跨通貨股酬交換之評價－遠期平賭測度法

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行政院國家科學委員會專題研究計畫成果報告

跨通貨股酬交換之評價－遠期平賭測度法

Pr icing Cr oss-Cur r ency Equity Swap – A For war d Mar tingale

Measur e Appr oach

計畫編號：NSC 89-2416-H-004-064

執行期限：89 年 8 月 1 日至 90 年 7 月 31 日

主持人：廖四郎 國立政治大學金融學系

I. Abstr act

This research derives the valuation formula for general cross-currency equity swaps. Under the HJM framework of international security economy, the valuation formula is derived by the use of domestic spot martingale measure. The pricing model is general in the sense that the settlement currency can be chosen to be any currencies. The study finds that the premium of exchange rate risk is important in the valuation process and this distinguishes it from domestic or cross-currency without exchange-rate risk swaps. Also, the hedging method of swaps is investigated.

Keywords: Equity Swaps, Domestic Spot Martingale Measure,

Cross-Currency Equity Swaps, Exchange-Rate Risk, Hedging Equity Swaps 中文摘要 本研究推導出股酬交換的定價公式。 在 HJM 隨機利率的國際金融架構下，利用 本國風險中立測度求出一般化的股酬交換 無套利價值。在交換契約中的結算通貨可 以是任意的貨幣。本文發現在跨通貨的股 酬交換定價公式中，匯率風險是決定其價 值的重要因素。同時本文亦探討了股酬交 換的避險方法。 關鍵詞：股酬交換、本國風險中立測度、 跨通貨股酬交換、匯率風險、股酬交換避 險

II. Pur poses of The Study

Equity swaps are important financial instruments in the over-the-counter markets.

They provide an efficient way of

international diversification and obtaining foreign equity returns without actual holding of foreign equities. Direct foreign equity investment may be tempered by some reasons, for example, tax consideration and the restriction of local government on the investment on equity of the foreigner.

There are relatively few papers

investigate the pricing formula of

international equity swaps. Rich (1995) decomposed an equity swaps into a portfolio of equity forward contracts and used a forward-start forward contract approach to value the basic equity swaps. Jarrow and Turnbull (1996) derived a formula for the fixed rate of a basic equity swap. Chance and Rich (1998) used arbitrage-free replicating portfolios to derive the valuation formulas for variety forms of equity swaps.

In this study, the valuation formula is derived for general equity swap in which the settlement currency can by set to be different

to the underlying equity markets.

Furthermore, the hedging strategies for the swaps are also investigated. The study sets up an international security economy under

Heath, Jarrow, and Morton (1992)

framework of stochastic interest rates. Then the dynamics are transformed to domestic spot martingale measure. The pricing formula can be derived in one of following three ways: domestic market method, foreign market method, and replication method. If we use the two former methods, we have to further transform the spot martingale measures to forward measure so as to

2 simplify the computation under stochastic interest rates. Here we use the replication method, since it also provides hedging strategies for cross-currency forward equity swap.

III. Results and Discussion

III.1. The International Security Economy Let the underlying probability space be

) ) ( , , , (ΩF P F_tw T_t=0 , where T o t w t F )₌ ( is the

natural filtration generated by the d-dimensional standard Brownian motion

) ,..., (W1 Wd

W= . We specify the stochastic

interest rates by the motion of forward rates:

t i i i dW T t dt T t T t df (, )=α(, ) +σ (, )• , (1) where i=h,k and h for domestic market and k for foreign market, respectively. Let the exchange rate of k currency in units of domestic currency be denoted by k

t Q and its dynamics are t k t k t k t dt dW dQ =α +σ • (2) Then the short term interest rates rti= fi(t,t)

and the prices of zero-coupon bonds are

) ) , ( exp( ) , ( = −

∫

T t i i du u t f T t B ; the savings accounts are =

∫

t i i t f u udu B 0 ) ) , ( exp( .

The dynamics of Bi(t,T) under P are

) ) , ( ) , ( )( , ( ) , ( i i t i i dW T t b dt T t a T t B T t dB = + • (3) where (, ) 2 1 ) , ( ) , (tT r * t T * t T a i i i t i = −α + σ , bi(t,T)=−σi*(t,T), and

∫

= T t i i(t,T) (t,u)du * α α ; t T t u du T t i i(, ) (, ) *

∫

= σ σ .

The other risky assets (called stocks) have

dynamics as ( t) i t i t i t i t S dt dW dS = µ +ξ • (4)

If we use domestic savings account as the numeraire such that all relative prices in terms of it are martingales under a new probability measure _P*_{, which is called the} domestic spot martingale measure. Then the Radon-Nikodym derivative is expressed as

∫

• −

∫

= • = T T u u u T W dW du dP dP 0 0 2 * ) 2 1 exp( ) (η η η ε ,

and η satisfies the following equations:

0 ) , ( ) , ( 2 1 ) , ( * 2 * * _{+ − • =} −αh t T σh t T σh t T ηt (5) 0 = • + − t h t h t h t r ξ η µ (6) 0 = • + − + t k t h t k t k Qt r r σ η µ (7) ) , ( ) , ( 2 1 ) , ( * 2 * * T t T t T t r_tk _{k k t}k _k k Qt α σ σ σ µ + − + − • 0 )) , ( ( − * • = + − k t k t k t t T r σ σ η (8) 0 ) ( + • = + − • + + t k t k t h t k t k t k t k Qt µ σ ξ r σ ξ η µ (9) Hence, under * P , we have * * ) , ( ) , ( ) , ( ) , ( h h h t h dW T t dt T t T t T t df =σ •σ +σ • dt T t T t T t dfk(, )=σ_k(, )•(σ*_k(, )−σ_tk) * ) , ( _t k tT •dW +σ * ) ( t k t k t h t k t k t Q r r dt dW dQ = − +σ • (10) * * )) , ( )( , ( ) , ( h _th _{h t} h dW T t dt r T t B T t dB = −σ • ( 1 1 ) ) ( t* h t h t h t h t S r dt dW dS = +ξ • (12) dt T t r T t B T t dBk(, )= k(, )[(_tk+σ_tk•σ_k*(, )) −σk*(t,T)•dWt*] (13) ) ) ((_tk _tk _tk _tk _t* k t k t S r dt dW dS = −σ •ξ +ξ • (14) where = −

∫

t u t t W du W 0 * _{η is a d-dimensional}

standard Brownian motion under *

P . III. Derivation of The Formula

Applying the technique of measure changes, we can also derive the dynamics under foreign spot, domestic forward, and foreign forward martingale measures.

We now consider the pricing of cross-currency equity swaps. Suppose the swap begins at time T₀≥t, and the payment dates are Tj, j=1,2,...n. For simplicity, we

assume Tj−Tj−1=δ. For each period, the

domestic investor receives NmRk(Tj−1,Tj)

and pays NmRg(Tj−1,Tj), where Nm is the

notional principal denominated in currency

m; R_k and R_g denote the equity returns of market k and g for the corresponding period, respectively. At time t, let its value be CFES_t(k,g;m) for N_m=1.

To derive CFESt(k,g;m), we may proceed

by using domestic forward martingale measure. Also, we can derive it by replication approach. Since the two methods lead to the same valuation formula, we prefer

3 here to present the method of replicating the payoff of a cross-currency equity swap ,due to this method also provides the way of hedging. The value of CFESt(k,g;m) is

∑

= − − − n j h T m T j j g j j k t P h t j j B Q T T R T T R E B 1 , 1 1 ] )) ( ) , ( ( [ *

∑

= − − − = n j m T j j g j j k t P h j j T R T T R T T Q E T t B 1 1 1, ) ( , )) ] ( [( ) , (

To derive CFES_t(k,g;m), we can replicate

m T j j g j j k T T R T T Q j R ( , ) ( , )) ( ₋₁ − ₋₁ under either *

P or PT_j , the resulting hedging strategies

are identical.

First, consider a foreign-equity linked foreign exchange call option. Its payoff at time Tj is k T m T j m k T Q j Q S j C ( )=( − )+ . For a put option: k T m T j m k T Q Qj S j P ( )=( − )+ .

Now, at time Tj₋₁, we buy one home

dollar stock k Tj S ₋₁, buy ) ( 1 1 1 k T m Tj S j Q _{− −} units

of put option and sell

) ( 1 1 1 k T m Tj S j Q _{− −} units of

call option. At Tj, this portfolio has payoff

k T k T m T j j j S S Q 1 −

. The cost of the portfolio at T_j−1

is ) ( )) ( ) ( ( 1 1 1 1 1 k T m T j k m j k m j j S Q T C T P − − − − − + . Do the

same trading strategy for security g. We then discount this value to time t.

Furthermore, we can use domestic and foreign zero-coupon bonds to replicate

) ( j−1 k m T P and ( j−1) k m T

C , thus we can express the value of cross-currency equity swap at t

asCFESt(k,g;m) ) , , ( )) , , ( 1 ){( , ( ₁ 1 1 j j n j m g j j g j m m t B tT f t T T H tT T Q ₋ = −

∑

+ = (1 (, ₁, )) (, j ₁, j)} m k j j k T T t H T T t f − − + − , (15)

where fi(t,T_j−1,T_j) is the forward rate at t

for the period [T_j−1,T_j], and H_im(t,T_j−1,T_j)

) , ( )( ) , ( ( exp{ Tj _ui _um _ui _{i j} t bi uTj − − +b uT =

∫

ξ σ σ −b_h(u,T_j))du− T 1( ( , ₁) _ui) t i j j T u b −ξ

∫

− − ( i( , j) i u m u −σ +b uT σ −b_h(u,T_j))du

IV. Evaluation of The Study

This study derives the valuation formula for the fair prices of cross-currency forward equity swaps. In this general setting, various forms of equity swaps can be applied by substituting proper conditions into (15). Furthermore, using the replication approach provides a practical way of hedging these financial instruments.

For future study, we may conduct numerical simulation. Also, we may derive the option prices on these swaps and consider the factor of default risks. After these works have been done, the models can be used efficiently for the financial industry.

V. Reference

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