第四章 brown 运动和 ito 公式. chapter 5 european option pricing ------ black-scholes formula

第四章

Brown运动和 Ito公式

Chapter 5

European Option Pricing

------Black-Scholes Formula

Chapter 5

European Option Pricing

------

Black-Scholes Formula

Introduction

In this chapter, we will describe the price movement of an underlying

asset by a continuous model --- geometrical Brownian motion.

we will set up a mathematical model for the option pricing (Black-Scholes PDE) and find the pricing formula (Black-Scholes formula).

We will discuss how to manage risky assets using the Black-Scholes formula and hedging technique.

History

In 1900, Louis Bachelier published his doctoral thesis ``Thèorie de la Spèculation", - milestone of the modern financial theory. In his thesis, Bachelier made the first attempt to model the stock price movement as a random walk. Option pricing problem was also addressed in his thesis.

History- In 1964, Paul Samuelson, a Nobel Economics

Prize winner, modified Bachelier's model, using return instead of stock price in the original model.

Let be the stock price, then is its return. The SDE proposed by P. Samuelson is:

This correction eliminates the unrealistic negative value of stock price in the original model.

tS /t tdS S

tt

t

dSrdt dW

S

History--

P. Samuelson studied the call option pricing problem (Ć. Sprenkle (1965) and J. Baness (1964) also studied it

at the same time). The result is given in the following. (V,etc as before)

2

1 2

2

1

/ 22 1

( ) ( ) ,

ln / 1/ 2,

1, ( )

2

c ST T

S

x

V e Se N d KN d

S K Td

T

d d T N x e d

History---

In 1973, Fischer Black and Myron Scholes gave the following call

option pricing formula

Comparing to Samuelson’s one, are no longer present. Instead, the risk-free interest r enters the formula.

1 2( ) - ( )rTV SN d Ke N d&S c

History----

The novelty of this formula is that it is independent of the risk preference of individual investors. It puts all investors in a risk-neutral world where the expected return equals the risk-free interest rate. The 1997 Nobel economics prize was awarded to M. Scholes and R. Merton (F. Black had died) for this brilliant formula and a series of contributions to the option pricing theory based on this formula.

Basic Assumptions

(a) The underlying asset price follows the geometrical Brownian motion:

μ – expected return rate (constant)σ- volatility (constant) - standard Brownian motion

tt

t

dSdt dW

S

tdW

Basic Assumptions -

(b) Risk-free interest rate r is a constant (c) Underlying asset pays no dividend (d) No transaction cost and no tax (e) The market is arbitrage-free

A Problem

Let V=V(S,t) denote the option price. At maturity (t=T),

where K is the strike price. What is the option's value during its lifetime

(0< t<T)?

( ) , call( , )

( ) , put

S KV S T

K S

Δ-Hedging Technique

Construct a portfolio

(Δ denotes shares of the underlying asset),

choose Δ such that Π is risk-free in (t,t+dt).

V S

Δ-Hedging Technique -

If portfolio Π starts at time t, and Δ remains unchanged in (t,t+dt), then the requirement Π be risk-free means the return of the portfolio at t+dt should be

. . ( )

t dt t

t

t t t t t

rdt

i e dV dS d dt r V S dt

Δ-Hedging Technique --

Since

where the stochastic process satisfies SDE*, hence by Ito formula

So

( , ),t tV S T

tS

22 2

2

1

2t t

V V V VdV S S dt S dW

t S S S

22 2

2

1

2

( )t

V V VS S S dt

t S S

VS S dW r V S dtS

Δ-Hedging Technique ---

Since the right hand side of the equation is risk-free, the coefficient of the random term on the left hand side must be zero. Therefore, we choose

tdW

V

S

Δ-Hedging Technique ----Black-Scholes Equation

Substituting it, we get the following PDE:

This is the Black-Scholes Equation that describes the option price movement.

22 2

2

10

2

V V VS rS rV

t S S

22 2

2

10

2

V V VS rS rV

t S S

Remark

The line segment {S=0, 0< t< T} is also a boundary of the domain . However, since the equation is degenerated at S=0, according to the PDE theory, there is no need to specify the boundary value at S=0.

Black-Scholes Equation in Cauchy Form

2 22

2

0

10,

2 2

( , ) (0, ]

, call

, put

x

x

V V Vr rV

x S

T

e KV

K e

Well-posed Problem

By the PDE theory, above Cauchy problem is well-posed. Thus the original problem is also well-posed.

Remark The expected return μ of the asset, a parameter in

the underlying asset model , does not appear in the Black-Scholes equation . Instead, the risk-free interest rate r appears it. As we have seen in the discrete model, by the Δ---hedging technique, the Black-Scholes equation puts the investors in a risk-neutral world where pricing is independent of the risk preference of individual investors. Thus the option price arrived at by solving the Black-Scholes equation is a risk-neutral price.

An Interesting Question 1) Starting from the discrete option price

obtained by the BTM, by interpolation, we can define a function on the domain

Σ={0< S<∞,0<t<T}; 2) If there exists a function V(S,t), such that

3) if V(S,t) 2nd derivatives are continuous in Σ,

What differential equation does V(S,t) satisfy?

0lim ( , ) ( , ), (( , ) ),tV S t V S t S t

( , )V S t

Answer of the Question

V(S,t) satisfies the Black-Scholes equation in Σ. i.e., if the option price from the BTM converges to a sufficiently smooth limit function as Δ 0, then the limit function is a solution to the Black-Scholes equation.

Black-Scholes Formula (call)

2

( )1 2

2

1

2 1

/ 2

( , ) ( ) ( )

ln( / ) ( / 2)( ),

,

1( )

2

r T t

x

V S t SN d Ke N d

S K r T td

T t

d d T t

N x e d

Black-Scholes Formula (put)

2

( )2 1

2

1

2 1

/ 2

( , ) ( ) ( )

ln( / ) ( / 2)( ),

,

1( )

2

r T t

x

V S t Ke N d SN d

S K r T td

T t

d d T t

N x e d

Generalized Black-Scholes Model (I) -----Dividend-Paying Options

Modify the basic assumptions as follows (a^)The underlying asset price movement

satisfies the stochastic differential equation

(b^) Risk-free interest rate r=r(t) (c^) The underlying asset pays continuous

dividends at rate q(t) (d) and (e) remain unchanged

( ) ( )tt

t

dSt dt t dW

S

Δ-hedging

Use the Δ-hedging technique to set up a continuous model of the option pricing, and find valuation formulas.

Construct a portfolio Choose Δ, so that Π is risk-neutral in [t,t+dt]. the expected return is

V S

t dt t tr dt

Δ-hedging -

Taking into account the dividends, the portfolio's value at is

Therefore, we have

t dt

t dt t dt t t t t t dtV S q dt S

t t t t t t t tdV dS r dt S q dt

B-S Equation with DividendApply Ito formula, and choose

we have

Thus, B-S Equation with dividend is

/t V S

2 22

2

( )( ) ( )

2

V t V V VS dt r t V S dt q t S dt

t S S S

2 2

22

( )( ) ( ) ( ) 0

2

V t V VS r t q t S r t V

t S S

2 2

22

( )( ) ( ) ( ) 0

2

V t V VS r t q t S r t V

t S S

Solve B-S Equation with dividend

Set

choose α&βto eliminate 0 & 1st terms:

( ) ( ),t tu Ve y Se

2 22

2

( )

2

( ) ( ) '( ) ( ) '( ) 0

u t uy

t y

ur t q t t y r t t u

y

Solve B-S Equation with dividend -

Let α&β be the solutions to the following initial value problems of ODE:

The solutions of the ODE are

( ) ( ) 0, ( ) 0,

( ) ( ) 0

d dr t q t r t

dt dtT T

( ) ( ) ( ) , ( ) ( )T T

t tt r q d t r d

Solve B-S Equation with dividend --

Thus under the transformation, and take the original problem is reduced to

^ ^

22

2

( )

0

( )t

t T t T

u uy

t y

u Ve y K

2 ^ 2

0 0( ) , ( )

t Td T d

Apply B-S Formula

Let σ=1, r=0, T=Tˆ, t=τ,^ ^

1 2

^^

1 ^

^ ^ ^2 1

( , ) ( ) ( )

ln / 1/ 2( )

u y yN d KN d

y K Td

T

d d T

European Option Pricing (call, with dividend)

Back to the original variables, we have

( ) ( ) ^ ^1 2

( ) ( )^ ^1 2

2

^1

2

^ ^ 22 1

( , ) ( ) ( )

( ) ( )

ln / ( ) ( ) ( ) / 2

( )

( )

T T

t t

t t

q d r d

T

t

T

t

T

t

V S t e Se N d KN d

Se N d Ke N d

S K r q dd

d

d d d

European Option Pricing (put, with dividend)

( ) ( )^ ^2 1

2

^1

2

^ ^ 22 1

( , ) ( ) ( )

ln / ( ) ( ) ( ) / 2

( )

( )

T T

t tr d q d

T

t

T

t

T

t

V S t Ke N d Se N d

S K r q dd

d

d d d

Theorem 5.1 c(S,t) - price of a European call option p(S,t) - price of a European put option,

with the same strike price K and expiration date T.

Then the call---put parity is given by

where r=r(t) is the risk-free interest rate, q=q(t) is the dividend rate, and σ= σ(t) is the volatility.

( ) ( )( , ) ( , )

T T

t tr d q d

c S t Ke p S t Se

Proof of Theorem 5.1

Consider the difference between a call and a put: W(S,t)=c(S,t)-p(S,t). At t=T,

W is the solution of the following problem

( , ) ( ) ( )W S T S K K S S K

2 2

22

( )( ) ( ) ( ) 0

2

t T

W t W WS r t q t S r t W

t S SW S K

Proof of Theorem 5.1-

Let W be of the form W=a(t)S-b(t)K, then

Choose a(t),b(t) such that

'( ) '( ) ( ) ( ) ( )

( ) ( ) ( ) 0

a t S b t K r t q t Sa t

r t a t S b t K

'( ) ( ) ( ) ( ) ( ) ( ) 0

'( ) ( ) ( ) 0

( ) ( ) 0

a t r t q t a t r t a t

b t r t b t

a T b T

Proof of Theorem 5.1--

The solution is

Then we get the call---put parity, the theorem is proved.

( ) ( )( ) , ( )

T T

t tq d r d

a t e b t e

Predetermined Date Dividend

If in place of the continuous dividend paying assumption (c^), we assume

(c~) the underlying asset pays dividend Q on a predetermined date t=t_1 (0<t_1<T)

(if the asset is a stock, then Q is the dividend per share).

After the dividend payday t=t_1, there will be a change in stock price:

S(t_1-0)=S(t_1+0)+Q.

Predetermined Date Dividend -

However, the option price must be continuous at t=t_1:

V(S(t_1-0),t_1-0)=V(S(t_1+0),t_1+0). Therefore, S and V must satisfy the boundary

condition at t=t_1:

V(S,t_1-0)=V(S-Q,t_1+0) In order to set up the option pricing model (take

call option as example), consider two periods [0,t_1], [t_1,T] separately.

Predetermined Date Dividend --

In 0≤ S<∞, t_1 ≤ t ≤ T, V=V(S,t) satisfies the boundary-terminal value problem

Obtain V=V(S,t) on t_1

2 22

2

( )( ) ( ) 0

2

t T

V t V VS r t S r t V

t S S

V S K

Predetermined Date Dividend ---

in 0 ≤ S<∞,0 ≤ t ≤ t_1,V=V(S,t) satisfies

By solving above problems, we can determine the premium V(S_0,0) to be paid at the initial date t=0 (S_0 is the stock price at that time).

1

2 22

2

1

( )( ) ( ) 0

2, 0

t t

V t V VS r t S r t V

t S SV V S Q t

Remark1

Note that there is a subtle difference between the dividend-paying assumptions (c^) and (c~) when we model the option price of dividend-paying assets.

Remark1- In the case of assumption (c^), we used the

dividend rate q=q(t), which is related to the return of the stock. Thus in [t_1,t_2], the dividend payment alone will cause the stock price By this model, if the dividend is paid at t=t_1 with the intensity d_Q,

then at t=t_1 the stock price Thus we can derive from the corresponding option pricing formula.

2

1 11

exp ( )t

t t tS S q t dt

1

10lim ( )

t t

Qttq t dt d

1 1

Qd

t tS S e

Remark1--

In the case of assumption (c~), we used the dividend Q, which is related to the stock price itself. So at the payday t=t_1, the stock price

Thus we have at t=t_1 the boundary condition for the option price.

We should be aware of this difference when solving real problems.

1 1t tS S Q

Remark 2

For commodity options, the storage fee, which depends on the amount of the commodity, should also be taken into account.

Therefore, when applying the Δ-hedging technique, for the portfolio:

where Δq^dt denotes the storage fee for Δ amount of commodity and period dt.

^,V S d dV dS q dt

Remark 2-

Similar to the derivation we did before, choose such that Π is risk-free in (t,t+dt). Then we get the terminal-boundary problem for the option price V=V(S,t)

This equation does not have a closed form solution in general. Numerical approach is required.

/V S

2 22 ^

2( ) 0

2

t T

V V VS rS q rV

t S S

V S K

Remark 2--

If the storage fee for Δ items of commodity and period dt is in the form of Δq^Sdt, proportional to the current price of the commodity, then

And the option price is given by the Black-Scholes formula.

2 22 ^

2( ) 0

2

V V VS r q S rV

t S S

Generalized B-S Model (II) ------Binary Options

There are two basic forms of binary option (take stock option as example):

Cash-or-nothing call

Cash-or-nothing call (CONC): In Case: t=T: stock price < strike price, the

option =0; In Case: t=T: stock price > strike price, the

holder gets $1 in cash.

Asset-or nothing call

Asset-or nothing call (AONC): In Case: t=T: stock price < strike price, the

option =0; In Case: t=T: stock price < strike price, the

option pays the stock price.

Modeling

If the basic assumptions hold, then the binary option can be modeled as

2 22

2( ) 0

2( ), ( )

( ), ( )t T

V V VS r q S rV

t S SH S K CONC

VSH S k AONC

Relation of CONC,AONC & VC

Consider a vanilla call option, a CONC and a AONC with the same strike price K and the same expiration date T. Their prices are denoted by V, V_C and V_A, respectively. On the expiration date t=T, these prices satisfy

( , ) ( , ) ( , )A CV S T V S T KV S T

Relation of CONC,AONC & VC - And V(S,t), V_A(S,t) and V_C(S,T) each

satisfies the same Black-Scholes equation. In view of the linearity of the terminal-boundary problem, therefore in Σ{0≤ S<∞,0 ≤ t ≤ T},

i.e throughout the option's lifetime, a vanilla call is a combination of an AONC in long position and K times of CONC in short position.

( , ) ( , ) ( , )A CV S t V S t KV S t

Theorem 5.2

^

^ 2

( , , , ) ( , , , )

where 2

A CV S t r q SV S t q r

r q r

Proof of Theorem 5.2

Let V_A(S,t)=Su(S,t). It is easy to verify that u(S,t) satisfies:

Define , then the above equation can be written as

2 22 2

2( ) 0

2

u u uS r q S qu

t S S

2 22 ^

2( ) 0

2

u u uS q r S qu

t S S

2^ 2r q r

Proof of Theorem 5.2 -

In Σ, compare the terminal-boundary problem for u(S,t), and the terminal-boundary problem for V_C(S,t). If the constants r, q are replaced by q and r^, then u(S,t) and V_C(S,t) satisfy the same terminal-boundary value problem. By the uniqueness of the solution, we claim

Thus the Theorem is proved.

^( , ) ( , ; , )Cu x t V S t q r

Solve pricing of CONC & AONC

Once the CONC price V_C(S,t;r,q) is found, the AONC price V_A(S,t;r,q) can be determined by Theorem 5.2.

In order to solve the CONC problem, make transformation

, ln( / )

( ) ( 1) ( )x

T t x S K

H S K H e H x

Solve pricing of CONC & AONC -

Then the Ori Prob. is reduced to a Cauchy problem

Analogous to the derivation of the Black-Scholes formula, we have

2 2 2

2( )

2 2( ,0) ( )

V V Vr q rV

x xV x H x

Solve pricing of CONC & AONC--

Back to the original variables (S,t), and by Theorem 5.2, we have

2( / 2)( , ) r x r qV x e N

2( )

2( )

ln( / ) ( / 2)( )( , ; , )

ln( / ) ( / 2)( )( , ; , )

r T tC

q T tA

S K r q T tV S t r q e N

T t

S K r q T tV S t r q Se N

T t

Generalized B-S Model (III) ------Compound Options

A compound option is an option on another option. There are many varieties of compound options. Here we explain the simplest forms of compound options.

A compound option gives its owner the right to buy (sell) after a certain days (i.e. t=T_1) at a certain price K^ a call (put) option with the expiration date t=T_2 (T_2>T_1) and the strike price K. There are following forms of compound options:

Compound Options1. At t=T_1 buy a call option on a call

option;2. At t=T_1 buy a call option on a put

option; 3. At t=T_1 sell a put option on a call

option;4. At t=T_1 sell a put option on a put

option;

Compound Options -

Three risky assets are involved: the underlying asset (stock), the underlying option (stock option) and the compound option.

First, in domain Σ_2{0≤ S<∞,0 ≤ t ≤ T_2}, define the underlying option price, which can be given by the Black-Scholes formula, denoted as V(S,t).

Compound Options --

Then on Σ_1{0≤ S<∞,0 ≤ t ≤ T_1}, set up a PDE problem for the compound option V_{co}(S,t). For this we again make use of the Δ-hedging technique to obtain the Black-Scholes equation for V_{co}(S,t).

Compound Options ---

At t=T_1, the corresponding terminal value is

For the case when r,q,σ are all constants,

we can obtain a pricing formula for this form of the compound option. Take a call on a call at t=T_1 as example.

^1

1^

1

( , ) , call( , )

( , ) , putco

V S T KV S T

K V S T

Compound Options ----

At t=0,

where

2

2 1

1 1 1 2

^2 2 1 2 2

( ,0) ( , ; / )

( , ; / ) ( ),

qTco

rT rT

V S Se M a b T T

Ke M a b T T e K N a

* 21

1 2 1 1

1

22

1 2 1 2

2

ln( / ) ( / 2),

ln( / ) ( / 2),

S S r q Ta a a T

T

S K r q Tb b b T

T

Compound Options ----- S^* is the root of the following equation:

and M(a,b;ρ) is the bivariate normal distribution

function: M(a,b;ρ)=Prob{X≤a,Y≤b}, where X~ N(0,1), Y~ N(0,1) are standard normal

distribution, Cov(X,Y)=ρ(-1<\rho<1).

2 1

2 1

* 22 1^ *

2 1

* 22 1

2 1

ln( / ) ( / 2)

ln( / ) ( / 2)

q T T

r T T

S K r q T TK S e N

T T

S K r q T TKe N

T T

Chooser Options (as you like it)Chooser option can be regarded as a

special form of compound option. The option holder is given this right:

at t=T_1 he can choose to let the option be a call option at strike price K_1, expiration date T_2 or let the option be a put option at strike price K_2, expiration date T^_2,(T_2,T^_2}>T_1>0).

Chooser Options -

Here four risky assets are involved: the underlying asset (stock), the underlying call option (stock option; strike

price K_1, expiration date T_2), the underlying put option (stock option; strike

price K_2, expiration dateT^_2) the chooser option.

Chooser Options --

Denote

both are solutions given by the Black-Scholes formula.

( , ) the underlying call option price

( , ) the underlying put option priceC

P

V S t

V S t

Chooser Options ---

In order to find the chooser option price V_{ch}(S,t) onΣ_1{0≤ S<∞,0 ≤ t ≤ T_1}, we need to solve the following terminal-boundary value problem:

1

2 22

2

1 1

( ) 0,2max ( , ), ( , )C Pt T

V V VS r q S rV

t S SV V S T V S T

Chooser Options ----

If the underlying call option V_C and put option V_P have the same strike price K and expiration date T_2, then by the call-put parity:

thus

2 1 2 1

1 1( , ) ( , ) r T T q T TP CV S T V S T Ke Se

1

2 1 2 1

1 1

1

max ( , ), ( , )

( , ) ( )

C Pt T

q T T r q T TC

V V S T V S T

V S T e Ke S

Chooser Options -----

Then by the superposition principle of the linear equations, we have

where

^( , ) ( , ) ( , )ch CV S t V S t V S t

2 1

2 1

( )^

( )( )1

( , )

put option( , )

q T T

r q T T

V S t e

T Ke

Numerical Methods (I) ------ Finite Difference Method

With the computation power we enjoy nowadays, numerical methods are often preferred, although for European option pricing closed-form solutions do exist. Especially for complex option pricing problems, such as compound options and chooser options, numerical methods are particularly advantageous.

BTM vs Finite Difference Method

The binomial tree method (BTM) is the most commonly used numerical method in option pricing.

Questions:

a. How to solve the Black-Scholes equation by finite difference method (FDM)?

b. What is the relation between FDM and BTM?

c. How to prove the convergence of BTM, which is a stochastic algorithm, in the framework of the numerical solutions of partial differential equations?

Introduction to Finite Difference Method

Finite difference method is a discretization approach to the boundary value problems for partial differential equations by replacing the derivatives with differences.

Types of Approaches

There are several approaches to set up finite difference equations corresponding to the partial differential equations.

Regarding the equation solving techniques, there are two basic types: 1. the explicit finite difference scheme, whose solving

process is explicit and solution can be obtained by direct computation;

2. the implicit finite difference scheme, whose solution can only be obtained by solving a system of algebraic equations.

Definition 5.1

Suppose is a finite difference equation obtained from discretization of the PDE Lu=0.

If for any sufficiently smooth function ω(x,t) there is

then the finite difference scheme

is said to be consistent with Lu=0.

0L u

, 0lim | | 0,t x

L L

L u

Lax's Equivalence Theorem

Given a properly posed initial-boundary value problem and a finite difference scheme to it that satisfies the consistency condition, then stability is the necessary and sufficient condition for convergence.

Implicit Finite Difference Scheme vs Explicit One

According to the numerical analysis theory of PDE, for the IB problem, the implicit FDS is unconditionally stable, whereas the explicit FDS is stable if a^2Δt/Δx^2=α≤1/2, and unstable if α>1/2. This result indicates, although the explicit FDS is relatively simple in algorithm, but in order to obtain a reliable result, the time interval must satisfy the condition Δt≤(1/2a^2)Δx^2. In contrast, although the implicit FDS requires solving a large system of linear equations in each step, the scheme is unconditionally stable, thus has no constraint on time interval Δt. That means if the computation accuracy is guaranteed, Δt can be large, and the result is still reliable.

Explicit FDS of the B-S Equation

We have shown the Black-Scholes equation can be reduced to a backward parabolic equation with constant coefficients under the transformation x=ln S:

2 2 2

2( ) 0,

2 2

( , ) ( )x

V V Vr q rV

t x x

V x T e K

Theorem 5.4

then the FD scheme of B-S is stable.

2 2

2

2

Set ( ) / ,

1if 1&1 | | 0,

2

t x

r q x

Numerical Methods (II) ------ BTM & FDM

BTM is essentially a stochastic algorithm. However, if S is regarded as a variable, and option price V=V(S,t) is regarded as a function of S,t, then BTM is an explicit discrete algorithm for option pricing. If the higher orders of Δt can be neglected, we will be able to show that it is indeed a special form of the explicit FDS of the Black-Scholes equation.

Theorem 5.5

If ud=1, ignoring higher order termsof Δt, then for European option, pricing the

BTM and the explicit FDS of the Black-Scholes equation

(ω=σ^2Δt/(ln u)^2=1) are equivalent.

Theorem 5.6(Convergence of BTM for E-

option) If

then as Δt→ 0, there must be

where V_Δ(S,t)$ is the linear extrapolation of V_m^n.

1 02

r qt

0lim ( , ) ( , ),tV S t V S t

Properties of European Option Price

European option price depends on 7 factors (take stock option as example): S (stock price), K (strike price), r (risk-free interest rate), q (dividend rate), T (expiration date) , t (time), σ (volatility).

Dependence on S

That is, as S increases, call option price goes up, and put option price goes down.

1

1

( ) ^

( ) ^

( ) 0

[1 ( )] 0

q T t

q T t

ce N d

Sp

e N dS

Dependence on K

For different strike prices, call option price decreases with K, and put option price increases with K.

2

2

( ) ^

( ) ^

( ) 0

[1 ( )] 0

r T t

r T t

ce N d

Kp

e N dK

Dependence on S & K Financially

When the stock price goes up or the strike price goes down, the call option holders are more likely to gain more profits in the future, thus the call option price goes up; In contrast, the put option holders have smaller chance to gain profits in the future, thus the put option price goes down.

Dependence on r

If the risk-free interest rate goes up, then the call option price goes up, but the put option price goes down.

( ) ^2

( ) ^2

( ) ( ) 0

( ) [1 ( )] 0

r T t

r T t

cK T t e N d

rp

K T t e N dr

Dependence on r Financially The risk-free interest rate raise has two effects: for

stock price, in a risk-neutral world, the expected return E(dS/S)=(r-q)dt will go up; For cash flow, the cash K received at the future time (t=T) would have a lower value Ke^{-r(T-t)} at the present time t. Therefore, for put option holders, who will sell stocks for cash at the maturity t=T, thus the above two effects result in a decrease of the put option price. For call option holders, the effects are just the opposite, and the option price will go up.

Dependence on q

If the dividend rate increases, then the call option price goes down, and the put option price goes up.

1

1

( ) ^

( ) ^

( ) ( ) 0

( ) [1 ( )] 0

q T t

q T t

cS T t e N d

q

pS T t e N d

q

Dependence on q Financially

The dividend rate directly affects the stock price. In a risk-neutral world, as the dividend rate increases, the expected return of the stock

E(dS/S)=(r-q)dt decreases, thus the call option price decreases, but the put option price increases.

Dependence on σ

when a stock has a high volatility σ, its option price (both call and put) goes up.

1

( ) ^'( ) 0q T tc pe N d T t

Dependence on σFinancially An increase of the volatility σ means an increase of the

stock price fluctuation, i.e., increased investment risk. For the underlying asset itself (the stock), since E(σdW_t)=0, the risks (gain or loss) are symmetric. But this is not true for an individual option holder.

Example (call): The holder benefits from stock price increases, but has only limited downside risk in the event of stock price decreases, because the holder's loss is at most the option's premium. Therefore the stock price change has an asymmetric impact on the call option value. Therefore the call option price increases as the volatility increases. Same reasoning can be applied to the put option.

Dependence on t&T

1 2

1

2

1

( ) ^ ( ) ^

( ) ^

( ) ^ ( ) ^1

( ) ^

( ) ( )

( )(sign is not fixed)

2

[1 ( )] [1 ( )]

[1 ( )](sign is not fixed)

2

q T t r T t

q T t

r T t q T t

q T t

cqSe N d rKe N d

t

e SN d c

TT tp

rKe N d qSe N dt

e S N d p

TT t

Dependence on t&T Financially

No matter how long the option's lifetime T is, a European option has only one exercise. A long expiration does not mean more gaining opportunity. So, European options do not become more valuable as time to expiration increases.

As for t, larger t, smaller T-t, means closer to the exercise day. Therefore, for European options we cannot predict whether the option price will go down or go up as the exercise day comes closer.

Dependence on t&T Financially-

However, there is an exception. In the case q=0,

i.e. with the expiration day coming closer, the call option on a non-dividend-paying stock will go down.

1

^( ) ^

2

( )( ) 0

2r T t

SN dcrKe N d

t T t

Table of European Option Price Changes

call put

S + -

K - +

r + -

q - +

σ + +

T ? ?

t ? ?

Risk Management—Δ(Sigma)

Δ is the partial derivative of the option or its portfolio price V with respect to the underlying asset price S. The seller of the option or its portfolio should buy Δ shares of the underlying asset to hedge the risk inherited in selling the option or portfolio.

V

S

Risk Management—Γ(Gamma)

Since Δ is a function of S & t, one must constantly adjust Δ to achieve the goal of the hedging. In practice, this is not feasible because of the transaction fee. Therefore in real operation one must choose the frequency of Δ wisely. This is reflected in the magnitude of Γ. A small Γ means Δ changes slowly, and there is no need to adjust in haste; Conversely, if Γ is large, then Δ is sensitive to change in S, there will be a risk if Δ is not adjusted in time.

2

2

V

S S

Risk Management—Θ(Theta)

Θ is the rate of change in the option or portfolio price over time. The Black-Scholes equation gives the relation between Δ, Γ and Θ:

22 ( )

2S r q S rV

Risk Management—(Vega)

is the partial derivative of the option or its portfolio price with respect to the volatility of the underlying asset.

The underlying asset volatility σ is the least known parameter in the Black-Scholes formula. It is practically impossible to give a precise value of σ Instead, we consider the sensitivity of the corresponding option price over σ This is the meaning of

VV

s¶

=¶

Risk Management—ρ(rho)

ρ is the partial derivative of the option or portfolio price with respect to the risk-free interest rate.

How to Manage Risk?

For European options, we have obtained the expressions of these Greeks in the previous section. Now we will explain how to use these parameters (especially Δ and Γ) in risk management.

A Specific Example

Suppose a financial institution has sold a stock option OTC, and faces a risk due to the option price change. Therefore it wants to take a hedging strategy to manage the risk. Ideally, a hedging strategy should guarantee an approximate balance of the expense and income, i.e., the money spent on hedging approximately equals the income from selling the option premiums.

How does the hedging strategy work?

At t=0 the seller buys Δ_0 shares of stock at S_0 per share, and borrows Δ_0 S_0 from the bank. At t=t_1, to adjust the hedging share to Δ_1 (S_1 is the stock price at t=t_1), the seller needs to buy Δ_1-Δ_0 shares at S_1 per share if Δ_1>Δ_0 and sell Δ_0- Δ_1 shares at S_1 per share if Δ_1<Δ_0; and borrow (save) the money needed (gained) for (from) buying (selling) the stocks, and at t=t_1 pay the interest Δ_0 S_0rΔt to the bank for the money borrowed at t=t_0. In general, at t=t_n, the seller owns Δ_n shares of stock, and has paid hedging cost D_n:

How does the hedging strategy work? -

On the option expiration day t=T, the seller owns Δ_N shares of stock, i.e. if S_T>K (i.e. the option is in the money), the seller

owns one share of stock, if S_{T}<K (i.e. the option is out of the money), the

seller owns no share of stock. If the option is in the money, the option holder will

exercise the contract to buy one share of stock S from the seller with cash K;

if the option is out of the money, the option holder will certainly choose not to exercise the contract.

How does the hedging strategy work?--

The above hedging strategy successfully hedges the risk in selling the option.

In this deal the sellor's actual profit is

profit=V_0e^{rT}-D_T,

where V_0 is the option premium. If there is a transaction fee for each hedging strategy adjustment, then the seller's profit is

profit=V_0e^{rT}-D_T-Σ_{i=0}^{N-1}e_i,

where e_i is the fee for the i-th adjustment.

Remark

In practical operation, hedging adjustment interval Δt is not a constant, and depends

on . If Γ is large, adjustment

is made more frequently; If Γ is small, adjustment can be made less frequently.

2

2

V

S S

Summary 1

Introduced a continuous model for the underlying asset price movement---the stochastic differential equation. Based on this model, using the Δ-hedging technique and the Ito formula, we derived the Black-Scholes equation for the option price, by solving the terminal value problem of the Black-Scholes equation, we obtained a fair price for the European option, independent of each individual investor's risk preference---the Black-Scholes formula.

Summary 2

As derivatives of an underlying asset, a variety of options can be set up in a various terminal-boundary problem for the Black-Scholes equation. To price these various options is to solve the Black-Scholes equation under various terminal-boundary conditions.

Summary 3

BTM is the most important discrete method of option pricing. When neglecting the higher orders of Δt, BTM is equivalent to an explicit finite difference scheme of the Black-Scholes equation. By the numerical solution theory of partial differential equation, we have proved the convergence of the BTM.

Summary 4The option seller can manage the risk in

selling the option by taking a hedging strategy. Since the amount of hedging shares Δ= Δ(S,t) changes constantly, the seller needs to adjust Δ at appropriate frequency according to the magnitude of Γ(S,t), to achieve the goal of hedging.