MULTIDIMENSIONAL MARKET MODEL - PDF Quantitative Methods in derivatives pricing

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.

market prices of risk, it is not possible to make these processes driftless, and we will not be able to express the price of a derivative as an expectation in the risk neutral measure.

■ We define a portfolio consisting of the assets, S_i,and the money market account. This portfolio is also a martingale in the risk neutral measure. We assume that this portfolio replicates the payoff of the derivative.

■ Using the martingale representation theorem we show that the quanti- ties of assets needed to define the portfolio that replicates the payoff of the derivative can be found if the market prices of risk can be deter- mined. This allows us to produce a meaningful formula for the derivative price in the form of an expectation.

Using Ito’s lemma, Equation 3.29 gives

(3.31)

We rewrite this as follows:

(3.32)

If we can determine a multidimensional market price of risk process, (t), by solving the linear system

(3.33)

then the discounted asset price can be written as follows:

(3.34) dS_i( )t

( )ⁱ --- S_i( )t ( )^t ---

--- (_i( )^t –r t( ))dt _i,j( )dW^t _j, i

j=1 j=M

∑

+ 1,…,N

= =

dS_i( )t ( )ⁱ --- S_i( )t ( )t ---

--- (i( )t –r t( ))dt i,j( )t j( )dtt

j=1 j=M

∑

–

i,j( )d Wt _j j( )s ds 0

∫

 + 

 ,i= 1,…,N

j=1 j=M

∑

_i,j( )^t _j( )^t j=1

j=M

∑

⁼ ⁱ^{( )}^t ^–^{r t}^{( )}^,ⁱ ⁼^1,^…^,^N

dS_i( )t ( )ⁱ --- S_i( )t ( )^t ---

--- _i,j( )dW^t ˆ_j,i

j=1

j=M

∑

^1,…,^N

= =

with

(3.35) where

sure.

If we cannot solve Equation 3.33, there is no risk neutral measure. If Equation 3.33 has more than one solution, there is more than one risk neutral measure.

To price a claim that pays V(T), we define a portfolio process, Y(t), , whose end value at t T matches the payoff of the claim. It is straightforward to show that this process, normalized with the money market account, is a martingale under the risk neutral measure. The ini- tial value of this process will then be equal to the value of the claim. The portfolio process is the value of an investment in the assets S_i and in a money market account that pays the risk-free rate r(t). The investor holds the amount i(t) of asset S_i over an infinitesimal time dt. During this interval, the investment changes in value for two reasons. There is a contribution due to change in the asset price equal to

and there is a contribution due to the return on the money market

account equal to . The portfolio process

follows

(3.36)

Using Equation 3.34 and a little algebra, we get

(3.37)

This shows that process Y(t) normalized with the money market account is a martingale under the risk neutral measure. Since the portfolio process matches the payoff of the derivative, we have

(3.38) d Wˆ

j( )t dW_j( )t d _j( )^s ds 0

∫

 

 

= Wˆ

j( )t ,j =

0≤ ≤t T =

Σ

i=1i=Ni( )dSt _i( ),t r t( )^[Y t( )^–

Σ

ⁱ⁼¹^i=Nⁱ^{( )S}^t ⁱ^{( )}^t ^]dt

dY t( ) i( )dSt _i( )t i=1

i=N

∑

^{Y t}^{( )} ⁱ^{( )S}^t ⁱ^{( )}^t

i=1 i=N

∑

– r t( )dt

dY t( ) ( )^t

--- _i( )^t

i=1

i=N

∑

^S---ⁱ( )^{( )}t^t _ij( )dW^t ˆ_j( )t j=1

j=M

∑

Y( )0 ( )0

--- E Y T( )

( )^T ---

E V T( ) ( )^T ---

1,…,M are Wiener processes in the risk neutral mea-

Since ⁽⁰⁾ 1, the value of the derivative is

(3.39) Now we need to see under which conditions the processes i(t) exist.

We can prove that this process exists through the martingale representation theorem in the same way as the one-dimensional case.

The following is a martingale such that M(0) 0:

(3.40)

The multidimensional version of the martingale representation theorem says that there exist processes f_i(t) such that

(3.41)

This gives a system of equations for the hedging ratios:

(3.42)

The pricing and hedging problem is now defined in terms of two sys- tems of equations: the market price of risk equations (3.33) and the hedging ratio equations (3.42).

We can make the following observations. If Equation 3.33 has a unique solution, namely, if there is a unique multidimensional process ^{(t), then} there is only one way to define the Radon-Nikodym derivative and, consequently, there is only one risk neutral measure.

If there is a unique solution for (t), there is a solution to the hedging Equation 3.42, and consequently, every derivative can be hedged. In this the market is said to be complete. Arbitrage is also ruled out because we can purchase or sell the replicating portfolio or the underlying assets in the right amounts and ensure a riskless profit.

If Equation 3.33 has multiple solutions for (t), the Radon-Nikodym derivative is not unique and we cannot use the equivalent martingale measure for pricing. In this case there is no arbitrage, but claims cannot be hedged. The market is said to be incomplete.

V( )0 E V T( ) ( )T ---

M t( ) i( )s S_i( )s ( )^s

--- ij( )s j=1

j=M

∑

^d^W^ˆ ^j^{( )}^s

∫

t i=1 i=N

∑

M t( ) f 0

∫

t ^j^{( )}^s j=1

j=M

∑

^d^W^ˆ ^j^{( )}^s

_i( )S^t _i( )t _ij( )^t i=1

i=N

∑

⁼ ^f^j^{( )}^t ^{( ),}^t ^j⁼^1,^…,^M

If Equation 3.33 does not have a solution for (t), there is arbitrage and the equivalent risk neutral measure does not exist.

To summarize,

■ Complete market: Absence of arbitrage means that the equivalent risk neutral measure is unique and every derivative can be hedged.

■ Incomplete market: Absence of arbitrage does not imply the equivalent risk neutral measure is unique and not every derivative can be hedged.

■ No risk neutral measure: There are arbitrage opportunities.

Extension to Other Normalizing Assets

The approach followed here can be extended to consider assets other than the money market account as normalizing assets. This is important because it often happens that the pricing problem can be better formulated when a different normalizing asset is used. We will illustrate the derivation with a numeraire or normalizing asset that does not pay dividends. For simplicity of the derivation, this time we will use vector notation.

We consider an underlying asset, S(t), and a normalizing asset, B(t), with processes

(3.43) (3.44)

where is a multidimensional Wiener process. We assume that these are risk neutral processes and that these assets don’t pay dividends. In order to operate with asset processes we must first make sure that all the Wiener processes involved are in a common measure. In this case it is just a matter of convenience to choose the risk neutral measure.

The process for the underlying asset, normalized with asset B(t) is

(3.45) dS t( )

S t( )

--- ⁼ r t( )dt⁺_S^d^W

dB t( ) B t( )

--- ⁼ r t( )dt⁺_B^d^W

dS B---- S B----

--- dS

---S dB

---B

– dS

---S dB ---B

– dB

---B

 

 ² +

r t( )dt⁺_S^d^W–^{r t}( )dt^–_B^d^W

= _S

– _Bdt+_B_Bdt

We can rewrite this as follows:

(3.46)

The Girsanov theorem tells us that the process

(3.47) is a Wiener process in a measure defined through the appropriate Radon- Nykodim derivative. We refer to the measure under which is a Wiener process as the measure induced by B(t). In order to obtain useful pricing formulas or pricing equations, we usually don’t need to compute this new measure transformation explicitly. All we need is the knowledge that in this new measure, is a Wiener process.

In the measure induced by B(t), the normalized asset process is

(3.48)

We now consider a replicating portfolio consisting of the underlying asset, S(t), and the normalizing asset, B(t). Following the same line of thought as in the previous section, we consider a portfolio Y(t) consisting of a hedge amount (t) invested in asset S(t), and the difference between the value of the portfolio and the value invested in the underlying asset is invested in the normalizing asset. The portfolio process is

(3.49) The fundamental difference between this portfolio and the one created with the money market account is that now the component of return due to asset B(t) is stochastic. Before, this component was r(t)dt, which is deter- ministic conditional on information at time t. As before, we now use Ito’s lemma to get a process for :

dS B---- S B----

--- = –_B(_S^–_B)dt+(_S^–_B)dW

_B(_S^–_B)dt

– +_B(_S^–_B)dt

+ (_S^–_B)d W _B( )^s ds 0

∫

( – )

W^B W B( )s ds 0

∫

t –

WB( )t

dS B---- S B----

--- = (_S^–_B)dW^B

dY t( ) ( )dS t^t ( ) [Y t( )–( )S t^t ( )]dB t( ) B t( ) --- +

Y B----

(3.50)

Using Equation 3.47, the process for the normalized replicating portfolio is

(3.51) This says that the portfolio price normalized with asset B(t) is a martingale. If we now consider a portfolio that matches the payoff of the derivative, we get a pricing equation in the form of an expectation taken with respect to the measure induced by B(t):

(3.52) Since Y(T) V(T), the pricing formula is

(3.53)

Deriving Risk-Neutralized Processes

As we will see in the next chapter, one of the main tasks in pricing is to transform all the processes involved to the measure where the pricing is done. In many cases this is the risk neutral measure. In other cases this measure is induced by an asset price other than the money market account. In most cases the first step for getting the process in the desired measure is getting the processes in the risk neutral measure.

If the underlying assets S_i don’t pay dividends, it is straightforward to get the risk-neutralized process for the assets. All we need to do is replace the Wiener process in the market measure with the Wiener process in the risk neutral measure. For example, for the case of a one-dimensional asset we have

dY B---- 1

B----dY Y B----dB

---B

– 1

B----dYdB ---B

– Y

B---- + dB

---B

 

 ²

= 1

B---- ( )^t dS (Y–( )^t S)dB ---B

+ Y

B---- dB ---B –

( )dS^t (Y–( )S^t )dB ---B +

– 1

B----dB ---B Y

B---- dB ---B

 

 ² +

( )^t S

B----[(_S^–_B)dW^–(_S^–_B)_B^dt]

( )^t S t( ) B t( )

---(_S^–_B)(dW^–_B^dt)

B---- ( )^t S t( ) B t( )

---(_S( )^t –_B( )^t )dW^B

Y( )0 B( )0

--- E₀^B Y T( )

B T( ) ---

V( )0 B( )E0 ₀^B V T( ) B T( ) ---

(3.54) (3.55) (3.56) (3.57) The risk-neutralized process for asset S is then

(3.58) If the asset pays a dividend rate y, however, the asset process in the market measure must reflect the dividend payments as a drop in value:

(3.59) In this case, the process that must be driftless is

(3.60) This is consistent with requiring that

(3.61) should be driftless. This equation results from regarding each infinitesimal dividend payment yS(t)dt as a payoff.

Equation 3.60 leads to the following expression for the risk-neutralized asset process:

(3.62) This last equation is also consistent with a martingale for the portfolio process Y(t).

Another simple example is the derivation of the risk neutral process for the foreign exchange rate, X(t), which represents the units of domestic currency needed to purchase one unit of foreign currency (notice that this definition is typically the inverse of the quoted foreign exchange rates, with the exception of the British Pound). We assume that the process for X(t) is of the form

dS t( ) = ^{S t}( )dt+( )^t S t( )dW t( )

^{S t}( )dt+( )S t^t ( )(dWˆ( )t ^–^dt)

^{S t}( )dt ( )S t^t ( ) dWˆ ( )t ^–^r ---dt

 – 

 

rS t( )dt+( )S t^t ( )dWˆ ( )t

dS t( ) = r t( )Sdt+( )S t^t ( )dWˆ ( )t

dS t( ) = (( )^t –y)S t( )dt+( )S t^t ( )dW t( )

dS t( ) ( )^t --- yS t( )

( )^t ---dt +

S t( ) ( )^t

--- yS s( )

( )s ---ds 0

∫

t +

dS t( ) = (r t( )–y)S t( )dt+( )^t S t( )dWˆ ( )t

(3.63) where is a multidimensional Wiener process. In order to derive the process for the foreign exchange, we must take into account that a Wiener process in the risk neutral measure in the foreign market (the one relevant to the exchange rate X(t)) is not in general a Wiener process in the risk neutral measure in the domestic market. In order to obtain a relationship for the drift we may consider the price process of a foreign asset, S_f(t), translated into the domestic currency, namely, d(X(t)S_f(t)). If we did this for an arbitrary asset S_f, we would introduce the Wiener process of S_f, whose risk neutral drift in the domestic currency is not known. If, however, we select the foreign money market account as the foreign asset, we don’t introduce any additional Brownian motions and we get

(3.64) The drift of the foreign exchange rate is obtained by requiring that the drift of the translated foreign money market account should be the risk-free rate. This gives us

(3.65) This says that in the domestic measure, the foreign exchange rate behaves like an asset that pays a dividend yield equal to the instantaneous foreign risk-free rate.

As a second example, we consider the relationship between Wiener processes in the foreign risk-free measure and Wiener processes in the domestic risk-free measure. Consider a foreign asset that in the foreign risk- free measure has the process

(3.66) The process for the foreign asset translated into the domestic currency is (3.67) where dt stands for cov( , ). This expression will produce the cor- rect domestic risk neutral drift if the following relationship is satisfied:

dX t( ) X t( )

--- = xdt+_xdW

d X t( ( )f( )t ) X t( )f( )t

--- = (x+r_f)dt+_xdW

x = r r– _f

dS_f( )t S_f( )t

--- ⁼ r_fdt⁺_f^d^W_f

d X t( ( )S_f( )t ) X t( )S_f( )t

--- ⁼ (r–r_f)dt⁺r_fdt⁺xdW⁺_f^d^W_f⁺_x_f^dt

_x_f ^dX---X dS_f

S_f ---

(3.68) where is a Wiener process in the domestic measure. This is the fundamental relationship that connects the two Wiener processes.

Dalam dokumen PDF Quantitative Methods in derivatives pricing (Halaman 66-75)