significant changes, a similar picture carries over to the case of derivatives with early exercise features, such as Bermudan or American options. We will discuss European derivatives first and derivatives with early exercise later on.
The most commonly used normalizing asset in continuous time is the money market account, (t), defined as the value of a continuously rein- vested unit of account.
(3.3) The measure under which the value of a derivative normalized with the money market account is a martingale is called the risk neutral measure.
Risk neutral pricing, where the money market account is the normaliz- ing asset, is the standard framework for pricing in continuous time. We will develop this framework in sufficient detail. Similar frameworks can also be set up for pricing with other normalizing assets.
We will start by pricing a derivative on a single asset in the risk neutral measure. We will then extend the same logic for the case where the normal- izing asset is something other than the money market account. After that, we will consider the multidimensional case of risk neutral pricing.
The change in value of this portfolio over an infinitesimal period of time, dt, is (3.6) In the next few paragraphs we will show that if W(t) is a Wiener pro- cess in the measure induced by (t), namely, the risk. neutral measure, the portfolio process Y(t) normalized with (t) is a martingale, and there is a hedging process, (t), that allows us to match the values of Y(T) with the payoff of the derivative at time T. Assuming for the moment that this is true, we get a pricing formula for the derivative as follows.
Since the normalized value of Y(t) is a martingale,
(3.7) Since we are saying that there is a (t) that allows us to match Y(T) with V(T), the value of Y(t) must be the value of the derivative at time t. This gives us the pricing formula:
(3.8) Since the portfolio Y(t) can be made to track the derivative price by properly choosing (t), this portfolio is said to replicate the value of the derivative as a function of time. Y(t) is called a hedge portfolio or a repli- cating portfolio.
The reader may infer that if instead of choosing the money market account as one of the components of Y(t) we had chosen another asset, say, asset B(t), we would arrive at a similar martingale expression, except that now the measure under which the Brownian motions in both S(t) and B(t) are expressed is determined, or induced, by B(t) (we will show this later).
Our pricing formula in this case would be
(3.9) We now prove the two necessary facts: The normalized replicating portfolio is a martingale, and the appropriate hedge ratio exists.
Using Ito’s lemma, the process for is
(3.10) dY t( ) ( )dS tt ( ) [Y t( )–( )S tt ( )]d( )t
( )t --- +
=
Y t( ) ( )t
--- Et Y T( )
( )T ---
=
V t( ) ( )t
--- Et V T( )
( )T ---
=
V t( ) B t( )
--- EtB V T( )
B T( ) ---
=
S t( ) ( )t ---
dS --- S
--- dS
---S d
---
– dS
---S d ---
– d
---
2 +
=
Replacing and and keeping first-order terms yields
(3.11)
We can rewrite this expression as
(3.12)
if we define
(3.13) Girsanov theorem tells us that
(3.14)
is a Wiener process in a world where the probability density of the out- comes of W(t), dP, is distorted (or rescaled) in the following way,
(3.15) where Z(t), known as the Radon-Nikodym derivative, is given by (see Equation 2.132)
(3.16) Equation 3.12 becomes
(3.17)
dS
---S d
---
dS t( ) ( )t --- S t( ) ( )t ---
--- = (( )t –r t( ))dt+( )t dW
dS t( ) ( )i --- S t( ) ( )t ---
--- = (( )t –r t( ))dt–(( )t –r t( ))dt+( )dWt +(( )t –r t( ))dt
( )t dW ( )t –r t( ) ( )t ---
+
dt
=
( )d W tt ( ) ( )s –r s( ) ( )s ---ds 0
∫
t +
=
( )t ( )s –r s( ) ( )s
---57 0≤ ≤t T
=
W( )t W t( ) ( )s ds 0
∫
t +=
W( )t W t( ) ( )s –r s( ) ( )s ---ds 0
∫
t +=
dP = Z T( )dP
Z t( ) ( )s dW s( ) 0
∫
t– 1
2--- 2( )s ds 0
∫
t –
exp
=
dS t( ) ( )t --- S t( ) ( )t ---
--- = ( )dWt ( )t
This means that in the measure defined by Equation 3.15, the asset price normalized with the money market account is a martingale:
(3.18) We can also express this result as follows: Equation 3.18 is a condition that changes the probability of outcomes of W(t), , in a way that we can define a new Wiener process W(t). In terms of this new Wiener process, the asset process becomes
(3.19) The process
(3.20) is expressed in the risk neutral measure. The quantity
(3.21) is called the market price of risk. The market price of risk is the amount by which the return of the asset exceeds the risk-free return per unit of volatility.
We will now show that the process for the normalized portfolio is also a mar- tingale when expressed in the risk neutral measure. Replace in Equation 3.6:
(3.22) Replacing for dS(t) and Y(t), we get
(3.23) S t( )
( )t
--- Et S T( )
( )T ---
=
0≤ ≤t T
dS
---S = ( )dtt +( )dW tt ( )
( )t dW( )t ( )s –r s( ) ( )s ---dt
–
=
r t( )dt+dW
=
dS
---S = r t( )dt+dW
( )t ( )t –r t( ) ( )t ---
=
d ---
dY t( ) = ( )dS tt ( )+[Y t( )–( )S tt ( )]r t( )dt
dY t( ) ( )t
--- 1
---dY Y
----d ---
– 1
---dYd
--- Y
---- d
---
2 +
–
= 1
---[dS+(Y–S)rdt] Y
----rdt –
= 1
---[dS–Srdt]
= 1
---[(rSdt+SdW)–Srdt]
=
---SdW
=
The process
(3.24) is a martingale, since it has no drift. Integrating this equation, we rearrange the solution as follows:
(3.25) The integral on the left-hand side is a martingale that starts at zero.
Call this martingale M(t):
(3.26) The martingale representation theorem says that there exists an adapted process f(t) such that
(3.27)
This says that there is the following hedge process:
(3.28) In summary, we have accomplished the following:
■ We defined the risk neutral measure by the condition that the process of the ratio of the underlying asset price to the money market account should be a martingale.
■ We constructed a portfolio consisting of a certain amount of the under- lying asset and the money market account.
■ We showed that this portfolio, normalized with the money market account, is also a martingale. If we assume that the terminal value of the portfolio equals the payoff of the derivative, this martingale gives us a pricing formula for the derivative.
■ We showed that the appropriate amount of the underlying asset in this portfolio can be found such that the portfolio value at maturity matches the value of the derivative at maturity.
dY t( ) ( )t
--- ( )t
( )t
---( )t S t( )dW( )t
=
( )s ( )s
---( )S ss ( )dW( )s 0
∫
t Y t---( )( )t Y( )0( )0 --- –
=
M t( ) ( )s ( )s
---( )S ss ( )dWˆ ( )s 0
∫
t=
M t( ) f s( )dWˆ 0
∫
t=
( )s ( )s
---( )S ss ( )dWˆ ( )s 0
∫
t=
( )t ( )f tt ( ) ( )S tt ( ) ---
=
The derivative security priced this way reflects absence of arbitrage. If portfolio Y(t) did not track the value of the derivative, one could sell whichever is more expensive and purchase whichever is less expensive and realize a riskless profit at maturity (since both have the same payoff value).
We can view this one-dimensional analysis in two ways. We can think of a market that contains only one underlying asset (not a very interesting market), or we can think of a pricing exercise in a multidimensional market where we consider an underlying asset in isolation.