Currency Options - Put-Call Parity - Research Paper Example

Research Paper Finance and Accounting Put call parity on currency options SECTION A: METHODOLOGY Estimate of Beta and Alpha 2. Test for Null hypothesis 3. Statistical Results of Alpha and Beta 4. Discussion SECTION B: APPLICATION 5. Application of the Put Pall Parity on the Data 6. Test Of Parity 7. Empirical Results 8. Conclusion 1. SECTION A: METHODOLOGY 1.1. Estimate of Beta and Alpha Considering the Canadian option on the currency stock, the payment of dividend is done at a constant rate q. The definition of the early premium uses the variables V (s, t), giving the Equation 1 presented below: V (s, t) = Cam(S, t) – C(S, t) EQ. 1 Where Cam(S, t) represents the value of the Canadian call option, while C(S, t) represents the call option for a different currency In the continuation, the two variables Cam(S, t) and C(S, t) are applied to satisfy the Black-Scholes equation presented by Hoque, Chan, and Manzur (2008) as shown in Equation 2 below: dv / dt +0.5σ^2S^2(d^2*v / dS^2) + bS(dv / dS) = rv EQ. 2 The carry cost is presented as b = r – q The alpha and beta are therefore calculated as: α = 2r / σ^2 β = 2b / σ^2 At least one option-pricing model for the put and the call prices is expected to satisfy the put-call parity for it to be accepted otherwise, it has to be rejected (Lung and Nishikawa, 2005). The put-call parity equation for the foreign exchange options is calculated as shown in equation 3 below: C (t) - P (t) = S (st) - Ke^ ((f-s)) EQ. 3 1.2. Test for Null hypothesis The null hypothesis to be tested states that α = 0 while β = 1. The price of the call option is obtained to be less than the strike price X because of the forward purchase. The value of call option is calculated as follows: C T = Max (S T – X, 0) EQ. 4 The price of the put option is expected to leverage on the forward sale to obtain values less than the strike price X. One unit of the foreign currency in the put option is worth: P T = Max (X – S T, 0) EQ. 5 1.2.1. Calculation of the Put call Parity The parity between the call and the put option is the difference obtained between the call and the put option prices. This is computed as shown below: C (t) - P (t) = S (st) - Ke^ ((f-s)) EQ. 6 It is also expected that the future pay offs are equal, therefore S = K. 1.2.2. Statistical Results of Alpha and Beta The regression results or the model was summarized in table 1 below. Table 1: Regression Results to Test PCP Currency Alpha (α) Beta (β) Auto-Correlation GARCH CAD Before After Before After 0.0003995 0.40 0.001008 1.008 ARMA(0, 1) GARCH(2, 1) Error (0.0007) (0.013) (0.0100) (0.0062) Standard Deviation 0.01 0.05 P - value 0.27 0.5137 T - value 0.5817 0.678 In the test of the null hypothesis (H0: α = 0, β = 1), the numbers enclosed in brackets are the standard errors of the regression. We then multiply the results of alpha (α) by 1,000. Table 1 above shows the values of alpha and beta before and after the multiplication (Poitras, 2002). After the multiplication, the empirical results show that the alpha (α) = 0.4, while beta (β) = 1.008. 1.3. Discussion In essence, the value of β is projected to be 1 and alpha (α) = 0. Considering the values obtained and the errors involved in each case, it is clear that α > 0, but β = 1. The conclusion from the empirical outcome of the regression model is that the null hypothesis (H: β = 1) is accepted. This is further supported by the p – value which measures 0.5137. It is expected that the p value p > = 0.05 at 95% confidence level. The standard deviation in this case is 0.01. This is very low, implying that the observations were close to the mean (Hull, 2005). However, the value of α is greater than 0, hence, the hypothesis (H: α = 0) fails and is rejected. The p - value measures 0.27 showing that the test is statistically significant at 95% confidence level. The standard deviation in this case is 0.05. This is very low, implying that the observations were close to the mean. 2. SECTION B: APPLICATION (Application of the Put Pall Parity on the Data) 2.1. Test Of Parity The parity test is conducted separately with the dataset for the put option and the call option. In each case, the common equation applied is C (t) - P (t) = S (st) – Ke^ (f - s) Where P (t) is the value of the put option at time t C (t) is the value of call at time t S (st) is the value of spot price of the asset F is the strike price R is the present rate of exchange The model is applied in the light of the stock price being less than the strike price. Calculation of the Put Price The stock price is 48 dollars, while the strike price measures 30 dollars with a risk-free rate of interest being 4.96% per year. The provided information is rearranged in the most convenient form as shown below: S (st) = 30 K= 48 r = 0.0496 T = 1.5 C (t) = 0.5 e = 3.9 The Put Call Parity equation is rearranged to calculate P as follows: P (t) = C (t) + K·e–rT – S0 P (t) = 0.5 + 48*3.9^-(0.0496*1.5) – 30 ≈ 13.8777 Calculation of the Call Price The stock price measures 50 Dollars while the strike price measures $65 with an annual risk-free rate of interest being 5%. Having the put option price as 0.8 dollars, the call option price is computed as: S0 = 50 K = 65, r = 0.05 T = 1 p = 0.8 e = 2.88 C = p + S0 – Ke^-rT Inputting numbers: C = 0.8 + 50 – 65*2.88^0.05*1 ≈ 10.852 In consideration of the call values by the end of the period, the variable C (T) and C (t) denote the call values. This leads to the application of the 1 - period analyses by the end of the first period. From C (T), the call option prices vary from 145.61 to 14800. In order to obtain the call price C (t), there has to be a construction of a portfolio to pay the call for the subsequent period. If for any reason the spot rate reduces, then the call price will also drop (Bodurtha & Courtadon, 1995). According to this procedure, we obtain the price of the call option. It can be realized that the discounting factor change from (1 + rd) to (1 + 0.5rd) considering the annual interest rate in which the periods have been reduced to half of a year. In this case, the estimated difference in the exchange rate, μ is obtained to be 15% per year, with a standard deviation, σ, of 1%, with the life of the option being 9 years. The summary of the variables and equations applied are shown in table 2 below: Table 2: Summary of Variables and Equations S 50 Present price X 45 Exercise price r 4 Risk-free Exchange Rate T 9 Time to of option maturity Sigma 5% Volatility (standard Deviation), d1 0.509 (LN(S/X)+(r+0.05*sigma^2)*T)/(sigma*SQRT(T)) d2 0.411 d1-sigma*SQRT(T) N(d1) 0.724 NormSDist(d1) N(d2) 0.621 NormSDist(d2) Call price 8.4 S*N(d1)-K*e^(-r*T)*N(d2) Put price 2.1 call price - S + K*e^(-r*T)   2.31 K*e^(-r*T)*N(-d2) - S*N(-d1) 2.2. Empirical Results From the data provided, the put price is calculated as Put Price Calculation Table 3: Put Price Calculation Expiration Strike Price Code Spot at High Price PT 96000.00 145.34 95854.66 P T = Max (X – S T, 0) P T = 96000.00 – 145.34 = 95,854.66 Call Price Calculation Table 4: Call Price Calculation Expiration Strike Price Code Spot at High Price CT 14800.00 145.61 14654.39 C T = Max (S T – X, 0) Referring to the call option Data set CT = 14800.00 - 145.61 = 14654.39 Calculation of the Put Call parity Put – Call Parity = PT – CT Put Call Parity = 95,854.66 - 14654.39 Put Call Parity = 81200.3 Table 5: Put Call parity and Pay Off Diagram PUT-CALL PARITY Payoff Diagram Inputs Call Price Now $95,854.66 Strike Price Now $96,000.00 Exercise Price $145.34 Risk free Rate 5.00% Time To Maturity 9.00 Dividend $2.00 Time To Dividend 0.10 Put call Parity 81200.3 Figure 1: Put Call parity – Pay off Diagram 2.3. Conclusion In the view of the assessment and analysis of the Put call parity applied to the data, the data cannot accept the call parity because the option and the currency positions do not have equal return or opportunities for arbitrage. The trade is thus, exposed to high risk and the put and the call parity may take too long to return. For arbitrage to occur, the purchase price of the currencies ought to reduce (Garay, Ordonez and Gonzalez, 2003). The Put options and the call options have a relationship such that they should yields the same amount of return, but the payoff diagram shows that the call pay off and the total pay off are not the same. The Put call parity is rejected by the data provided, for failure to meet the criteria of equal profit or loss from the two options. The preferred option is the put option because of the constant payoff realized throughout the period. References Bodurtha, J.N. & Courtadon, G.R. (1995). Probabilities and Values of Early Exercise: Spot and Futures Foreign Currency Options. The Journal of Derivatives, 3, 57-75. Garay, U., M. Ordonez and Gonzalez, M. (2003). Tests of the put-call parity relation using options on futures on the S&P 500 Index, Derivatives Use, Trading & Regulation 9: 59-80. Hoque, A., Chan, F. and Manzur, M. (2008). Efficiency of the foreign currency options markets, Global Finance Journal 19, 57-70. Hull, J. (2005). Options, Futures, and Other Derivatives, 6th edition, Englewood Cliffs, NJ: Prentice-Hall. Lung, P. P. & Nishikawa, T. (2005). The Implied Exchange Rates Derived from Option Premiums: A Test of the Currency Option Boundary. The Review of Futures Markets, 14, 67-98. Poitras, G. (2002). Risk Management, Speculation, and Derivative Securities. San Diego, CA: Academic Press. Read More