Journal Of Business & Economics Research September 2005 Volume 3, Number 9

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1 Opion Pricing And Mone Carlo Simulaions George M. Jabbour, ( George Washingon Universiy Yi-Kang Liu, George Washingon Universiy ABSTRACT The advanage of Mone Carlo simulaions is aribued o he flexibiliy of heir implemenaion. In spie of heir prevalence in finance, we address heir efficiency and accuracy in opion pricing from he perspecive of variance reducion and price convergence. We demonsrae ha increasing he number of pahs in simulaions will increase compuaional efficiency. Moreover, using a -es, we examine he significance of price convergence, measured as he difference beween sample means of opion prices. Overall, our illusraive resuls show ha he Mone Carlo simulaion prices are no saisically differen from he Black-Scholes ype closed-form soluion prices. INTRODUCTION M one Carlo simulaions (MCS) have recenly been an imporan echnique for opion pricing in finance. MCS avoid complicaed mahemaics and have a sraighforward implemenaion concepually and pracically. For example, o price a European down-and-ou call barrier opion 1 by MCS, jus rea i as a normal opion unless he underlying asse price reaches he pre-deermined level, as opposed o seing boundary condiions and solve a parial differenial equaion. In pracice, MCS are procedures of sampling random oucomes for a paricular process. However, while many academics and praciioners acknowledge he meris of MCS, some sudies discuss heir weaknesses in opion pricing. Clewlow and Srickland (1998) and Hull (000) poin ou ha MCS generae high variances ha lead o compuaional inefficiency. This problem can no be overlooked because such inefficiency may produce a biased esimaor of he opion price. In his paper, our focus is on he efficiency and accuracy of MCS in opion pricing. We demonsrae ha he esimaed sandard errors of MCS opion prices can be reduced by increasing he number of pahs in he simulaions. Addiionally, we use a -es o examine wheher MCS prices converge o Black-Scholes ype of closed-form soluion prices. The empirical evidence does no sugges any significan difference beween hose prices. Moreover, he resuls show ha hese wo ypes of prices converge as he number of pahs in simulaions increases. The layou of his paper is as follows: secion provides a quick lieraure review. Secion 3 examines variance reducion and price convergence of MCS. Secion 4 provides he conclusions. LITERATURE REVIEW Originaed from sudies in physics, MCS have been very successfully applied in finance. Hull and Whie (1987) use MCS o price opions wih sochasic volailiies. Schwarz and Torous (1989) apply MCS o he valuaion of morgaged-backed securiies. Boyle e al. (1997) use MCS o price American opions. On he oher hand, he disadvanages of MCS are also discussed in some sudies. Clewlow and Srickland (1998) and Hull (000) argue ha MCS are compuaionally inefficien due o he generaed high variances. THE EFFICIENCY AND ACCURACY OF MONTE CARLO SIMULATIONS Variance Reducion The efficiency of MCS increases wih he number of pahs used in he simulaions. Since MCS are sampling random variables, opion prices are random as well. The esimaed sandard error (ESE) is calculaed as he sample 1 A barrier opion is a coningen claim whose payoff depends on wheher he underlying asse has reached a cerain pre-deermined level for a specific pah. See Jackel (00) for a horough summary of he applicaions of MCS in finance. 1

2 sandard deviaion of MCS opion prices (SD) divided by he square-roo of he number of pahs (m): ESE SD / m (1) From equaion (1), ESE will decrease wih an increase in he number of pahs. Theoreically, if sample sandard deviaion of MCS opion prices (SD) is unchanged, when we increase m from 100 o 400, he ESE should be reduced by 50%. We demonsrae his negaive relaionship by using an example. The resuls are lised in Table 1. Table 1: Esimaed Sandard Errors of a European Pu Opion Price in Mone Carlo Simulaions The esimaed sandard error (ESE) is given by: ESE sample sandard deviaion ( SD) / number of pahs ( m) The inpu parameers are as follows: curren sock price s=10, exercise price x=10, ime o mauriy =0.5 year, risk-free rae r=0.1, sock reurn volailiy =0.4, and m=number of pahs. Sample sandard deviaion of Mone Carlo simulaion opion prices (SD) Esimaed sandard error of Mone Carlo simulaion opion prices (ESE) Case A (m=100) Case B (m=400) Case C (m=500) Theoreical sandard error reducion* Pracical esimaed sandard error reducion** * Theoreical sandard error reducion is he square roo of m in base case divided by he square roo of m in he referring case. For example, in case B, he heoreical sandard error reducion is 100 / ** Pracical esimaed sandard error reducion is he esimaed sandard error (ESE) in each case divided by he esimaed sandard error of he base case. For example, in case C, he pracical sandard error reducion is 0.048/0.187 = Table 1 shows he values of ESE for a hypoheical European pu opion for differen pahs. The heoreical sandard error reducion in each case is defined as he square roo of m in base case divided by he square roo of m in he corresponding case. In addiion, for comparison purposes, we calculae he pracical esimaed sandard error reducion, defined as he ESE of a case divided by he SD of he base case. Table 1 clearly shows how he errors of opion prices can be reduced as he number of pahs increases. For example, ESE can be effecively reduced from 1.87% in case A (wih 100 pahs) o only.48% in case C (wih 500 pahs). In addiion, he heoreical sandard error reducion (0% for case C) is very close o he pracical esimaed sandard error reducion (19.6%). The difference is due o he fac ha he SD is no he same in cases A and C. Price Convergence Price convergence is measured as he magniude of he differences in sample means of wo groups: MCS and Black-Scholes ype of closed-form soluion prices. If hese prices converge, he means of prices should be approximaely he same. For illusraion purposes, we choose pah dependen opions, specifically a down and ou call opion as our pricing arge. We will limi our example o he European syle o be consisen wih he assumpion of Black-Scholes ype of closed-form soluion. In a Black-Scholes framework, he sock price follows a geomeric Brownian moion. Tha is:

3 ds rs d S dw () where S is hesock price a ime, r is he risk free rae, σ is he volailiy, and W is a Wiener process a ime Using Wilmo (1998) approach, he value of a down and ou European call opion is given by: S V ( S, ) C( S, ) ( X ) ( k 1) X * C( S, ) (3) where k r /, V ( S, ) is he value of down and ou European call opion wih underlying asse price S and ime, C( S, ) is he value of vanilla European call opion wih underlying asse price S and ime, X is he barrier price, and C( X / S, ) is he value of vanilla European call opion wih underlying asse price X / S and ime. The European down and ou call opion is a pah dependen opion. In his sense, we need o check if he sock his he hreshold, a pre-se barrier price. If i does, he opion ceases and has zero value. If no, he opion survives and he final value of he underlying asse can be deermined. Once we know he final value of he underlying asse, he payoff and he price of he barrier opion can be calculaed. By repeaing he same procedure wih various realizaions, we can generae differen samples which allow us o compare he relaion beween price convergence and he number of pahs. Figure 1 compares he MCS opion prices wih he Black-Scholes ype of closed-form soluion prices in hree differen cases wih 100, 400, and 500 pahs. I also presens he inrinsic values excess of sock price over exercise price or zero. Because down and ou barrier opion has zero value unless he underlying sock exceeds he barrier price, i.e. 8 in his example, we esimae he opion value only when sock prices surpass he barrier. To examine he price convergence beween MCS and Black-Scholes ype prices, we apply a -es o compare he means. We assume an exercise price of $10, a ime o mauriy of six monhs, a barrier price of $8, a risk-free rae of 10%, and a volailiy of 40%. We use fory observaions of MCS prices wih respec o sock prices by changing he underlying asse price from $8 by $. unil i reaches $16. 3 As shown in figure 1, i is easy o see ha he MCS prices end o converge o he Black-Scholes ype of closed-form soluion prices as he number of pahs increases. To confirm his behavior, we es he following hypohesis: H0: mean of MCS prices = mean of closed-form soluion (Black-Scholes ype) prices H1: mean of MCS prices mean of closed-form soluion (Black-Scholes ype) prices The wo-ail null hypohesis es is rejeced if he -saisic is larger han he corresponding criical value, 1.96, under a 95% confidence inerval. Table summarizes he resuls. Firs we noice ha he sample sandard deviaion of MCS opion prices (SD) decreases wih he increase in he number of pahs. This is consisen wih he resuls in Table 1. Second, we find ha all he -saisics are less han he criical value This resul suggess ha we can no find saisically significan evidence o rejec he null hypohesis. Only a 10% level of significance (i.e. criical value 1.645) can he null hypohesis be rejeced in he case of 100 pahs. In pracice, since he number of pahs is usually larger han 100, i is reasonable o conclude ha he MCS prices are no significanly differen from he Black-Scholes ype prices. Furhermore, we find he p-values increasing wih he numbers of pahs in simulaions. The large p-value implies ha he likelihood of failing o rejec he false hypohesis (Type II error) is low. This indicaes 3 According o Griffihs e al. (1993), for applying a -es, a sample size of hiry observaions is considered large enough o saisfy he normal disribuion assumpion. 3

4 Figure 1: The Relaionship Beween Mone Carlo Simulaion Opion Prices and Black-Scholes Type Prices This figure depics he inrinsic value, he Mone Carlo simulaion opion prices (denoed by MC) and he Black-Scholes ype closed-form opion prices (denoed by closed-form soluion) of a European down and ou call opion wih 100, 400, and 500 pahs. The inpu parameers are: barrier price (x) =8, ime o mauriy () =0.5, exercise price (e) =10, risk-free rae (r) =0.1, volailiy (sigma) =0.4. Diagram A (Number of pahs =100) Diagram B (Number of pahs =400) Diagram C (Number of pahs =500) 4

5 ha he confidence level of our conclusion on no rejecing he null hypohesis increases wih he p-value. In oher words, he more simulaions are execued, he more accurae he null hypohesis is, and he more evidence on he convergence of MCS prices o Black-Scholes ype prices. Table : Two-ail Tes for Price Convergence (sample size = 40) This able shows any significan difference beween Mone Carlo simulaion prices and Black-Scholes ype prices. The pricing arge is a European down and ou call opion. The inpu parameers are: barrier price (x) =8, ime o mauriy () =0.5, exercise price (e) =10, risk-free rae (r) =0.1, volailiy (sigma) =0.4. Case 1, and 3 are characerized by various numbers of pahs in simulaion, m. Each case has fory-paired observaions. The null hypohesis for he wo-ail es is: H0: mean of MCS prices = mean of closed-form soluion (Black-Scholes ype) prices H1: mean of MCS prices mean of closed-form soluion (Black-Scholes ype) prices Case 1 (m=100) Case (m=400) Case 3 (m=500) Sample sandard deviaion of Mone Carlo simulaion opion prices (SD) Mean of he difference beween MCS and BS ype prices Variance of he difference beween MCS and BS ype prices Number of samples saisic/p-value* 1.79/ / /0.684 Hypohesis esing** canno rejec null hypohesis canno rejec null hypohesis * Under he null hypohesis, he -saisic is esimaed as follows: Mean of he difference beween MCS and BS ype prices Variance of he difference beween MCS and BS ype prices / Number of samples ** Based on a wo-ail 95% confidence inerval. canno rejec null hypohesis CONCLUSIONS In his paper, we address he issues of efficiency and accuracy of Mone Carlo simulaions in opion pricing from he perspecives of variance reducion and price convergence. We demonsrae ha increasing he number of pahs in simulaions will increase compuaional efficiency. Moreover, using -es, we examine he endency of price convergence, measured as he difference beween sample means of opion prices. Our resuls did no find significan evidence o rejec he null hypohesis ha he Mone Carlo simulaion prices and he Black-Scholes ype prices have he same mean. REFERENCES 1. Clewlow, L. and C. Srickland, Implemening derivaives models, (1998), John Wiley & Son Ld.. Black, F. and M. Scholes, (1973), The pricing of opions and corporae liabiliies, Journal of Poliical Economy, 81, Boyle, P., M. Broadie, and P. Glasserman, (1997), Mone Carlo mehods for securiy pricing, Journal of Economic Dynamics and Conrol, 1, Griffihs, W., R. C. Hill, and G. Judge, Learning and pracicing economerics, (1993), John Wiley & Son Ld. 5. Hull, J., Opions, fuures and oher derivaives, fourh ediion,(000), Prenice-Hall Inc. 6. Hull, J. and A. Whie, (1987), The pricing of opions on asses wih sochasic volailiies, Journal of Finance 4, Jackel, P., Mone Carlo mehod in finance, (00), John Wiley & Son Ld. 5

6 8. Schwarz, E. and W. Torous, (1989), Prepaymen and he valuaion of morgaged-backed securiies, Journal of Finance 44, Wilmo, P., Derivaives: he heory and pracice of financial engineering, (1998), John Wiley & Son Ld. NOTES 6