European Option Pricing

1
Chapter 5

• European Option Pricing
• ------
• Black-Scholes Formula

2
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.

3
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.

4
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.

5
History--

• P. Samuelson studied the call option pricing
  problem (C. Sprenkle (1965) and J. Baness (1964)
  also studied it at the same time). The result is
  given in the following. (V,etc as before)

6
History---

• In 1973, Fischer Black and Myron Scholes gave
  the following call
• option pricing formula
• Comparing to Samuelsons one, are no
  longer present. Instead, the risk-free interest r
  enters the formula.

7
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.

8
Basic Assumptions

• (a) The underlying asset price follows the
  geometrical Brownian motion
• µ expected return rate (constant)
• s- volatility (constant)
• - standard Brownian motion

9
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

10
A Problem

• Let VV(S,t) denote the option price. At maturity
  (tT),
• where K is the strike price.
• What is the option's value during its lifetime
  (0lt tltT)?

11
?-Hedging Technique

• Construct a portfolio
• (? denotes shares of the underlying asset),
  choose ? such that ? is risk-free in (t,tdt).

12
?-Hedging Technique -

• If portfolio ? starts at time t, and ? remains
  unchanged in (t,tdt), then the requirement ? be
  risk-free means the return of the portfolio at
  tdt should be

13
?-Hedging Technique --

• Since
• where the stochastic process satisfies
  SDE, hence by Ito formula
• So

14
?-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

15
?-Hedging Technique ----Black-Scholes Equation

• Substituting it, we get the following PDE
• This is the Black-Scholes Equation that describes
  the option price movement.

16
Remark

• The line segment S0, 0lt tlt T is also a
  boundary of the domain . However, since the
  equation is degenerated at S0, according to the
  PDE theory, there is no need to specify the
  boundary value at S0.

17
Black-Scholes Equation in Cauchy Form

18
Well-posed Problem

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

19
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 ?---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.

20
An Interesting Question

• 1) Starting from the discrete option price
  obtained by the BTM, by interpolation, we can
  define a function on the domain
• S0lt Slt8,0lttltT
• 2) If there exists a function V(S,t), such that
  
• 3) if V(S,t) 2nd derivatives are continuous in
  S,
• What differential equation does V(S,t) satisfy?

21
Answer of the Question

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

22
Black-Scholes Formula (call)

23
Black-Scholes Formula (put)

24
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 rr(t)
• (c) The underlying asset pays continuous
  dividends at rate q(t)
• (d) and (e) remain unchanged

25
?-hedging

• Use the ?-hedging technique to set up a
  continuous model of the option pricing, and find
  valuation formulas.
• Construct a portfolio
• Choose ?, so that ? is risk-neutral in t,tdt.
• the expected return is

26
?-hedging -

• Taking into account the dividends, the
  portfolio's value at is
• Therefore, we have

27
B-S Equation with Dividend

• Apply Ito formula, and choose
• we have
• Thus, B-S Equation with dividend is

28
Solve B-S Equation with dividend

• Set
• choose aßto eliminate 0 1st terms

29
Solve B-S Equation with dividend -

• Let aß be the solutions to the following initial
  value problems of ODE
• The solutions of the ODE are

30
Solve B-S Equation with dividend --

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

31
Apply B-S Formula

• Let s1, r0, TTˆ, tt,

32
European Option Pricing (call, with dividend)

• Back to the original variables, we have

33
European Option Pricing (put, with dividend)
34
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 rr(t) is the risk-free interest rate,
  qq(t) is the dividend rate, and s s(t) is the
  volatility.

35
Proof of Theorem 5.1

• Consider the difference between a call and a put
• W(S,t)c(S,t)-p(S,t).
• At tT,
• W is the solution of the following problem

36
Proof of Theorem 5.1-

• Let W be of the form Wa(t)S-b(t)K, then
• Choose a(t),b(t) such that

37
Proof of Theorem 5.1--

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

38
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 tt_1 (0ltt_1ltT)
• (if the asset is a stock, then Q is the dividend
  per share).
• After the dividend payday tt_1, there will be a
  change in stock price
• S(t_1-0)S(t_10)Q.

39
Predetermined Date Dividend -

• However, the option price must be continuous at
  tt_1
• V(S(t_1-0),t_1-0)V(S(t_10),t_10).
• Therefore, S and V must satisfy the boundary
  condition at tt_1
• V(S,t_1-0)V(S-Q,t_10)
• In order to set up the option pricing model
  (take call option as example), consider two
  periods 0,t_1, t_1,T separately.

40
Predetermined Date Dividend --

• In 0 Slt8, t_1 t T, VV(S,t) satisfies the
  boundary-terminal value problem
• Obtain VV(S,t) on t_1

41
Predetermined Date Dividend ---

• in 0 Slt8,0 t t_1,VV(S,t) satisfies
• By solving above problems, we can determine the
  premium V(S_0,0) to be paid at the initial date
  t0 (S_0 is the stock price at that time).

42
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.

43
Remark1-

• In the case of assumption (c), we used the
  dividend rate qq(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 tt_1 with the
  intensity d_Q,
• then at tt_1 the stock price
  Thus we can derive from the corresponding option
  pricing formula.

44
Remark1--

• In the case of assumption (c), we used the
  dividend Q, which is related to the stock price
  itself. So at the payday tt_1, the stock price
• Thus we have at tt_1 the boundary condition for
  the option price.
• We should be aware of this difference when
  solving real problems.

45
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 ?-hedging technique,
  for the portfolio
• where ?qdt denotes the storage fee for ? amount
  of commodity and period dt.

46
Remark 2-

• Similar to the derivation we did before, choose
  such that ? is risk-free in
  (t,tdt). Then we get the terminal-boundary
  problem for the option price VV(S,t)
• This equation does not have a closed form
  solution in general. Numerical approach is
  required.

47
Remark 2--

• If the storage fee for ? items of commodity and
  period dt is in the form of ?qSdt, proportional
  to the current price of the commodity, then
• And the option price is given by the
  Black-Scholes formula.

48
Generalized B-S Model (II) ------Binary Options

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

49
Cash-or-nothing call

• Cash-or-nothing call (CONC)
• In Case tT stock price lt strike price, the
  option 0
• In Case tT stock price gt strike price, the
  holder gets 1 in cash.

50
Asset-or nothing call

• Asset-or nothing call (AONC)
• In Case tT stock price lt strike price, the
  option 0
• In Case tT stock price lt strike price, the
  option pays the stock price.

51
Modeling

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

52
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
  tT, these prices satisfy

53
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 S0 Slt8,0 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.

54
Theorem 5.2

55
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

56
Proof of Theorem 5.2 -

• In S, 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.

57
Solve pricing of CONC AONC

• Once the CONC price V_C(S,tr,q) is found, the
  AONC price V_A(S,tr,q) can be determined by
  Theorem 5.2.
• In order to solve the CONC problem, make
  transformation

58
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

59
Solve pricing of CONC AONC--

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

60
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. tT_1) at a
  certain price K a call (put) option with the
  expiration date tT_2 (T_2gtT_1) and the strike
  price K. There are following forms of compound
  options

61
Compound Options

• 1. At tT_1 buy a call option on a call option
• 2. At tT_1 buy a call option on a put option
• 3. At tT_1 sell a put option on a call option
• 4. At tT_1 sell a put option on a put option

62
Compound Options -

• Three risky assets are involved the underlying
  asset (stock), the underlying option (stock
  option) and the compound option.
• First, in domain S_20 Slt8,0 t T_2, define
  the underlying option price, which can be given
  by the Black-Scholes formula, denoted as V(S,t).

63
Compound Options --

• Then on S_10 Slt8,0 t T_1, set up a PDE
  problem for the compound option V_co(S,t). For
  this we again make use of the ?-hedging technique
  to obtain the Black-Scholes equation for
  V_co(S,t).

64
Compound Options ---

• At tT_1, the corresponding terminal value is
• For the case when r,q,s are all constants, we can
  obtain a pricing formula for this form of the
  compound option. Take a call on a call at tT_1
  as example.

65
Compound Options ----

• At t0,
• where

66
Compound Options -----

• S is the root of the following equation
• and M(a,b?) is the bivariate normal
  distribution function
• M(a,b?)ProbXa,Yb,
• where X N(0,1), Y N(0,1) are standard normal
  distribution, Cov(X,Y)?(-1lt\rholt1).

67
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 tT_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_2gtT_1gt0).

68
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.

69
Chooser Options --

• Denote
• both are solutions given by the Black-Scholes
  formula.

70
Chooser Options ---

• In order to find the chooser option price
  V_ch(S,t) onS_10 Slt8,0 t T_1, we need to
  solve the following terminal-boundary value
  problem

71
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

72
Chooser Options -----

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

73
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.

74
BTM vs Finite Difference Method

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

75
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?

76
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.

77
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.

78
Definition 5.1

• Suppose is a finite difference
  equation obtained from discretization of the PDE
  Lu0.
• If for any sufficiently smooth function ?(x,t)
  there is
• then the finite difference scheme
  is said to be consistent with Lu0.

79
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.

80
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 a2?t/?x2a1/2, and unstable if
  agt1/2. 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
  ?t(1/2a2)?x2. 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 ?t. That means if the computation
  accuracy is guaranteed, ?t can be large, and the
  result is still reliable.

81
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
  xln S

82
Theorem 5.4

• then the FD scheme of B-S is stable.

83
Numerical Methods (II) ------ BTM FDM

• BTM is essentially a stochastic algorithm.
  However, if S is regarded as a variable, and
  option price VV(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 ?t 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.

84
Theorem 5.5

• If ud1, ignoring higher order terms
• of ?t, then for European option, pricing the BTM
  and the explicit FDS of the Black-Scholes
  equation
• (?s2?t/(ln u)21) are equivalent.

85
Theorem 5.6(Convergence of BTM for E- option)

• If
• then as ?t? 0, there must be
• where V_?(S,t) is the linear extrapolation of
  V_mn.

86
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),
  s (volatility).

87
Dependence on S

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

88
Dependence on K

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

89
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.

90
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.

91
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 (tT) 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 tT, 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.

92
Dependence on q

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

93
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.

94
Dependence on s

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

95
Dependence on sFinancially

• An increase of the volatility s means an increase
  of the stock price fluctuation, i.e., increased
  investment risk. For the underlying asset itself
  (the stock), since E(sdW_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.

96
Dependence on tT
97
Dependence on tT 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.

98
Dependence on tT Financially-

• However, there is an exception. In the case q0,
• i.e. with the expiration day coming closer, the
  call option on a non-dividend-paying stock will
  go down.

99
Table of European Option Price Changes
call put
S -
K -
r -
q -
s
T ? ?
t ? ?
100
Risk Management?(Sigma)

• ? 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 ? shares of the underlying
  asset to hedge the risk inherited in selling the
  option or portfolio.

101
Risk ManagementG(Gamma)

• Since ? is a function of S t, one must
  constantly adjust ? 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 ?
  wisely. This is reflected in the magnitude of G.
  A small G means ? changes slowly, and there is no
  need to adjust in haste Conversely, if G is
  large, then ? is sensitive to change in S, there
  will be a risk if ? is not adjusted in time.

102
Risk ManagementT(Theta)

• T is the rate of change in the option or
  portfolio price over time. The Black-Scholes
  equation gives the relation between ?, G and T

103
Risk ManagementV (Vega)

• V is the partial derivative of the option or its
  portfolio price with respect to the volatility of
  the underlying asset.
• The underlying asset volatility s is the least
  known parameter in the Black-Scholes formula. It
  is practically impossible to give a precise value
  of s Instead, we consider the sensitivity of the
  corresponding option price over s This is the
  meaning of V

104
Risk Management?(rho)

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

105
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 ? and G) in risk
  management.

106
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.

107
How does the hedging strategy work?

• At t0 the seller buys ?_0 shares of stock at S_0
  per share, and borrows ?_0 S_0 from the bank. At
  tt_1, to adjust the hedging share to ?_1 (S_1 is
  the stock price at tt_1), the seller needs to
  buy ?_1-?_0 shares at S_1 per share if ?_1gt?_0
  and sell ?_0- ?_1 shares at S_1 per share if
  ?_1lt?_0 and borrow (save) the money needed
  (gained) for (from) buying (selling) the stocks,
  and at tt_1 pay the interest ?_0 S_0r?t to the
  bank for the money borrowed at tt_0. In general,
  at tt_n, the seller owns ?_n shares of stock,
  and has paid hedging cost D_n

108
How does the hedging strategy work? -

• On the option expiration day tT, the seller owns
  ?_N shares of stock, i.e.
• if S_TgtK (i.e. the option is in the money), the
  seller owns one share of stock,
• if S_TltK (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.

109
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
• profitV_0erT-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
• profitV_0erT-D_T-S_i0N-1e_i,
• where e_i is the fee for the i-th adjustment.

110
Remark

• In practical operation, hedging adjustment
  interval ?t is not a constant, and depends
• on . If G is large,
  adjustment
• is made more frequently If G is small,
  adjustment can be made less frequently.

111
Summary 1

• Introduced a continuous model for the underlying
  asset price movement---the stochastic
  differential equation. Based on this model,
  using the ?-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.

112
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.

113
Summary 3

• BTM is the most important discrete method of
  option pricing. When neglecting the higher orders
  of ?t, 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.

114
Summary 4

• The option seller can manage the risk in selling
  the option by taking a hedging strategy. Since
  the amount of hedging shares ? ?(S,t) changes
  constantly, the seller needs to adjust ? at
  appropriate frequency according to the magnitude
  of G(S,t), to achieve the goal of hedging.