European currency options - Valuation Of Currency Options Under A Regime-Switching Gaussian HJM

Chapter 3 Valuation Of Currency Options Under A Regime-Switching Gaussian HJM Model

3.4 European currency options

In this section, we derive arbitrage-free prices for European currency options under the RSJD model via the forward Esscher-transform technique.

3.4.1 Regime-switching Esscher measure

The filtration of the spot-FX rate and the hidden Markov chains X and Y , respectively, by  and _t^S _T^{X Y}^, . The join filtration of the FX rate, the domestic and the foreign forward interest rates and Markov chain is denoted by a

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-algebra given below:





( , )t T  _T^{X Y}  _t^S  _t^W^F  _t^W^D

     .

A capital market described by a jump model is incomplete, under which the risk-neutral measure associated with change of probability measure is not unique.

Hence, we adopt the regime-switching Esscher transform. Two families of regime-switching parameters for the Esscher transform are denoted, respectively, by



^{, ( )}



‧

The domestic regime-switching Esscher transform under the RSJD model is the defined by: The martingale condition for the spot-FX rate under the domestic martingale measure

* * Proof (See Appendix A)

□

Hence, let S T^*( ) be the dynamic process of the FX rate at time T discounted by the current value of the domestic money-market account (^Din (3.6)) under the domestic measure ⁽^^C^*^,^^J^*⁾. The S T^*( ) process can be shown as follows:

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Hence, the new joint probability can be written as follows:

 

Note that the regime shift of the spot-FX rate represented by the jump variable (Z_n) in (2.2) has a differential impact on the risk premiums associated with jump intensities.

Trading the spot-FX rate may require risk premium for bearing undiversifiable regime-switching risk. This can be also observed from the fact that the moment-generating function of the jump variable adjusts the original intensity matrix

 for the risk premiums of jump term to arrive at a new intensity matrix ^^* with a

regime shift factor

2 2

3.4.2 Forward risk-adjusted measure via regime-switching forward Esscher transform In this subsection, we derive a regime-switching Gaussian HJM model under the

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 , the Markov-modulated domestic and foreign instantaneous forward rates are denoted, respectively, by f^D



t T X t, , ( )



and f^F



t T X t, , ( )



for any maturity

(cf. Valchev (2004))

For each state e_kI_X , under the domestic martingale measure ⁽^{ }^C^*^, ^J^*⁾, the dynamics of the domestic and the foreign ZCBs are given, respectively, as follows:

 

‧

By the following regime-switching forward Esscher transform, the domestic forward martingale measure, the T_-measure ^T^,(^^C^*^,^^J^*⁾, with respective to the numeraire

Note that the regime-switching Esscher transform can be used to observe regime-switching risk of interest rates.

Since the domestic ZCB numeraire in (4.8) does not have its own jump term, the

D( )

S t shares the same jump term of the spot-FX S t^*( ) under the domestic forward

measure ^T^,(^^C^*^,^^J^*⁾. Hence, we maintain the same asterisk (*) notation for the jump term of the FX rate S^D( )t . We use Ito’s lemma for a jump process to obtain the dynamics of the forward FX rate S^D( )t under the forward Esscher measure

* *

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   

 

  



+ σ_S V^F t T X t, , ( ) V^D t T X t, , ( ) dW  exp Z_n 1 d ( )t



t T X t, , ( )



d ^T



exp

 

Z_n^* 1



d ^*( )t

ξ  W    (3.4.9)

where ξ



t T X t^{, , ( )}



σS V^F



t T X t^{, , ( )}



V^D



t T X t^{, , ( )}



Clearly, the forward FX rate process S T^D( ) is a martingale process under the domestic forward measure ^T^,(^^C^*^,^^J^*⁾. In addition, the volatilities coefficient of the forward FX rate process accommodates not only the volatilities coefficient of the ZCBs, but also the corresponding regime-switching impact of the ZCBs. The dynamics in (4.9) facilitates valuing currency options next.

3.4.3 Valuation of European currency options

The arbitrage-free price, in units of domestic currency, of a European currency call option on a FX rate is derived using the risk-neutral valuation approach given below.

The final payoff of the currency call option is given by:

( ) ( ( ) )

C T N S T K ^ (3.4.10)

where S T( ) is the spot price of the deliverable currency at maturity T (i.e., the spot exchange rate at time T ), K is the strike price in units of domestic currency per foreign-currency unit, and N0 denotes units of the option. Without loss of generality, we assume N1.

Under the domestic martingale measure ⁽^{ }^C^*^, ^J^*⁾ with given Markov chain X and the joint probability Q T_ij^*( t n, ) of the MMPP, the currency-call option is given by:



^{, ( )} ^{( )}



^ ^C^*^, ^J^*^ ^exp



^t^T ^D^{( , ( ))}



^{( )} ^{( , )}

C t X t ^ t E^ ^ 



r u X u du C T ^ t T  . (3.4.11)

‧

With the occupation time of Markov chain X , the volatility parameter



t T X t, , ( )



V in the pricing model (4.12) can be expressed explicitly as follows:

   

² ,

 

‧

occupation time of state 1 and 2.

Since the currency call option given in (4.12) depends on the occupation times δ( ) and the jump number ( )n , the price notation C



 1, , 2 n ( , )t T



is used to replace the currency call option in (4.12). Hence, with the regime-switching of forward interest rates under the MMHJM model, a European currency call option can be expressed in terms of the occupation times given below Theorem 2:

Theorem 2 can be determined by its corresponding moment-generating function G Ψ( _X) given in (3.1).

Proof (See Appendix B)

□

Note that the existence of   ( , )₁ ₂ and Q^*( , ) n in (4.13) is attributed to the regime-switching risks of forward interest rates and the spot-FX rate. In addition, the currency option price given in (4.13) reflects not only the volatilities of the domestic and the foreign forward interest rates, but also the regime-switching risks of the forward interest rates. The currency-call option C t X t( , ( )) in (4.12) has two important reduced forms in corollary 1 and 2, depending on the conditions of regime switching and jump risk.

Corollary 1 Without regime-switching risk (X Y, ), but with jump risk

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Under the JDM given in (2.3) and the stochastic forward rates of the HJM model, the call price C t X t( , ( )) reduces to the C^JDM( )t given below: pricing model under the MMPP and the HJM model without being modulated by Markov chain X .

□

Corollary 2 Without regime switching(X Y, ) and jump risk

When the regime-switching features and the jump term are ignored, our pricing model becomes the currency-option pricing model given by Amin and Jarrow (1991, AJ). In addition, the currency option price C_JDM( )t in (4.15) becomes the Black-Scholes model price when the jump term is further ignored (n ), which is given by: 0

_BSM _JDM ( ) ( ; )_n 0

C t C t n _ (3.4.16)

‧

Thus, the RSJD model given in (4.12) or (4.13) is a general pricing model that can be reduced to several other cases for pricing currency options such as the JDM, the AJ model and the BSM, depending on the conditions with or without either the HJM model or the MMHJM model.

在文檔中馬可夫狀態轉換市場下之選擇權定價:雙重 Esscher trnasform 下馬可夫可調控高斯HJM 模型 - 政大學術集成 (頁 51-60)

European currency options

Chapter 3 Valuation Of Currency Options Under A Regime-Switching Gaussian HJM Model

3.4 European currency options

3.4 European currency options

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