Pricing Options and Convertible Bonds Based on an Actuarial Approach

Abstract

This paper discusses the pricing problem of European options and convertible bonds using an actuarial approach. We get the pricing formula of European options, extend the pricing results to the case with continuous dividend, and then derive the call-put parity relation. Furthermore, we get the general expression of convertible bond price. Finally, we conduct a comparative analysis of numerical simulation and make an empirical analysis between the B-S model and the actuarial model using the actual data in the Chinese stock market. The empirical results show that the efficiency of the actuarial model is superior to the B-S model.

1. Introduction

The contingent claim pricing has always been one of the core subjects in the field of financial engineering research. To price contingent claims correctly and scientifically is the basis of financial risk management as well as an indispensable component of modern finance. Black and Scholes [1] derived the well-known Black-Scholes (B-S) formula using arbitrage reasoning and stochastic analysis in their classic paper on option pricing and thus established the option pricing theory. The B-S pricing model has had a huge influence on the financial theory. However, it was based on multiple assumptions, such as a normal distribution for the underlying asset price process, no divided paying for the underlying asset, and the constant risk-free interest rate, which apparently do not fit in with the ever-changing financial market. Later, a number of researchers improved and popularized the B-S model. Merton [2] validated the B-S formula using the arbitrage theory and derived the analytic solution to the continuous time model for optimal consumption and investment decision using dynamic programming methods. Cox et al. [3] proposed the binominal option pricing model. Duffie [4] made further inference for the B-S formula with the conventional option pricing approach. Liu and Zhao [5] develop an efficient lattice approach for option pricing. Ammann et al. [6] propose and empirically investigate a pricing model for convertible bonds based on the enhanced Monte Carlo simulation. Xu [7] uses a lattice approach to pricing the convertible bond asset swap.

Most of the existent research on option pricing was based on martingale measure or numerical simulation theory in the framework of Black-Scholes option pricing approach and is only applicable in a complete financial market, where all contingent claims are capable of being replicated accurately with the asset portfolios available on the existent financial markets; that is to say, the market is in an equilibrium, arbitrage-free, and one and only equivalent martingale measure exists. But the complete market assumption may not be perfectly relevant to the actual investment environment. If the financial market is incomplete, assets cannot be accurately replicated or hedged like in a complete market. A common alternative practice is to seek a family of final wealth derived from self-financing strategies to approach the value of the asset, which naturally entails errors. For example, Follmer and Sondermann [8] proposed the “mean-variance” criterion to measure the error, but the solution process is rather complicated.

Bladt and Rydberg [9] first proposed an actuarial approach to pricing options, which transforms the option pricing into a problem equivalent to determining the fair insurance premium. As no economic assumptions are involved, this approach is valid for incomplete markets as well as for complete markets, and demonstrated that the price derived whereby is consistent with that from the B-S model in the continuous time case. Yan and Liu [10] used the actuarial approach to derive the European option pricing formula where the stock price is assumed to follow the Ornstein-Uhlenbeck (O-U) process, and Zhao and He [11] studied the option pricing model where the stock price is assumed to follow a fractional O-U process in a risk neutral market. The assumption of the stock price following the O-U process avoids the limitation that the stock price tends to change in one direction under the lognormal distribution assumption and weakens the tendency of the stock price rise. But the assumption for the above model that the interest rate is a deterministic function of time cannot satisfy the requirement of the actual conditions of the market. A number of empirical lines of evidence show that, in real financial markets, the interest rate has the property of mean-reversion, the volatility of the long-term interest rate is less than that of the short-term interest rate, and the volatility is greater when the interest rate is relatively higher [12, 13]. Liu et al. [14] studied the reload stock option pricing under the assumption of the interest rate following the Hull-White model with martingale pricing method. The stochastic interest rate model assumes that the interest rate converges with the time at a certain mean reversion level.

The content of this paper is arranged as follows. In Section 2, we make some basic assumption for the financial market where the stock prices are driven by O-U process and the interest rates are driven by Hull-White model. In Section 3, we consider the models in the continuous time and apply the actuarial approach to price the European option and the convertible bond. To show the role that the actuarial approach and the stochastic interest rates play, we conduct a comparative analysis of numerical simulation in Section 4. As in Section 5, we make an empirical analysis between the B-S model and the actuarial model, using the actual data in the Chinese stock market. Section 6 concludes the paper.

2. Basic Assumption

Suppose that the financial market is frictionless and continuous, and there are two assets. The risky asset is the stock and the risk-free asset is the bond. A complete probability space(Ω,𝐹,{𝐹_𝑡}_𝑡≥0,𝑃)describes the financial market where the filtration satisfies the usual conditions.

Assume that the stock price  𝑆(𝑡)follows the Ornstein-Uhlenback process as follows: 𝑑𝑆(𝑡)=𝜇(𝑡)−𝛼ln𝑆(𝑡)𝑆(𝑡)𝑑𝑡+𝜎_𝑠(𝑡)𝑆(𝑡)𝑑𝐵(𝑡),𝑆(0)=𝑆, (1) where  𝑆>0 and  𝜎_𝑆(𝑡)is the volatility of the stock price.𝛼is a constant and  𝜇(𝑡)and  𝜎_𝑆(𝑡)  are some determinate functions of  𝑡.

The short-term interest rate𝑟(𝑡)follows the Hull-White model as follows: 𝑑𝑟(𝑡)=(𝑎(𝑡)−𝑏(𝑡)𝑟(𝑡))𝑑𝑡+𝜎_𝑟(𝑡)𝑑𝑊(𝑡),𝑟(0)=𝑟, (2) where the parameters  𝑎(𝑡),  𝑏(𝑡),  and  𝜎_𝑟(𝑡)  are some determinate functions of time 𝑡. The Hull-White model is the mean-reversion model, where the parameter𝑎(𝑡)is the long-term average level, and  𝑏(𝑡)is the average reversion rate of interest rates. When the parameters𝑎(𝑡)  and  𝑏(𝑡)  are constant, the Hull-White model (2) becomes the Vasicěk model. The stochastic processes{𝐵(𝑡)∶𝑡≥0}  and  {𝑊(𝑡)∶𝑡≥0}are two standard Brownian Motions in the defined probability space(Ω,𝐹,{𝐹_𝑡}_𝑡≥0,𝑃), and their correlation coefficient is supposed to be𝜌.

Now, we give two definitions of the actuarial approach.

Definition 1 (see [9]). The expected yield rate of the stock price process{𝑆(𝑡),0≤𝑡≤𝑇}in the time interval[0,𝑇]is defined by ∫^𝑇₀𝛽(𝑡)𝑑𝑡which satisfies the following equation: 𝑒^{∫𝑇0𝛽(𝑡)𝑑𝑡}=𝐸[𝑆(𝑇)]𝑆. (3)

Definition 2 (see [9]). Suppose the expiration date of the European option is 𝑇 and the strike price is 𝐾. In the actuarial approach, the stock price is discounted by the expected yield rate defined in (3), and the strike price is discounted by the riskless interest rate on maturity. The European option value is defined as the expectation of the difference which is between the two discount values on the actual probability measure as the option is exercised. The sufficient and necessary condition for exercising the European call option on 𝑇 is exp−^𝑇₀𝛽(𝑡)𝑑𝑡𝑆(𝑇)>exp−^𝑇₀𝑟(𝑡)𝑑𝑡𝐾, (4) and the similar condition to the European put option is exp−^𝑇₀𝛽(𝑡)𝑑𝑡𝑆(𝑇)<exp−^𝑇₀𝑟(𝑡)𝑑𝑡𝐾. (5) Let𝐶(𝐾,𝑇)denote the call value and𝑃(𝐾,𝑇)denote the put value at the time 0. Then, in the actuarial approach, the two options values are defined as follows, respectively: 𝐶(𝐾,𝑇)=𝐸exp−^𝑇₀𝛽(𝑡)𝑆(𝑇)−𝑑𝑡−exp^𝑇₀𝑟(𝑡)𝑑𝑡𝐾×1_{exp{−∫^𝑇_{0𝛽(𝑡)𝑑𝑡}𝑆(𝑇)>exp{−∫}^𝑇_{0𝑟(𝑡)𝑑𝑡}𝐾}},𝑃(𝐾,𝑇)=𝐸exp−^𝑇₀𝑟(𝑡)𝐾−𝑑𝑡−exp^𝑇₀𝛽(𝑡)𝑑𝑡𝑆(𝑇)×1_{exp{−∫^𝑇_{0𝛽(𝑡)𝑑𝑡}𝑆(𝑇)<exp{−∫}^𝑇_{0𝑟(𝑡)𝑑𝑡}𝐾}}. (6)

3. Pricing Formula

In this section, we firstly consider the pricing problem of the European options in the financial market models described above. Then we extend the pricing result to the options whose underlying asset has continuous dividend and derive the call-put parity relation using the actuarial approach. Furthermore, we get the general expression of convertible bond price. At first, we offer two important lemmas.

Lemma 3 (see [15]). If the random variables 𝑊₁ and 𝑊₂ are both standard normally distributed with mean 0 and variance 1 notated as 𝑁(0,1), and their covariance is  Cov(𝑊₁,𝑊₂)=𝜌, then for any real numbers𝑎, 𝑏, 𝑐, 𝑑, and  𝑘 one has the following equation: 𝐸𝑒^{𝑐𝑊1+𝑑𝑊2}1_{{𝑎𝑊1+𝑏𝑊2≥𝑘}}=𝑒^{(1/2)(𝑐2+𝑑2+2𝜌𝑐𝑑)}𝑁𝑎𝑐+𝑏𝑑+𝜌(𝑎𝑑+𝑏𝑐)−𝑘𝑎²+𝑏².+2𝜌𝑎𝑏 (7)

Lemma 4. If the stock price S(t) is driven by the O-U process (1), then one gets the following equations: 𝑆(𝑡)=𝑆^{𝑒−𝛼𝑡}exp𝑒^{−𝛼𝑡𝑡}₀𝜇(𝑡)−12𝜎²_𝑠(𝑢)𝑒^𝛼𝑢𝑑𝑢+𝑒^{−𝛼𝑡𝑡}₀𝜎_𝑠(𝑢)𝑒^𝛼𝑢(𝑢,𝐸[𝑠(𝑡𝑑𝐵))]=𝑆^{𝑒−𝛼𝑡}exp𝑒^{−𝛼𝑡𝑡}₀𝜇(𝑡)−12𝜎²_𝑠(𝑢)𝑒^𝛼𝑢+1𝑑𝑢2𝑒^{−2𝛼𝑡𝑡}₀𝜎²_𝑠(𝑢)𝑒^2𝛼𝑢𝑑𝐵(𝑢). (8)

Proof. With theItŏformula, we get (𝑡)=1𝑑ln𝑆𝑆(𝑡)(𝑡)−1𝑑𝑆2𝑆²(𝑡)(𝑡)=𝜇(𝑡)(𝑡)−1𝑑𝑆−𝛼ln𝑆2𝜎²_𝑠(𝑡)𝑑𝑡+𝜎_𝑠(𝑡)(𝑡),𝑑(𝑡)𝑒𝑑𝐵ln𝑆^𝛼𝑡=𝜇(𝑡)−12𝜎²_𝑠(𝑡)𝑒^𝛼𝑡𝑑𝑡+𝜎_𝑠(𝑡)𝑒^𝛼𝑡𝑑𝐵(𝑡), (9) thus, we get ln𝑆(𝑡)𝑒^𝛼𝑡=ln𝑆+^𝑡₀𝜇(𝑢)−12𝜎²_𝑠(𝑢)𝑒^𝛼𝑢+𝑑𝑢_𝑡0𝜎_𝑠(𝑢)𝑒^𝛼𝑢𝑑𝐵(𝑢). (10) Then ln𝑆(𝑡)=𝑒^−𝛼𝑡ln𝑆+𝑒^{−𝛼𝑡𝑡}₀𝜇(𝑢)−12𝜎²_𝑠(𝑢)𝑒^𝛼𝑢+𝑑𝑢_𝑡0𝜎_𝑠(𝑢)𝑒^𝛼𝑢𝑑𝐵(𝑢), (11) so we have 𝑆(𝑡)=𝑆^{𝑒−𝛼𝑡}exp𝑒^{−𝛼𝑡𝑡}₀𝜇(𝑢)−12𝜎²_𝑠(𝑢)𝑒^𝛼𝑢𝑑𝑢+𝑒^{−𝛼𝑡𝑡}₀𝜎_𝑠(𝑢)𝑒^𝛼𝑢𝑑𝐵(𝑢). (12) Taking expectations on both sides of the above equation, then we get 𝐸[𝑆(𝑡)]=𝑆^{𝑒−𝛼𝑡}exp𝑒^{−𝛼𝑡𝑡}₀𝜇(𝑢)−12𝜎²_𝑠(𝑢)𝑒^𝛼𝑢+1𝑑𝑢2𝑒^{−2𝛼𝑡𝑡}₀𝜎²_𝑠(𝑢)𝑒^2𝛼𝑢𝑑𝐵(𝑢). (13)

3.1. European Options Pricing

We consider the European options, call and put, whose underlying assets are stocks. Suppose that the exercise date of the options is 𝑇 and the strike price is 𝐾.

Theorem 5. Assume that the short-term interest rate is described by the Hull-White model and the stock process{𝑆(𝑡),𝑡≥0}follows the O-U process, then, at the time 0, the pricing formulas of the call and put are as follows, respectively: 𝐶(𝐾,𝑇)=𝑆𝑁𝑑₁1−𝐾exp2𝜎²_𝑋−𝐺(0,𝑇)𝑁𝑑₂,𝑃()1𝐾,𝑇=𝐾exp2𝜎²_𝑋−𝐺(0,𝑇)𝑁−𝑑₂−𝑆𝑁−𝑑₁, (14) where 𝑛(𝑡)=^𝑡₀𝑏(𝑡)𝑑𝑡,𝑚(𝑡,𝑇)=^𝑇₀𝑒^{𝑛(𝑡)−𝑛(𝑠)}𝐺()()+𝑑𝑠,0,𝑇=𝑟𝑚0,𝑇^𝑇₀𝑎(𝑡)𝑚(𝑡,𝑇)𝑑𝑡,𝑋=^𝑇₀𝜎_𝑟(𝑡)𝑚(𝑡,𝑇)𝑑𝑊(𝑡),𝑌=𝑒^{−𝛼𝑇𝑇}₀𝜎_𝑠(𝑡)𝑒^𝛼𝑡(𝑡),𝜎𝑑𝐵_2𝑋=^𝑇₀𝜎²_𝑟(𝑡)𝑚²()𝜎𝑡,𝑇𝑑𝑡,_2𝑌=𝑒^{−2𝛼𝑇𝑇}₀𝜎²_𝑠(𝑡)𝑒^2𝛼𝑡𝑑𝑑𝑡,₁=ln(𝑆/𝐾)+𝐺(0,𝑇)+(1/2)𝜎²_𝑌+𝜌𝜎_𝑋𝜎_𝑌𝜎²_𝑋+𝜎²_𝑌+2𝜌𝜎_𝑋𝜎_𝑌,𝑑₂=ln(𝑆/𝐾)+𝐺(0,𝑇)−𝜎²_𝑋−(1/2)𝜎²_𝑌+𝜌𝜎_𝑋𝜎_𝑌𝜎²_𝑋+𝜎²_𝑌+2𝜌𝜎_𝑋𝜎_𝑌. (15)

Proof. By Definition 2, we get 𝐶(𝐾,𝑇)=𝐸𝑆(𝑇)exp−^𝑇₀𝛽(𝑡)𝑑𝑡⋅1_{{𝑆(𝑇)exp{−∫}^𝑇_{0𝛽(𝑡)𝑑𝑡}>𝐾exp{−∫}^𝑇_{0𝑟(𝑡)𝑑𝑡}}}−𝐸𝐾exp−^𝑇₀𝑟(𝑡)𝑑𝑡⋅1_{{𝑆(𝑇)exp{−∫}^𝑇_{0𝛽(𝑡)𝑑𝑡}>𝐾exp{−∫}^𝑇_{0𝑟(𝑡)𝑑𝑡}}}=𝐼₁−𝐼₂. (16) In the actuarial approach, the expected yield rate of the stock price satisfies (3), and, by Lemma 4, we have ^𝑇₀𝛽(𝑡)𝐸[𝑆(𝑇𝑑𝑡=ln)]𝑆=𝑒^−𝛼𝑇−1ln𝑆+𝑒^{−𝛼𝑇𝑇}₀𝜇(𝑡)−12𝜎²_𝑠(𝑡)𝑒^𝛼𝑡+1𝑑𝑡2𝑒^{−2𝛼𝑇𝑇}₀𝜎²_𝑠(𝑡)𝑒^2𝛼𝑡𝑑𝑡. (17) So, 𝑒^{−∫𝑇0𝛽(𝑡)𝑑𝑡}=exp1−𝑒^−𝛼𝑇ln𝑆−𝑒^−𝛼𝑇×^𝑇₀𝜇(𝑡)−12𝜎²_𝑠(𝑡)𝑒^𝛼𝑡−1𝑑𝑡2𝑒^{−2𝛼𝑇𝑇}₀𝜎_𝑠(𝑡)𝑒^2𝛼𝑡,𝑒𝑑𝑡_{−∫𝑇0𝛽(𝑡)𝑑𝑡}𝑆(𝑇)1=expln𝑆−2𝑒^{−2𝛼𝑇𝑇}₀𝜎²_𝑠(𝑡)𝑒^2𝛼𝑡𝑑𝑡+𝑒^{−𝛼𝑇𝑇}₀𝜎_𝑠(𝑡)𝑒^𝛼𝑡𝑑𝐵(𝑡). (18) ByItŏformula and [14], the interest rate model (2) satisfies ^𝑇₀𝑟(𝑡)𝑑𝑡=𝐺(0,𝑇)+^𝑇₀𝜎_𝑟(𝑡)𝑚(𝑡,𝑇)𝑑𝑊(𝑡). (19) The exercising condition𝑆(𝑇)exp{−∫^𝑇₀𝛽(𝑡)𝑑𝑡}>𝐾exp{−∫^𝑇₀𝑟(𝑡)𝑑𝑡}is equal to 1expln𝑆−2𝑒^{−2𝛼𝑇𝑇}₀𝜎²_𝑠(𝑡)𝑒^2𝛼𝑡𝑑𝑡+𝑒^{−𝛼𝑇𝑇}₀𝜎_𝑠(𝑡)𝑒^𝛼𝑡(𝑡𝐺()+𝑑𝐵)>𝐾×exp0,𝑇^𝑇₀𝜎_𝑟(𝑡)𝑚()(𝑡,⟺𝑡,𝑇𝑑𝑊)_𝑇0𝜎_𝑟(𝑡)𝑚(𝑡,𝑇)𝑑𝑊(𝑡)+𝑒^{−𝛼𝑇𝑇}₀𝜎_𝑠(𝑡)𝑒^𝛼𝑡(𝑡)𝐾𝑑𝐵>ln𝑆()+1−𝐺0,𝑇2𝑒^−2𝛼𝑇×^𝑇₀𝜎²_𝑠(𝑡)𝑒^2𝛼𝑡𝐾𝑑𝑡,⟺𝑋+𝑌>ln𝑆()+1−𝐺0,𝑇2𝜎²_𝑌. (20) Because{𝐵(𝑡)∶𝑡≥0}and{𝑊(𝑡)∶𝑡≥0}are two standard Brownian Motions in the probability space(Ω,ℱ,{ℱ_𝑡}_𝑡≥0,𝑃), so the defined normal random variables 𝑋 and 𝑌 are independent from ℱ_𝑡 and satisfy the following equations: 𝐸^𝑄[𝑋]=𝐸^𝑄[𝑌]=0,𝜎²_𝑋=^𝑇₀𝜎²_𝑟(𝑡)𝑚²()𝜎𝑡,𝑇𝑑𝑡,_2𝑌=𝑒^{−2𝛼𝑇𝑇}₀𝜎²_𝑠(𝑡)𝑒^2𝛼𝑡𝑑𝑡. (21) So, by Lemma 3, we get 𝐼₁−1=𝐸𝑆exp2𝑒^{−2𝛼𝑇𝑇}₀𝜎²_𝑠(𝑡)𝑒^2𝛼𝑡𝑑𝑡+𝑌⋅1_{{𝑋+𝑌>ln(𝐾/𝑆)−𝐺(0,𝑇)+(1/2)𝜎}²_𝑌}−1=𝑆exp2𝑒^{−2𝛼𝑇𝑇}₀𝜎²_𝑠(𝑡)𝑒^2𝛼𝑡𝑒𝑑𝑡𝐸⋅^𝑌⋅1_{{𝑋+𝑌>ln(𝐾/𝑆)−𝐺(0,𝑇)+(1/2)𝜎}²_𝑌}=𝑆𝑁𝑑₁,𝐼₂=𝐾𝑒^{−𝐺(0,𝑇)}⋅𝐸𝑒^−𝑋⋅1_{{𝑋+𝑌>ln(𝐾/𝑆)−𝐺(0,𝑇)+(1/2)𝜎}²_𝑌}1=𝐾exp2𝜎²_𝑋−𝐺(0,𝑇)𝑁𝑑₂. (22) Thus, the value of the call option is 𝐶(𝐾,𝑇)=𝑆𝑁𝑑₁1−𝐾exp2𝜎²_𝑋−𝐺(0,𝑇)𝑁𝑑₂. (23) Similarly, we get the value of put option as follows: 𝑃()1𝐾,𝑇=𝐾exp2𝜎²_𝑋−𝐺(0,𝑇)𝑁−𝑑₂−𝑆𝑁−𝑑₁. (24)

Furthermore, we deduce the following pricing inferences.

Inference  1. Suppose that the stock process{𝑆(𝑡),𝑡≥0}follows the O-U process (1), and the interest rate is described by the Hull-White model (2); then the call-put parity of the European options in the actuarial approach is 𝐶()1𝐾,𝑇+𝐾exp2𝜎²_𝑋−𝐺(0,𝑇)=𝑃(𝐾,𝑇)+𝑆. (25)

Inference  2. Under the market models (1) and (2), the underlying stock has continuous dividend yield marked by𝑞(𝑡). Then, respectively, the pricing formulas of the call and put option in the actuarial approach at time 0 are 𝐶^′(𝐾,𝑇)=𝑆exp−^𝑇₀𝑞(𝑡)𝑑𝑡𝑁𝑑^′₁1−𝐾exp2𝜎²_𝑋−𝐺(0,𝑇)𝑁𝑑^′₂,𝑃^′()1𝐾,𝑇=𝐾exp2𝜎²_𝑋−𝐺(0,𝑇)𝑁−𝑑^′₂−𝑆exp−^𝑇₀𝑞(𝑡)𝑑𝑡𝑁−𝑑^′₁, (26) where 𝑑^′₁=ln(𝑆/𝐾)+𝐺(0,𝑇)−∫^𝑇₀𝑞(𝑡)𝑑𝑡+(1/2)𝜎²_𝑌+𝜌𝜎_𝑋𝜎_𝑌𝜎²_𝑋+𝜎²_𝑌+2𝜌𝜎_𝑋𝜎_𝑌,𝑑^′₂=𝑆ln𝐾+𝐺(0,𝑇)−^𝑇₀𝑞(𝑡)𝑑𝑡−𝜎²_𝑋−12𝜎²_𝑌+𝜌𝜎_𝑋𝜎_𝑌×𝜎²_𝑋+𝜎²_𝑌+2𝜌𝜎_𝑋𝜎_𝑌⁻¹. (27)

3.2. Convertible Bond Pricing

We consider the pricing problem of convertible bond with the call provision. It is based on the actuarial approach where the stock price is driven by O-U process and the interest rate follows Hull-White model. For the callable convertible bonds, based on the classical theory of optimal investment strategy, Brennan and Schwartz [16] and Ingersoll [17], the investors will not exercise the right of conversion while the issuers will immediately exercise the right of redemption when the stock prices reach the callable price for the first time [18]. Let 𝑇 be the maturity date of the convertible bond, and let 𝜏^∗be the first time when the stock price goes up to the callable trigger price; then the optimal exercise time𝜏=𝜏^∗; otherwise,𝜏=𝑇.

Theorem 6. Suppose that the face value of the convertible bond is𝐵_𝐹, the conversion price is𝐾₁, and the callable trigger price is𝐾₂; then the general expression of convertible bond price𝐻is given by 𝐻=E𝑒^{−∫𝜏∗0𝛽(𝑡)𝑑𝑡}×𝐾₂𝐵_𝐹𝐾₁+𝑐𝜏^∗×1_{{𝜏∗≤𝑇}}+𝐵_𝐹𝐾₁𝑆𝑁𝑑₃(𝑇)1+𝑐exp2𝜎²_𝑋−𝐺(0,𝑇)𝑁𝑑₄×𝑃𝜏^∗+𝐵>𝑇_𝐹(𝑇)1+𝑐⋅exp2𝜎²_𝑋−𝐺(0,𝑇)𝑁−𝑑₄×𝑃𝜏^∗>𝑇, (28) where 𝑑₃=ln𝑆/𝐾₁+𝐺(0,𝑇)+(1/2)𝜎²_𝑌+𝜌𝜎_𝑋𝜎_𝑌𝜎²_𝑋+𝜎²_𝑌+2𝜌𝜎_𝑋𝜎_𝑌,𝑑₄=ln𝑆/𝐾₁+𝐺(0,𝑇)−𝜎²_𝑋−(1/2)𝜎²_𝑌+𝜌𝜎_𝑋𝜎_𝑌𝜎²_𝑋+𝜎²_𝑌+2𝜌𝜎_𝑋𝜎_𝑌. (29)

Proof. As𝜏^∗  is the first time when the stock price goes up to the callable trigger price, so𝜏^∗=inf{𝑡,𝑆_𝑡≥𝐾₂}. Under the optimal investment strategy, the optimal execution time𝜏is 𝜏𝜏=^∗,triggerbeforetheexpiry,𝑇,notriggerbeforetheexpiry. (30) Let𝑐(𝑡)express the time 𝑡 present value of the interest income before𝑡;then the theoretical value of the convertible bond at the time𝜏defined by𝑔(𝑟,𝑆,𝜏)is given by 𝑔()=𝐵𝑟,𝑆,𝜏^𝐹𝐾₁𝐾₂+𝑐𝜏^∗,𝜏^∗𝐵≤𝑇,_𝐹𝐾₁𝑆_𝑇+𝑐(𝑇),𝜏^∗>𝑇,𝑆_𝑇>𝐾₁,𝐵_𝐹+𝑐(𝑇),𝜏^∗>𝑇,𝑆_𝑇≤𝐾₁. (31) Then the actuarial price of convertible bond𝐻(𝑟,𝑆,t)can be expressed as follows: 𝐻(𝑟,𝑆,𝑡)=𝐸𝑒^{−∫𝜏∗0𝛽(𝑡)𝑑𝑡}×𝐾₂𝐵_𝐹𝐾₁+𝑐𝜏^∗×1_{{𝜏∗≤𝑇}}+𝐸𝑒^{−∫𝑇𝑡𝛽(𝑡)𝑑𝑡}×𝑆_𝑇𝐵_𝐹𝐾₁+𝑐(𝑇)×1_{{𝜏∗>𝑇,exp{−∫}^𝑇_{0𝛽(𝑡)𝑑𝑡}𝑆(𝑇)>exp{−∫}^𝑇_{0𝑟(𝑡)𝑑𝑡}𝐾1}}+𝐸𝑒^{−∫𝑇𝑡𝛽(𝑡)𝑑𝑡}×𝐵_𝐹+𝑐(𝑇)×1_{{𝜏∗>𝑇,exp{−∫}^𝑇_{0𝛽(𝑡)𝑑𝑡}𝑆(𝑇)≤exp{−∫}^𝑇_{0𝑟(𝑡)𝑑𝑡}𝐾1}}, (32) where𝑒^{∫𝜏∗0𝛽(𝑡)𝑑𝑡}=𝐸[𝑆(𝜏^∗)]/𝑆.
From Theorem 5, we get 𝑒^{−∫𝜏∗0𝛽(𝑡)𝑑𝑡}=exp1−𝑒^{−𝛼𝜏∗}ln𝑆−𝑒^{−𝛼𝜏∗}×^𝜏∗₀𝜇(𝑡)−12𝜎²_𝑠(𝑡)𝑒^𝛼𝑡−1𝑑𝑡2𝑒^{−2𝛼𝜏∗𝜏∗}₀𝜎_𝑠(𝑡)𝑒^2𝛼𝑡,𝐻𝑑𝑡₁≜𝐸𝑒^{−∫𝜏∗0𝛽(𝑡)𝑑𝑡}×𝐾₂𝐵_𝐹𝐾₁+𝑐𝜏^∗×1_{{𝜏∗≤𝑇}},𝐻₂≜𝐸𝑒^{−∫𝑇0𝛽(𝑡)𝑑𝑡}×𝑆_𝑇𝐵_𝐹𝐾₁+𝑐(𝑇)×1_{{𝜏∗>𝑇,exp{−∫}^𝑇_{0𝛽(𝑡)𝑑𝑡}𝑆(𝑇)>exp{−∫}^𝑇_{0𝑟(𝑡)𝑑𝑡}𝐾1}}=𝐵_𝐹𝐾₁𝑆𝑁𝑑₃(𝑇)1+𝑐exp2𝜎²_𝑋−𝐺(0,𝑇)𝑁𝑑₄×𝑃𝜏^∗,𝐻>𝑇₃≜𝐸𝑒^{−∫𝑇0𝛽(𝑡)𝑑𝑡}×𝐵_𝐹+𝑐(𝑇)×1_{{𝜏∗>𝑇,exp{−∫}^𝑇_{0𝛽(𝑡)𝑑𝑡}𝑆(𝑇)≤exp{−∫}^𝑇_{0𝑟(𝑡)𝑑𝑡}𝐾1}}=𝐵_𝐹(𝑇1+𝑐)⋅exp2𝜎²_𝑋−𝐺(0,𝑇)𝑁−𝑑₄×𝑃𝜏^∗>𝑇, (33)(34) where 𝑑₃ and 𝑑₄ are given by (29).

Then, the general expression of convertible bond price satisfies (28). So with (28), we can obtain the convertible bond price using the Monte Carlo simulation.

4. Numerical Examples

The general European options mainly take the classic B-S formula results as the reference prices. However, there are usually some differences between the actual transaction prices and the reference prices of the options, and it is difficult to obtain the data of the actual option prices. Therefore, we first consider some specific numerical examples under given parameters to compare the results of the actuarial pricing formula with the B-S formula and to reveal the impact of the stochastic interest rates on pricing results when the stock prices follow O-U process. Here, it should be pointed out that if there is no arbitrage in the market, there must be at least one risk-neutral martingale measure, and the stock price process can be transformed into a martingale. In this case, the O-U model is free of arbitrage, just as the method of B-S model is.

Now, we make numerical examples to the European options. The parameters in the financial market models are selected as follows: 𝐾=60,𝑇=1,𝑎=0.034,𝑏=0.016,𝜎_𝑠=0.395,𝜎_𝑟=0.023,𝛼=0.2,𝑟=0.0245,𝜌=−0.58, (35) and we obtain the option pricing results by Matlab.

As the underlying stock prices follow O-U process and the short-term interest rates are stochastic variable, we compare the European option values generated by actuarial approach with the values generated by the classic B-S model to the European call option and the put option, as Figure 1 shows. Figure 1 shows that there are more differences between the pricing results derived by the two different methods, actuarial approach and B-S model, for the out-the-money options, while the pricing differences between the two methods are smaller for in-the-money options; in particular, the results of deep-in-the-money options are almost the same. The differences are caused by the different methods used, the underlying stock prices of O-U process, and the stochastic short-term interest rates. Moreover, the using of different pricing models should have a larger impact which appears as negative on out-the-money options, as the exercise probabilities to out-the-money options are low, especially to deep-out-the-money options, and their values mainly are the time value. However, the changes of the stock prices have more impact on the prices of deep-in-the-money options, so the pricing differences between the two different methods are much smaller.

(a)

(b)

Figure 1 

Comparison of the options values from actuarial approach under stochastic interest rates and O-U process with those from B-S formula.

Then, we consider the impact of stochastic interest rates on the option pricing. To this end, we compare the pricing results of the European call option derived by the actuarial approach under the constant interest rates model with the results under the stochastic interest rates model. Table 1 shows that the pricing results of the call option tend to increase with the underlying stock prices rise for the two different models. However, if the underlying stock price is fixed, the pricing result of the call under constant stochastic interest rates is lower than the pricing result under stochastic interest rates. And with the stock price increasing, the difference between the two models increases at first and then decreases. The stochastic interest rates increase the uncertainty and may increase the prices of the call option.

Table 1 

Comparison of the results given by the actuarial approach with stochastic interest rates and constant interest rates.

5. Empirical Study

5.1. Data and Parameters

We analyze the differences of the two models, B-S model and actuarial model, using the actual data in the Chinese stock market. However, there are only three warrants that are similar to the call option. The only difference between the warrant and the option is that the option is a standardized contract but the warrant is not. However, it would not affect the results. So we choose the data of Changhong CWB1 which is equivalent to a European call option, as its exercise ratio is 1 : 1 and the underlying stock has not paid dividend during the chosen period; then we need not adjust the prices of the stock and the warrant. The data covers the period from August 19, 2009, when the Changhong CWB1 was listed to May 20, 2010, and the time interval between each observation is one day. This means that we end up with 181 observations, and this is the longest sample available to us. The data are taken from the Shanghai Stock Exchange and are collected through the Dazhihui quote software.

Next, we need to estimate the parameters in the models. Some parameters we can get from public information are as follows: the strike price of the Changhong CWB1 is 𝐾=5.23, the exercise date is 𝑇=2 years, the interval is Δ𝑡=1/250, and the interest rate at 0 time is 𝑟=0.0225 which is the official interest rate for one-year deposits during the period 2009-2010. As for the parameters in the interest rate model, we use the daily data of 7-day repo rates in the Shanghai Stock Exchange from August 19, 2009, to May 20, 2010, and the MLE method to estimate the parameters by Matlab. The result shows that the estimated parameters of the short-term interest rates driven by the Hull-White model in China market are𝑎=0.0322, 𝑏=0.1937, and 𝜎_𝑟=0.0863. In order to estimate the parameters of the underlying stock, we use the historical data of Changhong shares during the same time to compute the results: 𝜎_𝑆=0.6968and𝛼=0.1359by the MLE method. A large number of works in the literature have revealed that the correlation of the stock price and the interest rate is weak negative [19], and we indeed get the parameter𝜌=−0.284during the chosen time.

5.2. Results

Having specified the data and the parameters used in the simulation procedure, we now turn to the empirical results obtained from Matlab. Figure 2 shows the comparison results among the real values, actuarial model prices, and B-S model prices. The main features are that the actuarial prices are lower than the B-S prices mostly and the actual price points are much closer to the points computed by the actuarial model than the B-S model. Moreover, we analyze the goodness of fitting by the correlation coefficients. The results show that the correlation coefficient of the actual prices and the actuarial model prices is 0.9652, and the correlation coefficiency of the actual prices and the B-S model prices is 0.9607 which is lower than the former. It is an explicit evidence that the efficient of the actuarial model is superior to the B-S model. The analysis also shows that the correlation of the actuarial prices and the B-S prices is strong and positive as their correlation coefficient is 0.9996.

Figure 2 

Comparison of the values among the actual data, actuarial model, and B-S model.

6. Conclusions

In the martingale pricing method, the prices of the financial derivative securities (or contingent claims) are obtained by discounting their expectations of future cash flows which are computed with risk neutrality. It is valid for the complete market, because the unique equivalent martingale exists. However, if the market is not arbitrage-free and incomplete, the equivalent martingale measure does not exist or exists but is not unique; then it is hard to price by the martingale pricing method. As no economic assumptions are involved, the actuarial approach is valid for incomplete markets as well as for complete markets and needs not find an equivalent martingale measure for pricing. This paper discusses the pricing problem of the European options and convertible bonds using the actuarial approach. We get the pricing formula of the European options, extend the pricing results to the case with continuous dividend, and then derive the call-put parity relation. Furthermore, we get the general expression of convertible bond price. Finally, we conduct a comparative analysis of numerical simulation and make an empirical analysis between the B-S model and the actuarial model using the actual data in the Chinese stock market. The empirical results show that the efficiency of actuarial model is superior to the B-S model.

Acknowledgments

The authors would like to express gratitude for the support given by the National Natural Science Foundation of China (no. 71201013), the National Natural Science Innovation Research Group of China (no. 71221001), the National Science Fund for Distinguished Young Scholars of China (no. 70825006), and the Humanities and Social Sciences Project of the Ministry of Education of China (no. 12YJC630118).