]

(a) Time direction:

The European call option price is the payoff of the contract when time reaches maturity. This is also the corresponding boundary condition claimed in [5].

( , , ) max( , 0)

(b) Stock price direction:

If the stock price is zero, we also keep the choice in [5]:

( , 0, ) 0

u t v = .

A Neumann boundary condition is proposed in [5] for the maximum stock price:

max ( , , ) 1 u t S v S ∂ = ∂ .

We also considered other choices, such as the Black-Scholes formula in [1]. But the results from tests are not nice for those choices. Hence we use the values from the closed form solution with corresponding stock price and volatility.

(c) Volatility direction:

If volatility is zero, the boundary condition is:

( )

( , , 0) max( r T t , 0)

u t S = S−Ke− − .

It is much simpler comparing with the corresponding one proposed in [5].

( , , 0) ( , , 0) ( , , 0) ( , , 0) 0 u t S u t S rS ru t S u t S S κθ v ∂ ₊ ∂ _{− +} ∂ ∂ = .

If volatility reaches maximum, we use the same one as in [5]. It is intuitive to think that the option price is same as the spot price:

max

( , , )

Concrete value for the parameters:

After setting the boundary conditions, we should decide the value for some parameters, such as: maximum of stock price, maximum of volatility. In this thesis, we use four times of the strike price as the maximum value of stock price. Since there is no regulation for appropriate maximum volatility, the value of maximum volatility is determined by running tests based on the closed form solution.

As we noted, when the volatility reaches maximum, the option price should be equal to the spot price. It means that the curve of the option price should be a straight line which starts from origin and whose slope equals to one when the volatility reaches maximum. Then we will run some tests with the closed form solution using different groups of parameters and choose one appropriate maximum value of the volatility. The typical results that occurred during the tests runs are shown in Figure 5.

From the tests, we can conclude that when volatility is equal to 40 or bigger the curve of option price satisfies the requirement. Hence the maximum volatility for the finite difference method will be set to 40.

Iteration Process

As mentioned in Section 5.1.2, central difference is applied for approximating the first derivative with respect to price and volatility; formula (42) and (43) are given for approximating the second derivative and the cross derivative with respect to underlying asset or volatility; the Crank-Nicholson scheme is applied for solving the ordinary differential equation. After necessary substitutions, a part of the Heston PDE becomes as follow: 2 2 2 2 2 2 2 1 1, 2 1, 3 , 1 4 , 1 5 1, 1 6 , 1 1 ( ( ) ) 2 2 t t t t t t t j k j k j k j k j k j k u u u u rS v S v v S v ru s v s v a u a u a u a u a u a u κ θ λ σ σρ + − + − + + ∂ ∂ ∂ ∂ + − − + + + − ∂ ∂ ∂ ∂ ∂ = + + + + + u s v ∂ ∂ . (44) where 2 1 2 2 2 3 2 4 5 2 2 6 1 1 2 2 1 1 2 2 ( ) 2 2 ( ) 2 2 2 a Nr MN v MN a MN v Nr M v M v M a M v v M M v M v a v v a MN M a MN r MN v v σρ κ θ λ σ Nσρ σ κ θ λ σρ σ σρ = + Δ − = Δ − − Δ − Δ = + Δ Δ − Δ − Δ = − Δ Δ = = − − Δ − − Δ

M and N are the number of how many steps of the price and volatility are divided respectively. The index of price and volatility is represented by j and k respectively. Expression (44) also could be expressed as matrix form:

max max m 4 1, 0 6 3 4 6 3 1, 5 2, 4 2, 0 3 4 6 1 5 3 2, 1 5 1 2 2 2 1 2 3 1 2 , 3 1 1 2 3 v v v v v A A A A where A A M N M N A A a u a a a a a u a u A a u a a a a a a u a A and b a a a a A a = = ⎛ ⎞ ⎜ _⋅ ⎟ ⎜ ⎟ ⎜ ⎟ = ⋅ ⋅ ⋅ ⎜ ⎟ ⋅ ⋅ ⎜ ⎟ ⎜ ⎟ _{× ×} ⎝ ⎠ ⋅ ⎛ ⎞ ⎜ _⋅ ⎟ _⋅ ⎜ ⎟ + ⎜ ⎟ = ⋅ ⋅ ⋅ ⎜ _{⋅ ⋅} ⎟ ⎜ ⎟ ⎜ ⎟ _⋅ ⎝ ⎠ ⋅ ⎛ ⎞ ⎜ _⋅ ⎟ ⎜ ⎟ ⎜ ⎟ = ⋅ ⋅ = ⎜ ⎟ ⋅ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎛ ⎞ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ = ⋅ ⎜ ⎟ ⋅ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ax max max max max max max max

max max max max

5 3, 1 ,1 5 ,2 4 , 0 1 ,2 5 ,3 1 ,3 5 ,4 1 , 3 , 5 , v s s M v s s s s s N M v s v N N a u a u a u a u a u a u a u a u a u a u a u = ⎛ _{⎞ ⎫} ⎜ _{⎟ ⎪} ⎪ ⎜ ⎟ ⎬ ⎜ ⎟ ⎪ ⎜ ⎟ ⎪ ⎜ _{⎟ ⎭} ⎜ ⎟ ⎫ ⎜ ⎟ ⎪ ⎜ _{⎟ ⎪} ⎜ _{⎟ ⎬} ⎜ _{⎟ ⎪} ⎜ + ⎟ _⎪⎭ ⎜ ⎟ ⋅ ⎜ _{⎟ ⎫} ⎜ _⋅ ⎟ ⎪ ⎬ ⎜ ⎟ ⎪ ⋅ ⎜ _{⎟ ⎭} ⎜ _{+ +} ⎟ ⎜ ⎟ ⎫ ⎜ ₊ ⎟ ⎪ ⎜ ⎟ ⎪ ⎜ + ⎟ ⎪ ⎜ ⎟ ⎬ ⋅ ⎜ ⎟ ⎪ ⎜ _⋅ ⎟ ⎪ ⎜ ⎟ ⎜ _{+ +} ⎟ _⎭ ⎝ ⎠ M N N ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ _× ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ _⎪⎭

With the matrix form substituted in and the Crank-Nicholson scheme applied, the Heston PDE becomes:

1 1 1 1 1 1 1 ( ) ( ) 2 2 2 2 ( ) ( ) ( i i i i i i i i u u Au Au b b t A I u A I u b b t t − − − 0 ) i i − − − _{+ + + + =} Δ ⇒ − = − + − + Δ Δ . (45)

where i represents the index of time direction, tΔ is the time step, I is a

M× −N by−M×N identity matrix.

5.3 Results

In this section, the results obtained from the closed form solution and the finite difference method will be discussed and compared; convexity will also be discussed based on the results.

5.3.1 Results from the closed form solution

In this section, I will present the closed form solution graphically. Furthermore, inappropriate parameter choice will also be given and discussed. The convexity based on the closed form solution will be discussed at last.

Results and parameter

First, we choose two groups of parameters based on [10].

max max 20, 80, 0.1, 0.1, 0.04, 0, 0.5, 2, 2 20, 80, 0.2, 0.5, 0.02, 0.1, 0.3, 3, 2 K S r T K S r T ρ θ λ σ κ ρ θ λ σ κ = = = = − = = = = = = = = = − = = = = =

Figure 6 Results from the close form solution

The left hand side of Figure 6 is the result from the first group parameter, the right hand side of the figure is the result from second one. The pictures look nice and the curve of the option price is convex for difference volatility value.

max

20, 80, 0.1, 0.1, 0.04, 0, 0.5, 2, 2

K = S = r= ρ= − θ = λ = σ = κ = − T =

Figure 7 Results from the close form solution for negative κ

The left hand side of Figure 7 is the result when volatility equals to 2; the right hand side is the result when volatility equals to 0.1. Now we can clearly see that when volatility is 2, the price almost reaches the upper boundary. When volatility equals to 0.1, the price of option has negative value which is absolutely wrong. Those performances in Figure 6 are unreasonable. It is probably because of the negative κ . To verify my thought, we do tests using the first group of parameters. The results with volatility equals to 0.1 and 2 are shown in Figure 8:

Figure 8 Results for the first two groups of parameters when volatility = 0.1 and 2

Now we analyse the unreasonable choice of κ from theoretical point of view. As explained in Section 2.3.2, our volatility process has the form as follow:

2

( )

t t t

dv =κ θ −v dt+σ v dW .

The artificial process

2 2

4

t

V =σ B , where _t B is a n-dimensional Bessel process, has _t

the form as follow:

2 4 t t dV =σ ndt+σ V dW_t. If n 4κθ₂ σ

= is fulfilled, those two processes has similar performance near zero. First,

we have positive σ ; 2 θ is also positive since it is long term mean for volatility which is always positive. Hence when κ is negative, “n” will become negative which is unreasonable. Based on above analysis, any negative value for the term κ is not considered in other tests.

At last, we run the tests with many groups of appropriate parameters. The results are similar as Figure 6. We can say that the results shown in Figure 6 are the typical results. In all, the option price is convex based on the closed form solution with appropriate parameter.

Convexity based on closed form solution

Above, we can graphically see that the price for European call option is convex. On the other hand, we can show the convexity by illustrating the Γ .

form solution for the Heston model is always positive. We can also obtain numerical typical results for the density function of log-spot price which is shown in Figure 9:

Figure 9 Typical result of density function of log-spot price with different volatility

The numerical values are positive. Consequently, we can say that the convexity for European call option based on the closed form solution for the Heston model is preserved.

5.3.2 Results from the finite difference method

In the implementation part, we set the number of discretisation for the price and the volatility direction as 20, time maturity is 2, time step is 0.1, and maximum volatility is always 40 as claimed in Section 5.2.2. Other parameters, including ρ θ σ κ λ , , , , , ,r

max

20, 80, 0.1, 0.1, 0.04, 0, 0.5, 2

K = S = r= ρ= − θ = λ= σ = κ =

Figure 10 Typical results from FDM for different groups of parameters

The left hand side of Figure 10 is the picture with different volatilities. To illustrate the trend of the option price clearly, we only show the option price with four numbers of volatility. We could see that the curve of the option price is pulling up toward the boundary option price as long as the volatility increasing, also the option price is convex at least for those four volatility values. The right hand side of Figure 10 shows more precisely. We can conclude that the option price is convex for any volatility value between the interval [0, 40].

From the tests with different groups of parameters, it seems that the parameters ,

r ρ , θ and σ do not effect the results so much, we could always get the typical results as Figure 10. Also it is important to notice that:

(a) We did not change the sign of θ which is the long term mean for volatility. Since there is no negative volatility, θ couldn’t be negative.

(b) The termλ always equals to 0.

Next we have two groups of parameters with non-zero λ. The results obtained from the parameters below are shown in Figure 11.

max

20, 80, 0.1, 0.1, 0.04, 0.2, 0.5, 2

K = S = r= ρ = − θ = λ= σ = κ =

Figure 11 Results from λ= 0.2

We can see that the result after adding market price of volatility risk is still nice, and the option price is convex. We try larger market price of risk, the results obtained from the parameters below are shown in Figure 12.

max

20, 80, 0.1, 0.1, 0.04, 1, 0.5, 2

K = S = r = ρ = − θ = λ= σ = κ =

Figure 12 Results from λ= 1

Spot

price Option price with _{volatility = 2} λ= 0.2, Option price with _{volatility = 2} λ= 1,

4 0.26642 0.12408 8 1.3578 0.92477 12 3.146 2.4526 16 5.5315 4.6661 20 8.4403 7.5134 24 11.66 10.726 28 15.073 14.157 32 18.614 17.73 36 22.246 21.4 40 25.944 25.137 44 29.691 28.923 48 33.477 32.747 52 37.292 36.6 56 41.131 40.474 60 44.989 44.367 64 48.864 48.274 68 52.752 52.192 72 56.652 56.12 76 60.561 60.056

Table 5 Typical results of numerical comparison for the option price with different λ

From Table 5, we can clearly see that all the option price with larger λ when volatility equals to 2 is smaller to the one with smaller λ. This is the typical result during the tests. We conclude that the option price decreases when the market price of volatility risk increases. It also accords with the real situation.

The result for every group of parameters is nice and as expected. Based on all the results obtained from the finite difference method, we can conclude that the price for European call option is convex and increases when the volatility increases.

5.3.3 Comparison between the results of the previous sections

max

20, 80, 0.1, 0.1, 0.04, 0, 0.5, 2

K = S = r= ρ= − θ = λ= σ = κ =

Figure 13 Comparison of FDM and closed form solution

We can see that the results obtained from both methods are very close. For further comparison to see the difference of results, a statistical concept which is called the standard error regression (SER) is applied. We fix the volatility; calculate SER by following expression: 2 ( ( , ) ( , )) 1 n i i CFS t FDM t i C S v C S v SER n − = −

∑

where and represent the option price obtained from the

closed form solution and the finite difference method respectively when spot price is and fix volatility is . We calculate SER when volatility equals to 2, 4, 10 and 30

which are shown in Table 6. ( i, ) CFS t C S v CFDM(S vti, ) i t S v Volatility 2 4 10 30 SER 0.135 0.045 0.002 0.075

Table 6 Numerical comparison between FDM and closed form solution

Chapter 6 Conclusion

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