Risk preference, option pricing and portfolio hedging with proportional transaction costs

Chaos, Solitons and Fractals 95 (2017) 111–130

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Chaos, Solitons and Fractals Nonlinear Science, and Nonequilibrium and Complex Phenomena journal homepage: www.elsevier.com/locate/chaos

Risk preference, option pricing and portfolio hedging with proportional transaction costsR Xiao-Tian Wang a,∗, Zhe Li b, Le Zhuang c a

Department of Mathematics, South China University of Technology, Guangzhou 510640, Guangdong, PR China School of Business Administration, South China University of Technology, Guangzhou 510640, Guangdong, PR China c Department of Mathematics and Applied Mathematics, Guangdong University of Petrochemical Technology, Maoming, 525000, Guangdong, PR China b

a r t i c l e

i n f o

Article history: Received 7 August 2015 Revised 17 December 2015 Accepted 12 December 2016

Keywords: Scaling Option pricing with the transaction costs Leland’s strategy Risk preference Implied-volatility-frown Hedging performance

a b s t r a c t This paper is concerned in the option pricing and portfolio hedging in a discrete time case with the proportional transaction costs. Through the Monte Carlo simulations it has been shown that the fractal scaling and risk preference of traders have an important influence on the hedging performances in both option pricing and portfolio hedging in a discrete time case. In addition, the relation between preference of traders and implied volatility frown is discussed. We conclude that the risk preferences of traders play an important role in determining the shape of the implied-volatility-frown and the different options having the different hedging frequencies is another reason for the implied volatility frown. © 2016 Elsevier Ltd. All rights reserved.

1. Introduction Since the publication of the works of Black and Scholes [1] and Merton [2], the interest in option pricing has dramatically increased. An essential feature of the Black and Scholes as well as Merton approaches is that the trading is assumed to make in a continuous-time manner so that the price of any option does not depend on the time scaling and traders’ risk preferences (scalingfree pricing and preference-free pricing). However, in recent years many researchers have discovered that a number of financial market data display some complex and nonlinear characters. A series of studies have found that many financial market time series exist the scaling law [3–13]. Therefore, it has been suggested that one should consider the influence of the scaling and preference on option pricing and portfolio hedging. In this paper, through the Monte Carlo simulations for the independently 10 0 0 sample paths we show that the risk preference of traders and the fractal scaling [13] as well as the proportional transaction costs play an important role in option pricing and portfolio hedging. The problem of the option pricing and the portfolio hedging in a discrete time case with the proportional transaction costs has

R This work is supported by the National Natural Science Foundation of China (No.11071082, No.11271140). ∗ Corresponding author. E-mail addresses: [email protected], [email protected] (X.-T. Wang).

http://dx.doi.org/10.1016/j.chaos.2016.12.010 0960-0779/© 2016 Elsevier Ltd. All rights reserved.

been studied by many authors starting with Leland [14], Boyle and Emanuel [15], Lott [16], Wilmott [17,18], up to more recent works [19–29]. All those authors [13–29] show that the fractal scaling of traders has an important influence on option pricing and portfolio hedging, but they did not consider the effect of the risk preferences of traders on the hedging performances in both impliedvolatility-frown and option pricing. In fact, while there exist the proportional transaction costs the markets are incomplete. In those cases, the option prices are heavily dependent on the risk preferences of the traders. In the mean time, many econophysicists are also interested in analyzing financial time series through using different fractal scaling δ t to research the complex structures of economic systems. In particular, Mantegna and Stanley [5,6], Stanley and Plerou [7] and Stanley et al. [8] introduced the method of scaling invariance from complex science into economic systems for the first time. Since then, many researches on scaling laws in econophysics have taken place. Mandelbrot [3,4] and Mantegna and Stanley et al. [5–8] considered the problem of choosing the appropriate fractal scaling to analyses financial market data and price options. Bouchaud and Potters [9] and Potters et al. [10]. introduced an asymptotic method to tackle the residual risk and proposed to find the optimal strategies to price options. In this paper, on basis of the points of view of behavioral finance [30, 31] and econophysics [3–10,32] we will use the mixed hedging strategy [33] to price the options while there exist the proportional transaction costs.

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This paper is organized as follows. In Section 2, through the mixed hedging strategy X1 (t), a new option pricing formula is obtained with the proportional transaction costs and we show that the proportional transaction costs and the fractal scaling as well as the risk preference play an important role in option pricing. In Section 3, the Monte Carlo simulations for the independently 10 0 0 sample paths are given to show that the mixed hedging strategy X1 (t) is an improvement over the Leland hedging and the modified Leland hedging in high-frequency trading in the “real world” as X < S0 , even if this evidence is not absolutely conclusive. In Section 4, the relation between the risk preference of traders and the implied volatility frown is discussed. We conclude that the risk preferences of traders play an important role in determining the shape of the implied-volatility-frown and the different options having the different hedging frequencies is another reason for the implied volatility frown. Section 5 concludes. 2. Option pricing with proportional transaction costs In this section a new option pricing formula will be obtained while there are the proportional transaction costs. We show that the proportional transaction cost parameter k and the fractal scaling δ t as well as the risk preference parameter μ play an important role in option pricing while the continuous time trading assumption is given up. Leland [14] has derived a simple model for pricing options in the presence of transaction costs. He adopted the delta hedging strategy of rehedging at every time interval δ t. That is, every δ t the portfolio is rebalanced, whether or not the asymptotic replication error tends to zero in probability. In the proportional transaction cost option pricing model, we follow the other usual assumptions in the Black–Scholes model but with the following exception. (i) The portfolio is revised every δ t, where δ t is a finite and fixed, small time interval. (ii) Transaction costs are proportional to the value of the transaction in the underlying. Let k denote the round trip transaction cost per unit dollar of transaction. Suppose υ shares are bought (υ > 0) or sold (υ < 0) at the price S, then the transaction cost is given by 2k |υ|S in either buying or selling. The value of the constant k will depend on the individual investor. (iii) The hedged portfolio has an expected return equal to that from an option. This is exactly the same valuation policy as earlier on discrete hedging with no transaction costs. (iv) In the paper [33], in order to show that the residual risk and trade scaling play an important role in the Black–Scholes option pricing model, a mixed hedging strategy X1 (t), i.e.,

∂C μδt ∂ 2C X1 (t ) = + S ∂ St 1 + μδt ∂ St 2 t

(2.1)

has been proposed to price options in a frictionless financial market. Now we assume that traders make use of the mixed hedging strategy X1 (t) to price options while there exist proportional transaction costs. (v) Empirical findings [34,35] show that the price of a European option is a convex function of the underlying stock price. 2 Therefore, we assume that ∂∂SC >0, and ∂ C2 >0. t

∂ St

In addition, in our model where transaction costs are incurred at every time the stock or the bond is traded, the no arbitrage argument used by Black and Scholes no longer applies. The problem is that due to the infinite variation of the geometric Brownian motion, perfect replication incurs an infinite amount of transaction costs.

Now consider a simple financial market model with constant coefficients, which consists of a stock and a bond with price dynamics given by

St = S0 eμt+σ Bt ,

(2.2)

and

Dt = D0 ert ,

(2.3)

where μ, σ = 0, S0 > 0, r > 0, t ∈ [0, T], T ∈ R fixed, and {Bt }t ∈ [0, T] a standard one dimensional Brownian motion on a complete probability space (, Ft , P) which is equipped with the P − augmentation {Ft }t ∈ [0, T] of the natural Brownian filtration. After a small time interval δ t, the price changes in the bond and in the stock are

  δ Dt = rDt δt + O (δt )2 ,

(2.4)

  δ St = St eμδt+σ δBt − 1   σ2 = St μδt + σ δ Bt + (δ Bt )2 + G1 (δt ), 2

E[δ St ] = St [μδt +

σ 2 δt 2

] + E[G1 (δt )],

(2.5) (2.6)

and

E[(δ St )2 ] = St2 σ 2 δt + E[G2 (δt )], where



E[Gi (δt )] = O (δt )

2



i = 1, 2.

(2.7)

(2.8)

Let C = C(t, St ) be the value of a European call on the above underlying stock at time t with expiration date Tand exercise price X and the boundary conditions:

C ( T , ST ) = ( ST − X )+

at t = T ,

(2.9)

and

C (t, 0 ) = 0, where C(t, St ) is assumed to have continuous partial derivatives up to order three. Consider a replicating portfolio t with X1 (t) units of the stock and X2 (t) units of the bond. The value of the portfolio is given by

t = X1 (t )St + X2 (t )Dt ,

(2.10)

After the time interval δ t, the change in the value of the portfolio is

k 2

δ t = X1 (t )δ St + X2 (t )δ Dt − |δ X1 (t )|St+δt ,

(2.11)

Since C(t, St ) is assumed to have continuous partial derivatives up to order three, the change in the value of the option is

δC =

∂C ∂C 1 ∂ 2C δt + δ St + (δ St )2 + G3 (δt ), ∂t ∂ St 2 ∂ St 2

(2.12)

where

E[G3 (δt )] = o(δt )

(2.13)

Similar to Leland’s argument [14], if δ t is sufficiently small, from Eq. (2.1) we have

δ X1 (t, St ) =

∂ X1 (t, St ) ∂ X (t, St ) δ St + 1 δt ∂ St ∂t 1 ∂ 2 X1 (t, St ) + (δ St )2 + G4 (δt ) 2 ∂ St2

X.-T. Wang et al. / Chaos, Solitons and Fractals 95 (2017) 111–130

therefore



E[k|δ X1 (t )|St+δt ] ≈

and

∂ 2C 2 (δt ) 21 + O(δt ), kσ St 2 π ∂ St 2

(2.14)

d2 = d1 − σˆ



113

T − t,

where



where



σˆ = σ 2 + σ k

E |G4 (δt )| = o(δt ). From the assumptions (iii)–(v), we assume that the option can

be replicated by the portfolio t ,i.e.

E [δC − δ t ] = 0,

(2.15)

π

= X1 (t )St + X2 (t )Dt

(2.16)

 12 (δ t )

1 2

−

> 0.

(2.26)

On the other hand, assuming that the formula (2.25) holds, Lepinete [28] has proposed a modified Leland hedging strategy (M-L) in the “risk-neutral world” as follows



C (t, St ) = t ,

2



Dtni−1 = Cx ti−1 , Sti−1 −

i−2 



Cxt t j , St j





t j+1 − t j ,

i = 1, 2, . . . , n;

j=0

(2.27)

and

∂C μδt St ∂ 2C X1 (t ) = + , ∂ St 1 + μδt ∂ St 2

(2.17)

Hence from Eqs. (2.4)–(2.17) we get that





2

∂C ∂C σ σ + rS + + r−μ− ∂t ∂S 2 2 × S2

∂ 2C = rC. ∂ S2 

Let σˆ =

σ2 +

2

k μδt +σ 1 + μδt 2





2

π

where t0 = 0, tn = T, δ t = tj + 1 − tj , Cx (ti−1 , Sti−1 )



Cxt t j , St j =

1

(δt )− 2



2(r − μ − σ2 )μδt + σk 1 + μδt 2

2

π

and

 12 1

(δt )− 2

× as δt is small enough.

>0 (2.19)

1

 σˆ T − t j

ti−1

   ln St j /X r + σˆ 2 /2 − (d1 )2 1  − e 2 √ , 2 2π 2 T − tj (2.29)

(2.18)







= ∂ S∂ C ,

σˆ = σ 2 + σ k



2

π

 12 (δ t )−

1 2

> 0.

(2.30)

3. The Monte Carlo simulations for the X1 (t) hedging and Leland hedging in portfolio management with proportional transaction costs

From Eqs. (2.18) and (2.19), we obtain that

C (t, St ) = C0 (t, St )

= St N (d1 ) − X e−r (T −t ) N (d2 ),

(2.20)

where C0 (t, St ) denotes the Black–Scholes price of a European call with volatility σˆ , N(·) is the distribution function of a standard normal distribution,

ln (St /X ) + (r + σˆ2 )(T − t ) , √ σˆ T − t 2

d1 = and

d2 = d1 − σˆ



(2.21)

T − t.

In particular, if k = 0, and δ t = 0, from Eqs. (2.20) and (2.21) we have

C (t, St ) = C0 (t, St ) = St N (d1 ) − X e−r (T −t ) N (d2 ), ln (St /X ) + (r + σ2 )(T − t ) , √ σ T −t

(2.22)

2

d1 = and

d2 = d1 − σ



(2.23)

hedging cost = cumulative cost − exercise price if X < ST

T − t.

Furthermore if μ = 0, from Eq. (2.20) we get Leland ’s hedging strategy L and option pricing formula [14] as follows

L=

∂C , ∂ St

(2.24)

× (in − the − money ) or

hedging cost = cumulative cost if X > ST (out − the − money ), while the effective cost of the hedging strategy is

C (t, St ) = C0 (t, St ) = St N (d1 ) − X e d1 =

In this section, we examine the effect of the risk preferences of traders and the fractal scaling on the portfolio hedging performance p of the option pricing models with the proportional transaction costs. We consider to use low-frequency hedging, middlefrequency hedging, and high-frequency hedging to rebalance the portfolio and through the Monte Carlo simulations of the independently1,0 0 0 sample paths of the stock prices, each sample path containing 30,60,120,240,480,960,1920, and 3840 observations for δ t = 4/252, 2/252, 1/252, 1/504, 1/1008, 1/2016, 1/4032, and 1/8064, respectively; we get that the X1 (t) hedging is better than the Leland hedging and the modified Leland hedging in highfrequency trading as X < S0 . Mantegna and Stanley [1], Merton [2], Mandelbrot [3], Mandelbrot and Hudson [4], Stanley and Plerou [7] and Stanley et al. [8], Wilmott [17,18], and Grarcia et al. [36] discovered that the scaling law and the risk preference parameter μ play an important role in option pricing and pointed out that they must be measured in some cases. Now, we distinguish and analyze the final hedging costs, the effective costs, and the hedging performance of the hedging strategy. Recall that the final hedging cost of a call option is given by (see Refs. [35,37])

effective cost = discounted hedging cost − option price. −r (T −t )

N ( d2 ),

2 ln (St /X ) + (r + σˆ2 )(T − t )

√

σˆ T − t

,

(2.25)

On the other hand the hedging performance p is measured by

p=

the standard deviation of the discounted hedging cost , option price

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X.-T. Wang et al. / Chaos, Solitons and Fractals 95 (2017) 111–130

Fig. 3.1. Rebalance interval = 4/252.

Fig. 3.2. Rebalance interval = 2/252.

which is the ratio of the standard deviation of the cost of writing the option and hedging it to theoretical price. In our cases, the values of p are calculated by the independently 10 0 0 sample paths for the stock prices. Obviously, a perfect hedging would have a zero hedging performance. Throughout this section, the stock price data are obtained by means of the independently simulating 10 0 0 sample paths of a geometric Brownian motion with the different values of preference parameter μ and the same diffusion coefficient σ = 0.20. We use, as an example, a financial institution’s position that has sold a European call option on 10 0,0 0 0 shares of a non dividend paying stock. Assume that the stock price is $49, the riskfree interest rate is 5% per annum, the stock price volatility is 20% per annum, and the strike price X and the maturity T as well as the preference parameter μ are variable. With the notation above, S0 = 49, r = 0.05, and σ = 0.20.

Fig. 3.3. Rebalance interval = 1/252.

Fig. 3.4. Rebalance interval = 1/504.

Fig. 3.5. Rebalance interval = 1/1008.

3.1. The impact of the risk preferences on hedging performances in both middle-frequency trading and high-frequency trading Under the conditions above, we compare the X1 (t) hedging with the Leland hedging and the modified Leland hedging in hedging performance p with numerical experiment. We assume that δ t = 4/252, 2/252, 1/252, 1/504, 1/1008, 1/2016, 1/4032, and 1/8064; S0 = 49; σ = 0.20; k = 0.003; r = 0.05; and X = 40, 45, 50, 55, and 60. From Eqs. (2.17), (2.19)–(2.21), and (2.24) –(2.30), we obtained Figs. 3.1–3.8 and Tables A1–A7 in Appendix A. In Figs. 3.1–3.8, the hedging performances of those three hedgings have been listed as μ = 0.20. In addition, Tables A1–A6 report the hedging performances of those three hedging as μ = 0.20, 0.15, 0.10, 0.05, 0.00, and − 0.03 (see Appendix A). Figs. 3.1–3.8 and Tables A1–A6 show that the performances of those three strategies are sensitive to X and that they increase as the functions of X in both middle-frequency trading and highfrequency trading. All three hedging strategies exhibit qualitatively similar behavior: The hedging performance is best as X < S0 , and

Fig. 3.6. Rebalance interval = 1/2016.

the X1 (t) and Leland strategies are significantly better than the modified Leland hedging. From Figs. 3.1–3.7, we also know that the first two hedging strategies are practically identical, while the modified Leland hedging can give rise to a significantly higher hedging cost for a deep-out-of-money option.

X.-T. Wang et al. / Chaos, Solitons and Fractals 95 (2017) 111–130

Fig. 3.7. Rebalance interval = 1/4032.

115

Fig. 3.9. Rebalance interval = 1/52.

Fig. 3.8. Rebalance interval = 1/8064. Fig. 3.10. Rebalance interval = 2/52.

In the same way, in Tables A1–A6 we make a comparison of the hedging performances in three different strategies. For the different values of μ, Tables A1–A6 demonstrate the same tendency in the hedging performances as in Figs. 3.1–3.8. Furthermore, in terms of the hedging performances of three hedging strategies we obtain that those obtained with the X1 (t) hedging are the best even if the Leland hedging presents better hedging than that obtained with the modified Leland hedging (see Tables A1–A6 in Appendix A).The reason for this is that to reduce the hedging error the risk preference parameter μ and residual risk are considered in option pricing. On the other hand, Figs. 3.1–3.8 and Tables A1–A6 also illustrate the risk preference parameter μ has an important influence on the hedging performance in the case of the X1 (t) hedging, even if it does not play any role in the case of the Leland hedging and the modified Leland hedging. Another practical advantage of the X1 (t) hedging is that we propose a simple strategy in order to lower the option price as long as the seller is willing to take risks with a given risk preference parameter μ, for the empirical results show that the option prices of Leland’s strategy and the modified Leland’s strategy are all too high (see Table A7 in Appendix A). 3.2. The impact of risk preferences on hedging performances in low-frequency trading When we consider the hedging performances of the three hedging strategies with low frequency trading generally we obtain similar results as our previous analysis. We take δ t = 1/52, 2/52, 4/52, 5/52, and T = 20/52 years, while the other parameter values are the same as in the Section 3.1. From Eqs. (2.17), (2.19)–(2.21), and (2.24)–(2.30), we obtain Figs. 3.9–3.12 and Tables B1–B7. In Figs. 3.9–3.12, the hedging performances of those three hedgings have been listed as μ = 0.20. In addition, Tables B1–B6 report the hedging performances of three hedging as μ = 0.20, 0.15, 0.10, 0.05, 0.00, and − 0.03 (see Appendix B).

Fig. 3.11. Rebalance interval = 4/52.

Figs. 3.9–3.12 and Tables B1–B6 show that the performances of those three hedging are sensitive to X and that they increase as the functions of X in low-frequency trading. All three hedging strategies exhibit qualitatively similar behavior: The hedging performance is best as X < S0 . From Figs. 3.9 and 3.10, we know that the first two hedging strategies are better than the modified Leland hedging. However, from Figs. 3.11 and 3.12 we find that Leland hedging and the modified Leland hedging are better than the X1 (t) hedging. Furthermore, for the different values of μ with low frequency trading, in terms of the hedging performances of three hedging strategies we obtain that those obtained with the Leland hedging are the best (see Tables B1–B6 in Appendix B). Similar to the results in Section 3.1, we know that the risk preference parameter μ has an important influence on the hedging performance in the X1 (t) hedging and the option prices of the X1 (t) hedging are lowest as long as the seller is willing to take risks with a given risk preference parameter μ (see Table B7 in Appendix B).

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X.-T. Wang et al. / Chaos, Solitons and Fractals 95 (2017) 111–130

Remark 3.1. From Figs. 3.1– 3.10, Tables A1–A6 and Tables B1–B6 we know that in the real world the X1 (t) hedging and Leland hedging are better than the modified Leland hedging even if the asymptotic replication error in the modified Leland hedging [28] tends to zero in probability in a risk-neutral world. The reasons for this are that (i) since there exists the important distinction between the “real world” and the “risk-neutral world”, the modified Leland hedging [28] is only best in a “risk-neutral world”, and (ii) that in the mean-squared convergence sense we now use the hedging performance p to measure to the hedging error while the hedging error [20,23,27–29], in the modified Leland hedging is measured in the probability convergence sense. However, those two concepts of the convergences are not equivalent in general. Fig. 3.12. Rebalance interval = 5/52. Table 3.1 Hedging performance p of X1 (t) strategy, Leland strategy and the modified Leland strategy. p

Stock 1

Stock 2

Stock 3

X1 (t) L M-L

0.8568 0.8764 0.9773

0.9542 0.9837 1.1247

0.6494 0.6851 0.9964

Here, L denotes Leland strategy, and M-L denotes the modified Leland strategy, while stock1, stock 2, and stock 3 denote the stocks of Hang Seng Bank (Stock code: 0 0 011), HSBC Holdings PLC (Stock code: 0 0 0 05) and Ping-An Insurance Company of China, Ltd (Stock code: 601,318), respectively.

3.3. Empirical examples In order to explain our comparison of different strategies, we give the following examples. The data come from the Chinese Wind Financial Database (http://www.wind.com.cn/). The hedging performances of three strategies can be discovered in Table 3.1, and the hedging processes can been seen in Appendix C.

4. Relation between the risk preference and the implied volatility frown As mentioned in Refs. [35,38], one of the most intriguing anomalies reported in the derivatives literatures is the “implied volatility smile.” The name arose from the fact that, prior to the October 1987 market crash, the relation between the Black and Scholes [1] implied volatility of the index options and exercise price gave the appearance of a smile (see the figures in Refs. [35,36];and also see Fig. 4.1). Since October 1987, however, the index implied volatility function (hereafter, IVF), as we refer to it, decreases monotonically across exercise prices (see the figures in Refs. [35,36]; and also see Fig. 4.2), which is called as the “implied volatility smirk”. Under the assumptions of the Black–Scholes model, if markets are efficient and the option pricing model is correct, the IVF should be flat and constant through time.

Implied volatility

Volatility smile

Example 3.3.1. A comparison of hedging performances in X1 (t) strategy, Leland strategy and the modified Leland strategy for a call option underlying the stock of Hang Seng Bank (Stock code: 00,011). To illustrate the typical performance of X1 (t)hedging, we consider the half-daily X1 (t) hedging, Leland hedging and modified Leland hedging from January 14, 2015 to January 28, 2015. A 10day maturity European call option is given with the strike price X = $130. Example 3.3.2. A comparison of hedging performances in X1 (t) strategy, Leland strategy and the modified Leland strategy for a call option underlying the stock of HSBC Holdings PLC (Stock code: 0 0,0 05). To illustrate the typical performance of X1 (t) hedging, we consider the daily X1 (t)hedging, Leland hedging and modified Leland hedging from December 12, 2014 to January 14, 2015. A 20day maturity European option call is given with the strike price X = $75.

From Table 3.1, we can see that the X1 (t) hedging strategy is an improvement over the Leland hedging strategy and the modified Leland hedging strategy, which is in agreement with the results in both Sections 3.1 and 3.2.

Fig. 4.1. Image of the implied volatility smile.

Volatility skew

Implied volatility

Example 3.3.3. A comparison of hedging performances in X1 (t) strategy, Leland strategy and the modified Leland strategy for a call option underlying the stock of Ping-An Insurance Company of China, Ltd (Stock code: 601,318). To illustrate the typical performance of X1 (t) hedging, we consider the weekly X1 (t) hedging, Leland hedging and modified Leland hedging from July 18, 2014 to December 5, 2014. A 20-week maturity European call option is given with the strike price X = $40.

Strike e price

Strike price

Fig. 4.2. Image of the implied volatility skew.

X.-T. Wang et al. / Chaos, Solitons and Fractals 95 (2017) 111–130

117

Implied volatility

Vo olatility frown

Sttrike price

Fig. 4.4. Images of the implied volatility functions for different values of δ t, where r = 0.05, σ = 0.20 k = 0.003, S0 = 49, and T = 120/252.

Fig. 4.3. Image of the implied volatility frown.

Most attempts to explain the shape of the IVF focus on relaxing the Black–Scholes assumption of constant volatility by allowing the volatility of underlying security returns to evolve stochastically through time. The stochastic volatility models can generate the observed downward sloping IVF if innovations to volatility are negatively correlated with underlying asset returns. In addition, it has been shown that a jump-diffusion process can improve the model’s ability to generate IVFs consistent with market prices, but in order to do so parameters must be set to unreasonable values (for more details, see Ref. [38]). In particular, as far as we know, all those models cannot generate IVFs as Fig. 4.3 is shown (also see the figures in Refs. [35,36]), which is called the “implied volatility frown” or the “implied volatility cry” (hereafter, IVC).In the following, we show that the participants’ preferences and the residual risk may be the alternative explanations for the “implied volatility frown”. An avenue of investigation that may lead to a better understanding of the IVC is the study of option market participants’ preferences for different option series in the option markets. Let us consider the following primary problem: why do participants buy a European call option? One answer to this problem is that they are likely to use the option exercise price X as an anchor when evaluating the future stock price ST (see Refs. [30,31]). Therefore we make the following simple assumption

E[ST ] = X.

Fig. 4.5. Images of the implied volatility functions for different values of T, where k = 0.003, r = 0.05, σ = 0.20, S0 = 49, and δ t = 1/252.

(4.2)

From Eqs. (2.2) and (4.2), we have

μ=



1 X ln T S0



−

σ2 2

,

(4.3)

where S0 is the stock price at t = 0. Substituting μ in Eq. (4.3) into Eq. (2.19), we obtain that





σˆ = σ + 2

2 r−

 +

σk

2

π

1 T

X    − σ 2 T1 ln SX0 − S0     2 1 + T1 ln SX0 − σ2 δt  12

ln

1

(δt )− 2

> 0,

σ2 2



δt Fig. 4.6. Images of the implied volatility functions for different values of k, where r = 0.05, σ = 0.20, T = 120/252, S0 = 49, and δ t = 1/252.

(4.4)

as δ t is small enough. From Eq. (4.4), we get Figs. 4.4–4.9, which show that the traders’ preferences and the fractal scaling may be the alternative explanations for the “implied volatility frown”. The risk preference of traders plays an important role in determining the shape of the implied-volatility -frown and the different options having the different hedging frequencies is another reason for the implied volatility frown. Under the assumption of Eq. (4.4), Figs. 4.4–4.6 show that the phenomena of the “implied volatility frown” occur; and the implied volatility term structures are obtained through Figs. 4.7 and

4.8, which show that the implied volatilities tend to be an increasing functions of maturities when short-dated implied volatilities are historically low. This is because there are then the expectations that the implied volatilities will increase. The volatility surfaces combine volatility frown with the volatility term structure to tabulate the volatilities appropriate for pricing an option with any strike price and any maturity. Fig. 4.9 is an example of an implied volatility surface, which is obtained by Eq. (4.4). Figs. 4.10 and 4.11 show that the values of “implied volatility” decrease as the values of δ t increase, while Figs. 4.12 and 4.13 show that the values of “implied volatility” increase as the values of k increase .

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Fig. 4.7. Images of the volatility term structures for different values of σ , where X = 50, r = 0.05, S0 = 49, and k = 0.003.

Fig. 4.8. Images of the volatility term structures for different values of X, where σ = 0.20, r = 0.05, k = 0.003, S0 = 49, and δ t = 1/252.

Fig. 4.11. Images of the implied volatility with respect to δ t for different values of k, where r = 0.05 σ = 0.20, T = 120/252, S0 = 49, and X = 55.

Fig. 4.12. Images of the implied volatility with respect to k for different values of σ , where r = 0.05, T = 120/252, S0 = 49, and δ t = 1/252.

Fig. 4.9. Image of the implied volatility surface, where σ = 0.20, r = 0.05, k = 0.003, S0 = 49, and δ t = 1/252. Fig. 4.13. Images of the implied volatility with respect to k for different values of δ t, where r = 0.05, T = 120/252, σ = 0.20, S0 = 49, and X = 55.

Fig. 4.10. Images of the implied volatility with respect to δ t for different values of X, where r = 0.05 σ = 0.20, T = 120/252, S0 = 49, and k = 0.003.

Remark 4.1. As mentioned in Refs. [17,18], since 1973 no one has to measure μ, which, in a sense, is a good thing since it is very difficult to measure statistically. For a model to contain μ is seen as a very bad point against it. Nevertheless, sometimes it must be measured. For example, although the price of an American option is independent of μ, some properties of this option do depend on μ. The expected time to exercise is one of these. If you need to calculate the expected time to exercise then you must first determine μ, you cannot replace it by r. If you do use r, then you will get the wrong answer. The common use of “risk-neutral valuation” has led to many inaccuracies in the academic literature. Therefore μ must sometimes be measured. The discrete hedging of options is one of these times. In particular, in a real world sense, the X1 (t)

X.-T. Wang et al. / Chaos, Solitons and Fractals 95 (2017) 111–130

hedging strategy is proposed to price options, even if it is an imperfect hedging. Remark 4.2. On basis of the results of Section 2, the results of both Sections 3 and 4 are obtained, respectively; and they are connected through the risk preference parameter μ and the fractal scaling δ t. 5. Conclusion In this paper, we have proposed to use a mixed hedging strategy to price options with the proportional transaction costs. Through the Monte Carlo simulations it has been shown that risk preference parameter μ and fractal scaling δ t as well as resid-

119

ual risk have an important influence on the hedging performances in both option pricing and portfolio hedging, and that the mixed hedging strategy is better than both the Leland hedging strategy and the modified Leland hedging strategy in high-frequency trading in the “real world” as X < S0 , even if this evidence is not absolutely conclusive. In particular, on basis of the points of view of behavioral finance, we give an explanation for the implied volatility frown . Appendix A Hedging performances in both high-frequency trading and middle-frequency trading are as follows.

Table A1 Comparison of the hedging performances of Leland strategy, the modified Leland strategy and X1 (t) strategy for the different values of strike price (X) and rebalance interval (δ (t)), where risk-free rate r = 0.05 per annum (252 days), preference parameter μ = 0.20 per annum, volatility σ = 0.20 per annum, transaction cost k = 0.003, and T = 120/252 years. Strike price X

40

Hedging performance p (X1 (t) hedging) p (Leland hedging) p (Modified Leland hedging)

45

50

55

60

Rebalance interval = 4/252 0.01086 0.047972 0.011396 0.048546 0.027782 0.082747

0.139558 0.140101 0.217743

0.375132 0.369258 0.726389

0.936053 0.898553 2.348551

p (X1 (t) hedging) p (Leland hedging) p (Modified Leland hedging)

Rebalance interval = 2/252 0.009239 0.033375 0.009575 0.034151 0.027337 0.077676

0.093446 0.093808 0.19668

0.265502 0.262642 0.698293

0.6184 0.603508 2.159455

p (X1 (t) hedging) p (Leland hedging) p (Modified Leland hedging)

Rebalance interval = 1/252 0.006608 0.025484 0.006812 0.026072 0.02749 0.080036

0.076102 0.076635 0.197887

0.193229 0.190705 0.710348

0.481966 0.470504 2.197915

p (X1 (t) hedging) p (Leland hedging) p (Modified Leland hedging)

Rebalance interval = 1/504 0.007027 0.025636 0.007174 0.026013 0.028856 0.083761

0.063735 0.064216 0.193231

0.156146 0.15512 0.651522

0.410093 0.404006 1.976487

p (X1 (t) hedging) p (Leland hedging) p (Modified Leland hedging)

Rebalance interval = 1/1008 0.010623 0.02788 0.010715 0.02813 0.036053 0.085885

0.059529 0.059783 0.193115

0.154987 0.154267 0.625457

0.397109 0.394129 1.806557

p (X1 (t) hedging) p (Leland hedging) p (Modified Leland hedging)

Rebalance interval = 1/2016 0.012716 0.034334 0.012755 0.034462 0.037683 0.087363

0.065664 0.065792 0.190942

0.166431 0.166081 0.564971

0.441215 0.439821 1.612084

p (X1 (t) hedging) p (Leland hedging) p (Modified Leland hedging)

Rebalance interval = 1/4032 0.017499 0.044158 0.017522 0.044218 0.042152 0.099345

0.079771 0.079842 0.195644

0.190368 0.190206 0.536575

0.467625 0.467007 1.443194

p (X1 (t) hedging) p (Leland hedging) p (Modified Leland hedging)

Rebalance interval = 1/8064 0.023088 0.055799 0.023098 0.055824 0.045997 0.107735

0.095881 0.095911 0.191423

0.203934 0.203861 0.46918

0.503088 0.502832 1.239619

Table A2 Comparison of the hedging performances of Leland strategy, the modified Leland strategy and X1 (t) strategy for the different values of strike price (X) and rebalance interval (δ t), where risk-free rate r = 0.05 per annum (252 days), preference parameter μ = 0.15 per annum, volatility σ = 0.20 per annum, transaction cost k = 0.003, and T = 120/252 years. Strike price (X)

40

Hedging performance p (X1 (t) hedging) p (Leland hedging) p (Modified Leland hedging)

Rebalance interval =4/252 0.012062 0.049154 0.012308 0.049539 0.028885 0.084756 Rebalance interval = 2/252 0.008468 0.036343 0.008704 0.0369 0.029667 0.080445

p (X1 (t)hedging) p (Leland hedging) p (Modified Leland hedging)

45

50

55

60

0.138249 0.13834 0.220164

0.374989 0.371062 0.762287

0.838918 0.81688 2.148281

0.103136 0.103364 0.214289

0.251402 0.248806 0.714268

0.621655 0.61074 2.106283

(continued on next page)

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Table A2 (continued) Strike price (X)

40

p (X1 (t) hedging) p (Leland hedging) p (Modified Leland hedging)

45

50

55

60

Rebalance interval = 1/252 0.007052 0.02772 0.007206 0.028071 0.028781 0.078812

0.076879 0.077376 0.194575

0.188095 0.18656 0.684065

0.432168 0.42254 2.073437

p (X1 (t) hedging) p (Leland hedging) p (Modified Leland hedging)

Rebalance interval = 1/504 0.008127 0.024943 0.008231 0.025256 0.032414 0.082589

0.062087 0.062359 0.202576

0.156885 0.155754 0.682357

0.382828 0.379035 1.946061

p (X1 (t)hedging) p (Leland hedging) p (Modified Leland hedging)

Rebalance interval = 1/1008 0.010669 0.029071 0.010733 0.029245 0.036416 0.088541

0.059202 0.05932 0.196923

0.153424 0.152695 0.62603

0.40798 0.405737 1.790519

p (X1 (t) hedging) p (Leland hedging) p (Modified Leland hedging)

Rebalance interval = 1/2016 0.012747 0.035632 0.012776 0.035717 0.038183 0.094642

0.067582 0.06766 0.200473

0.167627 0.167303 0.598255

0.428748 0.427743 1.572525

p (X1 (t) hedging) p (Leland hedging) p (Modified Leland hedging)

Rebalance interval = 1/4032 0.017693 0.045313 0.017708 0.045355 0.044584 0.10152

0.077097 0.077138 0.192207

0.18662 0.186461 0.559065

0.474999 0.474571 1.446962

p (X1 (t) hedging) p (Leland hedging) p (Modified Leland hedging)

Rebalance interval = 1/8064 0.025664 0.056987 0.025672 0.057007 0.053415 0.109931

0.092578 0.09259 0.20319

0.20681 0.206745 0.515426

0.493082 0.492899 1.238469

Table A3 Comparison of the hedging performances of Leland strategy, the modified Leland strategy and X1 (t) strategy for the different values of strike price (X) and rebalance interval (δ t), where risk-free rate r = 0.05 per annum (252 days), preference parameter μ = 0.10 per annum, volatility σ = 0.20 per annum, transaction cost k = 0.003, and T = 120/252 years. Strike price (X)

40

Hedging performance p (X1 (t) hedging) p (Leland hedging) p (Modified Leland hedging)

45

50

55

60

Rebalance interval = 4/252 0.013961 0.05324 0.014187 0.053702 0.031373 0.084129

0.142676 0.142146 0.233497

0.349901 0.347966 0.773078

0.74395 0.735364 2.015974

p (X1 (t) hedging) p (Leland hedging) p (Modified Leland hedging)

Rebalance interval = 2/252 0.011142 0.039516 0.011235 0.03985 0.030028 0.084068

0.107695 0.107803 0.216595

0.240078 0.238915 0.700746

0.561312 0.555898 2.020437

p (X1 (t) hedging) p (Leland hedging) p (Modified Leland hedging)

Rebalance interval = 1/252 0.008665 0.030375 0.008788 0.03062 0.032631 0.079618

0.07568 0.075713 0.209807

0.189284 0.188281 0.687201

0.443094 0.43804 1.939035

p (X1 (t) hedging) p (Leland hedging) p (Modified Leland hedging)

Rebalance interval = 1/504 0.009516 0.027983 0.009581 0.028163 0.035694 0.087366

0.061419 0.061506 0.205363

0.16115 0.160337 0.697442

0.37418 0.371525 1.885909

p (X1 (t) hedging) p (Leland hedging) p (Modified Leland hedging)

Rebalance interval = 1/1008 0.011879 0.029917 0.011918 0.030016 0.039902 0.08614

0.061873 0.061918 0.20583

0.149493 0.14905 0.632081

0.372349 0.371045 1.687797

p (X1 (t) hedging) p (Leland hedging) p (Modified Leland hedging)

Rebalance interval = 1/2016 0.01567 0.036296 0.015691 0.036348 0.044881 0.093687

0.065662 0.065673 0.208245

0.168548 0.168326 0.630682

0.414361 0.413812 1.568561

p (X1 (t) hedging) p (Leland hedging) p (Modified Leland hedging)

Rebalance interval = 1/4032 0.019208 0.044639 0.019217 0.044663 0.047807 0.099809

0.077396 0.077406 0.199426

0.188925 0.188825 0.574093

0.446724 0.446469 1.38525

p (X1 (t) hedging) p (Leland hedging) p (Modified Leland hedging)

Rebalance interval = 1/8064 0.026526 0.054922 0.026531 0.054934 0.056125 0.108107

0.090938 0.090938 0.205302

0.212077 0.212034 0.539283

0.462017 0.461915 1.207687

X.-T. Wang et al. / Chaos, Solitons and Fractals 95 (2017) 111–130

121

Table A4 Comparison of the hedging performances of Leland strategy, the modified Leland strategy and X1 (t) strategy for the different values of strike price (X) and rebalance interval (δ t), where risk-free rate r = 0.05 per annum (252 days), preference parameter μ = 0.05 per annum, volatility σ = 0.20 per annum, transaction cost k = 0.003, and T = 120/252 years. Strike price (X)

40

Hedging performance p (X1 (t) hedging) p (Leland hedging) p (Modified Leland hedging)

45

50

55

60

Rebalance interval = 4/252 0.016557 0.057744 0.01663 0.057853 0.034818 0.088561

0.142541 0.142419 0.237612

0.328487 0.327294 0.741052

0.672586 0.669212 1.996365

p (X1 (t) hedging) p (Leland hedging) p (Modified Leland hedging)

Rebalance interval = 2/252 0.01087 0.040358 0.010907 0.040461 0.031327 0.084261

0.103951 0.103953 0.224076

0.24695 0.246789 0.714

0.450163 0.448987 1.900567

p (X1 (t) hedging) p (Leland hedging) p (Modified Leland hedging)

Rebalance interval = 1/252 0.009802 0.031146 0.009837 0.031226 0.035156 0.086281

0.074057 0.074029 0.216084

0.184077 0.183587 0.704697

0.401459 0.400124 1.983516

p (X1 (t) hedging) p (Leland hedging) p (Modified Leland hedging)

Rebalance interval = 1/504 0.010232 0.026973 0.010256 0.027045 0.037771 0.080819

0.060552 0.060544 0.213931

0.149475 0.149121 0.690598

0.344419 0.343443 1.766221

p (X1 (t) hedging) p (Leland hedging) p (Modified Leland hedging)

Rebalance interval = 1/1008 0.012485 0.030577 0.0125 0.030612 0.040126 0.087393

0.061026 0.061035 0.208976

0.152211 0.152027 0.624064

0.380737 0.380259 1.604079

p (X1 (t) hedging) p (Leland hedging) p (Modified Leland hedging)

Rebalance interval = 1/2016 0.0161 0.035358 0.016107 0.035378 0.045249 0.091492

0.067808 0.067802 0.210561

0.169825 0.169738 0.617391

0.402271 0.402063 1.470082

p (X1 (t) hedging) p (Leland hedging) p (Modified Leland hedging)

Rebalance interval = 1/4032 0.021374 0.045481 0.021378 0.04549 0.051917 0.09944

0.078155 0.078153 0.208472

0.197367 0.197328 0.592001

0.441874 0.441796 1.313629

p (X1 (t) hedging) p (Leland hedging) p (Modified Leland hedging)

Rebalance interval = 1/8064 0.02692 0.055791 0.026922 0.055795 0.05686 0.104989

0.090808 0.090807 0.209612

0.216774 0.216759 0.552293

0.441302 0.44127 1.160391

Table A5 Comparison of the hedging performances of Leland strategy, the modified Leland strategy and X1 (t) strategy for the different values of strike price (X) and rebalance interval (δ t), where risk-free rate r = 0.05 per annum (252 days), preference parameter μ = 0.00 per annum, volatility σ = 0.20 per annum, transaction cost k = 0.003, and T = 120/252 years. Strike price (X)

40

Hedging performance p (X1 (t) hedging) p (Leland hedging) p (Modified Leland hedging)

45

50

55

60

Rebalance interval = 4/252 0.016333 0.054342 0.016283 0.054333 0.035346 0.087762

0.144985 0.145003 0.247004

0.317188 0.317747 0.755672

0.669805 0.670018 1.88485

p (X1 (t) hedging) p (Leland hedging) p (Modified Leland hedging)

Rebalance interval = 2/252 0.012889 0.040293 0.012877 0.040232 0.034145 0.084464

0.102918 0.102959 0.226637

0.216728 0.217013 0.699827

0.43732 0.438464 1.83568

p (X1 (t) hedging) p (Leland hedging) p (Modified Leland hedging)

Rebalance interval = 1/252 0.011176 0.03175 0.011149 0.031693 0.037342 0.082278

0.075276 0.075295 0.22299

0.172337 0.172675 0.691696

0.326512 0.327144 1.657729

p (X1 (t) hedging) p (Leland hedging) p (Modified Leland hedging)

Rebalance interval = 1/504 0.01036 0.029117 0.010343 0.029085 0.038995 0.086175

0.059735 0.059757 0.220445

0.148633 0.148875 0.669189

0.332375 0.332839 1.597476

p (X1 (t) hedging) p (Leland hedging) p (Modified Leland hedging)

Rebalance interval = 1/1008 0.013731 0.03024 0.01372 0.030219 0.042119 0.087954

0.059046 0.059067 0.216954

0.151637 0.151758 0.648648

0.341352 0.341629 1.484804

p (X1 (t) hedging) p (Leland hedging) p (Modified Leland hedging)

Rebalance interval = 1/2016 0.01766 0.035678 0.017655 0.035668 0.047527 0.093533

0.068941 0.068949 0.220349

0.168198 0.168258 0.605769

0.363534 0.363649 1.341588

p (X1 (t) hedging) p (Leland hedging) p (Modified Leland hedging)

Rebalance interval = 1/4032 0.022985 0.044272 0.022983 0.044266 0.052939 0.100681

0.078501 0.078505 0.212925

0.197604 0.197631 0.581993

0.396678 0.39673 1.259576

p (X1 (t) hedging) p (Leland hedging) p (Modified Leland hedging)

Rebalance interval = 1/8064 0.029849 0.054557 0.029848 0.054554 0.060662 0.108157

0.093017 0.093019 0.213302

0.218993 0.219004 0.543219

0.422164 0.422184 1.106067

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Table A6 Comparison of the hedging performances of Leland strategy, the modified Leland strategy and X1 (t) strategy for the different values of strike price (X) and rebalance interval (δ t), where risk-free rate r = 0.05 per annum (252 days), preference parameter μ = −0.03 per annum, volatility σ = 0.20 per annum, transaction cost k = 0.003, and T = 120/252 years. Strike price (X)

40

50

55

60

Hedging performance p (X1 (t) hedging) p (Leland hedging) p (Modified Leland hedging)

Rebalance interval = 4/252 0.019663 0.060199 0.019583 0.060031 0.036645 0.090873

45

0.142661 0.142424 0.245494

0.319633 0.319456 0.754222

0.604769 0.604931 1.789104

p (X1 (t) hedging) p (Leland hedging) p (Modified Leland hedging)

Rebalance interval = 2/252 0.013625 0.043729 0.013547 0.043543 0.036217 0.085875

0.108906 0.109068 0.240897

0.207419 0.208072 0.690551

0.40824 0.409326 1.629185

p (X1 (t) hedging) p (Leland hedging) p (Modified Leland hedging)

Rebalance interval = 1/252 0.010583 0.031254 0.010515 0.031195 0.037541 0.082446

0.074047 0.074151 0.230299

0.168823 0.169757 0.691293

0.318664 0.320294 1.578668

p (X1 (t) hedging) p (Leland hedging) p (Modified Leland hedging)

Rebalance interval = 1/504 0.012509 0.02809 0.012467 0.028018 0.041312 0.085488

0.061298 0.06146 0.232513

0.147769 0.148426 0.674896

0.30241 0.303512 1.516036

p (X1 (t) hedging) p (Leland hedging) p (Modified Leland hedging)

Rebalance interval = 1/1008 0.013755 0.03 0.013729 0.029955 0.043873 0.087099

0.061938 0.061991 0.228963

0.14358 0.143876 0.630352

0.320064 0.320687 1.486041

p (X1 (t) hedging) p (Leland hedging) p (Modified Leland hedging)

Rebalance interval = 1/2016 0.018163 0.035485 0.018148 0.03546 0.051122 0.096413

0.067405 0.06743 0.225355

0.167711 0.167866 0.626522

0.361128 0.361429 1.345715

p (X1 (t) hedging) p (Leland hedging) p (Modified Leland hedging)

Rebalance interval = 1/4032 0.024528 0.043829 0.024522 0.043816 0.054942 0.097857

0.081826 0.081842 0.223573

0.193405 0.193465 0.574727

0.381306 0.381405 1.107624

p (X1 (t) hedging) p (Leland hedging) p (Modified Leland hedging)

Rebalance interval = 1/8064 0.030279 0.055291 0.030275 0.055285 0.060273 0.104678

0.093508 0.093515 0.223876

0.218402 0.218427 0.533785

0.401367 0.401407 1.006317

Table A7 Comparison of the option prices of Leland strategy, the modified Leland strategy and X1 (t) strategy for the different values of strike price (X) and rebalance interval (δ t) at time t = 0, where risk- free rate r = 0.05 per annum (252days), preference parameter μ = 0.10 per annum, volatility σ = 0.20 per annum, transaction cost k = 0.003, and T = 120/252 years. Strike price (X)

40

Option Option Option Option

price price of X1 (t) price of Leland price of modified Leland

45

50

55

60

Rebalance interval = 4/252 10.09733 5.926815 10.09835 5.929609 10.09835 5.929609

2.90078 2.904696 2.904696

1.175015 1.178373 1.178373

0.399094 0.401087 0.401087

Option price of X1 (t) Option price of Leland Option price of modified Leland

Rebalance interval = 2/252 10.11118 5.964145 10.11169 5.965509 10.11169 5.965509

2.952841 2.954734 2.954734

1.219802 1.221436 1.221436

0.425939 0.426929 0.426929

Option price of X1 (t) Option price of Leland Option price of modified Leland

Rebalance interval = 1/252 10.13084 6.015138 10.1311 6.015797 10.1311 6.015797

3.02313 3.024032 3.024032

1.280771 1.281557 1.281557

0.463344 0.463833 0.463833

Option price of X1 (t) Option price of Leland Option price of modified Leland

Rebalance interval = 1/504 10.15939 6.085642 10.15952 6.085955 10.15952 6.085955

3.118889 3.119311 3.119311

1.364706 1.365078 1.365078

0.516366 0.516605 0.516605

Option price of X1 (t) Option price of Leland Option price of modified Leland

Rebalance interval = 1/1008 10.20135 6.183137 10.20141 6.183283 10.20141 6.183283

3.248927 3.249119 3.249119

1.48016 1.480332 1.480332

0.591977 0.592092 0.592092

Option price of X1 (t) Option price of Leland Option price of modified Leland

Rebalance interval = 1/2016 10.26335 6.317205 10.26338 6.317271 10.26338 6.317271

3.423926 3.42401 3.42401

1.637926 1.638003 1.638003

0.699821 0.699874 0.699874

Option price of X1 (t) Option price of Leland Option price of modified Leland

Rebalance interval = 1/4032 10.35501 6.499856 10.35503 6.499883 10.35503 6.499883

3.656571 3.656606 3.656606

1.851329 1.851362 1.851362

0.852963 0.852987 0.852987

Option price of X1 (t) Option price of Leland Option price of modified Leland

Rebalance interval = 1/8064 10.4897 6.74566 10.48971 6.74567 10.48971 6.74567

3.961513 3.961527 3.961527

2.136317 2.136329 2.136329

1.068454 1.068464 1.068464

X.-T. Wang et al. / Chaos, Solitons and Fractals 95 (2017) 111–130

123

Appendix B Hedging performances in the low-frequency trading are as follows.

Table B1 Comparison of the hedging performances of Leland strategy, the modified Leland strategy and X1 (t) strategy for the different values of strike price (X) and rebalance interval (δ t), where risk- free rate r = 0.05 per annum (52weeks), preference parameter μ = 0.20 per annum, volatility σ = 0.20 per annum, transaction cost k = 0.003, and T = 20/52 years. Strike price (X)

40

50

55

60

H edging performance p (X1 (t)hedging) p (Leland hedging) p ( Modified Leland hedging)

Rebalance interval = 1/52 0.009913 0.052482 0.010459 0.053798 0.023479 0.085391

45

0.181348 0.181371 0.248508

0.537095 0.521832 0.888054

1.465834 1.397672 2.929348

p (X1 (t) hedging) p (Leland hedging) p ( Modified Leland hedging)

Rebalance interval = 2/52 0.013457 0.069607 0.013289 0.073151 0.022695 0.095961

0.252113 0.249327 0.281638

0.788621 0.75631 0.999078

2.310557 2.09636 3.224512

p (X1 (t) hedging) p (Leland hedging) p ( Modified Leland hedging)

Rebalance interval = 4/52 0.01972 0.093943 0.020893 0.099601 0.028535 0.114342

0.392429 0.377317 0.383454

1.174194 1.067815 1.158911

3.501003 2.879139 3.337888

p (X1 (t) hedging) p (Leland hedging) p ( Modified Leland hedging)

Rebalance interval = 5/52 0.016595 0.099818 0.019572 0.105375 0.026294 0.116644

0.406969 0.400626 0.394395

1.32444 1.188419 1.224161

4.622364 3.730525 3.752018

Table B2 Comparison of the hedging performances of Leland strategy, the modified Leland strategy and X1 (t) strategy for the different values of strike price (X) and rebalance interval (δ t), where risk- free rate r = 0.05 per annum (52weeks), preference parameter μ = 0.15 per annum, volatility σ = 0.20 per annum, transaction cost k = 0.003, and T = 20/52 years. Strike price (X)

40

Hedging performance p (X1 (t) hedging) p (Leland hedging) p ( Modified Leland hedging)

45

50

55

60

Rebalance interval = 1/52 0.010231 0.056231 0.010512 0.057106 0.023884 0.085022

0.180571 0.180364 0.244276

0.520291 0.512327 0.933756

1.402292 1.35585 3.113125

p (X1 (t) hedging) p (Leland hedging) p ( Modified Leland hedging)

Rebalance interval = 2/52 0.020605 0.075908 0.020282 0.077802 0.028352 0.096775

0.248223 0.246969 0.287229

0.728063 0.709091 1.001339

1.869161 1.751485 2.996597

p (X1 (t) hedging) p (Leland hedging) p ( Modified Leland hedging)

Rebalance interval = 4/52 0.0215 0.110626 0.021681 0.111714 0.027855 0.125078

0.367455 0.363547 0.374996

1.006494 0.965191 1.112546

2.716925 2.489254 3.043645

p (X1 (t) hedging) p (Leland hedging) p (Modified Leland hedging)

Rebalance interval = 5/52 0.023001 0.113159 0.024392 0.115406 0.031155 0.126245

0.420205 0.41579 0.420848

1.279621 1.218156 1.294057

3.624881 3.29321 3.66372

Table B3 Comparison of the hedging performances of Leland strategy, the modified Leland strategy and X1 (t) strategy for the different values of strike price (X) and rebalance interval (δ t), where risk- free rate r = 0.05 per annum (52weeks), preference parameter μ = 0.10 per annum, volatility σ = 0.20 per annum, transaction cost k = 0.003, and T = 20/52 years. Strike price (X)

40

Hedging performance p (X1 (t) hedging) p (Leland hedging) p ( Modified Leland hedging)

45

50

55

60

Rebalance interval = 1/52 0.014118 0.062327 0.014369 0.062694 0.027442 0.090944

0.179325 0.178961 0.261018

0.498916 0.49748 0.925509

1.187184 1.176726 2.768671

p (X1 (t)hedging) p (Leland hedging) p ( Modified Leland hedging)

Rebalance interval = 2/52 0.017715 0.080142 0.017421 0.080782 0.026032 0.099012

0.273461 0.273071 0.308025

0.734407 0.723465 1.047647

1.700343 1.6688 2.826956

p (X1 (t) hedging) p (Leland hedging) p ( Modified Leland hedging)

Rebalance interval = 4/52 0.026548 0.106975 0.0265 0.108104 0.031671 0.119506

0.366939 0.36609 0.379262

0.968493 0.949551 1.151632

2.41653 2.347225 3.171963

p (X1 (t) hedging) p (Leland hedging) p ( Modified Leland hedging)

Rebalance interval = 5/52 0.032251 0.127297 0.031622 0.127806 0.035463 0.137914

0.419134 0.422546 0.429256

1.192868 1.192498 1.241934

2.928743 2.906074 3.189408

124

X.-T. Wang et al. / Chaos, Solitons and Fractals 95 (2017) 111–130 Table B4 Comparison of the hedging performances of Leland strategy, the modified Leland strategy and X1 (t) strategy for the different values of strike price (X) and rebalance interval (δ t), where risk- free rate r = 0.05 per annum (52weeks), preference parameter μ = 0.05 per annum, volatility σ = 0.20 per annum, transaction cost k = 0.003, and T = 20/52 years. Strike price (X)

40

50

55

60

Hedging performance p (X1 (t) hedging) p (Leland hedging) p ( Modified Leland hedging)

Rebalance interval = 1/52 0.015066 0.059614 0.015065 0.059673 0.027716 0.08444

45

0.179153 0.179084 0.264049

0.470 0 06 0.469485 0.894806

1.218876 1.218795 2.724499

p (X1 (t)hedging) p (Leland hedging) p ( Modified Leland hedging)

Rebalance interval = 2/52 0.019727 0.086754 0.019793 0.086891 0.028387 0.103352

0.26239 0.262317 0.309301

0.649055 0.648158 0.97427

1.512851 1.506756 2.686532

p (X1 (t)hedging) p (Leland hedging) p( Modified Leland hedging)

Rebalance interval = 4/52 0.025097 0.118916 0.025398 0.119191 0.033586 0.130114

0.363394 0.364277 0.379468

0.939947 0.938614 1.104938

2.057714 2.045628 2.896376

p (X1 (t) hedging) p (Leland hedging) p ( Modified Leland hedging)

Rebalance interval = 5/52 0.033439 0.131177 0.033365 0.131249 0.037847 0.140927

0.39932 0.399482 0.415597

1.050648 1.050858 1.212823

2.451895 2.43419 3.135308

Table B5 Comparison of the hedging performances of Leland strategy, the modified Leland strategy and X1 (t) strategy for the different values of strike price (X) and rebalance interval (δ t), where risk-free rate r = 0.05 per annum (52weeks), preference parameter μ = 0.00 per annum, volatility σ = 0.20 per annum, transaction cost k = 0.003, and T = 20/52 years. Strike price (X)

40

Hedging performance p (X1 (t) hedging) p (Leland hedging) p ( Modified Leland hedging)

45

50

55

60

Rebalance interval = 1/52 0.015271 0.061638 0.015266 0.061542 0.02899 0.090449

0.182799 0.182773 0.268907

0.418675 0.419098 0.900883

0.958514 0.9607 2.468626

p (X1 (t) hedging) p (Leland hedging) p ( Modified Leland hedging)

Rebalance interval = 2/52 0.023398 0.094414 0.023333 0.094378 0.031577 0.106082

0.258939 0.259238 0.328399

0.57767 0.578587 0.955995

1.317951 1.314061 2.50853

p (X1 (t)hedging) p (Leland hedging) p ( Modified Leland hedging)

Rebalance interval = 4/52 0.034136 0.118909 0.034022 0.118988 0.038709 0.128767

0.35699 0.356322 0.385964

0.891776 0.887835 1.132689

2.337246 2.309556 2.974283

p (X1 (t) hedging) p (Leland hedging) p ( Modified Leland hedging)

Rebalance interval = 5/52 0.034376 0.132841 0.034424 0.132802 0.037298 0.137256

0.385617 0.384645 0.407565

0.998305 0.991728 1.169167

2.350563 2.318296 2.907691

Table B6 Comparison of the hedging performances of Leland strategy, the modified Leland strategy and X1 (t) strategy for the different values of strike price (X) and rebalance interval (δ t), where risk-free rate r = 0.05 per annum (52 weeks), preference parameter μ = −0.03 per annum, volatility σ = 0.20 per annum, transaction cost k = 0.003, and T = 20/52 years. Strike price (X)

40

Hedging performance p (X1 (t) hedging) p (Leland hedging) p ( Modified Leland hedging)

45

50

55

60

Rebalance interval = 1/52 0.016761 0.065667 0.016657 0.065624 0.030933 0.091818

0.182529 0.182331 0.282043

0.430509 0.43075 0.893513

0.877801 0.879018 2.283821

p (X1 (t) hedging) p (Leland hedging) p ( Modified Leland hedging)

Rebalance interval = 2/52 0.025026 0.096622 0.025019 0.096523 0.033499 0.110538

0.263027 0.262177 0.315532

0.615628 0.611329 0.932444

1.376277 1.356295 2.474382

p (X1 (t) hedging) p (Leland hedging) p ( Modified Leland hedging)

Rebalance interval = 4/52 0.03427 0.127839 0.034497 0.12759 0.039137 0.135562

0.366825 0.364915 0.398721

0.844182 0.82877 1.058245

2.038966 1.97281 2.740378

p (X1 (t) hedging) p (Leland hedging) p ( Modified Leland hedging)

Rebalance interval = 5/52 0.034239 0.142439 0.034329 0.143119 0.037576 0.146079

0.393835 0.393398 0.417895

0.906225 0.902288 1.136724

1.801149 1.747773 2.578343

X.-T. Wang et al. / Chaos, Solitons and Fractals 95 (2017) 111–130

125

Table B7 Comparison of the option prices of Leland strategy, the modified Leland strategy and X1 (t) strategy for the different values of strike price (X) and rebalance interval (δ t) at time t = 0, where risk-free rate r = 0.05 per annum (52 weeks), preference parameter μ = 0.10 per annum, volatility σ = 0.20 per annum, transaction cost k = 0.003, and T= 20/52 years. Strike price (X)

40

50

55

60

Option Option Option Option

price price of X1 (t) price of Leland price of modified Leland

Rebalance interval = 1/52 9.857898 5.559731 9.860788 5.569485 9.860788 5.569485

45

2.488562 2.502833 2.502833

0.867372 0.878696 0.878696

0.238678 0.244367 0.244367

Option price of X1 (t) Option price of Leland Option price of modified Leland

Rebalance interval = 2/52 9.849234 5.529685 9.854861 5.549343 9.854861 5.549343

2.444261 2.473305 2.473305

0.832448 0.855304 0.855304

0.221435 0.232668 0.232668

Option price of X1 (t) Option price of Leland Option price of modified Leland

Rebalance interval = 4/52 9.839856 5.495607 9.850754 5.535051 9.850754 5.535051

2.393344 2.452211 2.452211

0.792759 0.83869 0.83869

0.20241 0.224483 0.224483

Option price of X1 (t) Option price of Leland Option price of modified Leland

Rebalance interval = 5/52 9.836272 5.482096 9.849719 5.531402 9.849719 5.531402

2.372944 2.446806 2.446806

0.776998 0.834445 0.834445

0.195029 0.222409 0.222409

Appendix C The detailed hedging processes for X1 (t) strategy, Leland strategy, and the modified Leland strategy are as follows.

Table C1 X1 (t) hedging process in Example 3.3.1. The hedging occurs per half a day where strike price X = $130, risk-free rate r = 0.05 per annum, preference parameter μ = 0.8467 per annum, volatility σ = 0.0764 per annum, transaction cost k = 0.003, and T = 10/252 years. Half-a-day

Date

Minute

Stock price

X1 (t)

Shares purchased

Cost-of-shares purchased

Cumulative-cost including interest

Interest cost

Option price

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

2015/1/14 2015/1/14 2015/1/15 2015/1/15 2015/1/16 2015/1/16 2015/1/19 2015/1/19 2015/1/20 2015/1/20 2015/1/21 2015/1/21 2015/1/22 2015/1/22 2015/1/23 2015/1/23 2015/1/26 2015/1/26 2015/1/27 2015/1/27 2015/1/28

11:30 15:00 11:30 15:00 11:30 15:00 11:30 15:00 11:30 15:00 11:30 15:00 11:30 15:00 11:30 15:00 11:30 15:00 11:30 15:00 11:30

129.9 129.4 129.9 130.2 130.6 130.5 129.9 130.5 130.1 130.1 130.8 130.9 131.4 131.7 132.6 132.6 132.9 133 133.8 134.5 134.2

0.5709 0.4773 0.5696 0.6261 0.7011 0.6863 0.5670 0.6936 0.6117 0.6134 0.7717 0.8006 0.8936 0.9401 0.9924 0.9961 0.9996 1.0 0 0 0 1.0 0 0 0 1.0 0 0 0 1.0 0 0 0

0.5709 −0.0936 0.0922 0.0565 0.0750 −0.0148 −0.1193 0.1266 −0.0819 0.0016 0.1583 0.0290 0.0930 0.0464 0.0523 0.0038 0.0034 0.0 0 04 0.0 0 0 0 0.0 0 0 0 0.0 0 0 0

74.1655 −12.1150 11.9816 7.3618 9.7905 −1.9304 −15.4925 16.5222 −10.6511 0.2124 20.7075 3.7898 12.2203 6.1121 6.9347 0.4980 0.4582 0.0579 0.0013 0.0 0 0 0 0.0 0 0 0

74.1655 62.0578 74.0456 81.4147 91.2133 89.2919 73.8083 90.3378 79.6957 79.9160 100.6313 104.4312 116.6618 122.7855 129.7324 130.2433 130.7144 130.7853 130.7996 130.8126 130.8256

0.0074 0.0062 0.0073 0.0081 0.0090 0.0089 0.0073 0.0090 0.0079 0.0079 0.0100 0.0104 0.0116 0.0122 0.0129 0.0129 0.0130 0.0130 0.0130 0.0130 0.0130

0.9617 0.6908 0.9032 1.0396 1.2576 1.1608 0.7779 1.0949 0.8178 0.7819 1.1964 1.2347 1.6054 1.8428 2.6835 2.6676 2.9520 3.0387 3.8258 4.5129 4.20 0 0

Here hedging cost is $ (130.8256–130), discounted hedging cost is $ 0.8240, and hedging performance is 0.8568.

126

X.-T. Wang et al. / Chaos, Solitons and Fractals 95 (2017) 111–130 Table C2 Leland hedging process in Example 3.3.1. The hedging occurs per half a day where strike price X = $130, risk-free rate r = 0.05 per annum, preference parameter μ = 0.8467 per annum, volatility σ = 0.0764 per annum, transaction cost k = 0.003, and T = 10/252 years. Half-a-day

Date

Minute

Stock price

L

Shares purchased

Cost-of-shares purchased

Cumulative-cost including interest

Interest cost

Option price

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

2015/1/14 2015/1/14 2015/1/15 2015/1/15 2015/1/16 2015/1/16 2015/1/19 2015/1/19 2015/1/20 2015/1/20 2015/1/21 2015/1/21 2015/1/22 2015/1/22 2015/1/23 2015/1/23 2015/1/26 2015/1/26 2015/1/27 2015/1/27 2015/1/28

11:30 15:00 11:30 15:00 11:30 15:00 11:30 15:00 11:30 15:00 11:30 15:00 11:30 15:00 11:30 15:00 11:30 15:00 11:30 15:00 11:30

129.9 129.4 129.9 130.2 130.6 130.5 129.9 130.5 130.1 130.1 130.8 130.9 131.4 131.7 132.6 132.6 132.9 133 133.8 134.5 134.2

0.5283 0.4475 0.5253 0.5735 0.6396 0.6249 0.5182 0.6286 0.5537 0.5532 0.6965 0.7229 0.8218 0.8791 0.9700 0.9798 0.9943 0.9987 1.0 0 0 0 1.0 0 0 0 1.0 0 0 0

0.5283 −0.0808 0.0777 0.0482 0.0661 −0.0148 −0.1067 0.1105 −0.0749 −0.0 0 06 0.1434 0.0264 0.0989 0.0573 0.0909 0.0098 0.0146 0.0043 0.0013 0.0 0 0 0 0.0 0 0 0

68.6305 −10.4560 10.0967 6.2765 8.6389 −1.9250 −13.8580 14.4154 −9.7481 −0.0720 18.7534 3.4503 12.9924 7.5527 12.0472 1.2983 1.9379 0.5778 0.1756 0.0 0 03 0.0 0 0 0

68.6305 58.1812 68.2837 74.5670 83.2132 81.2965 67.4466 81.8686 72.1286 72.0638 90.8243 94.2837 107.2854 114.8487 126.9073 128.2182 130.1688 130.7595 130.9480 130.9613 130.9743

0.0068 0.0058 0.0068 0.0074 0.0083 0.0081 0.0067 0.0081 0.0072 0.0071 0.0090 0.0094 0.0106 0.0114 0.0126 0.0127 0.0129 0.0130 0.0130 0.0130 0.0130

1.1095 0.8332 1.0434 1.1739 1.3814 1.2823 0.9018 1.2076 0.9316 0.8908 1.2868 1.3164 1.6632 1.8829 2.6942 2.6743 2.9537 3.0391 3.8258 4.5129 4.20 0 0

Here hedging cost is $ (130.9743–130), discounted hedging cost is $ 0.9724, hedging performance is 0.8764, and L denotes the delta of the Leland strategy. Table C3 The modified Leland hedging process in Example 3.3.1.The hedging occurs per half a day where strike price X = $130, risk-free rate r = 0.05 per annum, preference parameter μ = 0.8467 per annum, volatility σ = 0.0764 per annum, transaction cost k = 0.003, and T = 10/252 years. Half-a-day

Date

Minute

Stock price

M-L

Shares purchased

Cost-of-shares purchased

Cumulative-cost including interest

Interest cost

Option price

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

2015/1/14 2015/1/14 2015/1/15 2015/1/15 2015/1/16 2015/1/16 2015/1/19 2015/1/19 2015/1/20 2015/1/20 2015/1/21 2015/1/21 2015/1/22 2015/1/22 2015/1/23 2015/1/23 2015/1/26 2015/1/26 2015/1/27 2015/1/27 2015/1/28

11:30 15:00 11:30 15:00 11:30 15:00 11:30 15:00 11:30 15:00 11:30 15:00 11:30 15:00 11:30 15:00 11:30 15:00 11:30 15:00 11:30

129.9 129.4 129.9 130.2 130.6 130.5 129.9 130.5 130.1 130.1 130.8 130.9 131.4 131.7 132.6 132.6 132.9 133 133.8 134.5 134.2

0.5283 0.4490 0.5303 0.5801 0.6465 0.6299 0.5216 0.6340 0.5569 0.5569 0.7008 0.7209 0.8115 0.8561 0.9320 0.9318 0.9369 0.9363 0.9355 0.9355 1.0 0 0 0

0.5283 −0.0793 0.0813 0.0498 0.0663 −0.0166 −0.1083 0.1124 −0.0771 0.0 0 0 0 0.1439 0.0201 0.0906 0.0446 0.0759 −0.0 0 02 0.0051 −0.0 0 06 −0.0 0 08 0.0 0 0 0 0.0645

68.6305 −10.2648 10.5644 6.4853 8.6649 −2.1706 −14.0683 14.6721 −10.0341 0.0032 18.8221 2.6288 11.9066 5.8715 10.0653 −0.0261 0.6775 −0.0755 −0.1063 −0.0011 8.6557

68.6305 58.3725 68.9427 75.4349 84.1073 81.9450 67.8848 82.5636 72.5377 72.5482 91.3775 94.0154 105.9313 111.8133 121.8896 121.8756 122.5652 122.5019 122.4077 122.4187 131.0865

0.0068 0.0058 0.0068 0.0075 0.0083 0.0081 0.0067 0.0082 0.0072 0.0072 0.0091 0.0093 0.0105 0.0111 0.0121 0.0121 0.0122 0.0122 0.0121 0.0121 0.0130

1.1095 0.8332 1.0434 1.1739 1.3814 1.2823 0.9018 1.2076 0.9316 0.8908 1.2868 1.3164 1.6632 1.8829 2.6942 2.6743 2.9537 3.0391 3.8258 4.5129 4.20 0 0

Here hedging cost is $ (131.0865–130), discounted hedging cost is $ 1.0843, hedging performance is 0.9773, and M-L denotes the delta of the modified Leland strategy.

X.-T. Wang et al. / Chaos, Solitons and Fractals 95 (2017) 111–130

127

Table C4 X1 (t) hedging process in Example 3.3.2.The hedging occurs per day where strike price X = $75, risk-free rate r = 0.05 per annum, preference parameter μ = −0.7064 per annum, volatility σ = 0.1521 per annum, transaction cost k = 0.003, and T = 20/252 years. Day

Date

Stock price

X1 (t)

Shares purchased

Cost-of-shares purchased

Cumulative-cost including interest

Interest cost

Option price

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

2014/12/12 2014/12/15 2014/12/16 2014/12/17 2014/12/18 2014/12/19 2014/12/22 2014/12/23 2014/12/24 2014/12/29 2014/12/30 2014/12/31 2015/1/2 2015/1/5 2015/1/6 2015/1/7 2015/1/8 2015/1/9 2015/1/12 2015/1/13 2015/1/14

75.20 0 0 73.90 0 0 72.4500 72.60 0 0 72.50 0 0 72.7500 73.50 0 0 74.10 0 0 74.0 0 0 0 74.3500 74.80 0 0 74.0 0 0 0 73.60 0 0 73.1500 71.80 0 0 70.50 0 0 71.0500 71.30 0 0 71.0 0 0 0 71.10 0 0 71.10 0 0

0.5429 0.3817 0.2149 0.2214 0.2025 0.2188 0.2974 0.3707 0.3481 0.3933 0.4610 0.3169 0.2390 0.1574 0.0305 0.0018 0.0020 0.0 0 08 0.0 0 0 0 0.0 0 0 0 0.0 0 0 0

0.5429 −0.1613 −0.1667 0.0064 −0.0188 0.0163 0.0786 0.0733 −0.0226 0.0451 0.0677 −0.1441 −0.0779 −0.0815 −0.1270 −0.0286 0.0 0 02 −0.0012 −0.0 0 08 0.0 0 0 0 0.0 0 0 0

40.8294 −11.9182 −12.0801 0.4673 −1.3653 1.1830 5.7768 5.4328 −1.6716 3.3566 5.0635 −10.6612 −5.7351 −5.9637 −9.1168 −2.0193 0.0144 −0.0856 −0.0582 −0.0 0 06 0.0 0 0 0

40.8294 28.9193 16.8449 17.3156 15.9537 17.1399 22.9201 28.3575 26.6916 30.0535 35.1229 24.4687 18.7384 12.7784 3.6642 1.6456 1.6603 1.5751 1.5171 1.5169 1.5172

0.0081 0.0057 0.0033 0.0034 0.0032 0.0034 0.0045 0.0056 0.0053 0.0060 0.0070 0.0049 0.0037 0.0025 0.0 0 07 0.0 0 03 0.0 0 03 0.0 0 03 0.0 0 03 0.0 0 03 0.0 0 03

1.5838 0.9085 0.4111 0.4171 0.3634 0.3905 0.5662 0.7415 0.6571 0.7506 0.9067 0.5136 0.3397 0.1932 0.0285 0.0014 0.0015 0.0 0 06 0.0 0 0 0 0.0 0 0 0 0.0 0 0 0

Here hedging cost is $ 1.5172, discounted hedging cost is $ 1.5112, and hedging performance is 0.9542. Table C5 Leland hedging process in Example 3.3.2.The hedging occurs per day where strike price X = $75, risk-free rate r = 0.05 per annum, preference parameter μ = −0.7064 per annum, volatility σ = 0.1521 per annum, transaction cost k = 0.003, and T = 20/252 years. Day

Date

Stock price

L

Shares purchased

Cost-of-shares purchased

Cumulative-cost including interest

Interest cost

Option price

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

2014/12/12 2014/12/15 2014/12/16 2014/12/17 2014/12/18 2014/12/19 2014/12/22 2014/12/23 2014/12/24 2014/12/29 2014/12/30 2014/12/31 2015/1/2 2015/1/5 2015/1/6 2015/1/7 2015/1/8 2015/1/9 2015/1/12 2015/1/13 2015/1/14

75.20 0 0 73.90 0 0 72.4500 72.60 0 0 72.50 0 0 72.7500 73.50 0 0 74.10 0 0 74.0 0 0 0 74.3500 74.80 0 0 74.0 0 0 0 73.60 0 0 73.1500 71.80 0 0 70.50 0 0 71.0500 71.30 0 0 71.0 0 0 0 71.10 0 0 71.10 0 0

0.5645 0.4159 0.2547 0.2618 0.2433 0.2605 0.3392 0.4104 0.3901 0.4342 0.4987 0.3646 0.2896 0.2066 0.0544 0.0057 0.0066 0.0036 0.0 0 02 0.0 0 0 0 0.0 0 0 0

0.5645 −0.1486 −0.1612 0.0071 −0.0185 0.0172 0.0786 0.0712 −0.0203 0.0440 0.0645 −0.1340 −0.0750 −0.0830 −0.1522 −0.0487 0.0 0 09 −0.0030 −0.0035 −0.0 0 02 0.0 0 0 0

42.4503 −10.9806 −11.6817 0.5141 −1.3378 1.2530 5.7805 5.2787 −1.5020 3.2743 4.8265 −9.9184 −5.5195 −6.0718 −10.9293 −3.4349 0.0644 −0.2132 −0.2451 −0.0120 0.0 0 0 0

42.4503 31.4781 19.8027 20.3207 18.9869 20.2437 26.0282 31.3120 29.8162 33.0965 37.9295 28.0187 22.5047 16.4374 5.5114 2.0775 2.1423 1.9296 1.6849 1.6732 1.6735

0.0084 0.0062 0.0039 0.0040 0.0038 0.0040 0.0052 0.0062 0.0059 0.0066 0.0075 0.0056 0.0045 0.0033 0.0011 0.0 0 04 0.0 0 04 0.0 0 04 0.0 0 03 0.0 0 03 0.0 0 03

1.6945 1.0134 0.4920 0.4971 0.4377 0.4656 0.6498 0.8282 0.7390 0.8317 0.9856 0.5830 0.3985 0.2381 0.0430 0.0031 0.0032 0.0015 0.0 0 0 0 0.0 0 0 0 0.0 0 0 0

Here hedging cost is $1.6735, discounted hedging cost is $1.6669, hedging performance is 0.9837, and L denotes the delta of the Leland strategy.

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X.-T. Wang et al. / Chaos, Solitons and Fractals 95 (2017) 111–130 Table C6 The modified Leland hedging process in Example 3.3.2.The hedging occurs per day where strike price X = $75, risk-free rate r = 0.05 per annum, preference parameter μ = −0.7064 per annum, volatility σ = 0.1521 per annum, transaction cost k = 0.003, and T = 20/252 years. Day

Date

Stock price

M-L

Shares purchased

Cost-of-shares purchased

Cumulative-cost including interest

Interest cost

Option price

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

2014/12/12 2014/12/15 2014/12/16 2014/12/17 2014/12/18 2014/12/19 2014/12/22 2014/12/23 2014/12/24 2014/12/29 2014/12/30 2014/12/31 2015/1/2 2015/1/5 2015/1/6 2015/1/7 2015/1/8 2015/1/9 2015/1/12 2015/1/13 2015/1/14

75.20 0 0 73.90 0 0 72.4500 72.60 0 0 72.50 0 0 72.7500 73.50 0 0 74.10 0 0 74.0 0 0 0 74.3500 74.80 0 0 74.0 0 0 0 73.60 0 0 73.1500 71.80 0 0 70.50 0 0 71.0500 71.30 0 0 71.0 0 0 0 71.10 0 0 71.10 0 0

0.5645 0.4164 0.2595 0.2743 0.2638 0.2897 0.3773 0.4563 0.4420 0.4931 0.5634 0.4325 0.3677 0.2994 0.1665 0.1336 0.1388 0.1417 0.1433 0.1437 0.0 0 0 0

0.5645 −0.1481 −0.1569 0.0148 −0.0105 0.0259 0.0876 0.0790 −0.0143 0.0511 0.0703 −0.1310 −0.0648 −0.0683 −0.1330 −0.0329 0.0052 0.0030 0.0015 0.0 0 04 −0.1437

42.4503 −10.9433 −11.3690 1.0722 −0.7583 1.8844 6.4381 5.8536 −1.0608 3.8027 5.2603 −9.6906 −4.7677 −4.9933 −9.5465 −2.3199 0.3690 0.2117 0.1099 0.0300 −10.2171

42.4503 31.5154 20.1526 21.2288 20.4747 22.3632 28.8058 34.6650 33.6111 37.4205 42.6881 33.0060 28.2449 23.2571 13.7152 11.3981 11.7693 11.9833 12.0956 12.1280 1.9133

0.0084 0.0063 0.0040 0.0042 0.0041 0.0044 0.0057 0.0069 0.0067 0.0074 0.0085 0.0065 0.0056 0.0046 0.0027 0.0023 0.0023 0.0024 0.0024 0.0024 0.0 0 04

1.6945 1.0134 0.4920 0.4971 0.4377 0.4656 0.6498 0.8282 0.7390 0.8317 0.9856 0.5830 0.3985 0.2381 0.0430 0.0031 0.0032 0.0015 0.0 0 0 0 0.0 0 0 0 0.0 0 0 0

Here hedging cost is $1.9133, discounted hedging cost is $1.9057, hedging performance is 1.1247, and M-L denotes the delta of the modified Leland strategy. Table C7 X1 (t) hedging process in Example 3.3.3.The hedging occurs per week where strike price X = $40, risk-free rate r = 0.05 per annum, preference parameter μ = 0.7855 per annum, volatility σ = 0.3359 per annum, transaction cost k = 0.003, and T = 20/52 years. Week

Date

Stock price

X1 (t)

Shares purchased

Cost-of-shares purchased

Cumulative-cost including interest

Interest cost

Option price

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

2014/7/18 2014/7/25 2014/8/1 2014/8/8 2014/8/15 2014/8/22 2014/8/29 2014/9/5 2014/9/12 2014/9/19 2014/9/26 2014/9/30 2014/10/10 2014/10/17 2014/10/24 2014/10/31 2014/11/6 2014/11/14 2014/11/21 2014/11/28 2014/12/5

39.18 39.59 42.5 43.65 42.75 43.65 42.54 42.35 43.83 42.55 42.2 41.98 41.5 41.15 41.39 40.56 43.2 43.21 45.85 44.88 53

0.5640 0.5832 0.7228 0.7725 0.7386 0.7802 0.7355 0.7299 0.8035 0.7494 0.7361 0.7294 0.7050 0.6863 0.7129 0.6503 0.8741 0.9044 0.9954 0.9998 1.0 0 0 0

0.5640 0.0193 0.1395 0.0497 −0.0339 0.0416 −0.0447 −0.0056 0.0736 −0.0541 −0.0133 −0.0067 −0.0244 −0.0186 0.0266 −0.0626 0.2238 0.0303 0.0909 0.0045 0.0 0 02

22.0958 0.7632 5.9302 2.1695 −1.4487 1.8162 −1.9017 −0.2360 3.2240 −2.3005 −0.5605 −0.2830 −1.0137 −0.7660 1.1008 −2.5397 9.6675 1.3102 4.1699 0.1998 0.0089

22.0958 22.8803 28.8325 31.0297 29.6108 31.4555 29.5841 29.3765 32.6287 30.3596 29.8283 29.5740 28.5888 27.8503 28.9779 26.4661 36.1590 37.5040 41.7100 41.9500 41.9992

0.0212 0.0220 0.0277 0.0298 0.0285 0.0302 0.0284 0.0282 0.0314 0.0292 0.0287 0.0284 0.0275 0.0268 0.0279 0.0254 0.0348 0.0361 0.0401 0.0403 0.0404

3.0088 3.1377 4.8636 5.5946 4.8358 5.3916 4.4734 4.2284 5.2089 4.1370 3.7662 3.4844 3.0231 2.6501 2.6535 1.9631 3.6970 3.5559 5.9380 4.9206 13.0 0 0 0

Here hedging cost is $(41.9992–40), discounted hedging cost is $1.9541, and hedging performance is 0.6494.

X.-T. Wang et al. / Chaos, Solitons and Fractals 95 (2017) 111–130

129

Table C8 Leland hedging process in Example 3.3.3.The hedging occurs per week where strike price X = $40, risk-free rate r = 0.05 per annum, preference parameter μ = 0.7855 per annum, volatility σ = 0.3359 per annum, transaction cost k = 0.003, and T = 20/52 years. Week

Date

Stock price

L

Shares purchased

Cost-of-shares purchased

Cumulative-cost including interest

Interest cost

Option price

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

2014/7/18 2014/7/25 2014/8/1 2014/8/8 2014/8/15 2014/8/22 2014/8/29 2014/9/5 2014/9/12 2014/9/19 2014/9/26 2014/9/30 2014/10/10 2014/10/17 2014/10/24 2014/10/31 2014/11/6 2014/11/14 2014/11/21 2014/11/28 2014/12/5

39.18 39.59 42.5 43.65 42.75 43.65 42.54 42.35 43.83 42.55 42.2 41.98 41.5 41.15 41.39 40.56 43.2 43.21 45.85 44.88 53

0.5398 0.5566 0.6865 0.7340 0.6999 0.7397 0.6947 0.6880 0.7596 0.7041 0.6892 0.6805 0.6543 0.6334 0.6553 0.5904 0.8143 0.8436 0.9814 0.9929 1.0 0 0 0

0.5398 0.0168 0.1299 0.0475 −0.0341 0.0398 −0.0450 −0.0067 0.0716 −0.0556 −0.0148 −0.0087 −0.0262 −0.0210 0.0220 −0.0649 0.2238 0.0293 0.1378 0.0116 0.0071

21.1483 0.6663 5.5199 2.0732 −1.4575 1.7393 −1.9163 −0.2830 3.1395 −2.3647 −0.6257 −0.3646 −1.0885 −0.8626 0.9102 −2.6338 9.6702 1.2676 6.3174 0.5190 0.3743

21.1483 21.8349 27.3758 29.4754 28.0462 29.8125 27.9249 27.6687 30.8348 28.4997 27.9014 27.5636 26.5016 25.6645 26.5994 23.9911 33.6844 34.9844 41.3355 41.8942 42.3087

0.0203 0.0210 0.0263 0.0283 0.0270 0.0287 0.0269 0.0266 0.0296 0.0274 0.0268 0.0265 0.0255 0.0247 0.0256 0.0231 0.0324 0.0336 0.0397 0.0403 0.0407

3.3059 3.4287 5.1347 5.8435 5.0875 5.6224 4.7096 4.4573 5.4069 4.3431 3.9657 3.6751 3.2069 2.8244 2.8120 2.1134 3.7923 3.6291 5.9483 4.9235 13.0 0 0 0

Here hedging cost is $(42.3087–40), discounted hedging cost is $2.2647, hedging performance is 0.6851, and L denotes the delta of the Leland strategy. Table C9 The modified Leland hedging process in Example 3.3.3.The hedging occurs per week where strike price X = $40, risk-free rate r = 0.05 per annum, preference parameter μ = 0.7855 per annum, volatility σ = 0.3359 per annum, transaction cost k = 0.003, and T = 20/52 years. Week

Date

Stock price

M-L

Shares purchased

Cost-of-shares purchased

Cumulative-cost including interest

Interest cost

Option price

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

2014/7/18 2014/7/25 2014/8/1 2014/8/8 2014/8/15 2014/8/22 2014/8/29 2014/9/5 2014/9/12 2014/9/19 2014/9/26 2014/9/30 2014/10/10 2014/10/17 2014/10/24 2014/10/31 2014/11/6 2014/11/14 2014/11/21 2014/11/28 2014/12/5

39.18 39.59 42.5 43.65 42.75 43.65 42.54 42.35 43.83 42.55 42.2 41.98 41.5 41.15 41.39 40.56 43.2 43.21 45.85 44.88 53

0.5398 0.5595 0.6919 0.7383 0.7017 0.7397 0.6913 0.6824 0.7517 0.6909 0.6723 0.6598 0.6295 0.6051 0.6242 0.5536 0.7762 0.7815 0.8851 0.8743 1.0 0 0 0

0.5398 0.0197 0.1324 0.0464 −0.0366 0.0380 −0.0483 −0.0089 0.0693 −0.0608 −0.0187 −0.0125 −0.0303 −0.0244 0.0191 −0.0706 0.2226 0.0053 0.1036 −0.0108 0.1257

21.1483 0.7817 5.6265 2.0249 −1.5656 1.6577 −2.0551 −0.3783 3.0364 −2.5855 −0.7875 −0.5242 −1.2575 −1.0029 0.7901 −2.8631 9.6169 0.2304 4.7478 −0.4825 6.6597

21.1483 21.9504 27.5980 29.6494 28.1123 29.7971 27.7706 27.4190 30.4818 27.9256 27.1649 26.6668 25.4350 24.4565 25.2702 22.4314 32.0698 32.3310 37.1099 36.6631 43.3581

0.0203 0.0211 0.0265 0.0285 0.0270 0.0287 0.0267 0.0264 0.0293 0.0269 0.0261 0.0256 0.0245 0.0235 0.0243 0.0216 0.0308 0.0311 0.0357 0.0353 0.0417

3.3059 3.4287 5.1347 5.8435 5.0875 5.6224 4.7096 4.4573 5.4069 4.3431 3.9657 3.6751 3.2069 2.8244 2.8120 2.1134 3.7923 3.6291 5.9483 4.9235 13.0 0 0 0

Here hedging cost is $(43.3581–40), discounted hedging cost is $3.2941, hedging performance is 0.9964, and M-L denotes the delta of the modified Leland strategy.

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