# Proactive Hedging European Option Pricing with a General Logarithmic Position Strategy

Authors: Lixin Qiao; Fangfang Sun; Xiaotuo Qiao; Meng Li; Xuefeng Wang
Journal: Discrete Dynamics in Nature and Society
 (2022)

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## Abstract

This study proposes an exotic option that extends the classical European option by requiring option holders to continuously trade in underlying assets according to a predesignated trading strategy with a general logarithmic position. The pricing formula for the exotic option with a general logarithmic strategy is derived from the Black–Scholes option pricing formula, and its price advantage is compared (based on simulations) to the classical European option and to the exotic option with a linear position. By varying key parameters, we found that the exotic option with a general logarithmic position has a significant price advantage (up to 34% under certain parameter settings) over the classical European option. Moreover, the exotic option with a general logarithmic strategy can save 5.5% more of the option premium than applying a linear position strategy. Our simulation results indicate that the price advantage of this proactive hedging option with a general logarithmic strategy depends heavily on the initial amount of capital; in particular, this exotic option is more suitable for traders with limited initial amounts of capital.


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## Body

## 1. Introduction

Options comprise a diverse group of indispensable trading products in the financial market, which make them effective tools for hedging risks. The options currently being traded include more than the standardized European and American options; in fact, a vast number of exotic options (e.g., barrier options, Asian options, lookback options, step options, rainbow options, binary options, and basket options, among others) are being changed, combined, and derived from the standard options [1–12]. These exotic options provide traders with more choices for outlining their trading portfolio strategies. In particular, they have the opportunity to finely match the nuances of traders’ market expectations and to satisfy their risk-return needs under various market conditions.In contrast to other financial instruments, the price of an option is not the value of the underlying asset, but rather, it is based on the hedging cost. In particular, the cheaper the option, the more favorable its price discovery. In addition, a portfolio strategy with cheaper options is easier to execute because the option price determines the break-even threshold. However, few studies have evaluated the price advantages of various options. Therefore, one of the main aims of this report is to propose an exotic option that can satisfy traders’ needs for a cheaper price.The pricing of options depends on the risk exposure of their underlying assets. The greater the risk exposure of the underlying asset, the more expensive the option. On this basis, Wang et al. proposed an exotic option that requires option holders to buy-in (or sell out) the underlying asset (stock) and allows them to adjust the holdings of the underlying asset according to its price changes within an option period [13–15]; this option was later named the proactive hedging European option by Li et al. [16]. For a given call option, the option buyer initially holds a certain amount of capital (A). When the price of the underlying assets exceeds a designated threshold (usually the exercise price), the option contract requires the option holder to continuously buy the underlying asset following the preagreed dynamic investment strategy. If the option is exercised, the seller of the call option only needs to bear the expected loss corresponding to the difference between the purchase price and the exercise price. Therefore, the pricing of this exotic option depends on the amount of residual exposure of the aforementioned dynamic position strategy. Simulations have indicated that the price of a proactive hedging European option is significantly lower than that of the analogous classical option. However, Wang et al. assumed that the dynamic position strategy was a continuous linear function, and this assumption is not perfect. For example, the trading behavior of traders is discontinuous, and therefore, it does not always satisfy the assumed continuous function. Li et al. improved the continuous linear function to establish a discrete linear dynamic position strategy [17], although a nonlinear dynamic position strategy might be more suitable for obtaining cheaper options. In an extreme case, if a buyer is purchasing the underlying asset with all of the capital at one time at the exercise price, then the risk exposure of the underlying asset would be completely covered, and the theoretical price of the option will be zero. It is, therefore, clear that for European call options, the greater the proportion of purchasing that occurs near the threshold (i.e., the exercise price), the greater amount of exposure that is covered. Therefore, a nonlinear dynamic position strategy should be considered.This study improves the linear position strategy by developing a nonlinear function and subjectively setting it as a general logarithmic function. The theoretical price of the option is derived, and its price advantage is evaluated. The remainder of this report is organized as follows. Section2 describes the exotic option with a general logarithmic position strategy in detail and derives its value function; Section 3 obtains the theoretical price; Section 4 discusses two special cases of the exotic option; and Section 5 compares the theoretical prices of the exotic options with those of the classical European options based on simulations. Since the derivation processes are very similar for call and put options, we only present the results for call options herein.


## 2. Proactive Hedging European Option with a General Logarithmic Position Strategy

### 2.1. Assumptions for the Proactive Hedging European Option

The proactive hedging European option is proposed based on the following assumptions:(1)
A call option holder should hold initial capital in the amount ofA=Q×Xe at the beginning of the option period for each part of the option contract, where Q is the underlying asset amount of one part of the option contract, and Xe is the exercise price.(2)
Option holders adjust the underlying asset holdings according to the price changes, which are subject to the general logarithmic position constraints of the option contract. The detailed dynamic position strategy will be introduced in Section2.2.(3)
The price of the underlying asset follows geometric Brownian motion (GBM).(4)
There are no other cash flows for the underlying assets during the option contract period. For example, stocks do not pay dividends.(5)
There are no transaction costs for buying or selling the underlying assets.(6)
The risk-free interest rate is constant over time.


### 2.2. General Logarithmic Position Strategy

When the underlying asset price reachesXe+δwhereδ≥0, the general logarithmic position strategy is activated. Then, the option holder spends β0⋅A to buy the underlying asset, where β00≤β0≤1 is the “initial capital utilization coefficient,” which denotes the proportion of capital spent.As the underlying asset price rises, the option holder will continuously increase their capital utilization, i.e., by buying the underlying asset in accordance with the general logarithmic function,β0⋅A, where β0≤β0≤β≤1 is the “highest capital utilization coefficient.” At this time, the underlying asset price reaches 1+αXe+δ, where α is the “investment strategy index,” which is a positive number. The greater the value of α, the larger the price range where traders can buy the underlying asset. The value of α is set within the range [0.5, 1] to encompass the various risk preferences of traders. Figure 1 illustrates the described process.Figure 1
Graphical illustration of the general logarithmic position strategy.The capital utilization coefficient function associated with this process,BS, can be expressed as follows:(1)BS=0,S<Xe+δ,β0+β−β0ln1+αlnSXe+δ,Xe+δ≤S<1+αXe+δ,β,1+αXe+δ≤S.


### 2.3. The Value Function for a Proactive Hedging European Option with a General Logarithmic Position Strategy

For the classical European option without a proactive hedging strategy, the option holder would suffer an expected loss,L, according to equation (2), for each part of the option contract as the underlying asset prices rise from Xe to S0, for S0 > Xe.(2)LbS0=QS0−Xe=AXeS0−Xe.In an exotic option with a proactive hedging strategy, the option holder is required to actively buy-in the underlying assets to hedge the risks. If we assume that the option holder trades using the general logarithmic position strategy described in Section2.2 and buys the underlying asset of the contract piece A⋅β0/Xe+δ when the underlying asset price S reaches Xe+δ, then, when the underlying asset price S changes to S + ∆S, the option holder increases their capital by Δq to buy shares, where Δq=ΔBΔA=AΔB′SΔΔS. Furthermore, when the underlying asset price S rises from Xe+δ to S0, the expected return obtained by the option holder is given by(3)RS0=0,St<Xe+δ,∫Xe+δS0A⋅B′SSS0−SdS+A⋅β0Xe+δS0−Xe−δ,Xe+δ≤S0<1+αXe+δ,∫Xe+δ1+αXe+δA⋅B′SSS0−SdS+A⋅β0Xe+δS0−Xe−δ,S0≥1+αXe+δ.Substituting equation (1) into equation (3) gives (4)RS0=0,St<Xe+δ,Aβ−β0ln1+αS0Xe+δ−lnS0Xe+δ−1+A⋅β0Xe+δS0−Xe−δ,Xe+δ≤S0<1+αXe+δ,A⋅α⋅β−β0⋅S0Xe+δ1+αln1+α−A⋅β−β0+A⋅β0Xe+δS0−Xe−δ,S0≥1+αXe+δ.If the expected loss born by the option seller is defined by(5)LS0=LbS0−RS0,then(6)LS0=AXeS0−Xe，S0<Xe+δ,AXeS0−Xe−A⋅β−β0ln1+αS0Xe+δ−lnS0Xe+δ−1+A⋅β0Xe+δS0−Xe−δ，Xe+δ≤S0<1+αXe+δ,AXeS0−Xe−A⋅α⋅β−β0⋅S0Xe+δ⋅1+α⋅ln1+α−Aβ−β0+A⋅β0Xe+δS0−Xe−δ，S0≥1+αXe+δ,Thus, the option contract valueVTSt,t corresponding to a unit of the underlying asset is given by(7)VTSt,t=LStA/Xe=VT1St,t,St<Xe,VT2St,t,Xe≤St<Xe+δ,VT3St,t,Xe+δ≤St<1+αXe+δ,VT4St,t,St≥1+αXe+δ,where(8)VT1St,t=0,VT2St,t=St−Xe,VT3St,t=St−Xe1−β0⋅XeXe+δ−β−β0⋅Xeln1+αStXe+δ−lnStXe+δ+β−β0⋅Xeln1+α+β0⋅Xe⋅δXe+δ,VT4St,t=St−Xe1−β0⋅XeXe+δ−α⋅β−β0⋅Xe⋅StXe+δ⋅1+α⋅ln1+α+Xe⋅β−β0+β0⋅Xe⋅δXe+δ.Corollary 1.
The proactive hedging European option with a general logarithmic position strategy will cover more risk exposure of the underlying assets than the same option with a general linear position strategy.Proof.
To simplify the proof process, the dynamic position strategy was reduced to the case whereδ=0 and β0=0. Then, the capital utilization coefficient function of the general logarithmic position strategy and the linear position strategy is denoted as BLGS and BLNS, which is expressed by equations (9) and (10), respectively, and depicted in Figure 2:(9)BLGS=0,St<Xe,βln1+αlnStXe,Xe≤St<Xe1+α,β,Xe1+α≤St,(10)BLNS=0,St<Xe,βα⋅XeSt−Xe,Xe≤St<Xe1+α,β,Xe1+α≤St.
In the same way, the expected returns of the general logarithmic and linear position strategies can be obtained based onRLGSt and RLNSt, respectively, where RLGSt (equation (11)) is derived from the special case where δ=0 and β0=0 equation (3), and RLGSt equation (12) is derived from the special case where δ=0 and β0=0, as in the study conducted by Wang et al. [13].(11)RLGSt=0，St<Xe,A⋅βln1+αStXe−lnStXe−1，Xe≤St<1+αXe,A⋅βα⋅StXe1+αln1+α−1，St≥1+αXe,(12)RLNSt=0,St<Xe,A⋅βStα⋅XelnStXe−St−Xeα⋅Xe,Xe≤St<1+αXe,A⋅βStα⋅Xeln1+α−1,St≥1+αXe.
To verify thatRLGSt≥RLNSt when St≥1+αXe, the following inequality should be satisfied:(13)α2−1+αln21+α≥0,where α∈0.5,1 (according to the aforementioned definition of the general logarithmic position strategy) is always true. The inequality in equation (13) is proved as follows.
Setx=1+α, then x∈3/2,2, and inequality (13) can be rewritten as x−12−xln2x>0. Set fx=x−12−xln2x, then f′x=2x−2−2lnx−ln2x, and f″x=2x−2−2lnx/x.
It is known that under the conditionx∈3/2,2, we can define x−1>lnx, and f″x>0 holds on 3/2,2, while f′x is a monotonically increasing function.
Therefore,(14)fmax′x=f′2=3−ln2+12>0,fmin′x=f′32=2−ln32+12>0.and f′x>0 is always true on 3/2,2, while fx>0 monotonically increases on 3/2,2.
As a result,(15)fminx=f32=14−32ln232>0,is sufficient.
WhenXe≤St<1+αXe, a similar procedure can be applied; therefore, a detailed verification process is omitted here.Figure 2
Capital utilization coefficients of the logarithmic strategy vs. the linear strategy.


## 2.1. Assumptions for the Proactive Hedging European Option

The proactive hedging European option is proposed based on the following assumptions:(1)
A call option holder should hold initial capital in the amount ofA=Q×Xe at the beginning of the option period for each part of the option contract, where Q is the underlying asset amount of one part of the option contract, and Xe is the exercise price.(2)
Option holders adjust the underlying asset holdings according to the price changes, which are subject to the general logarithmic position constraints of the option contract. The detailed dynamic position strategy will be introduced in Section2.2.(3)
The price of the underlying asset follows geometric Brownian motion (GBM).(4)
There are no other cash flows for the underlying assets during the option contract period. For example, stocks do not pay dividends.(5)
There are no transaction costs for buying or selling the underlying assets.(6)
The risk-free interest rate is constant over time.


## 2.2. General Logarithmic Position Strategy

When the underlying asset price reachesXe+δwhereδ≥0, the general logarithmic position strategy is activated. Then, the option holder spends β0⋅A to buy the underlying asset, where β00≤β0≤1 is the “initial capital utilization coefficient,” which denotes the proportion of capital spent.As the underlying asset price rises, the option holder will continuously increase their capital utilization, i.e., by buying the underlying asset in accordance with the general logarithmic function,β0⋅A, where β0≤β0≤β≤1 is the “highest capital utilization coefficient.” At this time, the underlying asset price reaches 1+αXe+δ, where α is the “investment strategy index,” which is a positive number. The greater the value of α, the larger the price range where traders can buy the underlying asset. The value of α is set within the range [0.5, 1] to encompass the various risk preferences of traders. Figure 1 illustrates the described process.Figure 1
Graphical illustration of the general logarithmic position strategy.The capital utilization coefficient function associated with this process,BS, can be expressed as follows:(1)BS=0,S<Xe+δ,β0+β−β0ln1+αlnSXe+δ,Xe+δ≤S<1+αXe+δ,β,1+αXe+δ≤S.


## 2.3. The Value Function for a Proactive Hedging European Option with a General Logarithmic Position Strategy

For the classical European option without a proactive hedging strategy, the option holder would suffer an expected loss,L, according to equation (2), for each part of the option contract as the underlying asset prices rise from Xe to S0, for S0 > Xe.(2)LbS0=QS0−Xe=AXeS0−Xe.In an exotic option with a proactive hedging strategy, the option holder is required to actively buy-in the underlying assets to hedge the risks. If we assume that the option holder trades using the general logarithmic position strategy described in Section2.2 and buys the underlying asset of the contract piece A⋅β0/Xe+δ when the underlying asset price S reaches Xe+δ, then, when the underlying asset price S changes to S + ∆S, the option holder increases their capital by Δq to buy shares, where Δq=ΔBΔA=AΔB′SΔΔS. Furthermore, when the underlying asset price S rises from Xe+δ to S0, the expected return obtained by the option holder is given by(3)RS0=0,St<Xe+δ,∫Xe+δS0A⋅B′SSS0−SdS+A⋅β0Xe+δS0−Xe−δ,Xe+δ≤S0<1+αXe+δ,∫Xe+δ1+αXe+δA⋅B′SSS0−SdS+A⋅β0Xe+δS0−Xe−δ,S0≥1+αXe+δ.Substituting equation (1) into equation (3) gives (4)RS0=0,St<Xe+δ,Aβ−β0ln1+αS0Xe+δ−lnS0Xe+δ−1+A⋅β0Xe+δS0−Xe−δ,Xe+δ≤S0<1+αXe+δ,A⋅α⋅β−β0⋅S0Xe+δ1+αln1+α−A⋅β−β0+A⋅β0Xe+δS0−Xe−δ,S0≥1+αXe+δ.If the expected loss born by the option seller is defined by(5)LS0=LbS0−RS0,then(6)LS0=AXeS0−Xe，S0<Xe+δ,AXeS0−Xe−A⋅β−β0ln1+αS0Xe+δ−lnS0Xe+δ−1+A⋅β0Xe+δS0−Xe−δ，Xe+δ≤S0<1+αXe+δ,AXeS0−Xe−A⋅α⋅β−β0⋅S0Xe+δ⋅1+α⋅ln1+α−Aβ−β0+A⋅β0Xe+δS0−Xe−δ，S0≥1+αXe+δ,Thus, the option contract valueVTSt,t corresponding to a unit of the underlying asset is given by(7)VTSt,t=LStA/Xe=VT1St,t,St<Xe,VT2St,t,Xe≤St<Xe+δ,VT3St,t,Xe+δ≤St<1+αXe+δ,VT4St,t,St≥1+αXe+δ,where(8)VT1St,t=0,VT2St,t=St−Xe,VT3St,t=St−Xe1−β0⋅XeXe+δ−β−β0⋅Xeln1+αStXe+δ−lnStXe+δ+β−β0⋅Xeln1+α+β0⋅Xe⋅δXe+δ,VT4St,t=St−Xe1−β0⋅XeXe+δ−α⋅β−β0⋅Xe⋅StXe+δ⋅1+α⋅ln1+α+Xe⋅β−β0+β0⋅Xe⋅δXe+δ.Corollary 1.
The proactive hedging European option with a general logarithmic position strategy will cover more risk exposure of the underlying assets than the same option with a general linear position strategy.Proof.
To simplify the proof process, the dynamic position strategy was reduced to the case whereδ=0 and β0=0. Then, the capital utilization coefficient function of the general logarithmic position strategy and the linear position strategy is denoted as BLGS and BLNS, which is expressed by equations (9) and (10), respectively, and depicted in Figure 2:(9)BLGS=0,St<Xe,βln1+αlnStXe,Xe≤St<Xe1+α,β,Xe1+α≤St,(10)BLNS=0,St<Xe,βα⋅XeSt−Xe,Xe≤St<Xe1+α,β,Xe1+α≤St.
In the same way, the expected returns of the general logarithmic and linear position strategies can be obtained based onRLGSt and RLNSt, respectively, where RLGSt (equation (11)) is derived from the special case where δ=0 and β0=0 equation (3), and RLGSt equation (12) is derived from the special case where δ=0 and β0=0, as in the study conducted by Wang et al. [13].(11)RLGSt=0，St<Xe,A⋅βln1+αStXe−lnStXe−1，Xe≤St<1+αXe,A⋅βα⋅StXe1+αln1+α−1，St≥1+αXe,(12)RLNSt=0,St<Xe,A⋅βStα⋅XelnStXe−St−Xeα⋅Xe,Xe≤St<1+αXe,A⋅βStα⋅Xeln1+α−1,St≥1+αXe.
To verify thatRLGSt≥RLNSt when St≥1+αXe, the following inequality should be satisfied:(13)α2−1+αln21+α≥0,where α∈0.5,1 (according to the aforementioned definition of the general logarithmic position strategy) is always true. The inequality in equation (13) is proved as follows.
Setx=1+α, then x∈3/2,2, and inequality (13) can be rewritten as x−12−xln2x>0. Set fx=x−12−xln2x, then f′x=2x−2−2lnx−ln2x, and f″x=2x−2−2lnx/x.
It is known that under the conditionx∈3/2,2, we can define x−1>lnx, and f″x>0 holds on 3/2,2, while f′x is a monotonically increasing function.
Therefore,(14)fmax′x=f′2=3−ln2+12>0,fmin′x=f′32=2−ln32+12>0.and f′x>0 is always true on 3/2,2, while fx>0 monotonically increases on 3/2,2.
As a result,(15)fminx=f32=14−32ln232>0,is sufficient.
WhenXe≤St<1+αXe, a similar procedure can be applied; therefore, a detailed verification process is omitted here.Figure 2
Capital utilization coefficients of the logarithmic strategy vs. the linear strategy.


## 3. Proactive Hedging Option Pricing Formula Based on GBM

### 3.1. Asset Price Behavior Based on GBM

LetSt be the asset price at the time t. If St follows GBM, then it satisfies the following equation:(16)dStSt=μdt+σdBt,where S0, the draft μ, and the volatility σ of the asset price are all positive constants for t≥0.By applying the fractional Wick–Itô formula, Hu and Øksendal [18] proved that equation (16) can be rewritten as follows:(17)St=S0expσBt+μt−12σ2t.Assuming risk-neutral conditions, i.e.,μ=r, then, for any two time points t1 andt2 where 0≤t1≤t2≤T, the relationship between St1 and St2 can be obtained from equation (17) such that(18)St2=St1exprt2−t1−12σ2t2−t1+σBt2−Bt1.


### 3.2. European Option Pricing Formula

Black and Scholes (1973) derived the famous B-S partial differential function for determining the theoretical price of a classical European option by applying Ito’s Lemma, as follows:(19)∂V∂t+rS∂V∂S+12σ2S2∂2V∂S2=rf,VS,T=fS,T,where f is the option price, t is the time, r is the risk-free return rate, σ is the volatility of the stock return, and S is the stock price.Theorem 1 (also Theorem 1 in ref. [19])..
Letf be a function, such that EfBt<∞. Then, for every 0≤t≤T,(20)E^fBt=∫−∞+∞12πT−texp−x−Bt22T−t⋅fxdx.Proposition 1.
The analytical solution to the B-S formula (19) under GBM is given by.(21)VSt,t=e−rT−t2π∫−∞+∞fSteσT−t⋅Z+T−tr−σ2/2⋅e−Z2/2dZ.where T is the option period, and f⋅ is the intrinsic value function of the option. The intrinsic value of an option is the value of the option at a maturity date. t=T.Proof.
According to the Feynman–Kac formula, we can define the function as follows:(22)VS,t=ESe−∫tTrdtfST.
Then, theVS,T satisfies the following partial differential equation:(23)∂V∂t+rS∂V∂S+12σ2S2∂2V∂S2=rf,with the terminal condition:(24)VS,T=fST=maxST−K,0.
To obtain the analytical solution of the option pricing formula in equation (22) under the GBM assumption, we apply Theorem 1 and combine equations (18) and (22), such that(25)VS,t=e−rT−tESfST=e−rT−tE^fS⋅exprT−t−12σ2T−t+σBT−Bt=e−rT−t2πT−t∫−∞+∞exp−x−Bt22T−tfS⋅exprT−t−12σ2T−t+σx−Btdx.
By settingZ=x−Bt/T−t, we can obtain the integral equation (20), and the proof is complete.


### 3.3. Pricing Formula of a Proactive Hedging Call Option with a General Logarithmic Position

The function in equation (7) is a stepwise function, and therefore, the pricing formula VCSt,t includes four components, as follows:(26)VCSt,t=V1St,t+V2St,t+V3St,t+V4St,t,where(27)V1St,t=e−rT−t2π∫S<XeVT1St⋅eσT−t⋅Z+T−tr−σ2/2,te−Z2/2dZ,V2St,t=e−rT−t2π∫Xe≤S<Xe+δVT2St⋅eσT−t⋅Z+T−tr−σ2/2,te−Z2/2dZ,V3St,t=e−rT−t2π∫Xe+δ≤S<1+αXe+δVT3St⋅eσT−t⋅Z+T−tr−σ2/2,te−Z2/2dZ,V4St,t=e−rT−t2π∫S≥1+αXe+δVT4St⋅eσT−t⋅Z+T−tr−σ2/2,te−Z2/2dZ.Therefore,V1St,t=0.For simplicity of identification, letU=S⋅eσT−t⋅Z+T−tr−σ2/2.WhenXe≤U<Xe+δ, then(28)lnXe/S−T−tr−σ2/2σT−t≤Z<lnXe+δ/S−T−tr−σ2/2σT−t.WhenXe+δ≤U<1+αXe+δ, then(29)lnXe+δ/S−T−tr−σ2/2σT−t≤Z<ln1+αXe+δ/S−T−tr−σ2/2σT−t.WhenU≥1+αXe+δ, then(30)Z≥ln1+αXe+δ/S−T−tr−σ2/2σT−t.To simplify these expressions, let(31)d1=lnXe/S−T−tr−σ2/2σT−t,d2=lnXe+δ/S−T−tr−σ2/2σT−t,d3=ln1+αXe+δ/S−T−tr−σ2/2σT−t,d4=d1−σT−t=lnXe/S−T−tr+σ2/2σT−t,d5=d2−σT−t=lnXe+δ/S−T−tr+σ2/2σT−t,d6=d3−σT−t=ln1+αXe+δ/S−T−tr+σ2/2σT−t.Then,(32)V2St,t=e−rT−t2π∫Xe≤S<Xe+δSt⋅eσT−t⋅Z+T−tr−σ2/2−Xee−Z2/2dZ=e−rT−t2π∫d1d2St⋅eσT−t⋅Z+T−tr−σ2/2⋅e−Z2/2dZ−Xe⋅e−rT−t2π∫d1d2e−Z2/2dZ=St2π∫d1d2eσT−t⋅Z+T−tr−σ2/2−rT−t−Z2/2dZ−Xe⋅e−rT−t2π∫d1d2e−Z2/2dZ=St2π∫d1d2e−1/2Z2−2σT−t⋅Z+σ2T−tdZ−Xe⋅e−rT−t2π∫d1d2e−Z2/2dZ=St2π∫d1d2e−1/2Z−σT−t2dZ−Xe⋅e−rT−t2π∫d1d2e−Z2/2dZ=St2π∫d4d5e−1/2Z−σT−t2dZ−σT−t−Xe⋅e−rT−tNd2−Nd1=St⋅Nd5−Nd4−Xe⋅e−rT−tNd2−Nd1.The expression ofVT3⋅ must be simplified for the sake of feasibility; therefore, let(33)Σ=σT−t⋅Z+T−tr−σ22.Then,(34)VT3⋅=St⋅eΣ−Xe1−β0⋅XeXe+δ−β−β0⋅Xeln1+αSt⋅eΣXe+δ−lnSt⋅eΣXe+δ+β−β0⋅Xeln1+α+β0⋅Xe⋅δXe+δ=1−β0⋅XeXe+δ−β−β0⋅Xeln1+α⋅Xe+δ⋅St⋅eΣ+β−β0⋅Xeln1+ασT−t⋅Z+β−β0⋅Xeln1+αlnSt+T−tr−σ22−Xe1−β0⋅XeXe+δ−β−β0⋅Xeln1+αlnXe+δ+β−β0⋅Xeln1+α+β0⋅Xe⋅δXe+δ=1−β0⋅XeXe+δ−β−β0⋅Xeln1+α⋅Xe+δ⋅St⋅eΣ+β−β0⋅Xeln1+ασT−t⋅Z+β0⋅Xe⋅δ+XeXe+δ−Xe+β−β0⋅Xeln1+α1+lnSt+T−tr−σ22−lnXe+δ=1−β0⋅XeXe+δ−β−β0⋅Xeln1+α⋅Xe+δ⋅St⋅eΣ+β−β0⋅Xeln1+ασT−t⋅Z+β0−1⋅Xe+β−β0⋅Xeln1+α1+lnStXe+δ+T−tr−σ22.If(35)M=St⋅1−β0⋅XeXe+δ−β−β0⋅XeXe+δ⋅ln1+αH=β−β0⋅Xeln1+α⋅σT−tI=β0−1⋅Xe+β−β0⋅Xeln1+α1+T−tr−σ22+lnStXe+δ,then VT3⋅ can be further simplified into (36)VT3⋅=M⋅eΣ+H⋅Z+I.Therefore,(37)V3St,t=e−rT−t2π∫Xe+δ≤St<1+αXe+δM⋅eΣ+H⋅Z+I⋅e−Z2/2dZ=e−rT−t2π∫d2d3M⋅eΣ⋅e−Z2/2dZ+e−rT−t2π∫d2d3H⋅Z⋅e−Z2/2dZ+e−rT−t2π∫d2d3I⋅e−Z2/2dZ=M2π∫d2d3e−1/2Z−T−t2dZ−e−rT−t⋅H2πe−d32/2−e−d22/2+e−rT−t⋅I⋅Nd3−Nd2=M⋅Nd6−Nd5−e−rT−t⋅H2πe−d32/2−e−d22/2+e−rT−t⋅I⋅Nd3−Nd2VT4⋅=St⋅eΣ−Xe1−β0⋅XeXe+δ−α⋅β−β0⋅Xe⋅St⋅eΣXe+δ⋅1+α⋅ln1+α+Xe⋅β−β0+β0⋅Xe⋅δXe+δ=1−β0⋅XeXe+δ−α⋅β−β0⋅XeXe+δ⋅1+α⋅ln1+α⋅St⋅eΣ−1−β⋅Xe.If(38)Q=1−β0⋅XeXe+δ−α⋅β−β0⋅XeXe+δ⋅1+α⋅ln1+αG=1−β⋅Xe,then VT4⋅ can be further simplified into(39)VT4⋅=Q⋅St⋅eΣ−G.Therefore,(40)V4St,t=e−rT−t2π∫St≥1+αXe+δ+∞Q⋅St⋅eΣ−G⋅e−Z2/2⋅dZ=e−rT−t2π∫d3+∞Q⋅St⋅eΣ⋅e−Z2/2⋅dZ−e−rT−t2π∫d3+∞G⋅e−Z2/2⋅dZ=Q⋅St2π∫d3−σT−t+∞e−Z−σT−t2/2⋅dZ−σT−t−G⋅e−rT−t2π∫d3+∞e−Z2/2⋅dZ=Q⋅St⋅1−Nd6−G⋅e−rT−t1−Nd3.Consequently, the final pricing formula for the proactive hedging call option is(41)VSt,t,α,β0,β,δ=VCSt,t=V1St,t+V2St,t+V3St,t+V4St,t=St⋅Nd5−Nd4−Xe⋅e−rT−tNd2−Nd1+M⋅Nd6−Nd5−e−rT−t⋅H2πe−d32/2−e−d22/2+e−rT−t⋅I⋅Nd3−Nd2+Q⋅St⋅1−Nd6−G⋅e−rT−t1−Nd3,where(42)d1=lnXe/S−T−tr−σ2/2σT−t,d2=lnXe+δ/S−T−tr−σ2/2σT−t,d3=ln1+αXe+δ/S−T−tr−σ2/2σT−t,d4=d1−σT−t=lnXe/S−T−tr+σ2/2σT−t,d5=d2−σT−t=lnXe+δ/S−T−tr+σ2/2σT−t,d6=d3−σT−t=ln1+αXe+δ/S−T−tr+σ2/2σT−t,M=St⋅1−β0⋅XeXe+δ−β−β0⋅XeXe+δ⋅ln1+α,H=β−β0⋅Xeln1+α⋅σT−t,I=β0−1⋅Xe+β−β0⋅Xeln1+α1+T−tr−σ22+lnStXe+δ,Q=1−β0⋅XeXe+δ−α⋅β−β0⋅XeXe+δ⋅1+α⋅ln1+α,G=1−β⋅Xe.At this point, the pricing formula for a proactive hedging call option with a linear position strategy is obtained.


## 3.1. Asset Price Behavior Based on GBM

LetSt be the asset price at the time t. If St follows GBM, then it satisfies the following equation:(16)dStSt=μdt+σdBt,where S0, the draft μ, and the volatility σ of the asset price are all positive constants for t≥0.By applying the fractional Wick–Itô formula, Hu and Øksendal [18] proved that equation (16) can be rewritten as follows:(17)St=S0expσBt+μt−12σ2t.Assuming risk-neutral conditions, i.e.,μ=r, then, for any two time points t1 andt2 where 0≤t1≤t2≤T, the relationship between St1 and St2 can be obtained from equation (17) such that(18)St2=St1exprt2−t1−12σ2t2−t1+σBt2−Bt1.


## 3.2. European Option Pricing Formula

Black and Scholes (1973) derived the famous B-S partial differential function for determining the theoretical price of a classical European option by applying Ito’s Lemma, as follows:(19)∂V∂t+rS∂V∂S+12σ2S2∂2V∂S2=rf,VS,T=fS,T,where f is the option price, t is the time, r is the risk-free return rate, σ is the volatility of the stock return, and S is the stock price.Theorem 1 (also Theorem 1 in ref. [19])..
Letf be a function, such that EfBt<∞. Then, for every 0≤t≤T,(20)E^fBt=∫−∞+∞12πT−texp−x−Bt22T−t⋅fxdx.Proposition 1.
The analytical solution to the B-S formula (19) under GBM is given by.(21)VSt,t=e−rT−t2π∫−∞+∞fSteσT−t⋅Z+T−tr−σ2/2⋅e−Z2/2dZ.where T is the option period, and f⋅ is the intrinsic value function of the option. The intrinsic value of an option is the value of the option at a maturity date. t=T.Proof.
According to the Feynman–Kac formula, we can define the function as follows:(22)VS,t=ESe−∫tTrdtfST.
Then, theVS,T satisfies the following partial differential equation:(23)∂V∂t+rS∂V∂S+12σ2S2∂2V∂S2=rf,with the terminal condition:(24)VS,T=fST=maxST−K,0.
To obtain the analytical solution of the option pricing formula in equation (22) under the GBM assumption, we apply Theorem 1 and combine equations (18) and (22), such that(25)VS,t=e−rT−tESfST=e−rT−tE^fS⋅exprT−t−12σ2T−t+σBT−Bt=e−rT−t2πT−t∫−∞+∞exp−x−Bt22T−tfS⋅exprT−t−12σ2T−t+σx−Btdx.
By settingZ=x−Bt/T−t, we can obtain the integral equation (20), and the proof is complete.


## 3.3. Pricing Formula of a Proactive Hedging Call Option with a General Logarithmic Position

The function in equation (7) is a stepwise function, and therefore, the pricing formula VCSt,t includes four components, as follows:(26)VCSt,t=V1St,t+V2St,t+V3St,t+V4St,t,where(27)V1St,t=e−rT−t2π∫S<XeVT1St⋅eσT−t⋅Z+T−tr−σ2/2,te−Z2/2dZ,V2St,t=e−rT−t2π∫Xe≤S<Xe+δVT2St⋅eσT−t⋅Z+T−tr−σ2/2,te−Z2/2dZ,V3St,t=e−rT−t2π∫Xe+δ≤S<1+αXe+δVT3St⋅eσT−t⋅Z+T−tr−σ2/2,te−Z2/2dZ,V4St,t=e−rT−t2π∫S≥1+αXe+δVT4St⋅eσT−t⋅Z+T−tr−σ2/2,te−Z2/2dZ.Therefore,V1St,t=0.For simplicity of identification, letU=S⋅eσT−t⋅Z+T−tr−σ2/2.WhenXe≤U<Xe+δ, then(28)lnXe/S−T−tr−σ2/2σT−t≤Z<lnXe+δ/S−T−tr−σ2/2σT−t.WhenXe+δ≤U<1+αXe+δ, then(29)lnXe+δ/S−T−tr−σ2/2σT−t≤Z<ln1+αXe+δ/S−T−tr−σ2/2σT−t.WhenU≥1+αXe+δ, then(30)Z≥ln1+αXe+δ/S−T−tr−σ2/2σT−t.To simplify these expressions, let(31)d1=lnXe/S−T−tr−σ2/2σT−t,d2=lnXe+δ/S−T−tr−σ2/2σT−t,d3=ln1+αXe+δ/S−T−tr−σ2/2σT−t,d4=d1−σT−t=lnXe/S−T−tr+σ2/2σT−t,d5=d2−σT−t=lnXe+δ/S−T−tr+σ2/2σT−t,d6=d3−σT−t=ln1+αXe+δ/S−T−tr+σ2/2σT−t.Then,(32)V2St,t=e−rT−t2π∫Xe≤S<Xe+δSt⋅eσT−t⋅Z+T−tr−σ2/2−Xee−Z2/2dZ=e−rT−t2π∫d1d2St⋅eσT−t⋅Z+T−tr−σ2/2⋅e−Z2/2dZ−Xe⋅e−rT−t2π∫d1d2e−Z2/2dZ=St2π∫d1d2eσT−t⋅Z+T−tr−σ2/2−rT−t−Z2/2dZ−Xe⋅e−rT−t2π∫d1d2e−Z2/2dZ=St2π∫d1d2e−1/2Z2−2σT−t⋅Z+σ2T−tdZ−Xe⋅e−rT−t2π∫d1d2e−Z2/2dZ=St2π∫d1d2e−1/2Z−σT−t2dZ−Xe⋅e−rT−t2π∫d1d2e−Z2/2dZ=St2π∫d4d5e−1/2Z−σT−t2dZ−σT−t−Xe⋅e−rT−tNd2−Nd1=St⋅Nd5−Nd4−Xe⋅e−rT−tNd2−Nd1.The expression ofVT3⋅ must be simplified for the sake of feasibility; therefore, let(33)Σ=σT−t⋅Z+T−tr−σ22.Then,(34)VT3⋅=St⋅eΣ−Xe1−β0⋅XeXe+δ−β−β0⋅Xeln1+αSt⋅eΣXe+δ−lnSt⋅eΣXe+δ+β−β0⋅Xeln1+α+β0⋅Xe⋅δXe+δ=1−β0⋅XeXe+δ−β−β0⋅Xeln1+α⋅Xe+δ⋅St⋅eΣ+β−β0⋅Xeln1+ασT−t⋅Z+β−β0⋅Xeln1+αlnSt+T−tr−σ22−Xe1−β0⋅XeXe+δ−β−β0⋅Xeln1+αlnXe+δ+β−β0⋅Xeln1+α+β0⋅Xe⋅δXe+δ=1−β0⋅XeXe+δ−β−β0⋅Xeln1+α⋅Xe+δ⋅St⋅eΣ+β−β0⋅Xeln1+ασT−t⋅Z+β0⋅Xe⋅δ+XeXe+δ−Xe+β−β0⋅Xeln1+α1+lnSt+T−tr−σ22−lnXe+δ=1−β0⋅XeXe+δ−β−β0⋅Xeln1+α⋅Xe+δ⋅St⋅eΣ+β−β0⋅Xeln1+ασT−t⋅Z+β0−1⋅Xe+β−β0⋅Xeln1+α1+lnStXe+δ+T−tr−σ22.If(35)M=St⋅1−β0⋅XeXe+δ−β−β0⋅XeXe+δ⋅ln1+αH=β−β0⋅Xeln1+α⋅σT−tI=β0−1⋅Xe+β−β0⋅Xeln1+α1+T−tr−σ22+lnStXe+δ,then VT3⋅ can be further simplified into (36)VT3⋅=M⋅eΣ+H⋅Z+I.Therefore,(37)V3St,t=e−rT−t2π∫Xe+δ≤St<1+αXe+δM⋅eΣ+H⋅Z+I⋅e−Z2/2dZ=e−rT−t2π∫d2d3M⋅eΣ⋅e−Z2/2dZ+e−rT−t2π∫d2d3H⋅Z⋅e−Z2/2dZ+e−rT−t2π∫d2d3I⋅e−Z2/2dZ=M2π∫d2d3e−1/2Z−T−t2dZ−e−rT−t⋅H2πe−d32/2−e−d22/2+e−rT−t⋅I⋅Nd3−Nd2=M⋅Nd6−Nd5−e−rT−t⋅H2πe−d32/2−e−d22/2+e−rT−t⋅I⋅Nd3−Nd2VT4⋅=St⋅eΣ−Xe1−β0⋅XeXe+δ−α⋅β−β0⋅Xe⋅St⋅eΣXe+δ⋅1+α⋅ln1+α+Xe⋅β−β0+β0⋅Xe⋅δXe+δ=1−β0⋅XeXe+δ−α⋅β−β0⋅XeXe+δ⋅1+α⋅ln1+α⋅St⋅eΣ−1−β⋅Xe.If(38)Q=1−β0⋅XeXe+δ−α⋅β−β0⋅XeXe+δ⋅1+α⋅ln1+αG=1−β⋅Xe,then VT4⋅ can be further simplified into(39)VT4⋅=Q⋅St⋅eΣ−G.Therefore,(40)V4St,t=e−rT−t2π∫St≥1+αXe+δ+∞Q⋅St⋅eΣ−G⋅e−Z2/2⋅dZ=e−rT−t2π∫d3+∞Q⋅St⋅eΣ⋅e−Z2/2⋅dZ−e−rT−t2π∫d3+∞G⋅e−Z2/2⋅dZ=Q⋅St2π∫d3−σT−t+∞e−Z−σT−t2/2⋅dZ−σT−t−G⋅e−rT−t2π∫d3+∞e−Z2/2⋅dZ=Q⋅St⋅1−Nd6−G⋅e−rT−t1−Nd3.Consequently, the final pricing formula for the proactive hedging call option is(41)VSt,t,α,β0,β,δ=VCSt,t=V1St,t+V2St,t+V3St,t+V4St,t=St⋅Nd5−Nd4−Xe⋅e−rT−tNd2−Nd1+M⋅Nd6−Nd5−e−rT−t⋅H2πe−d32/2−e−d22/2+e−rT−t⋅I⋅Nd3−Nd2+Q⋅St⋅1−Nd6−G⋅e−rT−t1−Nd3,where(42)d1=lnXe/S−T−tr−σ2/2σT−t,d2=lnXe+δ/S−T−tr−σ2/2σT−t,d3=ln1+αXe+δ/S−T−tr−σ2/2σT−t,d4=d1−σT−t=lnXe/S−T−tr+σ2/2σT−t,d5=d2−σT−t=lnXe+δ/S−T−tr+σ2/2σT−t,d6=d3−σT−t=ln1+αXe+δ/S−T−tr+σ2/2σT−t,M=St⋅1−β0⋅XeXe+δ−β−β0⋅XeXe+δ⋅ln1+α,H=β−β0⋅Xeln1+α⋅σT−t,I=β0−1⋅Xe+β−β0⋅Xeln1+α1+T−tr−σ22+lnStXe+δ,Q=1−β0⋅XeXe+δ−α⋅β−β0⋅XeXe+δ⋅1+α⋅ln1+α,G=1−β⋅Xe.At this point, the pricing formula for a proactive hedging call option with a linear position strategy is obtained.


## 4. Pricing Formula Discussion for Selected Special Cases

This section presents and discusses several special cases involving exotic options. The pricing formula in equation (41) has two sets of parameters. The first set includes β,β0,δ,andα, which describe the general logarithmic position strategy. The second set includes r, T, σ, and Xe, which are the basic parameters for the pricing of a classical European option. Herein, we focus on two special cases based on the first set of parameters:Special Case 1: whenβ=β0=0, δ=0, and α>0, this means that the option holder does not trade the underlying asset during the entire holding period. Theoretically, this exotic option pricing should be consistent with the classical European option pricing.Under this condition, we can define the following:M=St, H=0, I=−Xe, Q=1, G=XeThen,(43)VSt,t,α,0,0,0=VCSt,t=V1St,t+V2St,t+V3St,t+V4St,t=StNd5−Nd4−Xe⋅e−rT−tNd2−Nd1+St⋅Nd6−Nd5−Xe⋅e−rT−t⋅Nd3−Nd2+St⋅1−Nd6−Xe⋅e−rT−t1−Nd3=StNd5−Nd4+Nd6−Nd5+1−Nd6−Xe⋅e−rT−tNd2−Nd1+Nd3−Nd2+1−Nd3=StN−d4−Xe⋅e−rT−tN−d1.It is clear that equation (43) is consistent with the results obtained using the Black–Scholes formula.Special Case 2: whenβ=β0=1, δ=0, α=ε,andε>0 (where ε is a very small number), this means that traders use all of their money to buy underlying assets when the underlying asset price is above but very close to Xe. Then, theoretically, this position strategy almost completely covers the risk exposure of the underlying asset, which means that the value of this exotic option is close to 0.Under this condition, we can define the following:M=0, H=0, I=0, Q=0, G=0, d0=d1=d2, d=d3=d4Then,(44)VSt,t,α,1,1,0=VCSt,t=St⋅Nd5−Nd4−Xe⋅e−rT−tNd2−Nd1=StNd5−Nd4+Nd6−Nd5+1−Nd6=0.


## 5. Comparing Theoretical Prices of Proactive Hedging Call Options with General Logarithmic vs. General Linear Position Strategies

To verify that the exotic option proposed in this study has a price advantage, a simulation method was applied to compare the difference in the theoretical price of the European option with a general logarithmic position strategy, the European option with a linear strategy, and the classical European option. The pricing formula for the European option with the linear position strategy was described in a report by Wang et al. [15]. To facilitate this comparison, the parameter symbols used in this study are consistent with those used by Wang et al. [15] and in the classical Black–Scholes pricing formula. The calculation parameters used in this study are shown in Table 1. For simplicity, we assume t = 0 across all calculations in this section.Table 1
Calculation parameters.
SymbolValuesT0α50%, 70%, 100%β60%, 100%β030%, 50%δ$0, $7, $15 per shareT0.5 (half of one year)S0$20 per shareXe$20 per shareσ30%, 20%, 10%r6%The dynamic position strategy parameters and the basic parameters for the classic European option in Table1 were used to generate several potential parameter combinations. Then, the corresponding pricing formulas were used to calculate the theoretical prices of the European option with a general logarithmic position strategy, the European option with a linear position strategy, and the classical European option. The theoretical prices of these calculated options are listed in Table 2. To intuitively observe the price advantages of the exotic options proposed in this study, Table 2 includes the price ratios, which are defined by the following relationship:(45)ratio=Pexotic optionPclassical option.Table 2
Price ratios of exotic options considering the proactive hedging options relative to classical European options in terms ofα, β, β0, δ, and σ parameters.
Strategy parametersPrice ratios of exotic options with classical optionσ=30%σ=20%σ=10%αβ0βδLogarithmicLinearLogarithmicLinearLogarithmicLinear0.50.30.600.600.610.630.640.660.670.50.30.670.970.970.990.991.001.000.50.30.6151.001.001.001.001.001.000.50.50.600.470.470.480.480.490.490.50.50.670.960.960.990.991.001.000.50.50.6151.001.001.001.001.001.000.50.5100.340.360.390.400.440.450.50.5170.950.950.990.991.001.000.50.51151.001.001.001.001.001.000.70.30.600.630.640.650.660.670.680.70.30.670.970.970.990.991.001.000.70.30.6151.001.001.001.001.001.000.70.50.600.480.480.480.490.490.490.70.50.670.960.960.990.991.001.000.70.50.6151.001.001.001.001.001.000.70.5100.380.400.410.430.450.460.70.5170.950.950.990.991.001.000.70.51151.001.001.001.001.001.0010.30.600.640.660.660.670.680.6810.30.670.970.970.990.991.001.0010.30.6151.001.001.001.001.001.0010.50.600.480.490.490.490.490.5010.50.670.960.960.990.991.001.0010.50.6151.001.001.001.001.001.0010.5100.410.430.430.450.460.4710.5170.950.950.990.991.001.0010.51151.001.001.001.001.001.00The results in Table2 have several important indications:(1)
The theoretical prices of the proactive hedging European option are cheaper than those of the analogous classical European option under all tested parameter combinations. Moreover, the greater the market volatility is (i.e., the greater the value ofσ), the greater the price advantage of the exotic option is. As shown in Table 2, when the market volatility is σ=30%, the price of the proactive hedging European option with a general logarithmic position strategy (i.e., α=0.5,β=1,β0=0.5,δ=0) is only 34% of the price of the classical European option.(2)
The theoretical price of the proactive hedging European option with a general logarithmic position strategy is lower than that with a general linear position strategy. This result indicates that the general logarithmic position strategy would likely be more attractive to hedgers. This price advantage is especially pronounced whenδ=0. In this case, when the market volatility is σ=30% and the other strategy parameters are α=0.5,β=1,andβ0=0.5, the European option with a general logarithmic position strategy can save 5.5% more of the option premium than the linear position strategy. This means that the price advantage of the European option with the general logarithmic position strategy mainly depends on δ.(3)
Each of the four strategy parameters (α, β, β0, and δ) has a remarkable impact on the price advantage. The results in Table 2 show that when other variables are the same, the smaller the values of α and δ (where 0.5≤α≤1), or the larger the values of β and β00≤β0≤β≤1, and the greater the price advantage of the European option with the general logarithmic position strategy. This implies that the price advantage of these exotic options substantially depends on the amount of capital that is used to execute the dynamic position strategy.To evaluate the effects of changes in the strategy parameters (α, δ, β0, and β) on the theoretical price of options, the observed parameters were set as continuous variables, and all other strategy parameters were set as fixed values. According to the conclusions described above, the smaller the α and δ values, or the larger the β0 and β values, the greater the price advantage of this exotic option; therefore, we set the fixed parameter values to correspond to the situation with the greatest price advantage (α=0.5, δ=0, β=1) and compared this exotic option with the classical European call option.If the interval ofα∈0.5,1 holds true (Figure 3), the price advantage of this general logarithmic position strategy is the largest when α=0.5. As α increases, the price advantage of this exotic option decreases. However, the slope of the curve is not steep, which indicates that α has a relatively small influence on the price of the option.Figure 3
The theoretical price as a function ofα when δ=0, β=1, and β0 = 0.3, 0.6, or 0.9 (a, b, c), respectively.
(a)(b)(c)If the interval ofδ∈0,Xe=20 holds true (Figure 4), the price advantage of the option is the largest when δ=0. As δ increases, the price advantage of this exotic option decreases. In contrast to the α effect, the δ curve is clearly convex, which means that the closer δ is to 0, the more sensitive this exotic option price is to the δ parameter. This δ effect becomes extremely weak as δ approaches 1/2Xe=10.Figure 4
The theoretical price as a function ofδ when α=0.5, β=1, and β0 = 0.3, 0.6, or 0.9 (a, b, c), respectively.
(a)(b)(c)The impacts of theβ0 and β parameters are presented in Figures 3–6, and Figures 5 and 6, in particular, illustrate that the larger the values of β0 and β, the greater the price advantage of the exotic option. If 0≤β0≤β≤1, Figures 5 and 6 both indicate that β0andβ follow monotonic, linearly decreasing functions. If all other parameters are equal, the linear slope of β0 is steeper than that of β, which means that the decreasing effect of β0 is more significant. In addition, the trends in Figures 3–6 demonstrate that the smaller the value of β0, the more advantageous the general logarithmic position strategy.Figure 5
The theoretical price as a function ofβ0 when α=0.5, δ=0, and β = 0.3, 0.6, or 0.9 (a, b, c), respectively.
(a)(b)(c)Figure 6
The theoretical price as a function ofβ when α=0.5, δ=0, and β0 = 0.3, 0.6, or 0.9 (a, b, c), respectively.
(a)(b)(c)


## 6. Conclusions

This study details and discusses the pricing of an exotic option, which requires the call option buyer to actively cover the risk exposure by trading the underlying asset according to a preagreed general logarithmic position strategy. The option seller only bears the losses resulting from uncovered risk exposure. To test the pricing advantage of this exotic option, its pricing was compared with that of the linear position strategy proposed by Wang et al. [13–15] and that of the classical European option.The comparative analysis presented herein revealed that both of the described proactive hedging European options exhibited significant price advantages over the classical European option. For example, the theoretical price of applying the general logarithmic position strategy was as low as 34% of the classical European option price under certain parameter settings. The price advantage of this exotic option mainly depended on the capital utilization coefficients (β, β0), which were used to actively hedge the exposure risk of the underlying asset and the prices of the underlying asset Xe+δ when the proactive hedging strategy was initiated. Both β0 and β are monotonic, linearly decreasing functions of the exotic option, meaning that the larger the values of β0 and β, the greater the price advantage of this exotic option. Especially when β0 is very small, the European option with a general logarithmic position strategy has a considerable price advantage over that of the linear position strategy, and its implementation can save up to 5.5% of the option premiums. Additionally, this exotic option price is more sensitive to changes in the δ parameter as δ approaches zero; the δ effect becomes weaker as δ approaches 1/2Xe=10. These results indicate that it is more suitable for traders to use the European option with a general logarithmic position strategy when the initial amount of capital is limited.This study derives the theoretical price of the proactive hedging European option by applying a general logarithmic dynamic position strategy as a representative nonlinear position strategy. The described analyses indicate that the extension of this exotic option with the general logarithmic position strategy has important practical significance. Further studies should aim to develop more reasonable dynamic position strategies in the form of nonlinear functions.



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