Dynamic Programming and Hedging Strategies in ... - CiteSeerX

Dynamic Programming and Hedging Strategies in Discrete Time Shih-Feng Huang1 and Meihui Guo2 1

2

Department of Applied Mathematics, National University of Kaohsiung, Kaohsiung, Taiwan. [email protected] Department of Applied Mathematics, National Sun Yat-Sen University, Kaohsiung, Taiwan. [email protected]

1 Introduction A hedge is an important financial strategy used to reduce the risk of adverse price movements in an asset by buying or selling others. Recently, hedging becomes a more important issue consequent on the catastrophe for the global financial system caused by the bankruptcy of major financial-services firm such as the Lehman Brothers Holdings Inc.. In practice, investors are impossible to rebalance their hedging positions continuously, such as in the BlackScholes framework, and need to reduce the rebalance times to lower down their transaction costs. Thus how to set up hedging portfolios in discrete time is of more practical importance. Herein, we introduce four hedging strategies for path-independent contingent claims in discrete time- superhedging, local expected shortfall-hedging, local minimal variance and local expected squared risk-adjusted minimizing hedging strategies. Dynamic programming approach will be introduced to construct the hedging positions at each rebalance time. Normally, a hedge consists of taking an offsetting position in a related security, such as a derivative. In complete financial markets, contingent claims can be replicated by self-financing strategies, and the costs of replication define the prices of the claims. In incomplete financial markets, one can still eliminate the risk completely by using a “superhedging” strategy (or perfect hedge). However, from a practical point of view the cost of superhedging is often too expensive. Therefore investors turn to hedging strategies with less capitals by considering risk minimization criteria. Different hedging strategies are proposed based on different economic considerations such as minimizing the quadratic hedging risks or the the expected shortfall risks. In the following, several discrete time hedging strategies are discussed and the comparisons between these hedging strategies are studied. To simplify the illustration of the hedging strategies, we employ a trinomial model, which is a discrete time and discrete state incomplete market model, to introduce the construction of these hedging strategies. In addition, we will compare the hedging performances of

2

Shih-Feng Huang and Meihui Guo

these discrete time hedging strategies in the Black-Scholes and GARCH models. In the last section, we discuss the problem of hedging path-dependent contingent claims and introduce the hedging strategies against barrier options.

2 Discrete time hedging strategies In this section, several discrete time hedging strategies in incomplete market models are introduced. We illustrate four hedging strategies in a trinomial model, which is a discretized description of geometric Brownian motion often used to describe asset behavior. One can extend the results to multinomial market model analogously. In a trinomial tree the asset price at each node moves in three possible ways, up movement, down movement and jump movement. The general form of a one period trinomial tree is as shown in Figure 1 (a). Given the stock price St−1 at time t − 1, where t = 1, 2, · · ·, suppose that there are three possible stock prices at time t, St = uSt−1 , St = dSt−1 and St = jSt−1 , with probability p1 , p2 and p3 , respectively, where u > d > j, pi ’s are positive and p1 + p2 + p3 = 1. If we short a European call option with maturity date T and strike price K at the initial time, how to set up a hedging portfolio to hedge this short position? 2.1 Superhedging strategy First of all, we discuss the superhedging strategy, which was introduced by [1] in discrete time. They concluded that for a contingent claim FT the initial hedging capital of the corresponding superhedging strategy is identical to sup E Q (e−rT FT ),

Q∈Q

where Q is the set containing all the risk-neutral probability measures Q, and r is the continuously compounded riskless interest rate. In order to construct the superhedging strategy in the trinomial model, an easy method is introduced in the following. We use a one-period trinomial model for illustration. Denote the initial hedging capital by W0 and let W0 = h00 + h10 S0 , where h00 and h10 are the holding units of the riskless bond and the stock at the initial time, respectively. At the maturity date, the value of this hedging portfolio becomes W1 = h00 er + h10 S1 . Our aim is to search a hedging strategy ˆ0, h ˆ 1 ) such that H SH = (h 0 0 W0 (H SH ) = minH {W0 (H) : W1 (H) ≥ C1 = (S1 − K)+ , for S1 = uS0 , dS0 , and jS0 },

(1)

Dynamic Programming and Hedging Strategies in Discrete Time

uS0

3

C1

p1

S0

p2

(uS0 , uS0 K )

dS0

p3

W1 hˆ00er hˆ01S1

( jS0 ,0)

jS0

jS0

dS0

(a)

K

uS0

S1

(b)

Fig. 1. (a) One period trinomial model; (b) Superhedging of the one period trinomial model.

where C1 = (S1 − K)+ is the payoff function of a European call option with strike price K. Note that W1 is a linear function of S1 and C1 is convex in S1 . Thus the linear function passing through the two terminal points (uS0 , C1 (uS0 )) and (jS0 , C1 (jS0 )) is the optimal solution of (1). Hence, in the trinomial model with assuming K < uS0 , the superhedging strategy H SH is defined as  ˆ 0 = K−uS0 je−r h 0 u−j , ˆ 1 = uS0 −K h 0 (u−j)S0 and thus W0 (H SH ) =

(uS0 − K)(1 − je−r ) . u−j

In Figure 1 (b), the dash line represents the function of the superhedging strategy at time 1. Apparently, the values of dash line are all greater than the corresponding payoffs of the three possible stock prices jS0 , dS0 and uS0 at the expiration date. In other words, investors can eliminate all the risk of the short position of the European call option by setting the superhedging strategy H SH with initial capital W0 (H SH ). Example 1. If u = 1.1, d = 0.9, j = 0.8, r = 0, S0 = 100 and K = 100, then W0 (H SH ) = 20 3 . Furthermore, let q1 , q2 and q3 denote the risk-neutral probability measures of the events S1 = uS0 , S1 = dS0 and S1 = jS0 , respectively. Using the constraints of S0 = e−r E Q (S1 ), q1 + q2 + q3 = 1, and qi ’s are positive, we have 12 < q1 < 23 , q2 = 2 − 3q1 and q3 = 2q1 − 1. Hence, the

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no-arbitrage price of the European call option, C0 = e−r E Q (C1 ), is between 5 and 20 3 , where the upper bound is exactly the same as the hedging capital of the superhedge. This result is consistent with the conclusion in [1]. Although the superhedge can always keep investors staying on the safe side, it is often too expensive. Therefore, investors are unwilling to put up the initial amount of capital required by a perfect hedge and are ready to accept some risk with some risk minimizing criteria. In the following sections, we introduce several different risk minimizing criteria. 2.2 Local expected shortfall-hedging The expected shortfall of a self-financing hedging strategy H of a contingent claim with payoff function FT is defined as E{(FT − WT (H))+ }, where WT (H) is the terminal wealth of the self-financing hedging strategy H at the maturity date T . Investors want to know whether there exists an optimal hedging strategy, denoted by H ES , such that the expected shortfall risk is minimized with a pre-fixed initial hedging capital V0 , that is, H ES = arg min E{(FT − WT (H))+ }, H∈S where

S = {H | H is a self-financing hedging strategy with initial hedging capital V0 }.

[6] and [10] pioneered the expected shortfall-hedging approach and showed the existence of this hedging strategy. [16] proposed a searching algorithm to construct a hedging strategy which minimizes the expected shortfall risk in complete and incomplete discrete markets. But the searching algorithm often spends large of computation time. In order to overcome this time-consuming problem, [16], then, proposed a local expected shortfall-hedging strategy. The idea of the local expected shortfall-hedging strategy proposed in [16] is as follows. The first step is to find an optimal modified contingent claim X ∗ , which is a contingent claim that belongs to the set χ of all modified contingent claims, where χ ≡ {X | X < FT and E Q (X/B T ) ≤ V0 for all risk-netral probability measure Q}, and

X ∗ = arg min E(FT − X). X∈χ

(2)

In other words, the price of the superhedging strategy of any modified contingent claim is lower or equal than the initial hedging capital V0 , and Proposition


5

2 of [16] shows that the shortfall risk of the hedging strategy H ES is identical to the price of the superhedging strategy of a modifined contingent claim, that is, E[(FT − VT (H ES )+ ]) = E(FT − X ∗ ). Therefore, one can determine the desired hedging strategy H ES by the following two steps: [ Expected Shortfall Hedging ] 1. Find an optimal modified contingent claim X ∗ ∈ χ with criterion (2). 2. Construct a superhedging strategy for X ∗ . Since Step-2 can be accomplished by the method introduced in the previous section, the main concern is the first step. In complete markets, the optimal modified contingent claim X ∗ is a direct consequence of a slight modification of the Neyman-Pearson lemma (see [10] and [16]). Proposition 4 of [16] gives the solution of the optimal modified contingent claim as X ∗ (ω) = FT (ω)I{ P (ω)>c} + γI{ P (ω)=c} Q

Q

(3)

with cES = minω {P (ω)/Q(ω)} and . γ = [V0 BT − E Q (I{ P >c} )FT ] E Q (I{ P =c} ). Q

Q

If the market is incomplete, the construction of the optimal expected shortfall hedging strategy is much more complicated than that in complete markets due to the fact that the risk-neutral probability measures are not unique. For continuous time models, [10] showed that an optimal hedging strategy exists but didn’t provide an explicit algorithm to calculate it. As for the discrete models, Proposition 5 of [16] proposed an algorithm to construct the optimal expected shortfall hedging strategy. The basic idea of this algorithm is still based on equation (3). The main difficulty is to deal with the non-uniqueness of the equivalent martingale measures. Due to the fact that the set of mar˜ is a convex polyhedron, there exist a finite tingale measures, denoted by Q, ˜ where Q1 , · · · , QL are the number of martingale measures Q1 , · · · , QL ∈ Q, ˜ extreme points of the convex polyhedron Q. The optimal modified contingent claim X ∗ here has to satisfy max E Q (X ∗ /BT ) ≤ V0 . ˜ Q∈Q

However, this constraint consumes a huge computational effort. Therefore, [16] further proposed the following Local Expected Shortfall-hedging strategy, denoted by H LES :

6


[ Local Expected Shortfall Hedging ] Let FT be a European contingent claim and FtSH be the corresponding superhedging values at time t = 1, · · · , T . Then find sequentially a self-financing strategy H LES = (H1LES , · · · , HTLES ) with HtLES minimizing the local expected shortfall Et−1 {(FtSH − Wt (H))+ } for t = 1, · · · , T , where Et−1 (·) denotes the conditional expectation under the dynamic probability measure given the information up to time t − 1. In the following examples, we show how HtLES works. Example 2 gives a one-period trinomial case and Example 3 considers a two-period situation. Example 2. Consider the same one-period trinomial model as in Example 1. Let ω1 , ω2 and ω3 denote the states of S1 = uS0 , dS0 and jS0 , respectively, and P denote the dynamic probability measure with P (ω1 ) = 0.55, P (ω2 ) = 0.40 and P (ω3 ) = 0.05. As shown in Example 1, the set Q of risk-neutral probability measures can be expressed as Q = {(q1 , q2 , q3 ) :

1 2 < q1 < , q2 = 2 − 3q1 > 0 and q3 = 2q1 − 1 > 0}. 2 3

Let Q be the polyhedron correspond to Q, that is, Q = {(q1 , q2 , q3 ) :

2 1 ≤ q1 ≤ , q2 = 2 − 3q1 ≥ 0 and q3 = 2q1 − 1 ≥ 0}. 2 3

Then Q1 (ω1 , ω2 , ω3 ) = ( 12 , 12 , 0) and Q2 (ω1 , ω2 , ω3 ) = ( 32 , 0, 31 ) be two extreme points of this convex polyhedron Q. An investor is willing to set her initial hedging capital to be 6, which is less than the initial capital required by the superhedging strategy 20 3 . Our aim is to determine a trading strategy minimizing the expected shortfall with the initial hedging capital. By Proposition 5 of [16], since E Q2 (F1 ) > E Q1 (F1 ), we consider Q2 first. In order to determine the modified contingent claim, one can apply Proposition 4 of [16]. However, Proposition 4 of [16] can not be implemented directly to the trinomial model since trinomial model is not a complete market model. Nevertheless, due to the fact that Q2 (ω2 ) = 0, we can ignore the state ω2 temporarily and only determine the modified contingent claim by the states ω1 and ω3 : X(ω) = F1 (ω)I{ where c = min{

P (ω) >c} Q2 (ω)

+ γI{

P (ω) =c} Q2 (ω)

P (ω) : ω = ωi , i = 1, 3} Q2 (ω)

,


7

(S1, F1 , X ) (110, 10, 10)

C1

p1 0.55

S0 100

(110, 10)

p2

0.40

p3 0.05

(90, 0, 2)

LES 1

W1 ( H

) 34 0.4S1

(90, 2)

(80, 0, - 2)

80

90

100

110 S1

(80, - 2)

(a)

(b)

Fig. 2. (a) One period trinomial model in Example 2; (b) Local Expected Shortfallhedging strategy of the one period trinomial model.

and γ is chosen to ensure E Q2 (X) ≤ 6. By straightforward computation, we have X(ω1 ) = 10 and X(ω3 ) = −2. Next, determine the superhedging strategy for X(ωi ), i = 1, 3. By the same way introduced in Section 2.1, one can obtain the hedging portfolio ˜ 1 ) to be ˜0, h H0LES = (h 0 0 ( 0 ˜ = −34 h 0 , ˜ 1 = 0.4. h 0 which satisfies H1LES (ω1 ) = X(ω1 ) = 10 and H1LES (ω3 ) = X(ω3 ) = −1. Finally, we defined the value of the modified contingent claim of state ω2 by X(ω2 ) = H1LES (ω2 ) = 2. Note that for this modified contingent claim, we have E Q2 (X) = E Q1 (X) = 6 and since any risk-neutral probability measure can be expressed by Q = aQ1 + (1 − a)Q2 , 0 < a < 1, thus for all riskneutral probability measure Q ∈ Q we conclude that E Q (X) = 6, and the corresponding minimal expected shortfall is E[{F1 − W1 (H0LES )}+ ] = E[(F1 − X)+ ] = 0.1. Example 3. In this example, we extend the one-period trinomial model discussed in previous examples to two period. In each period, given the stock price St , t = 0, 1, let the stock prices at the next time point be St+1 = uSt , dSt and jSt , with dynamic probability 0.55, 0.40 and 0.05, respectively. In Figure 3, the values of S0 , S1 and S2 are set with S0 = 100, u = 1.1, d = 0.9 and j = 0.8. The payoff at time 2 is defined by F2 = (S2 − 100)+ , which is the

8


(S2 , F2 , X 2 ) (121, 21, 21)

(S1, F1, X1 )

(99, 0, 7)

(110, 14, 14)

(88, 0, 0) (99, 0, 0)

S0 (90, 0, - 2)

100

(81, 0, - 4) (72, 0, - 6) (88, 0, 0)

(80, 0, -10)

(72, 0, - 20)

Under Q2

§ 2 1· ¨ , 0, ¸ © 3 3¹

(64, 0, - 30)

Fig. 3. Local Expected Shortfall-hedging strategy of the two period trinomial model in Example 3.

payoff of a European call option with maturity date T = 2 and strike price K = 100. As in Example 2, the probability measures Q1 (ω1 , ω2 , ω3 ) = ( 21 , 21 , 0) and Q2 (ω1 , ω2 , ω3 ) = ( 32 , 0, 13 ) are the two extreme points of the convex polyhedron Q. Also assume that an investor’s initial hedging capital is set to be 6, which is still less than the initial capital required by the superhedging strategy max E Q (e−2r F2 ) = E Q2 (e−2r F2 ) = Q∈Q

28 , 3

where the riskless interest rate r is set to be 0. In the following, we illustrate how to obtain the modified contingent claim Xt , t = 1, 2 at each time point t and then construct the Local Expected Shortfall hedging strategy. Since E Q2 (e−2r F2 ) > E Q1 (e−2r F2 ), thus under the probability measure Q2 , compute the payoff F1 by the conditional expectation, E Q2 (e−r F2 |S1 ), given the stock price S1 . The first step is to find the one period optimal Expected Shortfall hedging with initial hedging capital 6 and payoff F1 in the one period trinomial model. This step can be solve by similar way as in Example 2. And we obtain the modified contingent claim X1 and the corresponding hedging strategy. For the second period, given any stock price S1 , the problem can be treated as another one period trinomial hedging task, that is, find the one period optimal Expected Shortfall hedging with initial hedging capital X1 (S1 ) and payoff F2 . Therefore, we can still adopt similar way as in Example 2 to obtain the modified contingent claim X2 and the corresponding hedging strategy.


9

The values of the modified contingent claim Xi , i = 1, 2, are given in Figure 3. Note that for the modified contingent claim X2 , we have E Q2 (X2 ) = Q1 E (X2 ) = 6 and since any risk-neutral probability measure can be expressed by Q = aQ1 + (1 − a)Q2 , 0 < a < 1, thus for all risk-neutral probability measure Q ∈ Q we conclude that E Q (X) = 6, and the corresponding expected shortfall is E[(F2 − X2 )+ ] = 1.235. 2.3 Local minimal variance hedging strategy In the following two sections, we introduce two different local squared risk minimizing hedging strategies. Both are not restricted to be self-financing trading strategies. That is, the traders are allowed to put or withdraw money at each rebalance time point. Let Wt−1 be the initial hedging capital of a hedging portfolio consisting of the riskless bond and the underlying stock at time t − 1, Wt−1 = h0t−1 Bt−1 + h1t−1 St−1 ,

(4)

where h0t−1 and h1t−1 are the holding units of the bond Bt−1 and the stock St−1 at time t − 1, respectively. Retain h0t−1 and h1t−1 constant till time t and denote the additional capital δt at time t as follows δt (St ) = Ft (St ) − (h0t−1 Bt + h1t−1 St ),

(5)

where Ft (St ) is the value of the contingent claim of stock price St and Ft (St ) = h0t Bt +h1t St . In general, the derivative value function Ft is a nonlinear function of the underlying asset St . For example, the payoff of a European call option is defined as FT = (ST − K)+ , where T is the expiration date and K is the strike price. If the underlying assets follows the Black-Scholes model (see [2]), then the call option value at time t is given by Ft = St N (d1 ) − Ke−r(T −t) N (d2 ), 0 ≤ t < T, where d1 =

ln(St /K) + (r + 0.5σ 2 )(T − t) √ , σ T −t

√ d2 = d1 − σ T − t, r is the riskless interest rate and N (·) is the cumulative distribution function of a standard normal random variable. Apparently, Ft is a nonlinear function of St for 0 ≤ t < T . Hence, in equation (5), the nonlinear function Ft would not be identical to any linear function of St . That is, one can not fully replicate the derivative value function by a linear hedging strategy as defined in (4) (see Figure 4). Therefore, we call the difference between the values of the derivative and the hedging portfolio by “additional capital”. At each rebalanced time point, if the additional capital is greater than zero, which means that the hedging portfolio can not hedge the derivative entirely,

10


h0t 1Bt h1t 1St Ft

a

b

St

Fig. 4. The values of the derivative and the hedging portfolio.

the trader, who wants to set up a hedging portfolio to hedge this derivative, needs to put more hedging capital for setting the hedging position for the next hedging period. If the additional capital is negative, then the trader can withdraw the difference back. (See Figure 4. The dash line is the hedging portfolio function and the curve represents the derivative values. When a < St < b, the hedging portfolio hedges Ft perfectly. But when St < a or St > b, the hedging portfolio loses money.) In practice, the trader have to pay transaction costs when she puts more money in the market or withdraw her money back. Hence, she may determine her hedging positions by some optimal criteria. In this section, we first introduce the local expected squared risk minimizing criterion. In this criterion, the holding units are determined by min

h0t−1 ,h1t−1

Et−1 [(δt (St )/Bt )2 ].

(6)

The closed-form expression of h0t−1 and h1t−1 for t = 1, · · · , T can be obtained by solving ³ ´ ³ ´ ∂ ∂ Et−1 [δt (St )/Bt ]2 = 0 and Et−1 [δt (St )/Bt ]2 = 0, 0 1 ∂ht−1 ∂ht−1 and the hedging strategy is given explicitly by [ Local Expected Squared Risk Minimizing Hedging ]  0 1 ˆ ,h ˆ ) = (F˜T , 0) (h    T T    ˜ ˜ ˆ 1 = Covt−1 (Ft , St ) , (7) h t−1  Vart−1 (S˜t )      ˆ0 ˆ 1 E(S˜t ) = Et−1 {h ˆ 0 + (h ˆ1 − h ˆ 1 )S˜t } ht−1 = Et−1 (F˜t ) − h t−1 t t t−1 where F˜t = Ft /Bt and S˜t = St /Bt for t = 1, · · · , T .


11

Security 1

. . .

adjusted

Security k

Riskless bond

discounted

0

T

Fig. 5. Risk-adjusted and discounted values.

In the following, we give an example to illustrate the local expected squared risk minimizing hedging strategy in a trinomial model. Example 4. Consider the same one-period trinomial model as in Example 2. By equation (7), the holding units of riskless bonds and the security are given ˆ 1 ) = (−40.26, 0.4553). The initial hedging capital is ˆ0, h by (h 0 0 ˆ 1 S0 = 5.5, ˆ0 + h W0 = h 0 0 which lies in the no-arbitrage region (5, 20 3 ). That is, investors can use an initial hedging capital of 5.5, which is less than the initial capital required by the superhedging strategy, to construct a hedging portfolio for minimizing the expected quadratic risks.

2.4 Local expected squared risk-adjusted minimizing hedging In contrast to the local expected squared risk minimizing strategy, we introduce another hedging strategy of squared risk minimization type in this section. Define the one-step-ahead risk-adjusted hedging cost as δ˜t (St ) ≡ δ(St e−λt ), where λt ≡ log{Et−1 (

St /Bt )} St−1 /Bt−1

is the risk premium. Figure 5 illustrates the concepts of risk-adjusted and discounted values. Instead of the criterion (6) in Section 2.3, the objective

12


here is set to find a hedging strategy which minimizes the one-step-ahead conditional expected squared discounted risk-adjusted hedging cost, min

h0t−1 ,h1t−1

Note that

Et−1 [(δ˜t (St )/Bt )2 ].

(8)

δ˜t (St )/Bt = F˜t∗ − (h0t−1 + h1t−1 S˜t∗ ), −λ

t e−λt where F˜t∗ = Ft (SBt et ) and S˜t∗ = St B are the discounted-adjusted values of t Ft and St , respectively. Statistically speaking, the criterion (8) is equivalent to find a best linear approximation of F˜t∗ with the shortest L2 -distance under the physical measure P . Thus, this best linear approximation will pass the point (Et−1 (S˜t∗ ), Et−1 (F˜t∗ )). By the definition of λt , we have

Et−1 (S˜t∗ ) = S˜t−1 , where S˜t−1 = SBt−1 . Hence, we denote the amount Et−1 (F˜t∗ ) to be the dist counted hedging capital for a given discounted stock price S˜t−1 at time t − 1, that is, ˜ t−1 = Wt−1 = Et−1 (F˜t∗ ). W Bt−1 Based on the optimal criterion (8), the closed-form expression of h0t−1 and can be obtained by solving

h1t−1

³ ´ ³ ´ ∂ ∂ 2 2 ˜ ˜ E [ δ (S )/B ] = 0 and E [ δ (S )/B ] = 0, t−1 t t t t−1 t t t ∂h0t−1 ∂h1t−1 and we then have the following “Local Expected Squared Risk-Adjusted Minimizing Hedging”, abbreviated by “LESRAM Hedging”, [ LESRAM Hedging ]  0 1 ˆ ,h ˆ ) = (F˜ ∗ , 0) (h  T T T      ˜ ∗ ˜∗ ˆ 1 = Covt−1 (Ft , St ) . (9) h t−1  Vart−1 (S˜t∗ )      ˆ0 ˆ 1 E(S˜∗ ) = Et−1 {h ˆ 0 + (h ˆ1 − h ˆ 1 )S˜∗ } ht−1 = Et−1 (F˜t∗ ) − h t−1 t t t t−1 t In the following, we give an example to illustrate the local expected squared risk-adjusted minimizing hedging strategy in a trinomial model. Example 5. Consider the same one-period trinomial model as in Example 2. First, we compute the risk premium


λ = log{E(e−r

13

St )} = e−r log(up1 + dp2 + jp3 ) = log(1.0005). St−1

The discounted risk-adjusted stock prices at time 1 are 22000 S˜1∗ (ω1 ) = S1 (ω1 )e−r−λ = > K = 100, 201 1800 S˜1∗ (ω2 ) = S1 (ω2 )e−r−λ = < K, 201 and

16000 S˜1∗ (ω3 ) = S1 (ω3 )e−r−λ = < K. 201

Hence, the corresponding discounted risk-adjusted option values are F˜1∗ (ω1 ) = 1900 ˜∗ ˜∗ 201 , and F1 (ω2 ) = F1 (ω3 ) = 0. By equation (9), the holding units of riskless ˆ 0, h ˆ 1 ) = (−38.06, 0.4326). Thus the bonds and the security are given by (h 0 0 initial hedging capital is ˆ 1 S0 = 5.199, ˆ0 + h W0 = h 0 0 which also lies in the interval of no-arbitrage prices, (5, 20 3 ). In other words, the criterion of minimizing the expected discounted adjusted risk costs not only provides a hedging strategy, but also gives investors a no-arbitrage price of the European call option in this incomplete market. If the market model is discrete time and continuous state type, such as the GARCH model (see [3]), Huang and Guo [13] show that Q Et−1 (F˜t∗ ) = Et−1 (F˜t ) = F˜t−1 , Ft where F˜t = B and the measure Q is the martingale measure derived by the t extended Girsanov change of measure proposed by Elliott and Madan [9]. In particular, if the innovation is assumed to be Gaussian distributed, then the GARCH martingale measure derived by the extended Girsanov principle is identical to that obtained by Duan [7]. Moreover, the formula of the optimal ˆ 1 ) can be expressed under the risk-neutral measure Q ˆ0 , h (h t−1 t−1

 Q Q ˜ ˜2 ˜ ˜ ˜   ˆ 0 = Ft−1 Et−1 (St ) − St−1 Et−1 (Ft St ) h   t−1 Q Vart−1 (S˜t ) , Q ˜ ˜  Cov ( F , t St )  t−1 1 ˆ  h =  t−1 VarQ (S˜t )

(10)

t−1

Q where CovQ t−1 and Vart−1 are the conditional covariance and variance given Ft−1 computed under the risk-neutral measure Q, respectively. Both (9) and (10) provide recursive formulae for building the LESRAM hedging backward from the expiration date. In practical implementation, investors may want to keep the holding units of the hedging portfolio constant

14


for ` units of time due to the impact of the transaction costs. If we denote the discounted hedging capital with hedging period ` at time t by ˆ0 + h ˆ 1 S˜t , F˜t,` = h t,` t,`

(11)

ˆ 0 and h ˆ 1 are the holding units of riskless bonds and the underlywhere h t,` t,` ing asset, respectively, and are determined instead by the following optimal criterion min1 EtQ [(δ˜t+` (St+` )2 ]. (12) 0 ht,` ,ht,`

ˆ0 , h ˆ 1 ) are functions of the hedging Note that the optimal holding units (h t,` t,` 0 ˆ and h ˆ 1 can be represented as period `. By similar argument as (10), h t,` t,`  ˆ 1 = CovQ (F˜t+` , S˜t+` )/VarQ (S˜t+` )  h t t t,` . (13)  h ˆ 0 = E Q (F˜t+` ) − ηˆ1 S˜t+` t t,` t,` Equation (13) is really handy in building LESRAM hedging. For example, suppose that a trader writes a European call option with strike price K and maturity date T , and she wants to set up a hedging portfolio consisting of the underlying stock and the riskless bonds at time t with the hedging capital Ft , which is the price of the European call option, to hedge her short position. And the holding units of the stock and bond remain constant till t + `, 0 < ` ≤ T − t, that is, the hedging period is `. The coefficients of this hedging are obtained as follows: [ `-period LESRAM Hedging ] 1. For a given stock price St at time t, generate n stock prices {St+`,j }nj=1 , at time t + ` conditional on St from the risk-neutral model. 2. For each St+`,j , derive the corresponding European call option prices, Ft+` (St+`,j ), by either the dynamic semiparametric approach (DSA) (see [11] and [12]) or empirical martingale simulation (EMS) method (see [8]) for t + ` < T . If t + ` = T , then FT (ST,j ) = (ST,j − K)+ . ˆ0 , h ˆ 1 ) are 3. Regress F˜t+` (St+`,j ) on S˜t+`,j , j = 1, · · · , n. Then (h t,` t,` the corresponding regression coefficients. In Step-2 of the above algorithm, we mention that the European call option values can be obtained by the DSA method. Herein, we give a brief introduction of this method. The DSA is proposed to solve the multi-step conditional expectation problems where the multi-step conditional density doesn’t have closed-form representation. It is an iterative procedure which uses nonparametric regression to approximate derivative values and parametric asset models to derive the one-step conditional expectations. The convergence order of


~

Assume FT is given. Start from. i

15

T

~

Fit the regression function of Fi : Fˆi

Compute the conditional expectation : E i 1 ( Fˆi )

If i 1 ! 0 , set i i 1

~

Define the approximate option value : Fi 1

If i 1

0

Stop

Fig. 6. Flow chart of the DSA.

the DSA is derived by ([11]) under continuity assumption on the transition densities of the underlying asset models. For illustration, suppose we want to compute the multi-step conditional expectation E0 (FT ). We transform the problem into E0 [E1 [· · · [ET −1 (FT )] · · ·]]. and then compute the one-step backward conditional expectation. Denote Ft = Et (Ft+1 ), t = 0, · · · , T − 1. In general, Ft is a nonlinear function of the underlying asset for t = 1, · · · , T − 1, and the conditional expectation Et−1 (Ft ) does not have closed-form representation, which makes the multi-step conditional expectation complexity. Therefore, Huang and Guo [11] proposed a semiparametric method to solve this problem. In each discrete time point t, using piecewise regression function to approximate the derivative value function Ft , denoted by Fˆt , and then compute the conditional expectation of Fˆt , that is, F˜t = Et (Fˆt+1 ) and F˜t is treated as an approximation of Ft . The procedure keeps iterating till the initial time to obtain the derivative price. See Figure 6 for the flow chart of the DSA.

3 The comparison of the discrete time hedging strategies In this section, we are interested in comparing commonly used delta hedging strategy with the discrete time hedging strategies introduced in Section 2.

16


The delta of a derivative is referred to as the rate of change in the price of a derivative security relative to the price of the underlying asset. Mathematically, the delta value ∆t at time t is defined as the partial derivative of the price of the derivative with respect to the price of the underlying, that is, ∂Vt ∆t = ∂S , where Vt and St are the prices of the derivative and the underlying t asset at time t, respectively. For example, considering a European call option with maturity time T and strike price K, the no-arbitrage option value at time t is Ct = e−r(T −t) EtQ [(ST − K)+ ], where r is the riskless interest rate. After simplit fying the partial derivative ∂C ∂St and exploiting the following property, Z i 1 ln(K/St ) h (St + h)ey − K dFt (y) = 0, h→0 h ln(K/(S +h)) t lim

where Ft (y) is the conditional distribution of ln SSTt given Ft under the martingale measure Q, one can show that the delta of the European call option can be expressed as ∆t (c) =

³S ´ ∂Ct T = e−r(T −t) EtQ I{ST ≥K} . ∂St St

(14)

And the delta value of the put option, ∆t (p), can also be derived from the following relationship based on the put-call parity: ∆t (c) − ∆t (p) = 1.

(15)

Since (15) is derived based on a simple arbitrage argument, the result is distribution-free, that is it does not depend on the distribution assumption of the underlying security. To calculate the delta value (14) of a European call option, one can either approximate the the conditional expectation, EtQ (ST I{ST ≥K} ), recursively by the DSA or approximate the partial derivaCt (St +h)−Ct (St ) t tive, ∆t (c) = ∂C , where h is a ∂St , by the relative rate of change h small constant and the option price Ct ’s can be obtained by the DSA. 3.1 LESRAM hedging and ∆ hedging Under Complete Markets In a complete market every contingent claim is marketable, and the risk neutral probability measure is unique. There exists a self-financing trading strategy and the holding units of the stocks and bonds in the replicating portfolio are uniquely determined. Thus we expect the LESRAM hedging will coincide with the ∆-hedging under the complete market models. In the following, we show directly the holding units of the stocks in an LESRAM hedging is the same as in the ∆-hedging for the two complete market models - the binomial tree model and the Black-Scholes model. For simplicity, let the bond price Bt = ert where r represents a constant riskless interest rate. First, consider


17

a binomial tree model. Assumes at each step that the underlying instrument will move up or down by a specific factor (u or d) per step of the tree, where (u, d) satisfies 0 < d < er < u. For example, if St−1 = s, then St will go up to su = us or down to sd = ds at time t, with the risk neutral probability q = P (St = su |St−1 = s) =

er − d = 1 − P (St = sd |St−1 = s). u−d

By straightforward computation, we have CovQ t−1 (Vt , St ) = q(1 − q)(su − sd )[Vt (su ) − Vt (sd )] and

2 VarQ t−1 (St ) = q(1 − q)(su − sd ) .

Thus by (10) the holding units of the stock in the η-hedging is Q Q ˜ ˜ ˆ 1 = Covt−1 (Ft , St ) = Covt−1 (Ft , St ) = Ft (su ) − Ft (sd ) , h t−1 ˜ su − sd VarQ VarQ t−1 (St ) t−1 (St )

which is consistent with the ∆-hedging of the binomial tree model. Next, consider the Black-Scholes model, dSt = rSt dt + σSt dWt ,

(16)

where r and σ are constants and Wt is the Wiener process. For a European call option with strike price K and maturity date T , the holding units of the stock in the ∆-hedging of the Black-Scholes model is ∆ = N (d1 (St )) at time t, where ln SKt + (r + 0.5σ 2 )(T − t) √ d1 (St ) = σ T −t and N (·) is the cumulative distribution function of the standard normal random variable. We claim in the following that h1t → N (d1 ) as dt → 0, where dt denotes the length of the time period [t, t+dt]. Denote the discounted stock price and option value at time t by S˜t and V˜t , respectively. Note that Q ˜ ˜ ˜ ˜ ˜ ˜ CovQ t (Ft+dt , St+dt ) = Et (Ft+dt St+dt ) − Ft St n h = EtQ e−r(t+dt) St+dt N (d1 (St+dt )) i o − Ke−r(T −t−dt) N (d2 (St+dt )) S˜t+dt − F˜t S˜t 2 )N (d1 (St )) − S˜t Ke−rT N (d2 (St )) ≈ EtQ (S˜t+dt h i −e−rt St N (d1 (St )) − Ke−r(T −t) N (d2 (St )) S˜t

˜ = N (d1 (St ))VarQ t (St+dt ),

18


√ where d2 (St ) = d1 (St ) − σ T − t and the approximation (≈) is due to k k EtQ [S˜t+dt N (di (St+dt ))] ≈ EtQ (S˜t+dt )N (di (St ))

for small dt, i = 1, 2 and k = 1, 2. Therefore, by (10) we have h1t → N (d1 ) as dt → 0. This result indicates that if investors are allowed to rebalance the hedging portfolio continuously, then η-hedging coincides with ∆-hedging. However, we should be aware that investors are not allowed to rebalance the hedging portfolio continuously and may want to reduce the number of the rebalance time as little as possible due to the impact of transaction costs in the real world. 3.2 LESRAM hedging and ∆ hedging under incomplete markets In this section, we consider that the log return of the underlying assets follows a GARCH model such as   R = r − 1 σ 2 + λσ + σ ε , ε ∼ D(0, 1) t t t t t 2 t , (17)  2 2 2 2 σt = α0 + α1 σt−1 εt−1 + α2 σt−1 where the parameters are set as in [7] Table 1. The relative values of the average squared hedging costs of ∆-hedging and LESRAM hedging in the GARCH(1,1) log-return model. GARCH-normal GARCH-dexp kur. K = 35 K = 40 K = 45 kur. K = 35 K = 40 K = 45 Case 1: α0 = 0.00001524, α1 = 0.1883, α2 = 0.7162, λ = 0.007452 G5 4.10 G10 4.33 G30 4.29

1.01 1.01 1.01

1.00 1.00 1.01

1.01 1.02 1.05

6.67 7.39 9.24

1.02 1.03 1.04

1.01 1.01 1.03

1.03 1.07 1.16

Case 2: α0 = 0.00002, α1 = 0.1, α2 = 0.8, λ = 0.01 G5 3.50 G10 3.53 G30 3.42

1.00 1.01 1.01

1.01 1.01 1.03

1.01 1.03 1.04

4.91 4.76 4.28

1.02 1.02 1.00

1.01 1.01 1.03

1.05 1.07 1.08

Case 3: α0 = 0.00002, α1 = 0.2, α2 = 0.7, λ = 0.01 G5 4.18 G10 4.42 G30 4.39

1.01 1.01 1.00

1.01 1.01 1.03

1.03 1.06 1.08

6.95 8.34 9.21

1.04 1.05 1.01

1.01 1.02 1.06

1.09 1.16 1.21

Case 4: α0 = 0.00002, α1 = 0.3, α2 = 0.6, λ = 0.01 G5 5.09 G10 6.06 G30 8.87

1.02 1.04 1.01

1.01 1.02 1.06

1.06 1.11 1.20

10.52 20.50 53.67

1.06 1.08 1.04

1.02 1.04 1.25

1.15 1.27 1.79


19

~ F

~ F ~ Ft

~ Ft

K(t,t 1)

'(t,t 1) ~ ~ (St , Ft )

~ ~ (St , Ft )

~ Ft 1

~ Ft 1

~ S

~ S

Fig. 7. One period ∆ hedging and LESRAM hedging.

λ = 0.007452, α0 = 0.00001524, α1 = 0.1883, α2 = 0.7162, q α0 σd = 1−α1 −β1 = 0.01263 (per day, i.e. 0.2413 per annum), K = 40, r = 0, and the innovation εt is assumed to be normal or double exponential distributed with zero mean and unit variance. Suppose that a trader writes a European call option with strike price K and maturity date T , and set up a delta hedging portfolio at the initial time, with the hedging capital F0 , that is, F 0 = h 0 + ∆0 S 0 , where F0 denotes the risk-neutral price derived by the extended Girsanov principle and thus the cash position h0 can be obtained by F0 − ∆0 S0 . Similarly, we can construct the LESRAM hedging portfolio by F0 = hη0 + η0 S0 . We simulate n = 10, 000 random paths to generate the stock price, {ST,i }ni=1 , under the physical model (17), and then compute the ratio of the average variations of the delta hedging and LESRAM hedging portfolios Pn [h0 erT + ∆0 ST,i − max(0, ST,i − K)]2 GT = Pi=1 , n η rT + η0 ST,i − max(0, ST,i − K)]2 i=1 [h0 e for T = 5, 10, 30 (days). The following table shows the simulation results of GT , T = 5, 10, 30, of the GARCH-normal and GARCH-dexp models with K = 35, 40, 45 and several different parameter settings. Note that the values of GT ’s in Table 1 are all greater than 1, which means the average variation of the LESRAM hedging is smaller than the corresponding delta hedging. Under the same parameter setting in both GARCH-normal

20


~ F

~ F ~ Ft

~ Ft

K(t,t 1) K(t,t ")

'(t,t 1)

~ ~ ( S t , Ft )

'(t,t ")

~ ~ (St , Ft )

~ Ft "

~ Ft"

~ S

~ S

Fig. 8. `-period ∆ hedging and LESRAM hedging.

and GARCH-dexp models, the kurtosis of the GARCH-dexp models is greater than the GARCH-normal model. The results shows that GT ’s seem to increase in the kurtosis of the log returns log ST /S0 , especially when the option is outof-the-money. In Figure 7, we plot the hedging strategies of ∆ hedging and LESRAM hedging for one period case, where F˜t is the discounted option value function at time t and the point (S˜t , F˜t ) denotes the time-t discounted stock price and discounted hedging capital. In the left-hand panel, the dash-line, ∆(t − 1, t), denotes the ∆ hedging values, which is the tangent of the curve F˜t at the point (S˜t , F˜t ). In the right-hand panel, the dot-line, η(t − 1, t), represents the LESRAM hedging, which is regression line of F˜t+1 under the risk-neutral probability measure derived by the extended Girsanov principle (see [9] and [14]). If the hedging period increases to `, ` > 1, then the ∆ hedging ∆(t, t + `) remains the same, that is, ∆(t, t + `) = ∆(t, t + 1), (see the left-hand panel of Figure 8). However, the LESRAM hedging, η(t, t + `) (see the red line in the right-hand panel of Figure 8), would be different from η(t, t + 1) since the hedging target is changed from F˜t to F˜t+` . This phenomenon states that the LESRAM hedging is more adapted to the hedging period ` than the ∆ hedging since the hedging strategy would adapt to the hedging target F˜t+` . This phenomenon also provides an explanation of why the GT ’s in Table 1 tends to increasing in the hedging period or in the kurtosis of the log returns in most cases. Because when the hedging period or the kurtosis of the log returns increases, the adapted property of the LESRAM hedging would reduce more variability between the hedging portfolio and the hedging target than the ∆ hedging.


21

4 Hedging strategies against barrier options In previous sections, we introduce several discrete time hedging strategies and illustrate the dynamic programming of the strategies for plain vanilla options. In this section, we consider the construction of hedging strategies against barrier options. Barrier option is a path-dependent contingent claim, that is, the payoff of a barrier option depends on the underlying asset values during the time interval [0, T ], where 0 and T stand for the initial and maturity dates, respectively. In general, the evaluation of path-dependent contingent claims is more complicated than path-independent ones since the randomness comes from not only the underlying asset value at maturity but also those before the expiration date. For example, the payoff of a down-and-in call option is defined as DICT = (ST − K)+ I(mint∈[0,T ] St ≤B) , where ST is the underlying asset value at time T , K is the strike price, B denotes the barrier and I(·) is an indicator function. By the definition of DICT , one can see that the payoff not only depends on ST but also depends on the minimum value of the underlying asset, mint∈[0,T ] St , in the time interval [0, T ]. Once the value of the underlying asset reaches or bellows the barrier B prior to maturity, the option becomes worthy and the payoff is then identical to the European call option with the same strike price and maturity date. Suppose that a trader shorts a down-and-in call option and wants to set up a hedging portfolio to hedge her short position. In addition, if the trader takes transaction costs into consideration, then the hedging strategies introduced in the previous sections may not be optimal since the increasing frequency of portfolio rebalancing costs more transaction fees. Due to the trade-off between risk reduction and transaction costs, Huang and Huang [15] proposed a hedging strategy which rebalances the hedging positions only once during the duration of the barrier option. In the following, we illustrate the hedging strategy for down-and-in call options when the underlying asset follows a geometric Brownian motion process. Assume that the underlying asset follows Model (16). We have the following results: (i) if B ≥ K, then the no-arbitrage price of the down-and-in call option is DIC0 (B, K) = P0 (K) · − K · DIB0B(B) σ2 ¡ log S0 +(r+ 2 )T ¢ 2r √ +B ( SB0 ) σ2 N + σ T

¡ log S0 BN

B S0

2

−(r+ σ2 )T √ σ T

¸ ¢

;

(18) (ii) if B < K, then the no-arbitrage price of the down-and-in call option is 2r

DIC0 (B, K) = ( SB0 ) σ2 BN 2r

−( SB0 ) σ2

2 B2 ¡ log KS +(r+ σ2 )T ¢ 0

KS0 B e

√ σ T ¡ log −rT

N

B2 KS0

2

+(r− σ2 )T √ σ T

¢ ,

(19)

22


where P0 (K) is the European put option price with the same strike price and maturity as the down-and-in call option, N (·) is the distribution function of a standard normal random variable, and DIB0 (B) denotes the no-arbitrage price of a down-and-in bond, which is defined as DIBT (B) = I( min St ≤ B ) , 0≤t≤T

and can be evaluated by · ¸ 2 ¡ B ¢( 2r2 −1) log SB0 +(r− σ22 )T log SB −(r− σ2 )T −rT 0 √ σ √ DIB0 (B) = e N( N( ) + S0 ) . σ T σ T (20) In particular, if the riskless interest rate equals zero, r = 0, then the above results can be simplified as: (i0 ) if B ≥ K, DIC0 (B, K) = P0 (K) + (B − K)DIB0 (B); B2 (ii0 ) if B < K, DIC0 (B, K) = K B P0 ( K ). By (i0 ) and (ii0 ), we can construct a perfect hedging strategy of a down-and-in call option. If B ≥ K, since the right-hand side of (i0 ) is a linear combination of a European put option and a down-and-in bond, thus the trader who shorts a down-and-in call option can hedge her short position via the following two steps: 1. At time 0, long a European put option with strike price K and maturity date T and also long (B − K) shares of a down-and-in bond with barrier B and maturity date T . Notice that the cost of this portfolio is exactly the same as the no-arbitrage price of the down-and-in call option. 2. Let τ denote the first hitting time when the underlying asset value reaches the barrier price, that is, τ = inf{t, St = B, 0 ≤ t ≤ T },

(21)

and let τ = ∞ if mint∈[0,T ] St > B. If τ = ∞, then the the down-and-in call option and the hedging portfolio set up in Step 1 are both worthless. If τ ≤ T , by the fact that DICτ (B, K) = Cτ (K) and put-call parity, Cτ (K) = Pτ (K) + B − K, one can then hedge the down-and-in call option perfectly via shorting the hedging portfolio set up in Step 1 and longing a European call option with strike price K and maturity date T at time τ . Similarly, if B < K, from (ii0 ), the trader can hedge her short position by longing K/B shares of a European put option with strike price B 2 /K and maturity date T at ¡time 0.¢ And then by put-call symmetry (see [5]), Cτ (K)I(τ ≤T ) = (K/B)P B 2 /K I(τ ≤T ) , the trader can short the European put option and long a European call option with strike price K and maturity date T without putting in or withdrawing any capital to hedge the downand-in call option perfectly at time τ (< T ). Moreover, if τ = ∞, then the down-and-in call option and the hedging portfolio set up at time 0 are both worthless. Therefore, in the case of r = 0, a down-and-in call option can be


23

perfectly hedged by plain vanilla options and down-and-in bond. This result is also obtained by [4]. How about the case of r > 0, which is more realistic than the case of r = 0? Recall the idea of constructing the perfect hedge when r = 0. If B ≤ K and at time τ (< T ), one can set up a perfect hedge by longing a European call option with the same strike price and maturity date as the down-and-in call option, that is, DICτ (B, K)I(τ ≤T ) = Cτ (K)I(τ ≤T ) . Then by put-call parity, Cτ (K)I(τ ≤T ) = (Pτ (K) + B − Ke−r(T −τ ) )I(τ ≤T ) , one can have the desired European call option by shorting a European put option and B − Ke−r(T −τ ) shares of down-and-in bond at time τ . Therefore, at time 0, the hedging portfolio comprises a European put option with strike price K and maturity date T and Be−rτ − Ke−rT shares of down-and-in bond. Since τ is a random variable, thus the above hedging portfolio can not be set up at time 0 in practice. On the other hand, if B > K and at time τ (< T ), by put-call symmetry, we have DICτ (B, K)I(τ ≤T ) = Cτ (K)I(τ ≤T ) =

K Ber(T −τ )

P

¡ B 2 e2r(T −τ ) ¢ I(τ ≤T ) . K

In this case, one can long Ke−rT /B shares of European put option with strike price B 2 e2r(T −τ ) /K and maturity date T to construct a perfect hedging portfolio at time 0. However, since the strike price of the European put option is now a random variable, this hedging strategy can’t be obtained in practice as well. In order to overcome this problem, Huang and Huang [15] proposed the following method. If B ≤ K, by using the inequalities e−rT I(τ ≤T ) ≤ e−rτ I(τ ≤T ) ≤ I(τ ≤T ) , we have the upper and lower bounds of the down-and-in call option value: L ≤ DIC0 (B, K) ≤ U, where L = P0 (K) + (B − K)DIB0 (B) and U = P0 (K) + (BerT − K)DIB0 (B) are both linear combinations of European put option and down-and-in bond. Next, we adopt a linear combination of L and U to construct a hedging portfolio, that is, V0 = (1 − α)L + αU, where V0 denotes the initial hedging capital and 0 < α < 1. Further let V0 be identical 0 −L to the price of the down-and-in call option and we then have α = DIC and U −L V0 = DIC0 (B, K) = P0 (K) + βDIB0 (B),

(22)

(B,K)−P0 (K) where β = DIC0DIB . The hedging strategy proposed by Huang and 0 (B) Huang [15] is to set up the portfolio (22), which comprises a European put option and β shares of down-and-in bond, at time 0 and hold the portfolio till time min(τ, T ). If τ ≤ T , then short the portfolio and long a European call option with strike price K and maturity date T . Notice that at time τ (≤ T ) the value of the hedging portfolio (22) becomes Pτ (K) + β, which may not be identical to the European call option price, Cτ (K). Therefore, the trader has to put in some additional capital for portfolio rebalancing and the total payoff of this hedging strategy is

24


[VT − DICT (B, K)]I(τ ≤T ) = [β − er(T −τ ) (B − Ke−r(T −τ ) )]I(τ ≤T ) ,

(23)

at maturity date, where the equality holds by the put-call parity, Cτ (K) = Pτ (K) + B − Ke−r(T −τ ) and the profit β comes from the down-and-in bond. On the other hand, if the underlying asset prices are never equal to or less than the barrier price, then the down-and-in call option and the hedging portfolio (22) are both worthless at maturity date. Similarly, if B > K, Huang and Huang [15] adopt the inequalities, L∗ ≤ 2 2rT e B2 ∗ ) are DIC0 (B, K) ≤ U ∗ , where L∗ = K = BeKrT P0 ( B K B P0 ( K ) and U both European put options. And then set up the hedging portfolio by a linear combination of L∗ and U∗ ∗ , V0 = (1 − α∗ )L∗ + α∗ U ∗ . Further let V0 = DIC0 0 −L and hence α∗ = DIC U ∗ −L∗ and the initial hedging portfolio is V0 =

(1 − α∗ )K B2 α∗ K B 2 e2rT P0 ( ) + P ( ), 0 B K BerT K ∗

(24) 2

)K which comprises (1−α shares of European put option with strike price BK B α∗ K and maturity date T and BerT shares of European put option with strike price (BerT )2 K

and maturity date T . Hold the hedging portfolio till time min(τ, T ), and if τ < T , then short the portfolio and long a European call option with strike price K and maturity date T . As in the case of B ≤ K, the value of the hedging portfolio (24) may not be identical to the European call option price at time τ and the trader needs some additional capital for the portfolio rebalancing. The total payoff of this hedging strategy is [VT − DICT (B, K)]I(τ ≤T ) ³ (1 − α∗ )K ´ (25) B2 α∗ K B 2 e2rT = er(T −τ ) Pτ ( ) + Pτ ( ) − Cτ (K) I(τ ≤T ) , rT B K Be K and [VT − DICT (B, K)]I(τ >T ) =

α∗ K B 2 e2rT PT ( )I(τ >T ) , rT Be K

at maturity. In the following, a simulation study is given to compare the hedging performance of the above hedging portfolio with non-hedging strategy for downand-in call options, where the non-hedging strategy means that the trader didn’t set up any hedging portfolio but put the money, obtained from shorting the down-and-in call option, in a bank. By generating N simulation paths, let N 1 X D0 = [V0 erT − DICT,j ]/DIC0 N j=1 and D1 =

N 1 X [VT,j − DICT,j ]/DIC0 N j=1


25

Table 2. The simulation results of hedging strategies (22) and (24) with parameters r = 0.05, µ = 0.10, σ = 0.20, S0 = 100 and N = 10, 000.

T(days) 90

(B/S0 , K/S0 ) (22)

¯0 D

¯1 D

0 1 q0.005 q0.005

1 q0.005 0 q0.005

(0.90,0.90)

0.0506 -0.0004 -20.4710 -0.3172 0.0155

(0.925,0.90)

0.0584 -0.0001 -10.9191 -0.1220 0.0112

(0.95,0.95)

-0.0723 -0.0002* -10.7767 -0.0973 0.0090

(0.975,0.95) -0.0692 -0.0001* -5.9533 -0.0290 0.0049 (24) (0.925,0.95)

0.0045 0.0251* -20.2215 -0.1753 0.0087

(0.95,0.975) -0.0809 0.0134* -14.1331 -0.0990 0.0070 180

(22)

(0.90,0.90)

-0.0536 -0.0018* -15.4581 -0.2480 0.0160

(0.925,0.90) -0.0085 -0.0003* -9.6246 -0.1155 0.0120 (0.95,0.95)

-0.1320 -0.0005* -9.1158 -0.0839 0.0092

(0.975,0.95) -0.1405 -0.0002* -6.1975 -0.0295 0.0048 (24) (0.925,0.95) -0.0645 0.0032* -13.7629 -0.1827 0.0133 (0.95,0.975) -0.1322 0.0013* -11.6451 -0.1039 0.0089 ¯1 > D ¯0 * denotes D

denote the average payoff of non-hedging strategy divided by DIC0 and the proposed hedging strategy divided by DIC0 , respectively, where DICT,j and VT,j are the values of the down-and-in call option and the proposed hedging strategy, respectively, obtained from the jth simulated path at maturity. Further let qα0 and qα1 are the α-quantiles derived from the empirical distributions of [V0 erT − DICT,j ]/DIC0 and [VT,j − DICT,j ]/DIC0 , j = 1, · · · , N , respectively. Table 2 gives the simulation results with parameters r = 0.05, µ = 0.10, σ = 0.20 and S0 = 100. In Table 2, one can see that D1 < D0 in most cases, which means that the average loss of the proposed hedging strat1 0 are and q0.005 egy is less than that of non-hedging strategy. Moreover, q0.005 adopted to measure the range of the risk of the hedging strategies. And the 0 1 /q0.005 is used to compare the hedging efficiency of the proposed ratio q0.005 hedging strategy with respect to the non-hedging strategy. In Table 2, the 0 1 decrease as the barrier price B increasing if T is fixed, /q0.005 values of q0.005 that is, the hedging performance of the proposed strategy becomes better 1 0 are all less than when B increasing. Furthermore, the values of q0.005 /q0.005 2%, which means that the proposed hedging strategy is able to hedge over 98% risk of the non-hedging strategy. The simulation study shows that the hedging performance of the proposed hedging strategy is much more better than the non-hedging one. Therefore, comparing with saving the money in

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the bank, the trader is suggested to adopt the proposed hedging strategy to hedge her short position of a down-and-in call option.

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