Valuation Of Reload Call Option With Binomial Tree

Transcription

1 Valuation Of Reload Call Option With Binomial Tree Yunita Wulan ari 1, Gunardi 2 1 Department of Mathematics, Universitas Gadah Mada, yunita-ws@ugmacid 2 Department of Mathematics, Universitas Gadah Mada Abstract Along with the rapid growth of investment and finance, the investors compete with each other to find a way to maximize pro_ts and minimize losses In option trading, one of the ways used is adding a reload feature on the option Reload option is option thatgive the right to its holder to exercise the option, then renew it at some time prior to maturity In this research, we study how to determine the pricing of reload call option with the Blac-choles stoc price model and the binomial tree, then applied to the stoc of Teleomoniasi Indonesia (TLMJ) From this study, we obtain the valuation of reload call option which is simple and easy in interpretation In addition we can also conclude that if the time to maturity is longer, the price of reload call option is more expensive and if the exercise price is higher, the price of reload call option is cheaper 2010 Mathematics ubect Classi_cation: 62P05 eywords: option, reload option, reload call option, Blac-choles stoc price model ection: -07 INTRODUCTION Along with the rapid growth of investment and finance, the investors compete with each other to find a way to maximaze profits and minimize losses Option is one of well nown alternative investments Traded option have added some features to mae it more attractive and desirable One of the famous features is reload Option with reload feature added is called reload option (also called restoration option, replacement option, or accelerated ownership option) Reload option gives the owner the right to execute his option and update it at one or several times before maturity (0<Ti<T) The number of new options granted is equal to the number of shares needed to pay the strie price using the stoc at its current value [1] Huddart et al (1999) added some modification to the algorithm of reload option price valuation, then applied the binomial option pricing model on its study case Further, he get Yunita Wulan ari & Gunardi 1

2 the result that option price valuation with binomial approach is easy on interpretation Besides that, it can be shown that the price of reload option is higher than the common option [2] Although reload option typically involve excercise at many dates, the optimal exercise policy is simple (always exercise when in the money) and surprisingly robust to the assumptions about the underlying stoc price and devidend process As a result, we obtain general reload option valuation formulas that can be evaluated numerically using integral as a formal valuation Furthermore, under the Blac-choles assumptions with or without continuous devidends, there are even simpler formulas for prices and hedge ratios [3] Whereas, the main factor that determine the reload option valuation is the underlying asset price [4] Reload feature added on option mae the difficult and complex option valuation This time, the reload option is unstudied and sold in Indonesia In this research, we will studied how to determine the value of reload option with the Blac-choles model and binomial tree method, then applied it on Indonesian stoc data CONTRACT TRUCTURE OF RELOAD CALL OPTION Reload option is designed to can be executed before maturity and automatically grants new option With a regular reload, the number of new options granted is equal to the number of shares needed to pay the strie price using the stoc at its current value If reload only done once before maturity, the value of option at reloaded time Ti (0<Ti<T) is and the value of reload option at maturity is where : strie price Ti T V T V max,0 Ti Ti max T T,0, if i T i Ti max T,0, if T i : maret stoc price at reload time T i : maret stoc price at maturity In general, the contract structure of reload call option with execution/reload time and the order of strie price on each reload time (, 1, 2,) is as follows : Yunita Wulan ari & Gunardi 2

3 At the time the contract was agreed (t=0), the option buyer has one option and no share At the first reload, the option buyer has 1 new reload options and 1 1 shares 1 At the second reload, the option buyer has new reload options and shares And at the -th reload, the option buyer will have shares new reload options and 1 TOC PRICE BINOMIAL TREE If the option is valid from the time signed (t=0) until the maturity (T), the time interval [0,T] can be devided into N descrete time, ie [(i-1)h, ih], i = 1,2,, N Each descrete T time period length is h In fact, the maret stoc price always turned up or down in line N with changing times The possibility of two-way change is used as the basic of binomial u d models For example, the stoc price at t=0 is 0 and it will be up to t 1 and down to t 1 with probability are p and (1-p) respectively at t+1 In the case of a constant interest rate, we assume the following ris-neutralized dynamics for stoc prices, : 1 2 d ln t s s dt sdwt 2 µs and σs are unnown parameters They are independent of time These two of parameters can be estimated by historical stoc price Wt is standard brownian motion [5] Based on this stoc price model, each descrete time period whose its length is h applies ln u d t1 ln t s h and ln 1 ln price at time t+1 if the stoc price at time t is nown as t t s h Therefore, it can be defined stoc u t1 s e t h and Yunita Wulan ari & Gunardi 3

4 d t t1 s h e Figure 1 shows the stoc price binomial tree at each node Figure 1 toc Price Binomial Tree RELOAD CALL OPTION BINOMIAL TREE Based on the stoc price binomial tree, it can be formed the reload call option The notation defined is as follows : i, : stoc price at node (i,) R i,, : -th value of the strie price at node (i,) V i : -th value of the reload call option at node (i,) r : ris-free interest rate (assume constant) The following are the strie price setting at each node, and we devide this setting into two situations : ituation 1 : i, R (in the money) i Option holder has two alternative, there are exercise the option and not i, and R will sent to 1, 1 and 1, ituation 2 : i, (out the money or at the money) Option holder will not exercise his option, R i, will sent to 1, 1 and 1, Define Ve and Vh as the early exercise value and the holding value respectively When i Yunita Wulan ari & Gunardi 4

5 i<n, we explain the approach which we calculate Vi,, as follows: The value of reload call option at maturity is V max R,0, for all n n, n Rn The value of reload call option at node (i,) is ituation 1 : R i, i V max V, V i e h where max,0 rh 1 V R e pv p V e i, i i1, 1, z i1 z z represents the z-th value of V e pv 1 p V 1, 1 rh h i1, 1, y i1 y or 1, which equals to i, y represents the y-th value of 1, 1 or i 1, R which equals to R i,, ituation 2 : i, R i rh i i1, 1, y i1 y R V e pv 1 p V where y represents the y-th value of i 1, 1 and 1, which equals to R i,, V 0,0,1 is the current reload call option RELOAD CALL OPTION OF TLMJ In study case, we will calculate the value of reload call option at beginning of August 2015 and the maturity date is three month It is assumed that option ust can be reload at the beginning of month until maturity Based on BI rate [6], ris-free interest rate paid per month is 0,6045% TLMJ s stoc prices at January 2014-August 2015 are used to predict the future stoc prices [7] Figure 2 shows stoc price, strie price, and the value of reload call option at each node The TLMJ s value of reload call option is IDR 161,994 per share Yunita Wulan ari & Gunardi 5

6 Figure 2 Reload Call Option Binomial Tree Figure 3 Reload Call Option of TLMJ in everal Maturity Date Yunita Wulan ari & Gunardi 6

7 Figure 4 Reload Call Option of TLMJ in everal trie Price Figure 3 explain that the length of time to maturity gives a positive influance to the value of reload call option However, the strie price gives a negative influence (Figure 4) REFERENCE [1] Ingersoll, J E Jr, 2007, Valuing Reload Options, Review of Derivatives Research, Vol9, pp [2] Huddart Jagannathan, R, dan aly, J, 1999, Valuing The Reload Features of Executive toc Options, Accounting Horizons, No13, pp [3] Dybvig, PH dan Loewenstein, M, 2003, Employee Reload Options : Pricing, Hedging, and Optimal Exercise, Review of Financial tudies, No16, pp [4] Cheng, WC, 2006, The Valuation of Employee Reload Option with tochastic Interest Rate, National Central University Electronic Theses and Dissertations, Taiwan [5] P Wilmott, Paul Wilmott Introduces Quantitative Finance, John Wiley and ons : England (2001) [6] [7] Yunita Wulan ari & Gunardi 7