Pricing Asian Options using Monte Carlo Methods

Transcription

1 U.U.D.M. Project Report 9:7 Pricing Asian Options using Monte Carlo Methods Hongbin Zhang Exaensarbete i ateatik, 3 hp Handledare och exainator: Johan Tysk Juni 9 Departent of Matheatics Uppsala University

2

3 Acknowledgeents First of all, I hereby would like to acknowledge the aster progra in Financial Matheatics at Uppsala University that enabled e to have such a wonderful experience in Sweden. Meanwhile, special thanks go to y advisor Johan Tysk, for his patient tutorship, in-depth coent and invaluable advice in writing this paper. It has been a great joy to study under his guidance and encourageent. Besides, I want to acknowledge all the teachers who gave e lectures during the two years, and I would like to express y gratitude for their enlightening instruction and war-hearted assistance. Finally, I want to thank y parents for all their sypathetic understanding and unfailingly support.

4 Abstract Asian options are of particular iportance for coodity products which have low trading volues (e.g. crude oil), since price anipulation is inhibited. Hence, the pricing of such options becoes one of the ost interesting fields. Since there are no known closed for analytical solutions to arithetic average Asian options, any nuerical ethods are applied. This paper deals with pricing of arithetic average Asian options with the help of Monte Carlo ethods. We also investigate ways to iprove the precision of the siulation estiates through the variation reduction techniques: the control variate and the antithetic variate ethods. We then copare the results fro these two ethods.

5 . Introduction. Financial Derivatives Derivatives are financial contracts, or financial instruents, whose values are derived fro the value of soething else which is known as the underlying. The underlying value on which a derivative is based can be a traded asset, such as a stock; an index portfolio; a futures price; a coercial real estate; or soe easurable state variable, such as the weather condition at soe location. The payoff can involve various patterns of cash flows. Payents can be spread evenly through tie, occur at specific dates, or a cobination of the two. Derivatives are also referred to as contingent clais. The ain types of derivatives are forwards, futures, options, and swaps.. Options An option is a contract between a buyer and a seller that gives the buyer the right but not the obligation to buy or to sell the underlying asset at an agreed price at a later date. There are two basic kinds of options: the call option and the put option. A call option gives the buyer the right to buy the underlying asset while a put gives the buyer to sell. The agreed price in the contract is known as the strike price; the date in the contract is known as the expiry date. The vast ajority of options are either European or Aerican options. There are any other types of options such as barrier options; Berudan options; Asian options; or look back options..3 European and Aerican Options European options are the foundations of the options universe. Presenting itself as the ost basic type of option contract, this type of option gives the holder or seller of the option the ability to exercise the option only at the expiry date. The pay-off is given by: ( ) Φ ( S) = S( T) for a European call option; and by ( ) Φ ( S) = S( T ) for a European put option,

6 where S(T) is the price of the underlying assets at the expiry date T and is the strike price. An Aerican option, in contrast to the European option, ay be exercised at any tie prior to the expiry date. The pay-off is given by ( τ ) Φ ( S) = ax S( ) for an Aerican call option; t τ T and by ( τ ) Φ ( S) = ax S( ) for an Aerican put option, t τ T where τ is the exercise tie. It is obvious that the Aerican option is uch ore coplicated and interesting than the European option..4 Why do people use options? Options have two ain uses: speculating and hedging. Investing in options has a leverage copared to investing directly in the corresponding underlying assets for the speculators. For instance, if you believe that Ericsson shares are due to increase then you ay speculate by becoing the holder of a suitable call option. Typically, you can ake a greater profit relative to your original payout than you would do by siply purchasing the shares. That is why the options becoe ore and ore popular in the financial arket. On the other hand, it is a very useful tool for hedging. Especially in the view of the option writers, they ay not be able to afford the huge potential risk just as the speculators could. They write options to gain soe certain and less risky profits. The echanis is like this: the writer can sell an option for ore than it is worth and then hedge away all the risk she or he ight be possible to take, and then ake a locked gain. This idea is central to the theory and practice of option pricing..5 Option Pricing Although options have existed at least in concept since antiquity, it wasnt until publication of the Black-Scholes (973) option pricing forula that a theoretically consistent fraework for pricing options becae available. Nowadays, option pricing plays a critical role in the research about the financial arket. Due to the narrow range the Black-Scholes forula can apply to, soe other option pricing ethods are introduced and used to analyze the coplicated options. There are three priary option pricing ethods widely used: binoial ethods, finite difference odels and

7 Monte Carlo odels. Binoial ethods involve the dynaics of the options theoretical value for discrete tie intervals over the options duration. The odel of this kind starts with a binoial tree of discrete future possible underlying stock prices. By constructing a riskless portfolio of an option and stock a siple forula can be used to find the option price at each node in the tree. Therefore they rely only indirectly on the Black-Scholes analysis through the assuption of risk neutrality. However, the binoial odels are considered ore accurate than Black-Scholes because they are ore flexible, e.g. discrete future dividend payents can be odeled correctly at the proper forward tie steps, and Aerican options can be odeled as well as European ones. The finite difference odel can be derived, once the equations used to value options can be expressed in ters of partial differential equations. The idea underlying finite difference ethods is to replace the partial derivatives occurring in partial differential equations by approxiations based on Taylor series expansions of functions near the point or points of interest. One should note that binoial ethods are particular cases of the explicit finite difference ethods. For any types of options, traditional valuation techniques are intractable due to the coplexity of the instruent. In these cases, a Monte Carlo approach ay often be useful. We will go into details in Section 4. 3

8 . Financial Background To begin with, we will enter the ost resplendent star in our financial universe the Black-Scholes world. We usually use the Black and Scholes odel to describe the price of an asset at tie t. The Black-Scholes odel consists of two assets with dynaics given by db() t = rb() t dt, (.) P ds() t = μs() t dt σs() t dw () t, (.) Where B(t) and S(t) are the prices of the risk-free asset and the risky asset respectively, W(t) is a standard Brownian otion, r, μ andσ are deterinistic constants. The stochastic differential equation (.) is defined on a certain probability space ( Ω, FP, ). According to the Girsanov theore, we know that there exists a probability easure such that under the easure ds() t = rs() t dt σ S() t dw () t, (.3) P where W ( t) = W ( t) (( μ r) / σ ) t which is a standard Brownian otion under. The solution of (.3) is ( σ σ ( )) ST ( ) = St ( )exp ( r /)( T t) W ( T) W ( t). (.4) Under we also have ( ) rt rt rt d e S() t = e ds() t S()( t de ) ( σ ) rt rt = e rs() t dt S() t dw () t S()( t re ) dt σ rt = e S() t dw () t. (.5) It can be seen fro (.5) that S(t)/B(t) is a artingale under the easure, so is called risk neutral easure. These results can be found in any textbook on stochastic analysis such as aratzas and Shreve (988), Oksendal (995) 4

9 Meanwhile, Black-Scholes equation gives us F( ) ( ) t ts, rsfs ts, Sσ FSS ( ts, ) = rf( ts, ), (.6) ( ) F T, S =Φ ( S). (.7) We also have df(, t S) = Ft t, S dt FS t, S ds FSS t, S ( ds) ( ) ( ) ( ) ( ) ( ) = Ft t, S rsfs t, S S σ FSS ( t, S ) dt σsfs ( t, S ) dw (, ) σ (, ) = rf t S dt SF t S dw. (.8) S Therefore, F ( tst) Hence,, ( ) / Btis ( ) a artingale under as well. (, ( )) (, ( )) Φ ( ( )) F tst F TST ST = EtS. ( t) = EtS. ( t), (.9) Bt ( ) BT ( ) BT ( ) (, ( )) exp ( ( )) ts. ( t) ( ( ( ))) F tst = rt t E Φ ST. (.) As for the vanilla European call option, we get by (.) (, ( )) exp ( ( )) ( ( ) ) C t S t r T t EtS. ( t) S T =, (.) where (, ()) C t S t denotes the price of the vanilla European call option. Then plug (.4) into (.), we get C ( t, S( t) ) rt ( t) ( r σ /)( T t) σ( W ( T) W ( t) ) = e E ts. ( t) Ste ( ) 5

10 x r σ ( T t) rt ( t) x σ ( T t) ( ) log S () t π( T t) σ = e Ste ( ) e dx x r σ ( T t) σ ( T t) x rt ( t) σ ( T t) St log S () t π( T t) σ = e ( ) e dx e x r σ ( T t) rt ( t) σ ( T t) e log S () t π( T t) σ dx x r σ ( T t) rt ( t) rt ( t) σ ( T t) St log S () t π( T t) σ = e ( ) e dx e x r σ ( T t) rt ( t) σ ( T t) e log S () t π( T t) σ dx y y ( ) rt t log r σ ( T t) log ( ) () r σ T t S t π S() t π ( T t) σ ( T t) σ = St () e dy e e dy = St () Φ( d) e Φ ( d), (.) rt ( t) where d St () log = ( T t) σ ( ) r σ T t, d St () log = ( T t) σ ( ) r σ T t. Note that ( ) ( ) ST ( ) ST ( ) = ST ( ). (.3) Take the expectation on both sides and discount back to tie t, we get (, ( )) (, ( )) C tst P tst ( ts. ( t) ts. ( t) ) ( rt t) E ( ST ) E ( ST ) = exp ( ) ( ) ( ) ( rt t) E ( ST ) = exp ( ) ( ) ts. ( t) 6

11 ( ) ( ) = exp rt ( t) E ST ( ) exp rt ( t) ts. ( t) ( ) = St () exp rt ( t), (.4) where P (t, S(t)) denotes the price of the vanilla European put option. This is the so-called put-call parity for the European options. Hence, we can directly derive the price of the European put option fro (.4) together with (.): (, ( )) P t S t (, ( )) ( ) exp ( ( )) = C tst St rt t ( ) ( ) ( ) = St () Φ( d) exp rt ( t) Φ( d) ( rt t) d St d = exp ( ) Φ( ) ( ) Φ( ) (.5) 7

12 3. Asian Options 3. What are Asian options? Asian options are options in which the underlying variable is the average price over a period of tie. Because of this fact, Asian options have a lower volatility and hence rendering the cheaper relative to their European counterparts (we will give the proof in 3.3). They are coonly traded on currencies and coodity products which have low trading volues. They were originally used in 987 when Bankers Trust Tokyo office used the for pricing average options on crude oil contracts; and hence the nae "Asian" option. There are soe different types of Asian options: Continuous arithetic average Asian call or put with T Φ ( S) = S( t) dt T or T Φ ( S) = S( t) dt T. Continuous geoetric average Asian call or put with T log S( t) dt T Φ ( S) = e T or Φ ( S) = e T log S( t) dt. Discrete arithetic average Asian call or put with it Φ ( S) = S( ) i= or it Φ ( S) = S( ) i=. Discrete geoetric average Asian call or put with it log S ( ) i= Φ ( S) = e i= or Φ ( S) = e it log S ( ). Denote the price of the arithetic average Asian call and put options at tie by a, a, C ( S T ) and P ( ), S T, and denote the price of the geoetric average Asian, 8

13 g, g, call and put options at tie by C ( S T ) and ( ), P S T., Proposition 3.. The following inequalities hold in the discrete case: a (, ) (, ) C S T C S T ; g,, a (, ) (, ) P S T P S T. g,, Proof: This directly follows fro the inequality between the geoetric and arithetic ean /( ) it it S( ) S( ) i, = i= together with the fact that ( X ) is increasing in X while ( X ) is decreasing in X. 3. Closed For Solution for the Geoetric Average Asian Options We assue the price of the asset follows a log-noral distribution continuous in tie, and the product of log-noral distributed rando variables is also log-noral distributed, however the su is not. Hence, we could expect that the pricing of geoetric average Asian options should be easy to deal with, while for arithetic average ones it ay be relatively ore coplicated to handle. In fact, the pricing forula of geoetric average Asian options can be derived in the Black-Scholes fraework., see [6]. The payoff function for the discrete geoetric average Asian call option is given by And we know it log S ( ) i= Φ ( S) = e /( ) it = S( ) i=. (3.) 9

14 3 St ( ) St ( ) St ( ) St ( i ) = i= St ( ) St ( ) St ( 3) St ( ) St ( ) St ( ) 3 St ( ) St ( ) S S (3.) where t = it / for i =,, 3,,. i Fro (.4) together with the property of the Brownian otion, it follows that (( r σ T σ T X) St ( ) = exp ( / ) / /, St ( ) (( r σ T σ T X ) St ( ) = exp ( / ) / /, St ( ) 3 (( r σ T σ T X3 ) St ( ) = exp ( / ) / /, St ( ) (( r σ T σ T X ) St ( ) = exp ( / ) / /, St ( ) (( r σ T σ T X ) St ( ) = exp ( / ) / /. S where { X i} i Furtherore, we have = are independent, N(,)-distributed rando variables. log St ( i ) i= = log St ( i ) i= /( ) S S /( ) = log ( ) Sti S i=

15 St ( ) ( ) ( ) ( ) = log St St St S( t ) S( t ) S( t) S St ( ) St ( ) = log log S( t ) S( t ) St ( ) St ( ) ( )log log St ( ) S ( ) = (( r σ /) T / σ T / X ( r σ T σ T X ) ( /) / / ( σ σ ) ( ) ( r /) T / T / X (( σ /) / σ / )) r T T X = ( )( r σ / ) T / σ T / ixi i= σ T / ix ( r σ /) T i= = i. (3.3) By the additive ean and variance property of independent noral rando variables, we know that σ T / i= ix i N σ T( 3 ),, ( ) i.e., σ T / ixi i= ( ) σ T N,. (3.4) 6( ) Therefore, we have σ T / ixi i= ( ) T = σ Z 6( ) (3.5)

16 where Z N(,). Plugging (3.5) into (3.3), we obtain that log St ( i ) i= /( ) S = ( ρ σ ) T σ TZ (3.6) Z Z where σ Z = σ 6( ) and ( r σ /) σ ρ = Z. Hence, we can obtain the price of the geoetric average Asian call option by using the risk-neutral ethod: /( ) g, it C ( S, T) = exp( rt) E S( ) i= ( ) = exp (( ρ ) ) exp( ρ ) exp (( ρ σz ) σz ) rt T E S T TZ exp ( ρ rt ) C ( S, T) = ( ) (3.7) where C ( S, T) denotes the price of a European call option with risk-free interest rate ρ and the volatility σ Z. By the Black-Scholes forula, we get g, C ( S T) = ( ρ r T) C S T, exp ( ) (, ) exp ( ρ ) Φ( ) exp( ρ ) Φ( ) = ( rt) S d T d

17 exp exp( ρ ) Φ( ) Φ( ), (3.8) = ( rt ) S T d d where d S log ρ σ T Z =, Tσ Z d S log ρ σ T Z =. Tσ Z If P ( S, T) we also get: denotes the price of a European call option with the sae paraeters, g P ( S T) = ( ρ r T) P S T, exp ( ) (, ) exp ( ρ ) exp( ρ ) Φ( ) Φ( ) = ( rt) T d S d exp rt Φ( d ) S exp( ρt ) Φ( d ) = ( ) (3.9) Since we know that the geoetric average Asian options can be siply priced according to the Black Scholes fraework whereas the arithetic ones cannot, we focus on the case of arithetic average Asian options and on the pricing of this kind by using soe other kind of pricing ethods. 3.3 An Inequality between the European and Asian Option Asian options are widely used in practice, and they are always the target which any scholars are focused on. There are ainly two reasons that could explain it: on one hand, the price of the underlying assets could be hard to anipulate near expiration date of the options due to their averaging effect; on the other, they are significantly cheaper than European options when used to hedge. The first reason is uch ore obvious than the second. Asian options could offer the protection against the price anipulation. For exaple, if a standard European call option is based on a stock which reains low in price during a large part of the final tie period and rises significantly at aturity, the option writer would have to face a assive loss; whereas the average option could avoid such anipulation. Copared to European options, it costs less to hedge by picking Asian options. Let us take a look at the discrete arithetic average Asian call option, please see [6]. Proposition 3.. We have the following inequality: 3

18 (, ( )) (, ( )) a, C t S t C t S t. Why could this be true? Is it intuitively reasonable without the foral proof? It sees to be difficult to copare directly since there is no explicit forula of the price of an Asian option. However, we ay look at the final value of the options. We know that rt ( t) ( ) = E ST ( ) e St ( ) it E S( ) E( S( T) ) ( i=,,,..., ) it. E S( ) E S( T) = E( S( T) ) i= i= One can guess that aybe the following holds: (, ( )) a, C t S t it = exp ( rt ( t) ) EtS. ( t) S( ) i= ( ) ( ) exp rt ( t) E ST ( ) ( t, S( t) ) = C. ts. ( t) Now we would like to give the foral proof the property of Asian options. Proof. Let us define it S( ) α i = ( i ) T S( ) cobined with (.4), it follows that and α = S(), T it ( i ) T αi = exp (( r σ / )( ) σ W ( ) W ( ), which is lognoral distributed with 4

19 rt Eα i = exp( ). Additionally, it is obvious that it S = α α α, ( ) i ST = α α α. ( ) In order to prove (, ( )) (, ( )) a, C t S t C t S t, we only have to prove that E ts. ( t) α α α α α α ( α α α ) E. ts. ( t) In fact, we have E ts. ( t) α α α α α α α α αα α α = EtS. ( t) = EtS. ( t), α α αα α α = with where = α E α αα α α = E( ) and Note that > if and only if > = ( ) 5

20 Now we have to distinguish two cases: (a), then we have EtS. ( t) = ( ) p( ) d ( ) p( ) d ts. ( t) ( ) = E. p ( ) d (b) For, EtS. ( t) = p ( ) d EtS. ( t) p( ) d = EtS. ( t) ( ) p( ) d ( ) ( ) EtS. ( t) p( ) d ( E ) ( ) = p( ) d ( ) p( ) d 6

21 ts. ( t) ( ) = E. Fro above, we can conclude that in both cases EtS. ( t) ts. ( t) ( ) E. Besides, it follows that ts. ( t) ( ) E α αα α α = EtS. ( t) ( α α α ) E, ts. ( t) where the inequality can be derived fro the induction on the nuber of rando variables which can be started at n=. Therefore, we can easily get (, ( )) (, ( )) a, C t S t C t S t, which akes Asian options uch ore popular when people want to hedge away the risk. 7

22 4. Monte Carlo ethods 4. The Monte Carlo Fraework Monte Carlo ethods were initially applied to option pricing by Boyle in 977. Nowadays, it has been ore and ore widely applied to price options with coplicated features. In this section Monte Carlo fraework will be described in a general setting. Suppose we want to estiate soeθ, and we have where ( ) ( ( )) θ = E g X. g X is an arbitrary function such that ( ( ) ) generate n independent rando observations X, X,, function f ( X ). The estiator of θ is given by E g X <, then we could X n fro the probability n θ = g( Xi ). n i= Since E( g( X ) ) <, we can get by strong law of large nuber n as.. g( Xi ) Eg( X) as n n i = i.e. as.. θ θ as n. The Monte Carlo siulation is never exact, and one always has to take the saple variance into account. It can be expressed as s n = n i= ( g( X ) ) i θ. Central liit theore tells us that 8

23 n ( θ θ) s d N(,) as n. We could also say that θ θ is approxiately an standard noral variable scaled by s/ n, i.e. for large n we have s s P θ zα < θ < θ zα α. n n For exaple, an estiate of the 95%-confidence interval for θ is given by s θ.96, θ.96 n s. Hence, we get a way to approxiate θ and its variation. n We refer to the estiation of θ by θ as the crude Monte Carlo ethod. The disadvantage of the crude Monte Carlo ethod is its slow rate of convergence. Since for large n we have s σ, we have to enlarge our n by a factor of to receive a reduction of the standard error s n by factor.. Thus ore accuracy is endowed by uch ore calculation. 4. Variance Reduction Techniques Instead of siply enlarging the nuber n, we can concentrate on reducing the size of s to narrow the confidence interval. Such techniques are known as variance reduction techniques. We would like to introduce soe of such techniques which are frequently used, see [9]. The Control Variate Method Suppose that we want to estiate θ: = E(Y) where Y = g(x). Suppose we can soehow find another rando variable Z with known ean E(Z). Then we can construct any unbiased estiators of θ:. θ = Y, which is our usual estiator;. θ c = Y c( Z E( Z) ) 9

24 where c is soe real nuber. And it is clear that E( θ ) = E( Y) c E( Z) E( Z) = E( Y) = θ, c ( ) hence we could apply Monte Carlo to the new estiator instead of the usual one. In this context, the rando variable Z is called the control variate. Now the question is whether it has a lower variance or not. We have that Var θ = Var Y c Var Z ccov Y Z ( c) ( ) ( ) (, ). Since c can be any real nuber, we have to choose it to iniize the quadratic. Siple calculus iplies that c in Cov( Y, Z) =. Var( Z) Plug this into the quadratic, then we get Var( θ ) = Var( Y ) c Var( Z) c Cov( Y, Z) cin in in Cov( Y, Z) = Var( Y ) Var( Z) Cov( Y, Z) = Var( θ ) Var( Z). As long as Cov(Y, Z), we do achieve a variance reduction. Of course, on a general proble we typically do not know Cov(Y, Z). However, it is possible to estiate Cov(Y, Z)) during a Monte Carlo siulation. That is all the echanis the control variate ethod involves. The introduction of an appropriate control variate provides a very efficient variance reduction technique, however, in soe probles it ay be difficult to find a suitable control variate. The alternative ethod we now discuss, the antithetic variate ethod, is often easier to apply since it concentrates on the procedure used for generating the rando deviates. The Antithetic Variate Method Suppose again that we would like to estiate θ = E(Y) = E[g(X)], and that we have

25 generated two saples Y and Y. Then an unbiased estiator of θ is given by Y θ = Y. Hence, we have VarY ( ) VarY ( ) CovY (, Y) Var( θ ) =. 4 It is obvious that we could get a variance reduction if we have the two saples negatively correlated. We now give the algorith for the noral case. Suppose that ( ) θ = E( Y) = E g( X), where X ~ N (, ), the crude Monte Carlo estiate is n θ = g( Xi ) n i=, with i.i.d. X i ~ N (, ), and the estiate by the antithetic variate ethod is n g( Xi) g( Xi) θa = n i=, with i.i.d. X i ~ N (, ), where X i and X i are the so-called antithetic variates. Clearly the two antithetic variates are negatively correlated. Thus, if the function g is onotonic, then we can achieve a variance reduction by using this ethod. For a discussion of this, please see [9]. 4.3 Pricing a European call option We are now in a position to use Monte Carlo for pricing the European option as an exaple. Fro (.4), we know that ( σ σ ) ST ( ) = St ( )exp ( r /)( T t) T tx, where X N(,).

26 For the sake of convenience, we siply let t =, then ( σ σ ) ST ( ) = S exp ( r /) T TX, where X N(,). Here we choose the European option, and then the relevant Monte Carlo algorith is as following: set su = for i = to n generate S(T) set su = su ax(, S(T) - ) end set C = exp( rt) su/ n To ake the result uch better, we would like to use the antithetic variate ethod to reduce the standard error. Table : Crude Monte Carlo versus control variate ethod for European call options with S =, r =.5, σ =., n = Option Option Standard Accurate Standard values by values by error by T option error by crude crude Monte antithetic antithetic values Monte Carlo Carlo variate variate

27 Table gives the results of the siulation ethod for the European call option with the given paraeters. We can easily see that the option values we get by both the crude Monte Carlo ethod and the antithetic variate ethod are very close to the corresponding accurate option values by the Black-Scholes forula, which deonstrates that Monte Carlo ethods can be used for option pricing. Besides, we get narrower confidence interval by the antithetic variate ethod. 3

28 5. Monte Carlo Methods for Arithetic Average Asian Options To get the price of an arithetic average price option we have to use Monte Carlo techniques again. Here we choose arithetic average price Asian call option as exaple. Firstly, we use the crude Monte Carlo ethod with siulated paths to price the Asian option. The algorith is as following: set su = for i = to n generate S(T / ), S(T / ),, S(T) end i= set su = su ax(, SiT ( / ) - ) set, a C = exp( rt) su/ n The results presented in Table, shows the liit of the crude Monte Carlo ethod. As increases, although the standard errors decrease, the siulations becoe less efficient for the reason that the estiates go down uch faster than the corresponding standard errors. On the other hand, when the tie step nuber gets larger, the standard error does not becoe saller at all. Since convergence of our schees is where our interest lies, variance reduction technique is necessary. Now we would like to use the control variate ethod in order to ake the approxiation ore efficient. In fact, there are any possible control variate choices we could use. It is a nature guess to use geoetric average Asian call option which is given by = it log S ( ) rt i= V e e as the control variate, and it ight be the best control variate we can find. Besides, European call option given by ( ( ) ) rt U = e S T is another choice. To ake things clear, we apply both to our schee. 4

29 Table : Crude Monte Carlo ethod for arithetic average Asian call options with S =, r =.5, σ =., T =, n = Accurate Option Standard Accurate geoetric values by error by European Asian call crude crude call option option Monte Monte values values Carlo Carlo Tables 3 shows the figures we get by using the control variate ethods. We can easily see that the option values by the two different control variates are approxiately the sae whereas the standard errors differ draatically. The standard deviations of the estiates with the European option as the control variate have been reduced by approxiately 5% - a very odest gain in efficiency. Observe that the introduction of the geoetric average Asian option as the control variate greatly reduces the standard deviation of the estiates, which is in agreeent with what we have guessed. For exaple the range of the 95 percent confidence liits in the case of the -tiestep Asian option with a current stock price of 9 have been reduced fro.466 to.4. To achieve the sae reduction by increasing the nuber of trials would require about 5,36, trials instead of,. The confidence liits by using the geoetric average Asian option as the control variate would appear to be sufficiently accurate for ost practical applications. 5

30 Table 3: different control variate ethods for arithetic average Asian call options with S =, r =.5, σ =., T =, n = Option Standard Option Standard values with error with values with error by control control control control variate U variate U variate V variate V It is of interest to exaine one ethod of using antithetic variates in the present proble as well. The results of one such set of calculations are displayed in Table 4. Coparison with the appropriate figures in Table together with Table 3 shows it is ore efficient when using the antithetic variate ethod than using European option as the control variate; however it only achieves a relatively low gain copared to the other control variate ethod. This low gain in efficiency ay be explained as follows. While X i and -X i have perfect negative correlation, this does not hold for the corresponding functions of the. It is not enough to effect a significant reduction in the variance of the revised estiate by such a ethod. 6

31 Table 4: antitheticl variate ethod for arithetic average Asian call options with S =, r =.5, σ =., T =, n = Option values by antithetic variate Standard error by antithetic variate Finally, siulation results in Table -4 together shows that no atter which ethod we choose aong the crude Monte Carlo, the control variate as well as the antithetic variate ethod, the relationship,, C g ( S, T) C a ( S, T) C ( S, T) always holds, which is in agreeent with what we showed in Proposition 3. and Proposition 3.. 7

32 6. Conclusion and Outlook We investigated the proble of pricing arithetic average Asian options using Monte Carlo siulation techniques. Before ipleenting such a ethod, we provided the background we ay need. In ters of Monte Carlo siulation, the option values we get are just estiators, thus finding a judicious choice of control variates to enhance the pricing perforance of siulation becoes the critical thing. Since it is quite easy to get the closed for solutions for the European option and the geoetric average Asian option, we used these two kinds of options as the control variate. Our results suggest applying the geoetric average Asian option as the control variate to the Monte Carlo approach, since it greatly iproves the standard deviation result to provide a narrower confidence interval. We also checked the antithetic variate ethod, and it turned out that this variance reduction technique was less attractive than the geoetric Asian option variate one in our case. In this paper, we have included an arguent showing that the arithetic average Asian call option has a value less than or equal to the corresponding European call. Our nuerical results show that this property holds in our siulations. A big disadvantage of using Monte Carlo ethods for path dependent options is the large nuber of calculations that are necessary to update the path dependent variables throughout the siulation. Even control variate ethods coe to the end of their capacity here. We should therefore also look for new innovations in the siulation theory, especially uasi Monte Carlo ethods as described in a financial setting by Joy, Boyle and Tan [5]. 8

33 Bibliography [] Toas Bjork. Arbitrage Theory in Continuous Tie. Oxford University Press, 4. [] P.P. Boyle. Options: A Monte Carlo Approach. Journal of Financial Econoics, 4:33-338, 977. [3] Allan Gut. Probability: A Graduate Course. Springer, 7. [4] Lishang Jiang. Matheatical Modeling and Methods of Option Pricing. World Scientific Publishing Copany, 5. [5] C.Joy, P.P. Boyle and. Seng Tan. uasi-onte carlo ethods in nuerical finance. Manageent Science, 4:96-938, 996. [6] A.G.Z. ena and A.C.F. Vorst. A Pricing Method for Options Based on Average Asset Values. Journal of Banking and Finance, 4:3-9, 99. [7] Bernt Oksendal. Stochastic Differential Equations: An Introduction with Applications. Springer, 7. [8] L.C.G. Rogers, and Z. Shi. The value of an Asian option. Journal of Applied Probability, 3:77 88, 995. [9] R.Y. Rubinstein, and D.P. roese. Siulation and the Monte Carlo Method. Wiley-Interscience, 7. [] P. Wilott, S. Howison, and J. Dewynne. The Matheatics of Financial Derivatives. Cabridge University Press,

34 Appendix. Matlab Code for 4.3 clear all %Crude Monte Carlo for European call options S=; =9; T=8; r=.5; siga=.; n=; Z=sqrt(T)*randn(n,); ST=S*exp((r-(siga^)/)*Tsiga*Z); Payoff=exp(-r*T)*ax(,ST-); C_ceu=ean(Payoff) Stderr= std(payoff)/sqrt(n) CI_95=[C_ceu-.96* Stderr,C_ceu.96* Stderr] %Antithetic variate for European call options ST_=S*exp((r-(siga^)/)*Tsiga*(-Z)); Payoff_=exp(-r*T)*ax(,ST_-); v=.5*(payoff Payoff_); C_aveu=ean(v) Stderrav= std(v)/sqrt(n) CI_95av=[ C_aveu-.96* Stderrav, C_aveu.96* Stderrav] %Black-Scholes forula for European call options d_ = (log(s/)(rsiga^/)*t)/(siga*sqrt(t)); d_= (log(s/)(r-siga^/)*t)/(siga*sqrt(t)); C_bseu= S*norcdf(d_)-*exp(-r*T)*norcdf(d_). Code for 4.4 clear all 3

35 S = ; = 9; siga =.; r =.5; T = ; Dt =.; = T/Dt; n = ; %Black-Scholes forula for European call options d_ = (log(s/)(rsiga^/)*t)/(siga*sqrt(t)); d_= (log(s/)(r-siga^/)*t)/(siga*sqrt(t)); C_bseu= S*norcdf(d_)-*exp(-r*T)*norcdf(d_) %Geoetric average Asian call options valuation by Black-Schole forula sigsqt= siga^*t*(*)/(6*6); ut =.5*sigsqT.5*(r -.5*siga^)*T; d=(log(s/) (ut.5*sigsqt))/(sqrt(sigsqt)); d=d - sqrt(sigsqt); geo=exp(-r*t)*( S*exp(uT)*norcdf(d)-*norcdf(d)); ranvec= randn(n,); Spath = S*[ones(n,),cuprod(exp((r-.5*siga^)*Dtsiga*sqrt(Dt)* ranvec),)]; ESpath= S*[ones(n,),cuprod(exp(r*Dt)*ones(n,),)]; % Crude Monte Carlo for arithetic average Asian call options arithave = ean(spath,); Parith = exp(-r*t)*ax(arithave-,); % payoffs Pean = ean(parith); Pstderr = std(parith)/sqrt(n) confc = [Pean-.96* Pstderr, Pean.96* Pstderr] % Control variate (European option) for arithetic average Asian call options europayoff = Spath(:,()); Peuro = exp(-r*t)*ax(europayoff -,); % European payoffs cov_euro=cov(parith, Peuro); c_euro=-cov_euro (,)/ cov_euro (,); E = Parith c_euro*( Peuro - C_bseu); % control variate version Eean = ean(e); Estderr = std(e)/sqrt(n); confcv = [Eean-.96* Estderr, Eean.96* Estderr] % Control variate (geoetric Asian option) for arithetic average Asian call options 3

36 geoave = exp((/())*su(log(spath),)); Pgeo = exp(-r*t)*ax(geoave-,); % geo payoffs cov_geo=cov(parith, Pgeo) c_geo=-cov_geo(,)/ cov_geo(,) Z = Parith c_geo*( Pgeo-geo); % control variate version Zean = ean(z); Zstderr = std(z)/sqrt(n); confcv_ = [Zean-.96* Zstderr, Zean.96* Zstderr] %Antithetic variate for arithetic average Asian call options Spath_= S*[ones(n,),cuprod(exp((r-.5*siga^)*Dtsiga*sqrt(Dt)* (-ranvec)),)]; Uarithave = ean(spath_,); Aarith =.5*exp(-r*T)*(ax(arithave-,) ax(uarithave-,)); % payoffs Aean = ean(aarith); Astderr = std(aarith)/sqrt(n) confav = [Aean-.96* Astderr, Aean.96* Astderr] 3