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In the mean time, potential losses cannot exceed the option price.
The effect of the risk-free rate is not straightforward. At a fixed
stock price, the rising risk-free rate increases the value of the call
option. Indeed, the option holder may defer paying for shares and
invest this payment into the risk-free assets until the option matures.
On the contrary, the value of the put option decreases with the risk-
free rate since the option holder defers receiving payment from selling
shares and therefore cannot invest them into the risk-free assets.
However, rising interest rates often lead to falling stock prices,
which may change the resulting effect of the risk-free rate.
Dividends effectively reduce the stock prices. Therefore, dividends
decrease value of call options and increase value of put options.
Now, let us consider the payoffs at maturity for four possible
European option positions. The long call option means that the in-
vestor buys the right to buy an underlying asset. Obviously, it makes
sense to exercise the option only if S > K. Therefore, its payoff is
PLC ¼ max [S À K, 0] (9:2:1)
The short call option means that the investor sells the right to buy an
underlying asset. This option is exercised if S > K, and its payoff is
PSC ¼ min [K À S, 0] (9:2:2)
The long put option means that the investor buys the right to sell an
underlying asset. This option is exercised when K > S, and its payoff
is
PLP ¼ max [K À S, 0] (9:2:3)
The short put option means that the investor sells the right to sell an
underlying asset. This option is exercised when K > S, and its payoff is
96 Option Pricing



PSP ¼ min [S À K, 0] (9:2:4)
Note that the option payoff by definition does not account for the
option price (also named option premium). In fact, option writers sell
options at a premium while option buyers pay this premium. There-
fore, the option seller™s profit is the option payoff plus the option
price, while the option buyer™s profit is the option payoff minus the
option price (see examples in Figure 9.1).
The European call and put options with the same strike price
satisfy the relation called put-call parity. Consider two portfolios.
Portfolio I has one European call option at price c with the strike
price K and amount of cash (or zero-coupon bond) with the present
value Kexp[Àr(T À t)]. Portfolio II has one European put option at
price p and one share at price S. First, let us assume that share does
not pay dividends. Both portfolios at maturity have the same value:
max (ST , K). Hence,
c þ Kexp[Àr(T À t)] ¼ p þ S (9:2:5)
Dividends affect the put-call parity. Namely, the dividends D being
paid during the option lifetime have the same effect as the cash future
value. Thus,
c þ D þ K exp [Àr(T À t)] ¼ p þ S (9:2:6)
Because the American options may be exercised before maturity, the
relations between the American put and call prices can be derived
only in the form of inequalities [1].
Options are widely used for both speculation and risk hedging.
Consider two examples with the IBM stock options. At market
closing on 7-Jul-03, the IBM stock price was $83.95. The (American)
call option price at maturity on 3-Aug-03 was $2.55 for the strike
price of $85. Hence, the buyer of this option at market closing on 7-
Jul-03 assumed that the IBM stock price would exceed $(85 þ 2.55) ¼
$87.55 before or on 3-Aug-03. If the IBM share price would reach say
$90, the option buyer will exercise the call option to buy the share for
$85 and immediately sell it for $90. The resulting profit4 is
$(90À87.55) ¼ $2.45. Thus, the return on exercising this option equals
2:45=2:55Ã 100% ¼ 96%. Note that the return on buying an IBM
share in this case would only be (90 À 83:95)=83:95Ã 100% ¼ 7:2%.
97
Option Pricing



(a) 20
Profit

15


10

Short Call
5
Stock price
0
0 5 10 15 20 25 30 35 40

’5
Long Call

’10


’15


’20



(b) 20
Profit

15
Long Put
10


5
Stock price
0
0 5 10 15 20 25 30 35 40

’5


’10
Short Put
’15


’20
Figure 9.1 The option profits for the strike price of $25 and the option
premium of $5: (a) calls, (b) puts.
98 Option Pricing



If, however, the IBM share price stays put through 3-Aug-03, an
option buyer incurs losses of $2.45 (i.e., 100%). In the mean time, a
share buyer has no losses and may continue to hold shares, hoping
that their price will grow in future.
At market closing on 7-Jul-03, the put option for the IBM share
with the strike price of $80 at maturity on 3-Aug-03 was $1.50. Hence,
buyers of this put option bet on price falling below $(80À1.50) ¼
$78.50. If, say the IBM stock price falls to $75, the buyer of the put
option has a gain of $(78:50 À 75) ¼ $3.50.
Now, consider hedging in which the investor buys simultaneously
one share for $83.95 and a put option with the strike price of $80 for
$1.50. The investor has gains only if the stock price rises above
$(83:95 þ 1:50) ¼$85:45. However, if the stock price falls to say $75,
the investor™s loss is $(80 À 85:45) ¼ À$5:45 rather than the loss of
$(75 À 83:95) ¼ À$8:95 incurred without hedging with the put
option. Hence, in the given example, the hedging expense of $1.50
allows the investor to save $(À5:45 þ 8:95) ¼$3:40.


9.3 BINOMIAL TREES
Let us consider a simple yet instructive method for option pricing
that employs a discrete model called the binomial tree. This model is
based on the assumption that the current stock price S can change at
the next moment only to either the higher value Su or the lower value
Sd (where u > 1 and d < 1). Let us start with the first step of the
binomial tree (see Figure 9.2). Let the current option price be equal to
F and denote it with Fu or Fd at the next moment when the stock price
moves up or down, respectively. Consider now a portfolio that con-
sists of D long shares and one short option. This portfolio is risk-free
if its value does not depend on whether the stock price moves up or
down, that is,
SuD À Fu ¼ SdD À Fd (9:3:1)
Then the number of shares in this portfolio equals
D ¼ (Fu À Fd )=(Su À Sd) (9:3:2)
The risk-free portfolio with the current value (SD À F) has the future
value (SuD À Fu ) ¼ (SdD À Fd ). If the time interval is t and the risk-
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Option Pricing



Su2

Fuu


Su

Fu


Sud
S
F
Fud


Sd

Fd


Sd2

Fdd
Figure 9.2 Two-step binomial pricing tree.


free interest rate is r, the relation between the portfolio™s present value
and future value is
(SD À F) exp(rt) ¼ SuD À Fu (9:3:3)
Combining (9.3.2) and (9.3.3) yields
F ¼ exp(Àrt)[pFu þ (1 À p)Fd ] (9:3:4)
where
p ¼ [ exp (rt) À d]=(u À d) (9:3:5)
The factors p and (1 À p) in (9.3.4) have the sense of the probabilities
for the stock price to move up and down, respectively. Then, the
expectation of the stock price at time t is
E[S(t)] ¼ E[pSu þ (1 À p)Sd] ¼ S exp (rt) (9:3:6)
This means that the stock price grows on average with the risk-free
rate. The framework within which the assets grow with the risk-free
rate is called risk-neutral valuation. It can be discussed also in terms of
the arbitrage theorem [4]. Indeed, violation of the equality (9.3.3)
100 Option Pricing



implies that the arbitrage opportunity exists for the portfolio. For
example, if the left-hand side of (9.3.3) is greater than its right-hand
side, one can immediately make a profit by selling the portfolio and
buying the risk-free asset.
Let us proceed to the second step of the binomial tree. Using
equation (9.3.4), we receive the following relations between the option
prices on the first and second steps
Fu ¼ exp (Àrt)[pFuu þ (1 À p)Fud ] (9:3:7)
Fd ¼ exp (Àrt)[pFud þ (1 À p)Fdd ] (9:3:8)
The combination of (9.3.4) with (9.3.7) and (9.3.8) yields the current
option price in terms of the option prices at the next step
F ¼ exp (À2rt)[p2 Fuu þ 2p(1 À p)Fud þ (1 À p)2 Fdd ] (9:3:9)
This approach can be generalized for a tree with an arbitrary number
of steps. Namely, first the stock prices at every node are calculated
by going forward from the first node to the final nodes. When the
stock prices at the final nodes are known, we can determine the
option prices at the final nodes by using the relevant payoff relation
(e.g., (9.2.1) for the long call option). Then we calculate the option
prices at all other nodes by going backward from the final nodes to
the first node and using the recurrent relations similar to (9.3.7) and
(9.3.8).
The factors that determine the price change, u and d, can be
estimated from the known stock price volatility [1]. In particular, it
is generally assumed that prices follow the geometric Brownian
motion
dS ¼ mSdt þ sSdW (9:3:10)
where m and s are the drift and diffusion parameters, respectively, and
dW is the standard Wiener process (see Section 4.2). Hence, the price
changes within the time interval [0, t] are described with the lognor-
mal distribution
p¬¬
ln S(t) ¼ N( ln S0 þ (m À s2 =2)t, s t) (9:3:11)
In (9.3.11), S0 ¼ S(0), N(m, s) is the normal distribution with mean
m and standard deviation s. It follows from equation (9.3.11) that the
expectation of the stock price and its variance at time t equal
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Option Pricing



E[S(t)] ¼ S0 exp (mt) (9:3:12)
Var[S(t)] ¼ S0 2 exp (2mt)[ exp (s2 t) À 1] (9:3:13)
In addition, equation (9.3.6) yields
exp (rt) ¼ pu þ (1 À p)d (9:3:14)
Using (9.3.13) and (9.3.14) in the equality (y) ¼ E[y2 ] À E[y]2 , we
obtain the relation
exp (2rt þ s2 t) ¼ pu2 þ (1 À p)d2 (9:3:15)
The equations (9.3.14) and (9.3.15) do not suffice to define the three
parameters d, p, and u. Usually, the additional condition
u ¼ 1=d (9:3:16)
is employed. When the time interval Dt is small, the linear approxi-
mation to the system of equations (9.3.14) through (9.3.16) yields
p ¼ [ exp (rDt) À d]=(u À d), u ¼ 1=d ¼ exp [s(Dt)1=2 ] (9:3:17)
The binomial tree model can be generalized in several ways [1]. In
particular, dividends and variable interest rates can be included. The
trinomial tree model can also be considered. In the latter model, the
stock price may move upward or downward, or it may stay the same.
The drawback of the discrete tree models is that they allow only for
predetermined innovations of the stock price. Moreover, as it was
described above, the continuous model of the stock price dynamics
(9.3.10) is used to estimate these innovations. It seems natural then to
derive the option pricing theory completely within the continuous
framework.

9.4 BLACK-SCHOLES THEORY
The basic assumptions of the classical option pricing theory are
that the option price F(t) at time t is a continuous function of time
and its underlying asset™s price S(t)
F ¼ F(S(t), t) (9:4:1)
and that price S(t) follows the geometric Brownian motion (9.3.10) [5,
6]. Several other assumptions are made to simplify the derivation of
the final results. In particular,
102 Option Pricing



. There are no market imperfections, such as price discreteness,
transaction costs, taxes, and trading restrictions including those
on short selling.
. Unlimited risk-free borrowing is available at a constant rate, r.
. There are no arbitrage opportunities.
. There are no dividend payments during the life of the option.
Now, let us derive the classical Black-Scholes equation. Since it is
assumed that the option price F(t) is described with equation (9.4.1)
and price of the underlying asset follows equation (9.3.10), we can use
the Ito™s expression (4.3.5)
 
@F @F s2 2 @ 2 F @F
dF(S, t) ¼ mS þ þS dt þ sS dW(t) (9:4:2)
@S2
2
@S @t @S
Furthermore, we build a portfolio P with eliminated random contri-
@F
bution dW. Namely, we choose À1 (short) option and shares of
@S
5
the underlying asset,
@F
P ¼ ÀF þ S (9:4:3)
@S
The change of the value of this portfolio within the time interval dt
equals
@F
dP ¼ ÀdF þ dS (9:4:4)
@S
Since there are no arbitrage opportunities, this change must be equal to
the interest earned by the portfolio value invested in the risk-free asset
dP ¼ rP dt (9:4:5)
The combination of equations (9.4.2)“(9.4.5) yields the Black-Scholes
equation
@F s2 2 @ 2 F
@F
þ rS þS À rF ¼ 0 (9:4:6)
@S2
2
@t @S
Note that this equation does not depend on the stock price drift
parameter m, which is the manifestation of the risk-neutral valuation.
In other words, investors do not expect a portfolio return exceeding
the risk-free interest.
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Option Pricing



The Black-Scholes equation is the partial differential equation with
the first-order derivative in respect to time and the second-order de-
rivative in respect to price. Hence, three boundary conditions deter-
mine the Black-Scholes solution. The condition for the time variable is
defined with the payoff at maturity. The other two conditions for the
price variable are determined with the asymptotic values for the zero
and infinite stock prices. For example, price of the put option equals
the strike price when the stock price is zero. On the other hand, the put
option price tends to be zero if the stock price approaches infinity.
The Black-Scholes equation has an analytic solution in some
simple cases. In particular, for the European call option, the Black-
Scholes solution is
c(S, t) ¼ N(d1 )S(t) À KN(d2 ) exp[Àr(T À t)] (9:4:7)
In (9.4.7), N(x) is the standard Gaussian cumulative probability
distribution
d1 ¼ [ ln (S=K) þ (r þ s2 =2)(T À t)]=[s(T À t)1=2 ],
(9:4:8)
1=2
d2 ¼ d1 À (T À t)
The Black-Scholes solution for the European put option is
p(S, t) ¼ K exp[Àr(T À t)] N(Àd2 ) À S(t)N(Àd1 ) (9:4:9)
The value of the American call option equals the value of the Euro-
pean call option. However, no analytical expression has been found
for the American put option. Numerical methods are widely used for
solving the Black-Scholes equation when analytic solution is not
available [1“3].
Implied volatility is an important notion related to BST. Usually,
the stock volatility used in the Block-Scholes expressions for the
option prices, such as (9.4.7), is calculated with the historical stock
price data. However, formulation of the inverse problem is also
possible. Namely, the market data for the option prices can be used
in the left-hand side of (9.4.7) to recover the parameter s. This
parameter is named the implied volatility. Note that there is no
analytic expression for implied volatility. Therefore, numerical
methods must be employed for its calculation. Several other functions
related to the option price, such as Delta, Gamma, and Theta (so-
called Greeks), are widely used in the risk management:
104 Option Pricing



@2F
@F @F
D¼ ,G¼ 2,Q¼ (9:4:10)
@S @S @t
The Black-Scholes equation (9.4.6) can be rewritten in terms of
Greeks
s2 2
Q þ rSD þ S G À rF ¼ 0 (9:4:11)
2
Similarly, Greeks can be defined for the entire portfolio. For example,

@P @S
the portfolio™s Delta is . Since the share™s Delta equals unity,
@S @S
Delta of the portfolio (9.4.3) is zero. Portfolios with zero Delta are
called delta-neutral. Since Delta depends on both price and time,
maintenance of delta-neutral portfolios requires periodic rebalancing,
which is also known as dynamic hedging. For the European call and
put options, Delta equals, respectively
Dc ¼ N(d1 ), Dp ¼ N(d1 ) À 1 (9:4:12)
Gamma characterizes the Delta™s sensitivity to price variation. If
Gamma is small, rebalancing can be performed less frequently.
Adding options to the portfolio can change its Gamma. In particular,
delta-neutral portfolio with Gamma G can be made gamma-neutral if
it is supplemented with n ¼ ÀG=GF options having Gamma GF .
Theta characterizes the time decay of the portfolio price. In add-
ition, two other Greeks, Vega and Rho, are used to measure the
portfolio sensitivity to its volatility and risk-free rate, respectively
@P @P
y¼ ,r¼ (9:4:13)
@s @r

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