Derivative Pricing and Financial Modelling
Wei Wei
Department of Actuarial Mathematics and Statistics
Heriot-Watt University
January 29, 2022
1
Introduction to financial derivatives
A financial derivative is a contract whose value is based on the performance of
an underlying asset. The underlying assets can be stock, commodities (metals,
oil, other physical products), currencies, etc. According to the flexibility of execution, financial derivatives can be generally divided into two categories: European
derivatives and American derivatives. A European derivative is a contract
which limits execution on its maturity date. An American derivative is a contract
which delivers its payoff at any time before or upon its maturity date. This course
is about how to price a derivative at any time before the maturity date. The key
to price a financial derivative is to tame the uncertainty of the derivative. Since the
uncertainty arises from its underlying asset(s), the key to pricing derivatives is to
use the underlying asset(s) to remove the risk from trading the derivative. The act
of removing the uncertainty is called hedging.
1.1
Value of money
Money has time value, that is, money available at the present time is worth more
than the identical sum in the future. The time value of money is charactered by
interest rates. There are several methods to calculating the value of money at
different dates. Here we briefly introduce simple interest, compound interest and
continuous interest. Suppose that the interest accumulation period is given by n,
the interest rate in each period is given by r and the principle is given by P . Then
under the simple interest accumulation rule, the value of the principle in the nth
period is given by P (1 + nr). Under the compound interest rule, the value of the
principle in the nth period is given by P (1 + r)n .
In this course, we use the “continuous interest accumulation rule”, which is an
extension of the compound interest rule, to compute the value of money. That is,
the increase of money during [t, t + dt] is given by rMt dt, where r is the interest rate
1
per time unit and Mt is the value of money at time t. On the other words, the value
of money evolves according to the differential equation given by
dMt
= rMt .
dt
Equivalently, for the principle M0 , the money value of money at time t is given by
Mt = M0 ert .
1.2
远
期
Forwards
A forward contract is an agreement to buy or sell an asset with a certain price at
a certain date. A forward contract is usually traded in the over-the-counter (OTC)
market. There are two parties in a forward contract. The one- who agrees to buy
(sell) the underlying asset is said to hold a long (short) position. Suppose that in
a forward contract an agent who takes a long position will buy the underlying asset,
which evolves according to the process S = {St }tØ0 , with price K, which is called
deliver price, at time T , which is called maturity, then the forward contract’s
payoff at the maturity is given by
多空
场外交易
ST ≠ K.
Similarly, from the perspective of a short position, the payoff of the forward contract
at the maturity described above is given by
K ≠ ST .
1.3
Futures
A futures contract is a standardised forward contract, traded on an organised
exchange,所
and such that, if a contract is traded at some time, the delivery price is
⼀
set to a special value Ft,T , called the futures price of the asset or the forward price
of the asset, chosen so that the value of the futures contract at initiation (that is,
at time t), zero. The two parties in a future contract should have faced a huge
default risk, as they usually do not know each other. To guarantee the contract is
honoured, a participant in a futures market is required to set up a so-called margin保证⾦
兑现
account as collateral,
and one’s daily profits and losses are reflected by adjustments
抵押品
in the account (the holder of a futures contract receives the change in value of the
futures price after each day, for each contract held). One also has to maintain the
balance in the margin account at some minimum value (the maintenance margin),
and receives a so-called margin call (a demand to top-up the margin account) if the
balance in the margin account falls below the maintenance margin. This mechanism
is designed to remove the default risk险
from the market, and hence futures markets机制
are very liquid. Futures markets have other specialised features that we will not go
further too much in this course. See Hull (2012) for more details.
贅
趟
inn
2
期权
1.4
Options
An option contract is an agreement that gives the buyer of the contract a right
(but not an obligation) to buy or sell an asset with a certain price at a certain date.
The specified price is the strike price or exercise price of the option and the specified
date is the maturity or expiration date of the option. There are two types of options
in terms of buying or selling the underlying asset. A call option is a right to buy
the underlying asset with the strike price at the maturity and a put option gives
the buyer the right to sell the underlying asset with the strike price at the maturity.
The payoff of a call option is given by
(ST ≠ K)+ ,
where ST is the price of the underlying asset at the maturity T and K is the strike
price.
Similarly, the payoff function of a put option is given by
(K ≠ ST )+
In terms of the flexibility of exercising the option, options can be categorised
into two types. A buyer of a European option can exercise the right only at the
maturity, while a buyer of an American option can exercise the right at or prior to
the maturity.
3
2
Arbitrage-free principle
In this section, we introduce the arbitrage-free principle, which is the foundation
of derivative pricing and valuation. Then we illustrate, using the arbitrage-free
principle, several properties of option price.
2.1
Arbitrage-free principle
Consider a financial market, which comprises n+1 financial securities, (S0 , S1 , ..., Sn ).
Here S0 is a risk-free asset (moeny account) and evolves according to
dS0t
= rS0t ,
dt
with r > 0. (S1 , ..., Sn ) represents risky assets (e.g., stocks, index...), which have
random returns on probability space ( , F, P).
A wealth process is given by a portfolio process („0t , „1t , ..., „nt ) and the underlying assets in the market, i.e.,
t
n
ÿ
=
„it Sit .
i=0
提取
注⼊
If during the entire period [0, T ] the investor does not infuse or withdraw fund in
the wealth, then the portfolio is said to be self-financing.
⾃筹
资⾦
Definition 2.1. A self-financing portfolio is said to have an arbitrage opportunity
in [0, T ], if there exits tú œ [0, T ), such that the wealth process
tú
Ø0
T
P(
=0
free
> 0) > 0.
T
Take
jake
Definition 2.2. If there exists no arbitrage opportunity for any self-financing portfolio in [0, T ], then the market is said to be arbitrage-free in the period [0, T ].
Theorem 2.3. If the market is arbitrage-free in [0, T ], and
processes, such that
1T
and
P(
1T
Ø
then for any t œ [0, T ),
1t
>
4
2t .
and
2
are wealth
(1)
2T
2T )
>
1
> 0,
(2)
(3)
Proof. Suppose that
such that
1t
2t ,
Æ
c
t
at time t œ [0, T ). Construct a portfolio process,
=
1t
≠
2t
+ M = 0,
where M is a money account. Then at time T , we have that
c
T
=
1T
≠
2T
+ M er(T ≠t) Ø
1T
≠
2T .
Then it follows from (1) and (2) that the portfolio constructing c yields an arbitrage
opportunity. This is a contradiction and thus completing the proof.
Corollary 2.4. If the market is arbitrage-free in [0, T ] and
2t , ’t œ [0, T ].
1T
=
2T ,
then
1t
=
Proof. Consider a wealth process constructed by
c
=
1
≠
2
⼆个正⽐⼀正it tMert
+ ‘M
where ‘ > 0, M is a bank account. Then at time T ,
c
T
= ‘M erT > 0.
1t
≠
正⽐如 不0
Then it follows Theorem 2.3 that
c
t
=
+ ‘M ert Ø 0.
2t
Let ‘ æ 0, then we have
1t
≠
Using the same method, we also prove
2.2
2t
2t
Ø 0.
≠
1t
Ø 0. Therefore,
1t
=
2t .
European options and call-put parity
In this section, we consider the relationship between the prices of call and put
options. We suppose that the market is arbitrage-free and no transaction cost. Also,
we suppose that the underlying stock pays no dividends. We define the underlying
stock process by St , the European call option price by ct , European put option price
by pt , the strike price by K, the expiration date by T and the constant risk-free
interest rate by r.
股息
Theorem 2.5. Suppose that P(ST ≠ K > 0) > 0 and P(ST ≠ K < 0) > 01 , then
(a)
1
(St ≠ Ke≠r(T ≠t) )+ < ct < St , ’t œ [0, T ).
Without any specification, this assumption will be added throughout the course.
5
Ct
St ru
q 19 K⼋
STST
PG k.to
a
so
T 1 71
Ctc St
1 知
51
1 9
9
0inch
11
Theorem
23
(b)
(Ke≠r(T ≠t) ≠ St )+ < pt < Ke≠r(T ≠t) , ’t œ [0, T ).
Proof. We only prove part (a) and the proof of part (b) is similar. At t = 0, we
construct a portfolio, such that 10 = S0 ≠ Ke≠rT and 20 = c0 . Then at time T ,
we have that 1T = ST ≠ K Æ (ST ≠ K)+ = 2T . As P(ST ≠ K < 0) > 0, then
P( 1T < 2T ) > 0. Therefore, Theorem 2.3 yields that ct > St ≠ Ke≠r(T ≠t) , ’t œ
[0, T ). Similarly, since cT = (ST ≠ K)+ and P(ST ≠ K > 0) > 0 , then it follows
from Definition 2.1 that ct > 0.As a result, we have (St ≠ Ke≠r(T ≠t) )+ < ct .
It is now suffices to prove ct < St , which follows from cT Æ ST .
Theorem 2.6 (Call-put parity).
ct + Ke≠r(T ≠t) = pt + St , ’t œ [0, T ].
(4)
Proof. Consider two portfolios, such that 1 = c + Ke≠rT , 2 = p + S. At t = T, we
have 1T = (ST ≠K)+ +K = max(K, ST ) and 2T = (K ≠ST )+ +ST = max(K, ST ).
Therefore, 1T = 2T . Then Corollary 2.4 yields 1t = 2t , ’t œ [0, T ].
2.3
American options and early exercises
In contrast to a European option holder, a holder of an American option has a right
to exercise the option early. Because of the early exercise right, it is easy to see that
an American option has higher value than the European option with the same strike
price, underlying asset and maturity. Define the American option call and put price
at time t by Ct and Pt respectively. Therefore,
Ct Ø ct , Pt Ø pt .
The following theorem illustrates that the early exercise right is of no use for an
American call option with a non-dividend paying underlying stock.
Theorem 2.7. If a stock has no dividend, then the American call option with the
stock being the underlying asset follows
Ct = ct .
Proof. It follows from Theorem 2.5 that
Ct Ø ct > (St ≠ Ke≠r(T ≠t) )+ Ø (St ≠ K)+ , ’t œ [0, T ).
Then it is to see that early exercise for the American option is unwise.
6
A natural question here is that: is the early exercise right for an American put
option useful? The answer is yes. In fact, suppose that the underlying asset falls
⼀⼀
below K(1 ≠ e≠r(T ≠t) ) and the price of the American option is the
same as the
European option. Then Theorem 2.5 yields that
nF
stckcte n
ke
Pt = pt Æ Ke≠r(T ≠t) < K ≠ St .
This contradicts Theorem 2.5.
Theorem 2.8. If C and P are the prices of American call option and put option
with a non-dividend underlying stock, and then
St ≠ K < Ct ≠ Pt < St ≠ Ke≠r(T ≠t) , ’t œ [0, T ).
Proof. We prove the left hand-side of the inequality. The right hand-side is similar
and thus we omit it.
At time t œ [0, T ), consider two portfolios, such that
1t
= Pt + St ,
2t
If the American option is not early exercised, then
2T
+
= (ST ≠ K) + Ke
1T
r(T ≠t)
c SK
= K + Ct .
= max(ST , K) + K(er(T ≠t) ≠ 1),
= (K ≠ ST )+ + ST = max(ST , K).
Therefore 2T > 1T , which implies 2t > 1t .
If the American option is early exercised at time · (t < · < T )
2·
= C· + Ker(· ≠t) ,
1·
t
= max(S· , K).
As a result, it follows from Theorem 2.5 that
2·
This completes the proof.
2.4
> (S· ≠ K)+ + Ker(· ≠t) >
Kerli
1· .
Option price and strike price
In this section, we study the dependence of option price on its strike price. For sake
of simplicity, we suppose that the underlying assets in this section are non-dividend
paying.
Theorem 2.9. Let ct (K) be the price of a European call option with the strike
price K. For two European call options c(K1 ), c(K2 ) with the same maturity date,
if K1 > K2 , then
0 Æ ct (K2 ) ≠ ct (K1 ) Æ K1 ≠ K2 , ’t œ [0, T ].
7
(5)
Proof. We will prove the right side of inequality (2.5). The left side is similar and
hence we omit the proof.
Construct two portfolios, such that
1
= c(K1 ) + K1 ,
2
= c(K2 ) + K2 .
On the maturity date t = T,
1T
= cT (K1 ) + K1 er(T ≠t) = (ST ≠ K1 )+ + K1 er(T ≠t) ,
2T
= cT (K2 ) + K2 er(T ≠t) = (ST ≠ K2 )+ + K2 er(T ≠t) .
• If ST > K1 . then
1T
• If ST < K2 , then
1T
= ST + K1 (er(T ≠t) ≠ 1) > ST + K2 (er(T ≠t) ≠ 1) =
2T .
• If K2 < ST < K1 , then 1T = K1 er(T ≠t) , 2T = ST +K2 (er(T ≠t) ≠1). Therefore
r(T ≠t)
+ K2 ≠ ST > (K1 ≠ K2 )(er(T ≠t) ≠ 1) > 0.
1T ≠ 2T = (K1 ≠ K2 )e
= K1 er(T ≠t) > K2 er(T ≠t) =
2T .
Then Theorem 2.3 yields the result.
Theorem 2.10. For two European put options with the same maturity date, if
K1 > K2 , then
0 Æ pt (K1 ) ≠ pt (K2 ) Æ K1 ≠ K2 .
盥
凸
Theorem 2.11. European call (put) option price ct (K)(pt (K)) is a convex function
of K, i.e., for K1 > K2 and K0 = ⁄K1 + (1 ≠ ⁄)K2 , (0 Æ ⁄ Æ 1). Then
ct (K0 ) Æ ⁄ct (K1 ) + (1 ≠ ⁄)ct (K2 ),
pt (K0 ) Æ ⁄pt (K1 ) + (1 ≠ ⁄)pt (K2 ).
Proof. We prove the convexity of call options. The convexity of put options is similar
hence the proof is omitted.
Consider two portfolios, such that
1
Then
1T
= c(K0 ),
= (ST ≠ K0 )+ ,
2T
2
= ⁄c(K1 ) + (1 ≠ ⁄)c(K2 ).
= ⁄(ST ≠ K1 )+ + (1 ≠ ⁄)(ST ≠ K2 )+
Then the convexity of the call option price follows from the convexity of the payoff
function (S ≠ K)+ with respect to K.
8
3
The Black-Scholes-Merton model
In this section we introduce the Black-Scholes-Merton model.
3.1
Assumptions
• The underlying asset follows from a geometric Brownian motion, i.e.,
dSt
= µdt + ‡dWt
St
(6)
where W = {Wt }tØ0 is a standard Brownian motion. µ and ‡ are constant
and represent the expected return rate and volatility of the underlying asset
respectively.
预期回报率
• The risk-free interest rate r is constant,
• The underlying asset pays no dividend,
• No transaction cost and tax.
• The market is arbitrage-free.
3.2
3.2.1
Black-Scholes equation
Delta hedging
We introduce the celebrated Black-Scholes’ delta hedging rule (Black and Scholes
1973), which is based on the arbitrage-principle in the market (see Merton et al.
1973). The uncertainty of the price of options come from the randomness of the
underlying asset. The delta hedging is to use the underlying asset to remove the
uncertainty of the option price.
Suppose the value of the option is defined by V . Construct a portfolio at time
t œ [0, T ),
t
= Vt ≠
t St
Here denotes the shares of the underlying asset. Choose
Holding t during [t, t + ”t], we have
t+”t
≠
t
= Vt+”t ≠ Vt ≠
t (St+”t
such that
is risk-free.
≠ St )
Letting ”t æ 0, then we have that
d
t
=r
t dt
= r(Vt ≠
9
t St )dt
(7)
dVt ≠
ˆV
1
ˆ2V
ˆV
(t, St ) + ‡ 2 St2 2 (t, St ) + µSt
(t, St ))dt
ˆt
2
ˆS
ˆS
ˆV
+ ‡St
(t, St )dWt ≠ t (µSt dt + ‡St dWt )
ˆS
t dSt = (
To remove the uncertainty, we let
(
t
=
ˆV
(t, St ),
ˆS
(8)
(9)
then (7) and (8) yield that
ˆV
1
ˆ2V
ˆV
(t, St ) + ‡ 2 St2 2 (t, St ) + µSt
(t, St ))dt ≠
ˆt
2
ˆS
ˆS
t µSt dt
= r(Vt ≠
t St )dt
which implies
ˆV
1 2 2 ˆ2V
ˆV
(t, S) + ‡ S
(t, S) + rS
(t, St ) ≠ rV = 0,
2
ˆt
2
ˆS
ˆS
(10)
for ’(t, S) œ [0, T ) ◊ (0, Œ). Moreover, at t = T , the payoff function yields that
V (T, S) =
I
(S ≠ K)+
(K ≠ S)+
(call option)
(put option).
(11)
Equation (10) shows the dynamics of a European contingent. This equation is first
established by Black and Scholes (1973) and thus called the Black-Scholes (BS)
equation. A distinctive feature of BS equation is that it does not depend on the
drift of the underlying asset. This feature sparks the so-called risk neutral pricing,
which opens a new way to study the derivative pricing theory and the financial
market.
3.2.2
Replication
Another way to derive the Black-Scholes equation is replication, that is, replicating
the option by constructing a portfolio comprising the underlying asset and the riskfree asset.
We now try to replicate the option by a portfolio, which means we are looking
for a self-financing portfolio process („0 , „1 ), such that
Vt = „0t Mt + „1t St ,
(12)
and
d(„0t Mt + „1t St ) = „0t dMt + „1t dSt ,
or equivalently
Mt d„0t + d„0t dMt + St d„1t + dSt d„1t = 0,
where Mt is the risk free asset2 .
2
Condition (13) is called self-financing condition.
10
(13)
Consider the dynamics of Vt , then we have
ˆV
1
ˆ2V
ˆV
(t, St ) + ‡ 2 St2 2 (t, St ) + µSt
(t, St ))dt
ˆt
2
ˆS
ˆS
ˆV
+ ‡St
(t, St )dWt
ˆS
dVt = (
(14)
(15)
On the other hand, the dynamics of „0t Mt + „1t St is given by
d(„0t Mt + „1t St ) = „1t (µSt dt + ‡St dWt ) + „0t rMt dt
Then it follows from (12) that
d(„0t Mt + „1t St ) = „1t (µSt dt + ‡St dWt ) + r(Vt ≠ „1t St )dt
(16)
Choosing „1t = ˆV
(t, St ) and building up the equation between (14) and (16), we
ˆS
obtain the Black-Scholes equation (10).
3.3
The Black-Scholes Formula
We solve the BS equation (10) with the terminal condition (11) in this section. We
consider the call option case, the put option case is similar and we thus omit it.
Let x = ln S, · = T ≠ t, then (10) becomes
ˆV
1 2 ˆ2V
1
ˆV
≠
+ ‡
+ (r ≠ ‡ 2 )
≠ rV = 0,
2
ˆ·
2 ˆx
2
ˆx
(·, x) œ (0, T ] ◊ (≠Œ, Œ)
(17)
with the initial condition
V (0, x) = (ex ≠ K)+ ,
x œ (≠Œ, Œ).
(18)
Equation (17) and initial condition (3.3) are called Cauchy problem in partial differential equation (PDE) theory. To solve the Cauchy problem, we set
V = ue–· +—x ,
(19)
where –, — are chosen so that equation (17) is reduced to a heat equation. By simple
calculation, we have
V· = e–· +—x (u· + –u)
Vx = e–· +—x (ux + —u)
Vxx = e–· +—x (uxx + 2—ux + — 2 u).
11
Substitute the above three equations into (17) and eliminate e–· +—x , then we have
1
1
‡2
‡2
u· ≠ ‡ 2 uxx ≠ (—‡ 2 + r ≠ ‡ 2 )ux + (r ≠ —(r ≠ ) ≠ — + –)u = 0
2
2
2
2
We choose
1
r
≠ 2,
2 ‡
2
‡
‡2
1
‡2
– = ≠r + —(r ≠ ) + — 2 = ≠r ≠ 2 (r ≠ )2 .
2
2
2‡
2
The by transformation (19), the equation (17) is reduced to
—=
1
u· ≠ ‡ 2 uxx = 0
2
(20)
u(0, x) = e≠—x (ex ≠ K)+
(21)
with the initial condition
As we know the solution of the Cauchy problem corresponding to the heat equation
(20) is given by the Poisson formula
u(·, x) =
⁄ Œ
≠Œ
K(x ≠ ›, · )„(›)d›,
where „(›) is the initial value, K(x ≠ ›, · ) is the fundamental solution of the heat
equation (20):
(x≠›)2
1
K(x ≠ ›) = Ô
e≠ 2‡2 · .
‡ 2fi·
Therefore, the solution u can be written as
⁄ Œ
(x≠›)2
1
Ô
e≠ 2‡2 · e≠—x (ex ≠ K)+ d›
≠Œ ‡ 2fi·
⁄ Œ
(x≠›)2
1
Ô
=
e≠ 2‡2 · e≠—x (ex ≠ K)d›
ln K ‡ 2fi·
u(·, x) =
2
Note that — = 12 ≠ ‡r2 , and – = ≠r ≠ 2‡1 2 (r ≠ ‡2 )2 . Then back to the original function
V (·, x), we have
1
1
V (·, x) = e( 2 ≠ ‡2 )x+(≠r≠ 2‡2 (r≠
r
‡2 2
) )·
2
u(·, x) = I1 + I2 ,
where
I1 = e
≠r·
⁄ Œ
ln K
2
(x≠›+(r≠ ‡2
1
2‡ 2 ·
Ô
e≠
‡ 2fi·
12
)· )2
+›
d›.
Denote
÷ = x ≠ › + (r ≠
then
‡2
)·,
2
‡2
ex ⁄ x≠ln K+(r≠ 2 )· ≠ (÷+‡22 · )2
I1 = Ô
e 2‡ · d÷.
‡ 2fi· ≠Œ
Define Ê =
2
÷+‡
Ô · , N (x)
‡ ·
=
Ô1
2fi
sx
Ê2
≠ 2
dÊ, then
≠Œ e
x ≠ ln K + (r +
Ô
I1 = e N (
‡ ·
x
‡2
)·
2
).
Similarly, we have that
I2 = ≠Ke
≠r·
x ≠ ln K + (r ≠
Ô
N(
‡ ·
‡2
)·
2
).
Back to the original variables (t, S), we have
V (t, S) = SN (d1 ) ≠ Ke≠r(T ≠t) N (d2 ),
(22)
where
2
S
ln K
+ (r + ‡2 )(T ≠ t)
Ô
d1 =
‡ T ≠t
2
ln S + (r ≠ ‡2 )(T ≠ t)
Ô
d2 = K
‡ T ≠t
Equation (22) is the celebrated Black-Scholes formula for a call option.
Moreover, by the call-put parity (4), we obtain the value of the put option
p(t, S) = c(t, S) + Ke≠r(T ≠t) ≠ S
= Ke≠r(T ≠t) N (≠d2 ) ≠ SN (≠d1 ).
(23)
Equation (23) is the Black-Scholes formula for a put option.
3.4
Derivation of the Black-Scholes formula: a probabilistic
perspective
In this subsection, we derive the Black-Scholes formula using the Feynman-Kac
formula, which provides a linkage between PDEs and conditional expectations.
Let us focus on the European call option. The European put option price can be
obtained then by the call-put parity. It follows from the Feynman-Kac formula that
13
the solution to the PDE problem (10) and (11) can be represented by the following
conditional expectation
c(t, S) = E[e≠r(T ≠t) (XT ≠ K)+ |Xt = S],
where
dXt = rXt dt + ‡Xt dWt .
Solving the SDE, we have that
1
XT = S exp((r ≠ ‡ 2 )(T ≠ t) + ‡(WT ≠ Wt )).
2
Let
WT ≠ Wt
Y =≠ Ô
.
T ≠t
Then
Ô
1
c(t, S) = E[e≠r(T ≠t) (S exp((r ≠ ‡ 2 )(T ≠ t) ≠ ‡Y T ≠ t) ≠ K)+ ].
2
It is easy to see that
Ô Y follows the standard normal distribution and S exp((r ≠
1 2
‡ )(T ≠ t) ≠ ‡Y T ≠ t) ≠ K > 0 if and only if y < d2 . Therefore,
2
Ô
y2
1 ⁄ d2 ≠r(T ≠t)
1
c(t, S) = Ô
e
(S exp((r ≠ ‡ 2 )(T ≠ t) ≠ ‡y T ≠ t) ≠ K)e≠ 2 dy
2
2fi ≠Œ
⁄ d2
Ô
1
1
1
=Ô
(S exp(≠ ‡ 2 (T ≠ t) ≠ ‡y T ≠ t ≠ y 2 ))dy ≠ e≠r(T ≠t) KN (d2 )
2
2
2fi ≠Œ
⁄ d2
Ô
S
1
=Ô
exp(≠ (y + ‡ T ≠ t)2 )dy ≠ e≠r(T ≠t) KN (d2 )
2
2fi ≠Œ
⁄ d2 +‡ÔT ≠t
1 2
S
=Ô
e≠ 2 z dz ≠ e≠r(T ≠t) KN (d2 )
2fi ≠Œ
= SN (d1 ) ≠ Ke≠r(T ≠t) N (d2 ).
3.5
Sensitivity parameters (Greeks)
Delta. The derivatives of the call option pricing function c(t, s) with respect to
various variables are called sensitivity parameters, or Greeks. A straightforward
computation yields that
t
=
ˆc
(t, s) = N (d1 )
ˆs
It is to see that the delta of a call option is positive. It means that if one sells a call
option, it is hedged with a dynamically adjusted long position in the stock.
14
Similarly, the delta of a put option can be obtained by the call-put parity, which
ˆc
implies that ˆp
= ˆx
≠ 1, then we have
ˆs
t
=
ˆp
(t, s) = ≠N (≠d1 ).
ˆs
It is to see that the
of a put option is negative. It means that if one sells a put
option, it is hedged with a dynamically adjusted short position in the stock.
Theta. The theta of a call option is
ˆc
‡s
= ≠rKe≠r(T ≠t) N (d2 ) ≠ Ô
N Õ (d1 ).
ˆt
2 T ≠t
As N (·) and N Õ (·) are both positive, theta is always negative, meaning that the
price of an option declines as we approach the maturity. This is true regardless of
whether the option is a call or put, which you can easily verify using put-call parity.
Gamma. The gamma of a call option is
ˆ2c
1
Ô
(t,
s)
=
N Õ (d1 ),
2
ˆs
‡s T ≠ t
2
which is always positive and is equal to ˆˆs2p , the put gamma.
Gamma is closely related to the risk introduced into the BS hedging program if
trading is not continuous. Note that Gamma measures how quickly the delta of an
⼀ as the stock price changes. If the magnitude of gamma is small, then
option changes
delta changes slowly, so a trader will not have to re-hedge very often in order to
maintain delta neutrality. On the other hand, if the magnitude of gamma is large,
then the trader must re-hedge very often to maintain delta neutrality.
Vega. The vega of an option is the derivative of the option price with respect
to volatility, and given by
Ô
ˆc
(t, s) = s T ≠ tN Õ (d1 ).
ˆ‡
对称
不
The positivity of vega shows the asymmetric
change of the option price with respect
to the volatility. Take a call option as an example. The holder benefits from the
increase of the stock price, but has only limited downside risk in the decrease of the
stock. Therefore the call option price increases as the volatility increases.
15
4
Risk neutral pricing
It is worth noting that the return rate µ of the stock price does not appear in the BS
equation. In this section, we introduce a probabilistic way to solve the BS equation
based on this observation.
4.1
Two mathematical theorems
Theorem 4.1 (Girsanov, one dimension). Let {Wt }tœ[0,T ] be a Brownian motion on
a probability space ( , F, P), and let F = {Ft }tœ[0,T ] be a filtration for the Brownian
motion. Let {◊t }tœ[0,T ] be an adapted process. Define
Zt = e
≠
st
0
◊s dWs ≠ 12
⁄ t
WtQ = Wt +
0
st
0
◊s2 ds
◊s ds,
and assume that
1
E[e 2
sT
0
◊s2 ds
] < Œ.
(24)
Then E[ZT ] = 1 and under the probability measure Q given by
Q(A) =
⁄
A
ZT dP,
the process W Q is a Brownian motion.
Condition (24) is Novikov’s condition, which ensures that Z is a martingale under
P.
Theorem 4.2 (Martingale representation, one dimension). Let {Wt }tœ[0,T ] be a
Brownian motion on a probability space ( , F, P), and filtration F = {Ft }tœ[0,T ]
is generated by the Brownian motion. Let {Mt }tœ[0,T ] be a martingale with respect
to this filtration. Then there is an adapted process { t }tœ[0,T ] , such that
Mt = M0 +
4.2
⁄ t
0
s dWs ,
’t œ [0, T ].
Stock price under the risk neutral measure
Let {Wt }tœ[0,T ] be a Brownian motion on a probability space ( , F, P), and let
F = {Ft }tœ[0,T ] be a filtration for the Brownian motion. Consider the stock price
governed by
dSt = µt St dt + ‡t St dWt ,
16
where µ and ‡ are allowed to be adapted processes. We assume that ‡t > 0, ’t œ
[0, T ]. In addition, we suppose that the interest rate is an adapted process {rt }tœ[0,T ] .
We define the discount process by
Dt = e≠
Note that
st
0
rs ds
.
dD(t)St = (µt ≠ rt )Dt St dt + ‡t Dt St dWt
= ‡t St Dt (◊t dt + dWt ),
where we define the market price of risk to be
◊t =
µt ≠ r t
.
‡t
(25)
We introduce the probability measure Q defined in Theorem 4.1, with the market
price of risk ◊t given by (25). Then we rewrite the dynamics of the stock price as
dDt St = Dt St ‡t dWtQ
where W Q is define in Theorem 4.1 with the market price of risk ◊ given by (25).
We call Q the risk neutral measure because it is equivalent to the original P
and it renders the discounted stock price Dt St into a martingale. The undiscounted
stock price St follows the dynamics
dSt = rt St dt + ‡t St dWtQ .
4.3
(26)
Portfolio process under the risk neutral measure
In this section, we study portfolio processes under the risk neutral measure Q.
Consider the portfolio comprising shares of stocks and a certain amount of riskfree asset. The the instantaneous change of the portfolio is given by
dXt = t dSt + rt (Xt ≠ t St )dt
= t (µt St dt + ‡t St dWt ) + rt (Xt ≠
= rt Xt dt + t St ‡t (◊t dt + dWt ).
t St )dt
(27)
Then it follows from Ito’s formula that
dDt Xt =
t St ‡t (◊t dt
+ dWt ) =
t d(Dt St ).
Therefore, combined with the dynamics of the stock price (26), we have that
dDt Xt =
Q
t ‡t Dt St dWt .
17
(28)
4.4
Pricing under the risk neutral measure and hedging
In Section 3, we introduced the pricing mechanism based on the delta hedging
strategy. If a contingent claim can be replicated by the underlying asset and the
risk-free asset, then the price of the claim can be obtained by solving a PDE problem.
The discussion in Section 3 depends on the Markov property of the payoff function.
In this section, we extend the pricing mechanism to a more general case, which
removes the Markov property of the payoff function.
In section 3.2.2, we see that the key to price a contingent claim is to replicate
the claim by the underlying asset and the risk-free asset. Following this idea, we
first suppose that the contingent claim yields a payoff VT at the maturity T , where
VT is FT measure. Note that we here do not suppose thatVT is a function of ST and
thus not imposing the Markov property on the payoff function.
Second, we suppose that the contingent claim can be replicated by the portfolio
process X defined by (27). (Roughly speaking, if every contingent claim can be
replicated by the stock and the risk free asset in the market, then we say the market
is complete.) Suppose that there exists a portfolio process X, such that
XT = VT
Then it follows Corollary 2.4 that
Xt = Vt .
Moreover, the discounted portfolio process (28) tells that Dt Xt is a martingale under
the risk neutral measure Q. Then
Dt Vt = Dt Xt = EQ [DT XT |Ft ] = EQ [DT VT |Ft ],
(29)
which yields the risk neutral pricing formula for a European contingent claim
Vt =
1 Q
E [DT VT |Ft ]
Dt
(30)
A natural question here is that: how can we find the hedging strategy ? We now
answer this question by the martingale representation.
Following (29), it is easy to see that Dt Vt is a martingale under the risk neutral
measure Q. The the martingale representation theorem (Theorem 4.2) yields that
Dt Vt = V0 +
⁄ t
Q
s dWs , 0
0
On the other hand, for any portfolio process
is given by
Dt Xt = X0 +
⁄ t
0
18
t,
Æ t Æ T.
the discounted wealth process X
Q
s ‡s Ds Ss dWs
In order to have Xt = Vt , ’t œ [0, T ], we choose
X0 = V0
and choose
t
such that
t ‡t Dt St
=
t,
which yields
t
4.5
=
t
‡t Dt St
,
’t œ [0, T ],
t œ [0, T ].
Forwards and futures
In this subsection, we study the application of the risk neutral pricing introduced
in subsection 4.4 on the valuation of forwards and futures. We assume that every
contingent claim can be hedged and the underlying assets are non-dividend paying. We consider all problems in this subsection in the filtered probability space
( , F, {Ft }0ÆT̂ , P). Here, T̂ is a large number so that all financial securities expire
before time T̂ . We suppose that there is a unique risk-neutral measure Q.
Forwards. Let {St }tœ[0,T̂ ] be an asset price process and let {rt }tœ[0,T̂ ] be an
st
interest rate process. As usual, we define the discount factor by Dt = e 0 rs ds . It
follows from the risk neutral pricing theory, the price at time t of a zero-coupon
bond paying 1 at time T is given by
≠
B(t, T ) =
1 Q
E [DT |Ft ].
Dt
(31)
Definition 4.3. A forwards contract is an agreement to pay a specified delivery
price K at a delivery date T , for the asset price {St }tœ[0,T̂ ] . The T ≠forward price
F orS (t, T ) of this asset at time t, where t œ [0, T ], is the value of K such that the
forwards contract has no-arbitrage price zero at time t.
Theorem 4.4.
F orS (t, T ) =
St
, t œ [0, T ].
B(t, T )
Proof. It follows from the risk neutral pricing theory that,
0=
1 Q
E [DT (ST ≠ K)|Ft ],
Dt
which implies
1 Q
1 Q
E [DT ST |Ft ] =
E [DT K|Ft ],
Dt
Dt
19
(32)
which yields that
St = KB(t, T ).
This completes the proof.
Futures. A futures contract is a standardised forward contract, traded on an
organised exchange, and such that, if a contract is traded at some time, the delivery
price is set to a special value Ft,T , called the futures price of the asset or the forward
price of the asset, chosen so that the value of the futures contract at initiation (that
is, at time t), zero. A distinctive feature of futures is the margin mechanism, which
differs futures from forwards. In contrast to a forward contract which yields a lumpsum payoff at the maturity, a futures contract provides a cash flow. To see this, let
us consider a time interval [0, T ], which we divide into subintervals using partition
points 0 = t0 < t1 · · · < tn = T. We shall refer to each subinterval [tk .tk+1) as a day.
Suppose that the interest rate is constant within each day. Then the discount
process is given by D0 = 1 and, for k = 1, 2, ·, n ≠ 1,
Dtk+1 = exp{≠
k
ÿ
j=0
rtj (tj+1 ≠ tj )},
which is Ftk ≠measurable.
Consider a futures holder who takes a long position between time tk and tk+1 .
At time tk , he receives Ftk+1 ,T ≠ Ftk ,T , which is called marking to margin. At each
time tk , the future contract price equals zero, thus the risk neutral pricing theory
yields that
1 Q
E [Dtk+1 (Ftk+1 ,T ≠ Ftk ,T )|Ftk ] = 0.
Dtk
As Dtk+1 is Ftk ≠measurable, we have
EQ [Ftk+1 ,T ≠ Ftk ,T |Ftk ] = 0,
which implies that Ft,T , t = t0 , t1 , · · · , tn is a martingale under Q.
Since FT,T = ST , then we have that
Ft,T = EQ [ST |Ftk ],
k = 0, 1, 2, · · · .n.
We now extend this idea into the continuous time case.
Definition 4.5. The futures price of an asset whose value at time T is ST is given
by
Ft,T = EQ [ST |Ft ],
20
t œ [0, T ].
Theorem 4.6. The futures price is a martingale under the risk neutral measure Q,
it satisfies FT,T = ST , and the value of a long (or a short) futures position to be held
over an interval of time is always zero.
Proof. It is easy to see that Ft,T is a martingale under the risk neutral measure Q,
it satisfies FT,T = ST . Therefore, it suffices to prove that the long or short position
of a future contract values 0.
Consider an agent who takes
shares of future contract. Suppose the profit
which the shares of future contract create is defined by X. Then we have that
dXt =
t dFt,T
+ rXt dt,
which implies that
dDt Xt = Dt
t dFt,T
This suggests that {Dt Xt }tœ[0,T ] is a martingale under Q, as Ft,T is a martingale
under Q. Moreover, since holding any position ofs a futures does not trigger any
payment, we have that X0 = 0. Therefore Dt Xt = 0t Ds s dFs,T , which implies that
EQ [Dt Xt ] = 0. Let t = 1(≠1), ’t œ [0, T ], we have that the long (short) position
of a future contract values 0.
In the end of this subsection, we compare the forward and futures prices. If the
interest rate process r is deterministic, then it is to see that
1 Q
E DT [ST |Ft ]
DT
1
St
=
Dt St =
= F orS (t, T ).
DT
B(t, T )
Ft,T = EQ [ST |Ft ] =
If r is a general stochastic process, then the forward-future spread is given by
F orS (0, T ) ≠ Ft,T =
=
S0
Q
E DT
≠ EQ [ST ] =
1
EQ DT
1
cov Q (DT , ST ),
B(0, T )
[EQ [DT ST ] ≠ EQ DT EQ ST ]
where cov Q is the covariance under the risk neutral measure Q.
4.6
4.6.1
Fundamental theorems of asset pricing
Two mathematical theorems
In this subsection, we study the financial market theory using risk neutral pricing
theory. The mathematical tools are the multi-dimensional Girsanov theorem and
the mult-dimensional martingale representation theorem.
21
In this subsection, let Wt = (W1t , W2t , · · · , Wnt ) be a n≠ dimensional Brownian
motion3 defined on a probability space ( , F, {Ft }tØ0 , P), and Ft }tØ0 the filtration
generated by W.
Theorem 4.7 (Girsanov, multi-dimension). Let {(◊1t , ◊2t , · · · , ◊nt )}tœ[0,T ] be an adapted
process. Define
Zt = e≠
s t qn
◊ dWjs ≠ 12
j=1 js
0
WitQ = Wit +
⁄ t
0
st
0
||◊||2s ds
i = 1, 2, · · · , n,
◊is ds,
and assume that
E[e
1
2
sT
0
||◊s ||2 ds
] < Œ.
(33)
Then E[ZT ] = 1 and under the probability measure Q given by
Q(A) =
⁄
A
ZT dP,
the process W Q is a Brownian motion.
Condition (33) is Novikov’s condition, which ensures that Z is a martingale.
Theorem 4.8. [Martingale representation, multi-dimension] Let {Mt }tœ[0,T ] be a
martingale with respect to {Ft }tœ[0,T ] . Then there is an adapted process {( 1t , 2t , · · · ,
such that
Mt = M 0 +
⁄ tÿ
n
0 j=1
js dWjs ,
’t œ [0, T ].
We now consider the model of the market. Suppose that there are m stocks in
the market, each with the dynamics given by
n
ÿ
dSit
= µit dt +
‡ijt dWjt ,
Sit
j=1
i = 1, 2, · · · , m.
where µi and Ò
‡ij are adapted processes.
qn
2
Let ‡it =
j=1 ‡ijt , which we assume is never 0. and we define
Bit =
⁄ tÿ
n
0
‡ijs
dWjs ,
j=1 ‡is
3
i = 1, 2, · · · , m.
We say W is a n≠ dimensional Brownian motion, if (i) each Wi is one dimensional Brownian
motion; (ii) if i ”= j, then Wi and Wj are independent.
22
nt )}tœ[0,T ] ,
It is easy to see that Bi is a continuous martingale and furthermore,
2
‡ijt
dBit dBit =
dt = dt,
2
j=1 ‡i
n
ÿ
i = 1, 2, · · · , m.
Then it follows from Levy’s theorem that Bi is a Brownian motion. We may rewrite
the stock price in terms of the Brownian motion Bi ,
dSit = µit Sit dt + ‡it Sit dBit ,
i = 1, 2, · · · , m.
For i ”= k, Bi and Bk are typically not independent. To see this, let us consider
dBit dBkt = flikt dt
where
flikt =
qn
‡ijt ‡kjt
,
‡it ‡kt
j=1
We call flit the instantaneous correlation between Bi and Bk .
Then Ito’s formula yields that
Bit Bkt =
which implies
⁄ t
0
⁄ t
Bis dBks +
0
Bks dBis +
cov(Bit , Bkt ) = E
Note that
⁄ t
0
⁄ t
0
fliks ds,
fliks ds.
dSit dSkt = ‡it ‡kt Sit Skt dBit dBkt = flikt ‡it ‡kt Sit Skt dt
which can be rewritten as
dSit dSkt
= flikt ‡it ‡kt dt.
Sit Skt
The volatility processes ‡i and ‡k are the respective instantaneous standard derivations of the relative changes in Si and Sk , and the process flik is the instantaneous
correlation between these relative changes.
Define a discount process
Dt = e
≠
st
0
rs ds
,
where r is an adapted process.
The discounted stock process is given by
dDt Sit = Dt Sit ((µi ≠ rt )dt + ‡it dBit ),
23
i = 1, 2, · · · , m.
(34)
4.6.2
Existence of the risk-neutral measure
Definition 4.9. A probability measure Q is said to be risk neutral if
(i) Q and P are equivalent (i.e., for every A œ F, P(A) = 0 if and only if
Q(A) = 0), and
(ii) under Q, the discounted stock price Dt Sit is a martingale for every i = 1, 2. · · · , m.
In order to make discounted stock prices be martingales, we would like to rewrite
the stock price as
dDt Sit = Dt Sit
n
ÿ
‡ijt (◊jt + dWjt ),
j=1
i = 1, 2, · · · , m.
(35)
If we can find (◊1 , ◊2 , · · · , ◊n ), such that (35) holds, then by the Girsanov theorem,
we can construct a measure Q, which is risk neutral. In other works, we are looking
for a solution to the following equation in the unknown process (◊1 , ◊2 , · · · , ◊n ),
µit ≠ rt =
n
ÿ
i = 1, 2, 3, · · · , m, t œ [0, T ].
‡ijt ◊jt ,
j=1
(36)
We call these equations market price of risk equations.
We now consider the portfolio process X, which comprises = ( 1 , 2 , · · · , n )
shares of stocks and amount of money. The dynamics of the portfolio process can
be given by
dXt =
m
ÿ
it dSit
i=1
= rt Xt dt +
+ rt (Xt ≠
m
ÿ
i=1
it
Dt
m
ÿ
it Sit )dt
i=1
d(Dt Sit ),
which implies
dDt Xt =
m
ÿ
it d(Dt Sit ),
(37)
i=1
Lemma 4.10. Let Q be a risk neutral measure, and let X be the value of a portfolio.
Under Q, the discounted portfolio value DX is a martingale.
Proof. The result follows from the fact that DS is a martingale if Q exists.
Definition 4.11. An arbitrage is a portfolio value process X satisfying X0 = 0 and
also satisfying for some time T > 0
P(XT Ø 0) = 1,
P(XT > 0) > 0.
24
Theorem 4.12 (First fundamental theorem of asset pricing). If a market model has
a risk neutral measure, then it does not admit arbitrage.
Proof. Consider a portfolio value process X, such that for some time T > 0
P(XT Ø 0),
P(XT > 0) > 0.
Suppose that there exists a risk neutral measure Q, then we have that
X0 = EQ [DT XT ] > 0.
This completes the proof.
4.6.3
Uniqueness of the risk-neutral measure
In this subsection, we consider the hedging strategies under the assumption that
there exists at least one risk neutral measure Q. That is, the systems of equations
(36) admits at least one solution ◊ = (◊1 , ◊2 , · · · , ◊n ). It follows from the multidimensional Girsanov theorem (4.7) that we can construct the risk neutral measure
Q with this ◊. Under the risk neutral measure Q,
{{WjtQ
:=
⁄ t
0
◊js ds + Wjt }tœ[0,T ] }j=1,2,··· ,n
is a martingale.
In the previous subsection, we obtained the result that the market is arbitrage
free if and only if there exists a risk neutral measure, though we only proved one
side. The existence of a risk neutral measure makes the risk-neutral pricing possible.
Before applying the risk neural pricing theory, we need to verify that if the hedging
strategy, which backs up the corresponding risk neutral price, exists.
Mathematically, we define a security terminated at terminal T by a FT ≠ random
variable VT . If we define the risk neutral price of VT by the expectation of discounted
payoff under Q, that is,
Vt =
1 Q
E [DT VT |Ft ],
Dt
then a natural question here is that, if there is a hedging strategy supporting the
price Vt ?
To answer this question, we first consider Dt Vt , which is a martingale under Q.
Then it follows from the multidimensional martingale representation 4.8 that
Dt Vt = V0 +
or
dDt Vt =
⁄ tÿ
n
0 j=1
n
ÿ
Q
js dWjs ,
Q
jt dWjt ,
j=1
25
t œ [0, T ]
t œ [0, T ].
(38)
We would like to find a portfolio value process X, which can be used to replicate V .
In the other words, we are looking for a X, such that Xt = Vt , t œ [0, T ]. Consider
the stochastic differential of Dt Xt , which is given by (37), then we have that
dDt Xt =
=
=
m
ÿ
i=1
m
ÿ
it d(Dt Sit )
it Dt Sit
i=1
n ÿ
m
ÿ
n
ÿ
‡ijt (◊jt + dWjt )
j=1
Q
it Dt Sit ‡ijt dWjt .
(39)
j=1 i=1
We let V0 = X0 , then it follows from (38) and (39) that seeking for a hedging
strategy = ( 1 , 2 , · · · , m ) (and hence portfolio value process X) is equivalent
to looking for the solution to the following system of equations
jt
Dt
=
m
ÿ
i=1
it Sit ‡ijt ,
j = 1, 2, · · · , n, t œ [0, T ].
The solvability of the above system of equations actually is equivalent to the uniqueness of the solutions to the market price of risk equations (36) or the uniqueness
of the risk neutral measure. This equivalence is called second fundamental asset
pricing theorem, which is summarised as follows.
Definition 4.13. A market model is complete if every derivative security can be
hedged.
Theorem 4.14 (Second fundamental theorem of asset pricing). Consider a market
model that has a risk-neutral probability measure. The model is complete if and only
if the risk-neutral probability measure is unique.
26
5
5.1
Change of numeraire
Market model
In this section, we will work with the multidimensional market model, where the
uncertainty is given by n≠ dimensional Brownian motion. In particular, we let Wt =
(W1t , W2t , · · · , Wnt ) be a n≠ dimensional Brownian motion defined on a probability
space ( , F, {Ft }tØ0 , P), and {Ft }tØ0 the filtration generated by W. There is an
adapted interest rate process rt , 0 Æ t Æ T. This can be used to create a money
market account whose price per share at time t is
st
Mt = e
And the discount process is given by
Dt = e≠
0
rs ds
st
0
.
rs ds
.
There are m primary assets in the market and their prices are given by
n
ÿ
dSit
= µit dt +
‡ijt dWjt ,
Sit
j=1
i = 1, 2, · · · , m.
(40)
where µi and ‡ij are adapted processes.
We suppose that there is a unique risk-neutral measure Q, that is, there is a
unique n≠dimensional adapted process ◊ = (◊1 , ◊2 , · · · , ◊n ) satisfying the market
price of risk equations (36). The risk neutral measure is constructed using the
multidimensional Girsanove Theorem (4.7). Under Q, the Bronian motion
WjtQ = Wjt +
⁄ t
0
◊js ds, j = 1, 2, · · · , n,
are independent of one another.
Moreover, the uniqueness of the risk neutral measure, according to the second
fundamental asset pricing theorem, implies the completeness of the market model,
which means that every security can be hedged. Thus, the discounted process of
any security in the complete market is a martingale under the risk neutral measure.
5.2
Numeraire
A numeraire is the unit of account in which other assets are denominated. For example, We say the measure Q is risk-neutral for the money market account numeraire.
We shall see that sometimes it is convenient to change the numeraire because of
modeling considerations as well. A model can be complicated or simple, depending
on the choice of the numeraire for the model.
In principle, we can take any positively priced asset as a numeraire and denominate all other assets in terms of the chosen numeraire. The asset we take as
27
numeraire could be one of the primary assets given by (40) or it could be a derivative asset. Regardless of which asset we take, it has the stochastic representation
provided by the following theorem.
Theorem 5.1 (Stochastic representation of assets). Let N be a strictly positive
price process for a non-dividend-paying asset, either primary or derivative, in the
multidimensional market model. Then there exists a vector volatility process
‹ = (‹1 , ‹2 , · · · , ‹n )
such that
dNt = rt Nt dt + Nt
n
ÿ
‹jt dWjtQ .
(41)
j=1
The equation is equivalent to each of the equations
dDt Nt = Dt Nt
n
ÿ
‹jt dWjtQ ,
(42)
j=1
Dt Nt = N0 exp{
⁄ tÿ
n
0 j=1
⁄ tÿ
n
Nt = N0 exp{
where ||‹t || =
qn
j=1
0 j=1
1⁄ t
||‹s ||2 ds},
2 0
(43)
1
(rs ≠ ||‹s ||2 )ds},
2
(44)
Q
‹js dWjs
≠
Q
‹js dWjs
+
⁄ t
0
|‹jt |2 , ’t œ [0, T ].
Proof. Under the risk neutral measure Q, DN is a martingale. Therefore, the martingale representation theorem yields that, there exists an adapted process
=
( 1 , 2 , · · · , n ), such that
dDt Nt =
n
ÿ
Q
jt dWjt .
j=1
Since Nt > 0, ’t œ [0, T ], we let ‹jt =
see that Dt Nt follows (42).
Note that Nt = Mt (Dt Nt ), then
jt
Dt Nt
, ’j = 1, 2, · · · , n, t œ [0, T ]. Then it is to
dNt = rt Mt Dt Nt dt + Mt dDt Nt ,
(41) follows from (42) and the fact Mt Dt = 1.
Finally, (43) and (44) follows from the application of Ito’s formula on (42) and
(41) respectively.
28
We can use multidimensional Girsanov theorem to change the measure. Define
WjtN = ≠
⁄ t
0
‹js ds + WjtQ , j = 1, 2, · · · , n,
and a new probability measure
1 ⁄
P (A) =
DT NT dQ, ’A œ F.
N0 A
N
Then it follows from the multidimensional Girsanov theorem that
W N = (W1N , W2N , · · · , WnN )
is a Brownian motion under measure PN .
Moreover, for an arbitrary random variable X, the expectation of X under PN
is given by
EN [X] =
1 Q
E [DT NT X].
N0
More generally,
Dt Nt
DT NT
= EQ [
|Ft ],
N0
N0
0 Æ t Æ T,
is the Radon-Nikodym derivative process, which is similar to Zt in the Girsanov theorem. Therefore, for a random variable Y , which is Ft ≠ measurable and integrable,
we have that
EN [Y |Fs ] =
1
EQ [Y Dt Nt |Fs ],
Ds Ns
0 Æ s Æ t Æ T.
Theorem 5.2. Let St and Nt be the prices of two assets and let ‡ = (‡1 , ‡2 , · · · , ‡n )
and ‹ = (‹1 , ‹2 , · · · , ‹n ) denote their respective volatility vector processes:
dDt St = Dt St
n
ÿ
‡jt dWjtQ ,
j=1
dDt Nt = Dt Nt
n
ÿ
‹jt dWjtQ ,
j=1
where ‡ and nu are adapted processes.
Take Nt as the numeraire, then the price of St becomes StN =
measure PN , the process S N is a martingale. Moreover,
dStN
=
StN
n
ÿ
j=1
(‡jt ≠ ‹jt )dWjtN .
29
St
.
Nt
Under the
Proof. We have that
⁄ tÿ
n
Dt St = S0 exp{
Dt Nt = N0 exp{
0 j=1
Q
‡js dWjs
≠
⁄ tÿ
n
0 j=1
1⁄ t
||‡s ||2 ds},
2 0
Q
‹js dWjs
≠
1⁄ t
||‹s ||2 ds}.
2 0
and hence
StN
⁄ tÿ
n
S0
1⁄ t
Q
=
exp{
(‡js ≠ ‹js )dWjs ≠
(||‡s ||2 ≠ ||‹s ||2 )ds}.
N0
2 0
0 j=1
Define
Xt =
⁄ tÿ
n
0 j=1
Q
(‡js ≠ ‹js )dWjs
≠
1⁄ t
(||‡s ||2 ≠ ||‹s ||2 )ds.
2 0
Then it is to see that
dXt =
1
(‡jt ≠ ‹jt )dWjtQ ≠ (||‡t ||2 ≠ ||‹t ||2 )dt,
2
j=1
n
ÿ
and
dXt dXt =
n
ÿ
j=1
With f (x) =
S0 x
e ,
N0
(‡jt ≠ ‹jt )2 dt.
we have that StN = f (Xt ), then
dStN = StN (
n
ÿ
j=1
(‡jt ≠ ‹jt )(dWjtQ ≠ ‹jt )) = StN (
n
ÿ
j=1
(‡jt ≠ ‹jt )dWjtN ).
This completes the proof.
5.3
Forward measure
Although there may be multiple Brownian motions driving the model of this section,
in order to simplify the notation, we assume in this section that there is only one.
It is not difficult to rederive the results presented here under the assumption that
there are n≠Brownian motions.
1311
Blt 不⼆六 a strictly positive
Consider a zero coupon bond terminated at T . The bond isEQIDHFD
asset, and the discounted price Dt B(t, T ) given by (31) is a martingale under the
risk neutral measure Q. It follows from Theorem 5.2 that there exists an adapted
process ‡t (T ) in t such that
d(Dt B(t, T )) = Dt B(t, T )‡t (T )dWtQ .
30
Definition 5.3. Let T be a maturity date. We define the T ≠forward measure PT
by
PT (A) =
⁄
A
DT
dQ,
B(0, T )
’A œ F.
It follows from the Girsanov theorem that
WtT = WtQ ≠
⁄ t
0
‡t (T )dWtQ
is a Brownian motion under PT .
For a contingent claim VT , which is FT ≠measurable, we have that
1
EQ [VT DT |Ft ]
Dt B(t, T )
1
=
Vt ,
B(t, T )
ET [VT |Ft ] =
which yields that
Vt = B(t, T )ET [VT |Ft ].
(45)
Equation (45) illustrates that if we can find a simple dynamics of the underlying
asset under the T ≠forward measure, then we only need to consider the estimate
of VT , and hence we do not have to consider the correlation between the discount
factor DT and VT .
5.4
The Black-Scholes formula with random interest rates
We present a generalized Black-Scholes option pricing formula that permits the interest rate to be random. The classical Black-Scholes assumption that the volatility
of the underlying asset is constant is here replaced by the assumption that the
volatility of the forward price of the underlying asset is constant. Because the forward price is a martingale under the forward measure, and W T (t) is the Brownian
motion used to drive asset prices under the forward measure, the assumption of
constant volatility for the forward price is equivalent to the assumption
dF orS (t, T ) = ‡F orS (t, T )dWtT ,
(46)
where ‡ is a constant. The bond maturity T is chosen to coincide with the expiration
time T of the option.
Theorem 5.4. Let S be the price of an asset denominated in currency, and assume
the forward price of this asset satisfies (46) with a positive constant ‡. The value at
31
time t œ [0, T ] of a European call on this asset, expiring at time T with strike price
K, is
Vt = St N (d1 ) ≠ KB(t, T )N (d2 ),
where
1
F orS (t, T ) 1 2
d1,2 = Ô
(ln
± ‡ (T ≠ t)).
K
2
‡ T ≠t
Proof. It follows from the risk neutral pricing theory that
Vt =
1 Q
E [DT (ST ≠ K)+ |Ft ].
Dt
Then (45) yields that
Vt = B(t, T )ET [(ST ≠ K)+ |Ft ].
Note that ST = F orS (T, T ), then the representation above becomes
Vt = B(t, T )ET [(F orS (T, T ) ≠ K)+ |Ft ].
Since the dynamics of FS (t, T ) follows (46), then the standard Black-Scholes formula
(22) yields that
ET [(F orS (T, T ) ≠ K)+ |Ft ] = F orS (t, T )N (d1 ) ≠ KN (d2 )
Moreover, it follows from (32) that St = F orS (t, T )B(t, T ). Then
Vt = B(t, T )(F orS (t, T )N (d1 ) ≠ KN (d2 ))
= St N (d1 ) ≠ KB(t, T )N (d2 ).
32
6
6.1
Interest rate models
Short rate models
Consider a probability space ( , F, P). Suppose that W is a one dimensional Brownian motion and the filtration generated by the Brownian motion is {Ft }tØ0 . We
assume the completeness of the market model, i.e., there exists a unique risk measure Q, such that for every asset, either primary or derivative, the discounted price
is a martingale under Q.
Suppose that the interest rate rt is governed by the following stochastic differential equation
drt = b(t, rt )dt + ‡(t, rt )dWtQ ,
(47)
where W Q is a Brownian motion under Q and b, ‡ are continuous deterministic
functions such that SDE (47) admits a unique strong solution.
6.1.1
Vasicek model
In the Vasicek model (Vasicek 1977), the interest rate rt follows the differential
equation given by
drt = (b + —rt )dt + ‡dWtQ ,
(48)
where b, —, ‡ are constant and — is negative.
We consider the zero coupon bond price under Vasicek model. It follows from
the risk neutral pricing theory that the zero coupon bond price P (t, r; T ) can be
given by
P (t, r; T ) = EQ [e≠
sT
t
rs ds
|rt = r]
where t, r, T mark the current time, current value of the interest rate and the maturity date of the bond.
It follows from the Fyenman-Kac formula that the price function P (t, r; T ) solves
the partial differential equation problem as follows.
ˆP
1 ˆ2P
ˆP
+ ‡ 2 2 + (b + —r)
≠ rP = 0 r œ (≠Œ, Œ),
ˆt
2 ˆr
ˆr
with the terminal condition
P (T, r; T ) = 1.
Conjecture that
P (t, r; T ) = exp(≠A(t) ≠ B(t)r).
33
(49)
Plug the representation of the bond price P into the PDE, then we have
dA(t)
‡2
= B(t)2 ≠ bB(t),
dt
2
dB(t)
= ≠—B(t) ≠ 1,
dt
t œ [0, T ),
t œ [0, T ).
In addition, the terminal condition of the PDE problem yields the the terminal
conditions of the ODEs above, i.e.,
A(T ) = 0,
B(T ) = 0.
Therefore, solving the above ODE system, we have that
B(t) =
A(t) =
1 —(T ≠t)
(e
≠ 1),
—
‡ 2 (4e—(T ≠t) ≠ e2—(T ≠t) ≠ 2—(T ≠ t) ≠ 3)
e—(T ≠t) ≠ 1 ≠ —(T ≠ t)
+
b
.
4— 3
—2
Substitute the representation of A(t) and B(t) into (49), then we obtain the closed
form formula for a zero coupon bond under Vasicek model.
In addition, it follows from the dynamics of the Vasicek model (48) and Ito’s
formula that
b
EQ [rt |r0 = x] = xe—t + (e—t ≠ 1),
—
V arQ (rt |r0 = x) =
6.1.2
‡ 2 2—t
(e ≠ 1)
2—
Cox-Ingersoll-Ross model (CIR) modelú
In the CIR model (Cox et al. 2005), the dynamics of the interest rate rt is given by
Ô
drt = Ÿ(◊ ≠ rt )dt + ‡ rt dWtQ .
(50)
where the constant parameters Ÿ Ø 0, ◊ Ø 0, ‡ > 0.
In contrast to the Vasicek model in which the interest rate will take values in
(≠Œ, Œ), the CIR model can keep the interest rate process always above 0, which
makes the model more realistic. In particular, if 2Ÿ◊ > ‡ 2 , then rt will never reach
0, for any r0 > 0. Otherwise, rt will occasionally touch 0.
34
It follows from risk neutral pricing theory that the bond price P (t, r; T ) can be
given by
P (t, r; T ) = EQ [e≠
Then the Feynman-Kac formula yields that
sT
t
rs ds
|rt = r].
ˆP
1
ˆ2P
ˆP
+ ‡ 2 r 2 + Ÿ(◊ ≠ r)
≠ rP = 0 r œ (≠Œ, Œ),
ˆt
2
ˆr
ˆr
with the terminal condition
P (T, r; T ) = 1.
Conjecture that
P (t, r; T ) = exp(≠A(t) ≠ B(t)r).
(51)
Plug the representation of the bond price P into the PDE, then we have
dA(t)
= ≠Ÿ◊B(t),
dt
t œ [0, T ),
dB(t)
‡2
= B(t)2 + ŸB(t) ≠ 1, t œ [0, T ).
dt
2
In addition, the terminal condition of the PDE problem yields the the terminal
conditions of the ODEs above, i.e.,
A(T ) = 0,
B(T ) = 0.
Therefore, solving the above (Riccati) ODE system, we have that
B(t) =
2(e“(T ≠t) ≠ 1)
,
(“ + Ÿ)e“(T ≠t) + “ ≠ Ÿ
(“+Ÿ)(T ≠t)
2
≠2Ÿ◊
2“e
A(t) =
log(
),
2
“(T
≠t)
‡
(“ + Ÿ)(e
≠ 1) + 2“
Ô
where “ = Ÿ2 + 2‡ 2 .
Substitute the representation of A(t) and B(t) into (51), then we obtain the
closed form formula for a zero coupon bond under CIR model.
In addition, it follows from the dynamics of the CIR model (50) and Ito’s formula
that
EQ [rt |r0 = x] = xe≠Ÿt + ◊(1 ≠ e≠Ÿt ),
V arQ (rt |r0 = x) = x
‡2◊
‡ 2 ≠Ÿt
(e ≠ e≠2Ÿt ) +
(1 ≠ e≠Ÿt )2 .
Ÿ
2Ÿ
35
6.2
Heath-Jarrow-Morton (HJM) model
Short rate models are not always flexible enough to calibrating them to the observed
initial term-structure. In the late eighties, Heath, Jarrow and Morton (Heath et al.
1992) proposed a new framework for modelling the entire forward curve directly.
This section provides the essentials of the HJM framework.
6.2.1
Forward rates
Let us fix a time horizon T̄ . All bonds in the following discussing will mature at
or before time T̄ . We assume that all bonds bear no risk of default. We denote
that price at time t of a zero coupon bond maturing at time T Æ T by B(t, T ). We
assume further that for every t and T such that 0 Æ t Æ T Æ T̄ , the bond price
B(t, T ) is defined. In addition, we suppose that the interest rate is strictly positive
between times t and T , then B(t, T ) is strictly less than one whenever t < T.
Definition 6.1. We define the forward rate at time t for investing at time T to be
f (t, T ) = ≠
ˆ log B(t, T )
.
ˆT
intuitively, f (t, T ) is the instantaneous interest rate at time T that can be locked
in at the earlier time t. Moreover, it is easy to see that
B(t, T ) = e≠
sT
t
f (t,v)dv
,
which implies that it does not appear to matter (at least theoretically) whether we
build a model for forward rates or for bond prices.
In addition, the interest rate at time t is
rt = f (t, t).
6.2.2
Dynamics of forward rates and bond prices
Assume that f (0, T ) is known at time 0. We call this the initial forward rate curve.
In HJM model, the forward rate at later times t for investing at still later times T
is given by
f (t, T ) = f (0, T ) +
or
⁄ t
0
–(u, T )du +
⁄ t
0
‡(u, T )dWu ,
df (t, T ) = –(t, T )dt + ‡(t, T )dWt ,
(52)
where W is a Brownian motion under the actual measure P and –, ‡ are adapted
stochastic processes in t, representing the drift and volatility of f under the actual
measure P.
36
Consider the dynamics of ≠
d(≠
⁄ t
0
st
0
f (t, v)dv. It follows from simple calculation that
f (t, v)dv) = f (t, t)dt ≠
⁄ T
= rt ≠
t
⁄ T
t
df (t, v)dv
(–(t, v)dt + ‡(t, v)dWt )dv
We reverse the order of the integration, writing
⁄ T
t
⁄ T
–(t, v)dtdv =
‡(t, v)dWt dv =
t
⁄ T
t
⁄ T
t
–(t, v)dvdt = –ú (t, T )dt,
‡(t, v)dvdWt = ‡ ú (t, T )dWt ,
where
–ú (t, T ) =
⁄ T
–(t, v)dv,
‡ ú (t, T ) =
⁄ T
‡(t, v)dv.
t
t
In conclusion, we have that
d(≠
⁄ T
t
Note that B(t, T ) = e
≠
f (t, v)dv) = rt dt ≠ –ú (t, T )dt ≠ ‡ ú (t, T )dWt .
sT
t
f (t,v)dv
. Then Ito’s formula yields that
1
dB(t, T ) = B(t, T )(rt ≠ –ú (t, T ) + ‡ ú (t, T )2 )dt ≠ ‡ ú (t, T )B(t, T )dWt .
2
6.2.3
No arbitrage condition
The HJM model has a zero coupon bond with maturity T , for every T Æ T̄ . We
need to make sure there is no arbitrage opportunity in the market. According the
Girsanov theorem that it is equivalent to guarantee that the risk neutral measure Q
exists.
Consider the dynamics of Dt B(t, T ). It follows from Ito’s formula that
1
dDt B(t, T ) = Dt B(t, T )((≠–ú (t, T ) + ‡ ú (t, T )2 )dt ≠ ‡ ú (t, T )dWt )
2
(53)
We want to write the above equation in the form of
dDt B(t, T ) = ≠‡ ú (t, T )Dt B(t, T )(◊t + dWt ).
37
(54)
In other words, we will find a process ◊, such that
WtQ =
⁄ t
0
◊s ds + Wt ,
(55)
is a Brownian motion under the measure Q, which is defined by
Q(A) =
⁄
A
1
e≠ 2
s T̄
0
◊s2 ds≠
s T̄
0
◊s dWs
, ’A œ FT̄ .
Comparing (53) and (54), we must solve the equation
1
≠‡ ú (t, T )◊t = ≠–ú (t, T ) + ‡ ú (t, T )2
2
(56)
Recall the definitions of –ú (t, T ) and ‡ ú (t, T ) and differentiate the above equation
with respect to T , we then have
–(t, T ) = ‡(t, T )(‡ ú (t, T ) + ◊t ).
(57)
Here ◊t is called market price of risk.
Theorem 6.2 (HJM no arbitrage condition). With the above notations, a termstructure model for zero coupon bond prices of all maturities in [0, T̄ ] and driven
by a one dimensional Brownian motion does not admit arbitrage if there exists a
process ◊, such that (57) holds for all 0 Æ t Æ T Æ T̄ .
Proof. It remains to check that if ◊t solves (56), then we can use Girsanov theorem
to construct the risk neutral measure. The existence of the risk neutral measure
guarantees the absence of arbitrage.
Suppose that ◊t solves (57), then integrating with respect to the second variable
from t to T , we have that
1
1
–ú (t, T ) ≠ –ú (t, t) = (‡ ú (t, T ))2 ≠ (‡ ú (t, t))2 + ‡ ú (t, T )◊t ≠ ‡ ú (t, t)◊t .
2
2
Note that –ú (t, t) = ‡ ú (t, t) = 0, we have that
1
–ú (t, T ) = ‡ ú (t, T )2 + ‡ ú (t, T )◊t ,
2
which completes the proof.
As long as ‡(t, T ) is non-zero, we can solve (57) for ◊t :
◊t =
–(t, T )
≠ ‡ ú (t, T ),
‡(t, T )
0 Æ t Æ T.
This shows that ◊t is unique and hence the risk neutral measure is unique. In this
case, it follows from the second fundamental theorem of asset pricing that the market
model is complete.
38
6.2.4
HJM under risk neutral measure
In this subsection, we consider the dynamics of f (t, T ) and B(t, T ). For the forward
rate f (t, T ), whose dynamics is defined by (52), we have that
df (t, T ) = –(t, T )dt + ‡(t, T )dWt
= ‡(t, T )‡ ú (t, T )dt + ‡(t, T )(◊t + dWt )
= ‡(t, T )‡ ú (t, T )dt + ‡(t, T )dWtQ
where W Q is defined by (55).
Turning to the bond price B(t, T ), we know that
dDt B(t, T ) = ≠‡ ú (t, T )Dt B(t, T )(◊t + dWt ) = ≠‡ ú (t, T )Dt B(t, T )dWtQ .
Note that B(t, T ) =
1
(Dt B(t, T )),
Dt
then it follows from Ito’s formula that
dB(t, T ) = rt B(t, T )dt ≠ ‡ ú (t, T )B(t, T )dWtQ .
The following theorem summarises this discussion.
Theorem 6.3 (Term structure evolution under risk neutral measure). Under the
HJM no arbitrage condition (57), the foward rates evolve according to the equation
df (t, T ) = ‡(t, T )‡ ú (t, T )dt + ‡(t, T )dWtQ ,
and the zero coupon bond prices evolve according to the equation
dB(t, T ) = rt B(t, T )dt ≠ ‡ ú (t, T )B(t, T )dWtQ ,
or equivalently,
B(t, T ) = B(0, T ) exp{
⁄ t
0
rs ds ≠
⁄ t
0
‡
ú
(s, T )dWsQ
1⁄ t ú
≠
(‡ (s, T ))2 ds}.
2 0
(58)
where W Q is a Brownian motion under the risk neutral measure Q.
6.3
Forward LIBOR models
In this section, we derive the Black formula for the interest rate cap, a common
fixed income derivative underlain on the LIBOR (London Interbank Offered Rate).
The idea is similar to the one introduced in Section 5.4, that is, We simplifies the
structure of the pay-off function by introducing a forward measure, so that the
pricing formula of the fixed income derivative can be obtained within the standard
Black-Scholes framework.
39
6.3.1
Motivation
Consider the HJM model that satisfies the no arbitrage condition,
df (t, T ) = ‡(t, T )‡ ú (t, T )dt + ‡(t, T )dWtQ ,
where W Q is a Brownian motion under the risk neutral measure Q and ‡(t, T ), ‡ ú (t, T )
are defined in the previous section.
We want to choose a ‡(t, T ) such that f (t, T ) Ø 0. A natural candidate is a
geometric Brownian motion, in which ‡(t, T ) = ‡fs(t, T ), where the constant ‡ > 0.
In this case, the drift ‡(t, T )‡ ú (t, T ) = ‡ 2 f (t, T ) tT f (t, v)dv and the dynamics of
the forward interest rate f becomes
df (t, T ) = ‡ 2 f (t, T )
⁄ T
t
f (t, v)ddt + ‡f (t, T )dWtQ ,
It is worth noting that the fast growth of the drift term will cause the explosion
of stochastic differential equation. This difficulty with continuously compounding
forward rates causes us to introduce forward LIBOR.
6.3.2
LIBOR and forward LIBOR
Let 0 Æ t Æ T and ” > 0 be given. Suppose that B(t, T ) is the price of a bond
with maturity date T and principal 1 at time t œ [0, T ]. Then at time t, in order to
lock in the interest rate during [T, T + ”], one can hold a short position of size 1 in
B(t,T )
a T ≠zero coupon bond and a long position of size B(t,T
in a T + ” zero coupon
+”)
bond. This position can be created at zero cost at time t, it calls for ”investment”
B(t,T )
of 1 at time T to cover the short position, and it repays B(t,T
at time T + ”.
+”)
Suppose that money will be accumulated according to the simple interest rate rule,
then during the time interval t, T + ”, the simple interest rate is given by
L(t, T ) =
B(t, T ) ≠ B(t, T + ”)
”B(t, T + ”)
(59)
Definition 6.4. We call L(t, T ) defined by (59) forward LIBOR, ’0 Æ t < T .
Particularly, when t = T , L(T, T ) is LIBOR set at time T . ” is called the tenor of
the LIBOR.
6.3.3
Pricing a backset LIBOR contract
An interest rate swap is an agreement between two parties A and B that A will make
fixed interest rate payments on some “notional amount” to B at regularly spaced
dates and B will make variable interest rate payments on the same notional amount
on these same dates. The variable rate is often backset LIBOR, defined on one
payment date to be the LIBOR set on the previous payment date. The no-arbitrage
price of a payment of backset LIBOR on a notional amount of 1 is given by the
following theorem.
40
Theorem 6.5 (Price of backset LIBOR). Let ” > 0, 0 Æ t Æ T be given. The no
arbitrage price at time t of a contract that pays L(T, T ) at time T + ” is given by
S(t) = B(t, T + ”)L(t, T ).
Proof. It follows from the risk neutral pricing theory that
S(t) = EQ [e≠
s T +”
t
rs ds
L(T, T )|Ft ].
Then the definition of LIBOR (59) yields that
≠ B(T, T + ”)
|Ft ]
”B(T, T + ”)
s T +”
s T +”
1
1
= EQ [e≠ t rs ds
|Ft ] ≠ EQ [e≠ t rs ds |Ft ]
”B(T, T + ”)
”
sT
s T +”
s T +”
1
1
= EQ [e≠ t rs ds EQ [e≠ T rs ds |FT ]
|Ft ] ≠ EQ [e≠ t rs ds |Ft ].
”B(T, T + ”)
”
(60)
S(t) = E [e
Q
≠
s T +”
t
rs ds 1
As B(T, T + ”) = EQ [e≠
yields that
s T +”
T
rs ds
S(t) =
|FT ], B(t, T + ”) = EQ [e≠
B(t, T ) ≠ B(t, T + ”)
.
”
s T +”
t
rs ds
|Ft ], then (60)
(61)
Combining (61) with the definition of L(t, T ), we obtain
S(t) = L(t, T )B(t, T + ”).
This concludes the proof.
6.3.4
Black caplet formula
A common fixed income derivative security is an interest rate cap, a contract that
pays the difference between a variable interest rate applied to a principal and a fixed
interest rate (a cap) applied to the same principal whenever the variable interest
rate exceeds the fixed rate. let the tenor ”, the principal (also called the notional
amount) P, and the cap K be fixed positive numbers. An interest rate cap pays
(”P L(”j , ”j ) ≠ K)+ at time ”(j + 1), for j = 0, 1, · · · , n. To determine the price at
time zero of the cap, it suffices to price one of the payments, a so-called interest rate
caplet, and then sum these prices over the payments. We show here how to do this
and obtain the Black caplet formula. We also note that each of these payments is
K
of the form ”P (L(”j , ”j ) ≠ K Õ )+ , where K Õ = ”P
. Thus, it suffices to determine the
+
time zero price of the payment (L(T, T ) ≠ K) at time T + ” for an arbitrary T and
K > 0.
41
It follows from Theorem 6.5 that L(t, T ) =
us to introduce the forward measure
P
T +”
S(t)
.
B(t,T +”)
This observation motivates
⁄
1
(A) =
DT +” dQ, ’A œ FT +” .
B(0, T + ”) A
under which
WtT +” =
⁄ t
0
‡(s, T + ”)ds + WtQ
(62)
is a Brownian motion.
S(t)
Theorem 5.2 implies that B(t,T
is a martingale under PT +” . According to the
+”)
martingale representation, there exits an adapted process “(t, T ), a process in t,
such that
dL(t, T ) = “(t, T )L(t, T )dWtT +” .
(63)
Theorem 6.6 (Black caplet formula). Consider a caplet that pays (L(T, T ) ≠ K)+
at T + ”, where K is a non-negative number. Suppose “ in (63) is non-random, then
the price of the caplet at time zero is given by
B(0, T + ”)(L(0, T )N (d1 ) ≠ KN (d2 ))
where
d1,2
L(0, T ) 1 ⁄ T
= sT Ò
(log
±
“(t, T )2 dt).
2
K
2
0
“(t, T ) dt
1
0
Proof. It follows from the risk neutral pricing theorem that the price of the caplet
is given by
DT +”
(L(T, T ) ≠ K)+ ]
B(0, T + ”)
= B(0, T + ”)ET +” [(L(T, T ) ≠ K)+ ]
EQ [DT +” (L(T, T ) ≠ K)+ ] = B(0, T + ”)EQ [
Then the Conclusion follows from the dynamics of L(t, T ) (63) and the standard
Black-Scholes formula.
6.3.5
Forward LIBOR and zero coupon bond volatilities
In the end of this section, we derive the process “(t, T ).
According to the definition of the fowrd LIBOR, L(t, T ) can be rewritten as
L(t, T ) =
B(t, T )
1
≠ .
”B(t, T + ”) ”
42
It follows from (58) that
⁄ t
B(t, T )
= exp{≠ (‡ ú (s, T ) ≠ ‡ ú (s, T + ”))dWsQ
B(t, T + ”)
0
⁄ t
1
≠
((‡ ú (s, T ))2 ≠ (‡ ú (s, T + ”))2 )ds}
2 0
Then Ito’s formula yields that
1
dL(t, T ) = (L(t, T ) + )(‡ ú (t, T + ”) ≠ ‡ ú (t, T ))(‡ ú (t, T + ”)dt + dWtQ ).
”
Together with (62), we have that
1
dL(t, T ) = (L(t, T ) + )(‡ ú (t, T + ”) ≠ ‡ ú (t, T ))dWtT +” .
”
Comparing with (63), we obtain
“(t, T ) =
1 + ”L(t, T ) ú
(‡ (t, T + ”) ≠ ‡ ú (t, T )).
”L(t, T )
43
7
7.1
Foreign exchange models
The market model
The model is driven by a two dimensional Brownian motion (W1 , W2 ) defined on the
probability space ( , F, P), where P is a probability measure in the actual world.
We suppose that there are two currencies in the market. Particularly, the stock
price in the domestic currency follows the stochastic differential equation
dSt = µt St dt + ‡t St dW1t .
The domestic money account is accumulated with the continuously compounding
interest rate rt , hence the domestic money account and the domestic discount factor
can be respectively given by
st
Mt = e
0
rs ds
,
Dt = e
≠
st
rs ds
0
.
We assume that the foreign interest rate is rtf , which leads to the foreign money
account and foreign discount factor are respectively given by
Mtf
st
=e
0
rsf ds
,
Dtf
=e
≠
st
0
rsf ds
.
Finally, we suppose that foreign exchange rate Q, which gives the units of domestic
currency per unit of foreign currency, satisfies
dQt = “t Qt dt + ‡2t Qt (flt dW1t +
Ò
1 ≠ fl2t dW2t )
If we measure the foreign account by the domestic currency, then the value of the
foreign money account is given by
M̄tf := Qt Mtf .
Assume that fl, ‡1 , ‡2 “, µ, r, rf are adapted processes with respect to the filtration
{Ft }tØ0 generated by the Brownian motion (W1 , W2 ). Moreover, we suppose that
‡1 > 0, ‡2 > 0, ≠1 < fl < 1.
Define
W3t =
⁄ t
0
fls dW1s +
⁄ tÒ
0
1 ≠ fl2s dW2s
Then it follows Levy’s characterisation that W3 is a Brownian motion under P. In
addition, by simple calculation, we have that
dW1t dW3t = flt dt,
and hence
dSt dQt
= flt ‡1t ‡2t dt.
St Qt
44
7.1.1
Domestic risk neutral measure
There are three assets that can be traded: the domestic money market account,
the stock, and the foreign money market account. We shall price each of these in
domestic currency and discount at the domestic interest rate. The result is the
price of each of them in units of the domestic money market account. Under the
domestic risk-neutral measure, all three assets priced in units of the domestic money
market account must be martingales. We use this observation to find the domestic
risk-neutral measure.
We note that the first asset, the domestic money market account, when priced
in units of the domestic money market, has constant price 1. This is always a
martingale, regardless of the measure being used.
We consider the stock price St . It is to see that the stock price satisfies that
dSt = rt St dt + ‡1t St (◊1t dt + dW1t ),
where ◊1t is the solution to the following market price of risk equation
‡1t ◊1t = µt ≠ rt ,
which implies,
µt ≠ r t
.
‡1t
◊1t =
To obtain the domestic risk neutral measure Q, we construct W1Q , which will be the
Brownian motion under Q, as follows.
W1tQ =
⁄ t
0
◊1s ds + W1t .
(64)
The third asset that can be traded is foreign currency money account, whose dynamics is given by
dM̄tf
=
dMtf Qt
= (“t +
rtf )Mtf Qt dt
+
Mtf Qt ‡2t (flt dW1t
Substitute (64) into the equation above, we then obtain
dM̄tf
=
rt M̄tf dt
+ (“t +
rtf
≠ rt ≠
‡2t flt ◊1t )M̄tf dt
+
+
Ò
1 ≠ fl2t dW2t ).
M̄tf ‡2t (flt dW1tQ
+
Ò
1 ≠ fl2t dW2t )
(65)
To construct the risk neutral measure, we would like to choose ◊2t , such that
W2tQ
and
dM̄tf
=
rt M̄tf dt
+
=
⁄ t
0
◊2s ds + W2t ,
‡2t M̄tf (flt dW1tQ
45
+
(66)
Ò
1 ≠ fl2t dW2tQ )
(67)
Combining (65), (66) and (67), we obtain the second market price of risk equation
as follows,
“t +
rtf
≠ rt = ‡2t (flt ◊1t +
Solving the equation above, we obtain that
◊2t =
Ò
1 ≠ fl2t ◊2t ).
“t + rtf ≠ rt ≠ ‡2t flt ◊1t
Ò
‡2t 1 ≠ fl2t
.
We are now using the Girsanov theorem to construct the risk neutral measure.
Define
Zt = exp{≠
⁄ t
0
⁄ t
⁄ t
1 2
2
(◊1s + ◊2s
)ds ≠
◊1s dW1s ≠
◊2s dW2s }.
2
0
0
Construct the risk neutral measure Q, as follows,
Q(A) =
⁄
A
ZT dP,
A œ FT ,
Then the Girsanov theorem yields that (W1Q , W2Q ) is a Brownian motion under Q
and hence the discounted prices Dt Mtf , Dt St are martingales under Q.
In addition, it is easy to verify that
W3t =
⁄ t
0
Q
fls dW1s
+
⁄ tÒ
0
Q
1 ≠ fl2s dW2s
is a Brownian motion under Q. Thus, we can write the dynamics of the foreign
exchange rate Q in the following form,
dQt = (rt ≠ rtf )Qt dt + ‡2t Qt dW3tQ
(68)
Particularly, we also have that
dW3tQ dW1tQ
7.1.2
= flt dt,
dW3tQ dW2tQ
=
Ò
1 ≠ fl2t dt
Foreign risk neutral measure
Similar to the previous subsection, we here would like to choose a risk neutral
measure, such that the discounted domestic money account, discounted stock price
and discounted foreign money account are martingales, where the discount factor is
given by Dtf and the currency unit is the foreign.
First, it is easy to see that the foreign money account, when priced and discounted
at the foreign currency, is always 1. Therefore, the discounted foreign money account
at the foreign currency is a martingale.
46
We consider the stock price and the domestic money account. We note that at
the foreign currency, the stock price and the domestic money account can be given
by
Dtf St
St
= f ,
Qt
Mt Qt
Dtf Mt
Mt
= f .
Qt
Mt Qt
Note that Dt Mtf Qt , Dt St and Dt Mt = 1 are martingales under Q. This observation
Df S
Dtf Mt
and Theorem 5.2 imply that Qt t t and Q
are martingales under Pf , where
t
Pf (A) =
⁄
DT M f QT
dQ.
Q0
A
Particularly, the process W f = (W1f , W2f ) given by
W1tf
=≠
W2tf = ≠
⁄ t
0
⁄ t
‡2s fls ds + W1tQ
0
Ò
‡2s 1 ≠ fl2s ds + W2tQ
is a two dimensional Brownian motion under Pf . Here Pf is said to be the foreign
risk neutral measure.
Similarly, we can introduce the Brownian motion (under Pf ) W3f given by
W3tf =
⁄ t
0
f
fls dW1s
+
It is easy to verify that
W3tf = ≠
⁄ t
0
⁄ tÒ
0
f
1 ≠ fl2s dW2s
.
‡2s ds + W3tQ ,
and
dW3tf dW1tf = flt dt,
dW3tf dW2tf =
Finally, it follows from (5.2) or Ito’s formula that
Ò
1 ≠ fl2t dt
Mt Dtf
Mt Dtf
d
=≠
‡2t dW3tf ,
Qt
Qt
Dtf St
Dtf St
d
=
(‡1t dW1tf ≠ ‡2t dW3tf ).
Qt
Qt
47
7.2
Garman-Kohlhagen formula
In this subsection, we assume the domestic and foreign interest rates r and rf and
the volatility ‡2 are constant. Consider a call on a unit of foreign currency whose
payoff in domestic currency is (QT ≠ K)+ . It follows from the risk neutral pricing
theory that at time zero, the value of the option is given by
EQ [e≠rT (QT ≠ K)+ ].
In this context, the dynamics of Q given by (68) becomes
dQt = (r ≠ rf )Qt dt + ‡2 Qt dW3tQ
Using the same techniques in the standard Black-Scholes framework, we obtain that
EQ [e≠rT (QT ≠ K)+ ] = e≠r T Q0 N (d1 ) ≠ Ke≠rT N (d2 ),
f
(69)
where
d1,2 =
1
Q
1
Ô (log 0 + (r ≠ rf ± ‡22 )T ),
K
2
‡2 T
and N (·) is the distribution function of a standard normal distribution. Equation
(69) is the Garman-Kohlhagen formula.
References
Black, F. and M. Scholes (1973): “The Pricing of Options and Corporate
Liabilities,” The Journal of Political Economy, 81, 637–654.
Cox, J. C., J. E. Ingersoll Jr, and S. A. Ross (2005): “A theory of the term
structure of interest rates,” in Theory of valuation, World Scientific, 129–164.
Heath, D., R. Jarrow, and A. Morton (1992): “Bond pricing and the term
structure of interest rates: A new methodology for contingent claims valuation,”
Econometrica: Journal of the Econometric Society, 77–105.
Hull, J. (2012): Options, Futures, and Other Derivatives, Pearson, Eighth Editon.
Merton, R. C. et al. (1973): “Theory of rational option pricing,” Theory of
Valuation, 229–288.
Vasicek, O. (1977): “An equilibrium characterization of the term structure,” Journal of financial economics, 5, 177–188.
48