The Differential Algorithm for American Put Option with Transaction Costs under CEV Model

2.1 Introduction to the CEV model

Constant Elasticity of Variance model, which referred to as CEV model, means that elasticity of variance is constant. It is a natural extension of Geometric Brownian Motion[12].

Let σ(S(t)) be variance of stock price S(t), we may have S(t)dσ(S(t))σ(S(t))d(S(t))=α, where α is a constant which denotes elastic factor. Thus, σ(S(t)) = σ[S(t)]^α (σ is positive).

Under CEV process the volatility of stock price takes the form

where 0 ⩽ α ⩽ 1, µ is the drift rate, σ² is the variance of stock price volatility and B(t) deotes Brownian Motion.

When α = 1, volatility of stock price satisfies dS(t) = µSdt + σSdB(t), the Geometric Brownian Motion. When α = 1/2 and α = 0, it satisfies dS(t)=μSdt+σS12dB(t) (Square Root model) and dS(t) = µSdt + σdB(t)(Absolute model) respectively.

The differential equation of the price of put option with transaction costs under the CEV process takes the form

Equation (1) is a second-order nonlinear parabolic partial differential equation.

2.2 Finite difference method for numerical solution of the model

With stock price S on the horizontal axis and time t on the vertical, we segment maximum of stock price S_max and the contract period (from the current zero moment to the expiration date T). That is, the gridding method is used to the solution area of stock price Ω = {0 ⩽ S(t) ⩽ S_max, 0 ⩽ t ⩽ T}[13].

Assuming that there are M + 1 points in time (0, ∆T, 2∆T, · · ·, T, where ΔT=TM) and N + 1 points in stock price (0, ∆S, 2∆S, · · ·, S_max, where ΔS=SmaxN). The constructed grid is as shown in Figure 1.

Figure 1

The option price solving diagram

The point (i, j) corresponds the time point j∆T and stock price i∆S, and V_ij denotes the option price at point (i, j). For points inside the grid, there are

Substituting (2), (3) and (4) into equation (1) (note that S = i∆S) yields

where

(5) denotes the Crank-Nicolson form of numerical solution of option pricing model with transaction costs. We can rewrite (5) it in matrix form as follows:

Since the cash flow of American put option is max{X – S(T), 0} at time T, we can easily find

Combining (6) with (5) yields the solving equation boundary with initial conditions. That is,

where i = 1, 2, · · · , N, j = 1, 2, · · · , M.

Since the option price at time M∆T can be solved from (6), substituting this result into (7) at time (M – 1)∆T yields

We can solve V_i,M–1, i = 1, 2, ···, N – 1 from (7) and (8). Then comparing V_i,M–1 with X – i∆S. If V_i,M–1 < X – i∆S, the best option is to implement option at time (M – 1)∆T. This implies V_i,M–1 equals X – i∆S. According to the same method, the American option price V_i,0, i = 1, 2, · · ·, N – 1 can be solved. In all of these values, there is an option price is required. In general, it takes a lot of time to use time steps and space steps to get reasonable estimate of the option price. In order to make the values of M and N small as far as possible, we will use the controlling variable technology. In particular, it is to use finite difference method to calculate the American put option price V_A and the corresponding price of European put option V_E, and also use the Black-Scholes formula to calculate the price of European put option V_BS[14]. Finally, American put option price is estimated as V_A + V_BS – V_E using the controlling variable technology.

3.1 The selection of data

As of Jan. 31, 2013, Hong Kong Exchanges and Clearing Limited (HKEx) launched 65 Hong Kong stock options. In this paper we select 1 month, 2 months and 5 months of American stock option (CKH) issued by Cheung Kong (Holdings) Limited as the research object. Its underlying stock is Cheung Kong Holdings (00001). The data used is from the website of Hong Kong Exchanges and Clearing Limited.

3.2 The estimate of volatility

Choosing a total of 247 trading days’ closing price data of Cheung Kong Holdings (00001) from Jan. 1, 2012 to Dec. 31, 2012 as estimation samples, estimate the historical volatility of stock returns of the contract period.

To calculate the volatility of stock price, set

where S is the daily closing price, i = 1, 2, · · ·, n, σ_E is the standard deviation of stock returns within each contract period and τ denotes the number of sample days of contract period. The calculation results obtained by MATLAB language programming are shown in Table 1.

3.3 The estimate of risk-free interest rate

Risk-free interest rate refers to the an ideal investment gains obtained from the capital invested in an item without any risk. In Hong Kong, the risk-free interest rate is usually refers to HIBOR (Hong Kong interbank offered rate). As the duration of the American option of this paper is 1 month, 2 months and 5 months respectively, the corresponding HIBOR rates for 1 month, 2 months and 5 months on January, 31, 2013 as shown in Table 2 are selected.

3.4 Example analyses

We take the CKH option on January 2, 2013 for example. Its contract period is 1 month, and the price is 150 (HKD). When calculating the theoretical price of HK stock option, the stock price is the closing price of the underlying stock on the trading day. By comparison, take the maximum S_max = 150(HKD), historical volatility σ_E = 0.068415, the risk free rate r = 0.22786%, the transaction cost k = 0.3% and contract period 1/12, 2/12 and 5/12 respectively.

Using the above data, three values of α respectively corresponding to the lognormal model (α = 1), square root model (α = 0.5) and absolute model (α = 1) represent their impact on the option price. Table 3 shows the resulting option price on January 2, 2013 using different steps M and N by MATLAB language programming.

Thus, the estimate of Hong Kong option price on January 2, 2013 is obtained. We can see that as time step length and price step length decreases, and stock price option prices gradually convergence and tend to be stable. Actually by the analysis principle of numerical solution of partial differential equation, although it takes a lot of time and calculation to solve a linear equations with a big order in each layer of the finite difference method, format is very stable. When α = 1/2, M = 200, N = 300, the convergence effect of the option price is very good. Therefore, in this paper we select α = 1/2, M = 200, N = 300 as model parameters.

By modifying parameters in M file that has been written in MATLAB software, we deal with financial data of the Cheung Kong Holdings to get the estimated option price from Jan. 1, 2013 to Jan. 15, 2013 (10 days). The results are shown in Table 4.

To analyse the error between theoretical and actual prices of CKH option, we take two metrics: average relative percentage error (RPE) and average absolute percentage error (ARPE). The two measures can be used to measure the deviation of model, but the former is more focused on measuring systematic bias, judging model of pricing is overvalued or undervalued, while the latter can measure either the pricing deviation or pricing efficiency.

Combined with the result of Table 4, we calculate the average relative percentage error and average absolute percentage error according to the different classification of CKH expiration period. Because the two indicators of the calculation results are exactly the same, the only lists the average relative percentage error are shown in Table 5.

From the Table 5, we find that average relative percentage error of CKH option price for a month is 32.735% on Jan. 4, 2013, on which day its share price compared to the previous two days did not fluctuate wildly, but the real option price is low. This lead to a higher theoretical option price determined by the stock price relative than the real option price. If we get rid of the calculation result on Jan. 4, 2013, the RPE value of CKH options for 1 month is 13.201%. Therefore, we can think that the pricing efficiency of the CEV model becomes low by the extension of expiration time.

According to the results of Table 4, we use least squares method to fit theoretical and actual prices of different expiration periods respectively. The comparison results are shown in Figure 2 below.

Figure 2

Theoretical and actual option prices curves of different expiration periods

From the images of the fitting and error results we can see that there are certain errors between theoretical and actual prices calculated by CEV model with transaction cost, to a certain extent, close to and have the same trend of fluctuations. It is worth to note that the theoretical and actual prices are close to a certain extent, and have the same trend of fluctuations. It is clear that the American option pricing model under the CEV process with transaction costs has the rationality.