期权定价理论的应用与投资策略分析
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摘要
本文主要利用无套利原理,随机动态规划方法和PDE理论就期权定价理论的应用和投资策略的分析展开研究.全文分为四章,主要内容如下.
在第二章,我们利用无套利原理建立了上市公司同时发行债券和权证时,权证和交易期权定价的数学模型.结果显示权证和交易期权的实际价值比用Black-Scholes公式得到的价格要小.这是前人在研究权证时没有考虑到的情况.本章的第二个工作是：利用PDE方法研究了美式封顶看涨和美式保底看跌期权的对称性.我们证明了与其他学者用随机分析方法给出的结论是一致的.
在第三章和第四章,我们从投资产品的价格出发,利用随机动态规划方法分别研究了均值-方差和安全第一投资策略模型.本文的模型与前人的收益率模型是不同的：本文是基于投资品的价格研究投资策略,而前人的工作是把投资产品的收益率作为已知来研究投资策略.与前人的工作相比,我们的研究有更大的优势,如产品的价格更易观测的到,可以用不同的价格规律反映不同类别的投资产品,以及通过参数变动反映投资环境的变化等.最后,我们得到了不带交易费用时的解析解,并且探讨了交易费用、期望收益率以及波动率等一些外生变量变化时对两种投资策略的影响.在第四章,还特别给出了在相同条件下,当外界投资环境变化时,两种投资策略的相应变化幅度.
This paper use the theory of no-arbitrage, stochastic dynamic programming method and PDE theory to study the application of option theory and the investment strategy. And it falls into four chapters. The main contents are as follows.
Warrant is an important tool for the listed firm to finance itself. Since the Ameri-can light power firm issued the first Warrant in 1911, many people began to have much interest in studing the pricing of Warrant. At the start of issuing Warrant, the methods of B-S[14] and Merton[83], [85] were very porpular. Subsequently, Galai, Schneller[49], Hanke, Potzelberger[53], How, Howe[56] and so on did much research on Warrant pric-ing. In Chapter two, we study what happened to the pricing of Warrant and traded option when the listed firm issued both bonds and Warrants. For this purpose, We construct the mathematical model on stock of the listed firm. Then we get the change of the pricing of Warrant and traded option by the option pricing theory. The results show clearly that actual value is lower than the price by B-S formula and the wealth by which the actual value decreases is transfered to the issuer of Warrant. Similarly, the actual value of traded option is also diluted and the issuer of traded option will get extra wealth. The main results we obtained are as follows:
Theorem 0.1 The listed firm had only common stocks of N shares and the market value of the firm's equity at time t is Vt. The Warrants Wt(St,x,tO,TW)of M shares with strike price x and maturity TW, and a corporate bond D with face value F and maturity TD(TD> TW). are issued at time tO by the listed firm. Then there exists K1>0,such that where a=1+(?),and PPt(Vt,K2,TW)is the price of the put option,with strike price K2 and maturity TW,written on the put option Pt(Vt,F,TD).
Theorem 0.2 The listed firm had only common stocks of N shares and the market Value of the firm's equity at time t is Vt.The Warrants Wt(St,χ,tO,TW)of M shares with strike priceχand maturity Tw,and a corporate bond D with face value F and maturity TD(TD>TW),are issued at time to by the listed firm.A third party issues the traded option with strike priceχ0 and maturity TC(TC0,such that where where CCt(Ut,K4,Tc)is the price of the call option,with strike price K4 and ma-turity TC,written on the call option Ct(Ut,K1,TW); CPPt(Vt,K5,TC)is the price of a call option,with strike price K5 and maturity TC,written on the compound option PPt(Vt,K2,TD);PPt(Vt,K6,TC)is the price of a put option,with strike price K6 and maturity TC,written on the put option Pt(Vt,F,TD).
Most people paid much attention on the Warrant pricing or studing the effects of issuing Warrants on stocks.Our results are differential from the previous work and are very significant for both investors and listed firms.
For European and American options, Put-call symmetry is very important in capital markets, which can provide some useful investment information for investors. In 1969, Stoll[112] first proposed the relationship between the values and optimal exercise boundaries of the European call and put option. Merton[83], Castelli[22], Detemple[29] and so on did lots of research about the symmetry. In the second part of Chapter two, we use PDE method to study the symmetry between the the values and the optimal exercise boundaries of American capped call and floored put option. Our results agree with the research by the method of stochastic analysis. The main results we obtained are as follows:
If the strike price of American capped call option is the same as the strike price of American floored put option, we have
Theorem 0.3 Let Vc(S,t;r,q,L(t)) and S= Sc(t;r,q,L(t)) are the value and the op-timal exercise boundary of the American call option with cap L(t), where L(t)> K for any t∈[0, T]. And let Vp(S,t;r,q,m(t)) and S= Sp(t;r,q,m(t)) are the value and the optimal exercise boundary of the American put option with floor m(t), where 0< m(t)< K for any t∈[0,T].If /the same underlying asset pays the divi-dend continuously, the two options own the same maturity T, the strike price K and L(t)m(t)= K2. Then If the strike price of American capped call option is not equal to the strike price of American floored put option, we have
Theorem 0.4 Under the assumptions of Theorem.0.3, and the two options own the same maturity T, and L(t)m(t)= KcKP. where Kc, Kp are respectively the strike prices of American capped call option and American floored put option. Then
Corollary 0.5 Under the assumptions of Theorem 0.3, and L(t)m(t)= S2. Then The results of Corollary 0.5 are consistent with Corollary 9 of Detemple [29].
The classical Markowitz model proposes the porfolio selcetion problem which max-imizes the expectation under given variance or minimizes the variance under given expectation in terms of the rate of return of asset. And Markowitz model is on the hypothesis that the rate of asset return follows the normal distribution and the capital market is frictionless and without transaction costs. In reality, it is difficult to measure the rate of asset return. However, the price of asset can be got easily from the market and the various asset follows the different law because of the characters of asset, for example some assets follow the model of jump-diffusion, and another assets follow the model of mean-reverting. And the investment model in terms of the rate of asset re-turn have not these merits. Therefore, in Chapter three, we use the stochastic dynamic programming method to study the mean-variance strategy model in terms of the price of asset. And we get the analytic solution without transaction costs and study the effects of the exogenous variables such as transaction costs, expected return rate and volatility on the investment strategy. By the investment model in terms of the price of asset, we can learn the according change of investment strategy with the change of investment environment. The main results we obtained are as follows:
We suppose a capital market with n risky assets and 1 riskless asset. The return of riskless asset is equal to one and the price of the risky assets follows the stochastic process S= (St)t=0,...,T= (S1t,…,Snt)t=0,…,T,T= 2. The investment can be permitted during the time interval of [0,2] and be allowed to short sell the risky assets. The .transaction costs are proportional to the volume of transaction. Then
Theorem 0.6 The mean-variance portfolio selection model in two srage is as follows: whereχt is the wealth of the investor at the beginning of the t-th period,St=(St1,St2,…,Stn)' denotes the price of risky assets at time t,ξ≥0,0≤η<1 are constats and are respectively the proportions of transaction costs when the investor buy or sell. share of asset,and lt=(lt1,lt2,…,lnt)'and mt=(mt1,mt2,…,mtn)'are the number of asset the investor buys or sells at time t,σ>0 is a preselected risk level. And Pt=St+1-(1+ξ)St,Qt=(1-η)St-St+1.Then,whenξ=η=0,the optimal investment strategy is as follows: whereΛ11,Λ12,Γ11,Γ12,L1,L2,L3,M1,M2,M3 and a,b,c,d,e are given by mathematical deriva-tion.
At last,we show the investment strategy analysis when the transaction costs,expected return rate and volatility change.By some examples,we can get the following results:
(1) The transaction costs affect the trading frequency but not the trading volume.
(2) Ifμ1≈μ2，σ1<σ2, whereμi,σi,i= 1.2 are respectively the expected return rate and the volatility of asset i. Then the investor prefer the asset 1 to the asset 2. With the increase ofμ2, the investor come to prefer the asset 2.
(3) Ifμ2<μ2,σ1≈σ2. Then the investor prefer the asset 2 to the asset 1. With the increase ofσ2, the investor come to prefer the asset 1.
The safety-first investment model use another idear to study the portfolio selection problem and was first proposed by Roy[103] in 1952. The objective of the investor is to minimize the probability that the teminal wealth is below a preselected threshold, which is different from the mean-variance model. Many people study the safety-first investment model in terms.of the rate of return of asset, and it is similar to the mean-variance model. In Chapter four, we use the method and assumptions of the Chapter three to construct the mathematical model of safety-first with the assumption that the law of the price of asset is given. The main results we obtained are as follows:
Theorem 0.7 Under the assumptions of Chapter three, the safety-first portfolio selec-tion model in two stage is as follows: whereμdenotes the preselected threshold. Then, whenξ=η= 0, the optimal invest-ment strategy is as follows: where Λ11,Λ12,Γ11,Γ12, L1, L2, L3, M1, M2, M3 and a, b, c, d, e are given by mathematical deriva-tion.
At last, we show the investment strategy analysis when the transaction costs, expected return rate and volatility change. By some examples, we can get the following results:
(1) The transaction costs affect the trading frequency but not the trading volume.
(2)Ifμ1≈μ2,σ1＜σ2, whereμi,σi,i= 1,2 are respectively the expected return rate and the volatility of asset i. Then the investor prefer the asset 1 to the asset 2. With the increase ofμ2, the investor come to prefer the asset 2.
(3) Ifμ2＜μ2,σ1≈σ2. Then the investor prefer the asset 2 to the asset 1. With the increase ofσ2, the investor come to prefer the asset 1.
(4) If the investment environment changes, the effections of different investment strategy(mean-variance investment or safety-first investment strategy) on the investor are varying.
在第二章,我们利用无套利原理建立了上市公司同时发行债券和权证时,权证和交易期权定价的数学模型.结果显示权证和交易期权的实际价值比用Black-Scholes公式得到的价格要小.这是前人在研究权证时没有考虑到的情况.本章的第二个工作是：利用PDE方法研究了美式封顶看涨和美式保底看跌期权的对称性.我们证明了与其他学者用随机分析方法给出的结论是一致的.
在第三章和第四章,我们从投资产品的价格出发,利用随机动态规划方法分别研究了均值-方差和安全第一投资策略模型.本文的模型与前人的收益率模型是不同的：本文是基于投资品的价格研究投资策略,而前人的工作是把投资产品的收益率作为已知来研究投资策略.与前人的工作相比,我们的研究有更大的优势,如产品的价格更易观测的到,可以用不同的价格规律反映不同类别的投资产品,以及通过参数变动反映投资环境的变化等.最后,我们得到了不带交易费用时的解析解,并且探讨了交易费用、期望收益率以及波动率等一些外生变量变化时对两种投资策略的影响.在第四章,还特别给出了在相同条件下,当外界投资环境变化时,两种投资策略的相应变化幅度.
This paper use the theory of no-arbitrage, stochastic dynamic programming method and PDE theory to study the application of option theory and the investment strategy. And it falls into four chapters. The main contents are as follows.
Warrant is an important tool for the listed firm to finance itself. Since the Ameri-can light power firm issued the first Warrant in 1911, many people began to have much interest in studing the pricing of Warrant. At the start of issuing Warrant, the methods of B-S[14] and Merton[83], [85] were very porpular. Subsequently, Galai, Schneller[49], Hanke, Potzelberger[53], How, Howe[56] and so on did much research on Warrant pric-ing. In Chapter two, we study what happened to the pricing of Warrant and traded option when the listed firm issued both bonds and Warrants. For this purpose, We construct the mathematical model on stock of the listed firm. Then we get the change of the pricing of Warrant and traded option by the option pricing theory. The results show clearly that actual value is lower than the price by B-S formula and the wealth by which the actual value decreases is transfered to the issuer of Warrant. Similarly, the actual value of traded option is also diluted and the issuer of traded option will get extra wealth. The main results we obtained are as follows:
Theorem 0.1 The listed firm had only common stocks of N shares and the market value of the firm's equity at time t is Vt. The Warrants Wt(St,x,tO,TW)of M shares with strike price x and maturity TW, and a corporate bond D with face value F and maturity TD(TD> TW). are issued at time tO by the listed firm. Then there exists K1>0,such that where a=1+(?),and PPt(Vt,K2,TW)is the price of the put option,with strike price K2 and maturity TW,written on the put option Pt(Vt,F,TD).
Theorem 0.2 The listed firm had only common stocks of N shares and the market Value of the firm's equity at time t is Vt.The Warrants Wt(St,χ,tO,TW)of M shares with strike priceχand maturity Tw,and a corporate bond D with face value F and maturity TD(TD>TW),are issued at time to by the listed firm.A third party issues the traded option with strike priceχ0 and maturity TC(TC
Most people paid much attention on the Warrant pricing or studing the effects of issuing Warrants on stocks.Our results are differential from the previous work and are very significant for both investors and listed firms.
For European and American options, Put-call symmetry is very important in capital markets, which can provide some useful investment information for investors. In 1969, Stoll[112] first proposed the relationship between the values and optimal exercise boundaries of the European call and put option. Merton[83], Castelli[22], Detemple[29] and so on did lots of research about the symmetry. In the second part of Chapter two, we use PDE method to study the symmetry between the the values and the optimal exercise boundaries of American capped call and floored put option. Our results agree with the research by the method of stochastic analysis. The main results we obtained are as follows:
If the strike price of American capped call option is the same as the strike price of American floored put option, we have
Theorem 0.3 Let Vc(S,t;r,q,L(t)) and S= Sc(t;r,q,L(t)) are the value and the op-timal exercise boundary of the American call option with cap L(t), where L(t)> K for any t∈[0, T]. And let Vp(S,t;r,q,m(t)) and S= Sp(t;r,q,m(t)) are the value and the optimal exercise boundary of the American put option with floor m(t), where 0< m(t)< K for any t∈[0,T].If /the same underlying asset pays the divi-dend continuously, the two options own the same maturity T, the strike price K and L(t)m(t)= K2. Then If the strike price of American capped call option is not equal to the strike price of American floored put option, we have
Theorem 0.4 Under the assumptions of Theorem.0.3, and the two options own the same maturity T, and L(t)m(t)= KcKP. where Kc, Kp are respectively the strike prices of American capped call option and American floored put option. Then
Corollary 0.5 Under the assumptions of Theorem 0.3, and L(t)m(t)= S2. Then The results of Corollary 0.5 are consistent with Corollary 9 of Detemple [29].
The classical Markowitz model proposes the porfolio selcetion problem which max-imizes the expectation under given variance or minimizes the variance under given expectation in terms of the rate of return of asset. And Markowitz model is on the hypothesis that the rate of asset return follows the normal distribution and the capital market is frictionless and without transaction costs. In reality, it is difficult to measure the rate of asset return. However, the price of asset can be got easily from the market and the various asset follows the different law because of the characters of asset, for example some assets follow the model of jump-diffusion, and another assets follow the model of mean-reverting. And the investment model in terms of the rate of asset re-turn have not these merits. Therefore, in Chapter three, we use the stochastic dynamic programming method to study the mean-variance strategy model in terms of the price of asset. And we get the analytic solution without transaction costs and study the effects of the exogenous variables such as transaction costs, expected return rate and volatility on the investment strategy. By the investment model in terms of the price of asset, we can learn the according change of investment strategy with the change of investment environment. The main results we obtained are as follows:
We suppose a capital market with n risky assets and 1 riskless asset. The return of riskless asset is equal to one and the price of the risky assets follows the stochastic process S= (St)t=0,...,T= (S1t,…,Snt)t=0,…,T,T= 2. The investment can be permitted during the time interval of [0,2] and be allowed to short sell the risky assets. The .transaction costs are proportional to the volume of transaction. Then
Theorem 0.6 The mean-variance portfolio selection model in two srage is as follows: whereχt is the wealth of the investor at the beginning of the t-th period,St=(St1,St2,…,Stn)' denotes the price of risky assets at time t,ξ≥0,0≤η<1 are constats and are respectively the proportions of transaction costs when the investor buy or sell. share of asset,and lt=(lt1,lt2,…,lnt)'and mt=(mt1,mt2,…,mtn)'are the number of asset the investor buys or sells at time t,σ>0 is a preselected risk level. And Pt=St+1-(1+ξ)St,Qt=(1-η)St-St+1.Then,whenξ=η=0,the optimal investment strategy is as follows: whereΛ11,Λ12,Γ11,Γ12,L1,L2,L3,M1,M2,M3 and a,b,c,d,e are given by mathematical deriva-tion.
At last,we show the investment strategy analysis when the transaction costs,expected return rate and volatility change.By some examples,we can get the following results:
(1) The transaction costs affect the trading frequency but not the trading volume.
(2) Ifμ1≈μ2，σ1<σ2, whereμi,σi,i= 1.2 are respectively the expected return rate and the volatility of asset i. Then the investor prefer the asset 1 to the asset 2. With the increase ofμ2, the investor come to prefer the asset 2.
(3) Ifμ2<μ2,σ1≈σ2. Then the investor prefer the asset 2 to the asset 1. With the increase ofσ2, the investor come to prefer the asset 1.
The safety-first investment model use another idear to study the portfolio selection problem and was first proposed by Roy[103] in 1952. The objective of the investor is to minimize the probability that the teminal wealth is below a preselected threshold, which is different from the mean-variance model. Many people study the safety-first investment model in terms.of the rate of return of asset, and it is similar to the mean-variance model. In Chapter four, we use the method and assumptions of the Chapter three to construct the mathematical model of safety-first with the assumption that the law of the price of asset is given. The main results we obtained are as follows:
Theorem 0.7 Under the assumptions of Chapter three, the safety-first portfolio selec-tion model in two stage is as follows: whereμdenotes the preselected threshold. Then, whenξ=η= 0, the optimal invest-ment strategy is as follows: where Λ11,Λ12,Γ11,Γ12, L1, L2, L3, M1, M2, M3 and a, b, c, d, e are given by mathematical deriva-tion.
At last, we show the investment strategy analysis when the transaction costs, expected return rate and volatility change. By some examples, we can get the following results:
(1) The transaction costs affect the trading frequency but not the trading volume.
(2)Ifμ1≈μ2,σ1＜σ2, whereμi,σi,i= 1,2 are respectively the expected return rate and the volatility of asset i. Then the investor prefer the asset 1 to the asset 2. With the increase ofμ2, the investor come to prefer the asset 2.
(3) Ifμ2＜μ2,σ1≈σ2. Then the investor prefer the asset 2 to the asset 1. With the increase ofσ2, the investor come to prefer the asset 1.
(4) If the investment environment changes, the effections of different investment strategy(mean-variance investment or safety-first investment strategy) on the investor are varying.
引文
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