AbstractPricing Asian options in the case of a discrete time model is quite a simple task. On the other hand, calculating the price of Asian options in the case of continuous time model becomes complicated because of summation of dependent random variables with an uncommon distribution function.

This essay will seek to make it easier to calculate pricing of Asian option with respect to continuous time models. In the same line, this essay will define a comonotonic random variable for an inverse Cumulative Distribution Function (CDF) as well as their main properties on the foundation of research conducted by Dhaene et al. Other theorems explained by Dhaene et al will add value to this study in determining the actual value of Asian Options.Keywords: Comonotonic, random variables, continuous time, Asian options, TheoremContentsPricing Asian Options under Continuous Time Model 1Abstract 12. My Second Section 32.1. Similar to section above, following lecturer feedback 62.

2. Theorems of Derivation of lower bound 1 from Albrecher et al 92.3. Proof of proposition in Albrecher et al 102.4. LB_1 is very close to the true value of the arithmetic Asian Option 11References 15 2. My Second SectionComonotonic Random VectorFor a random variable of type V = such that all are mutually dependent, and the distribution of random vector is unspecific and the marginal distribution for is known. = 3, p.

16Therefore, to make a decision, we define the dependence of the random vector which develops an unfavorable aggregate claim V, using the provided marginal distribution.However, using a joint distribution with possibly the largest sum yields random vector with a comonotonic distribution that provides two possible outcomes and . 3, p.16Therefore, for a set H (a thin set with all dimensions 1) of -vectors in is comonotonic provided that it satisfies the following properties:1. The set H is comonotonic given that for and in H, either or holds. 3, p.16Therefore, a set is comonotonically increasing in each component and given that for any . A subset of a comonotonic set is also comonotonic.

A set H within (,) is denoted as and defined as = {(, ) ? H 3, p.162. The set H is comonotonic defined as is comonotonic for in {1, 2, …, n}3, p.163. The random vector = is comonotonic given that it has comonotonic support. Comonotonic distributions are characterized by high dependence. 3, p.

174. For = , () = min {), ), …, )} 3, p.175. Uniform distribution U ~ Uniform (0, 1) we get that ((U), (U), (U), …, (U));3, p.

17Definition of an inverse cumulative distribution function and its key propertiesThe Cumulative Distribution Function of a random variable is F (X) = PX ? xF (X) = PX ? x is a right-continuous increasing function such that, 3, p.12 (+ 3, p.12The Inverse Distribution Function is comonotonically increasing and left continuous function such that inf { x }, r 0, 1 3, p. 12By definition, inf =+ for all x ? r 0, 1, 3, p. 12then 3, p. 12 We can define inverses of distribution functions for in r such that r 0, 1 and r is a value in the closed interval 0, 1.

3, p. 12Therefore: inf {x ? ? ? (x) ? r}, sup {x ? ? ? (x) ? r}, whereby, inf = + and sup = – 3, p. 12The value of the inverse cdf at r is (r) for the left hand dimension of the intervalThe value of the inverse cdf at r is (r) for the right hand dimension of the borderThen, given that r 0, 1 is a monotonically increasing right continuous function(r) = sup {x ? ? ? (x) ? r} 3, p.12The probability mass for yields (0)=-? and (1)=+? for the interval (0), (1) (r) and (r) are finite values for all r (0, 1). 3, p. 13Any variable d 0, 1, we can define the d-mixed inverse function of such that:(r) = d (r) + (1-d)(r), r (0, 1), which is a monotonically increasing function.

3, p. 13If d = 0, (r) = (r) and (r) = (r). Then, d ? (0, 1), 3, p.

13 (r) ? (r) ? (r), r ? (0, 1) 3, p. 13Proof of Lemma 2Lemma 2 states that the set H is comonotonically defined as is such that it is comonotonic for in {1, 2, …, n}For random variables and and If g is a monotonically increasing left continuous function, (r) = g ( (r)) 3, p. 13If g is a monotonically increasing right continuous function, (r) = g ( (r)) 3, p. 13If g is a monotonically decreasing left continuous function, (r) = g ( (1- r)) 3, p. 14If g is monotonically decreasing and right continuous function, (r) = g ( (1- r)) 3, p. 14 Support BUnder the assumption that X has comonotonic support B, x ? and be such that= {y ? B ? ? }, for j= 1, 2, …, n. 3, p.18Based on comonotonicity of B, i is such that = 3, p.

18Hence, (x) = Pr (X ?) = Pr (X ? ) = () 3, p. 18 = min {(), (),… ()} 3, p. 18= so that () ? () holds for any j. 3, p. 18Since (x) = min {(), (),… ()} 3, p.

18we get from 3, p. 18that Pr (U) , , …, (U) 3, p. 18PrU (), (), …, () 3, p. 18=PrU ()} 3, p. 18= ()} 3, p.

182.1. Similar to section above, following lecturer feedbackA set H ?n is comonotonic given that is comonotonic for in {1, 2, 3, …, n}3, p.16According to feedback, the derivation from the definition of comonotonicity is as seen above, for (1) in Theorem 2. A comonotonic random vector is such that = {() ? H} a set H ?n is comonotonic give that is comonotonic for any in {1, 2, 3, …, n} 3, p.

16In (2), considering comonotonic support B,= {y ? B ? ? }, 3, p.16such that j= 1, 2, …, n for ?n because B is a comonotonic set which is non-decreasing, we find i exists such that = 3, p.16Hence, () = Pr ( = Pr ( ) = () min {(), (), (), …, ()} 3, p.

16 so that () ? () holds for all values of For (3), section 2, we get that () = Pr ( 3, p.16= Pr ( ) = () and=Pr (U) , …, (U) 3, p.16= Pr (), …, U ()= Pr U ()} 3, p.

16=()} 3, p.16For and a non- increasing function and left continuous function g, so, for any real variable x, we find from . Since g is left continuous, we have g(z) x z sup { y ? g(y) x} holds for all real values of z and x.Therefore, (1- r) (X) r sup {y ? g(y) x} 3, p. 15If sup {y ? g(y) x} is finite we find from 3, p.15and (1- r) (X) (1- r) sup {y ? g(y) x} that (1- r) sup {y ? g(y) x} sup {y ? g(y) x} 3, p.

15However, if sup {y ? g(y) x} is +? or -? we cannot use it is possible to verify that(1- r) (X) (1- r) sup {y ? g(y) x} holds. 3, p.15If the sup equals -?, then (1- r) (X) (1- r) sup {y ? g(y) x} becomes r r -? 3, p. 15If the sup equals +?, then r (X) r sup {y ? g(y) x} becomes r r +? 3, p. 15Now that g is a non-increasing and left continuous, the results is that?sup {y ? g(y)}g()? x 3, p. 15Combining the equations gives us ?x g()? x for all x meaning that (1- r) = g ( (1- r)) holds. 3, p.

15Again, for and a non-increasing right continuous function g such that for any real variable x, we find from . Since g is left continuous, we have g(z) x z sup { y ? g(y) x} holds for all real values of z and x. Therefore, (1- p) (X) r Fx sup {y ? g(y) x} 3, p. 15If sup {y ? g(y) x} is finite we find from 3, p. 15and (1- ) (X) (1- r) sup {y ? g(y) x} 3, p. 15that (1- ) sup {y ? g(y) x} sup {y ? g(y) x} 3, p. 15However, if sup {y ? g(y) x} is +? or -? we cannot use but we can verifythat(1- ) (X) (1- ) sup {y ? g(y) x} holds.

3, p. 15If the sup equals -?, then(1- ) (X) (1- ) sup {y ? g(y) x} becomes -? 3, p. 15If the sup equals +?, then p (X) p sup {y ? g(y) x} becomes +? 3, p. 15Now that g is a monotonically increasing and left continuous, the results is that?sup {y ? g(y)?}g()? x 3, p.

15Combining the equations gives us ?x g ()? x for all x meaning that (1- ) = g ( (1- )) holds. 3, p. 152.2. Theorems of Derivation of lower bound 1 from Albrecher et alAccording to Dhaene et al, these theorems are required in deriving Lower Bound 1 are:i) Using the inverse function the cumulative distribution function of = + + … + 3, p.28 () = sup {() } 3, p.28=sup {() } 3, p.28=sup { } 3, p.

28ii) For (0) the probability of ()(()) =, (0) () 3, p.29iii) + = ( – (())+, ((0), (1)) 3, p.31Equation (i) is used to derive cdf of follows such that() = sup { } 3, p.

44Equation (ii) ((0), (1)), 3, p.45 (()) = , or ? ? = (1 – ()) = 3, p.45Equation (iii) is used to determine the stop-loss premiums of E ( – )+ = ? ? – E ? ? = (1 – ()))+In mathematics and finance, it is possible to derive lower bound for sums of random variables and present value functions. In Dhaene et al, Theorem 6 and 7 require derivation of lower bound1.Theorem 6 states that the inverse distribution function of a sum of comonotonic random variables (, , , …, ) derived by:() = (), , 3, p.26Calculation of lower bound quantiles for Theorem 6() = () = () = = , ? (0, 1)2.3. Proof of proposition in Albrecher et alThe process , t ? 0 is a standard Brownian motionGiven that is a cumulative logreturn for a risky asset, then its price = {, t ? 0} in the time interval n, n+t, n, t ? 0 the increment becomes – which is normally distributed.

Since, Brownian motion is defined as for a probability space (k, l, m), for every (k, l, m) given a continuous function V(t) in t ? 0, that gives V(0) = 0 and depends on , then V(t), t ? 0, is a Brownian motion for all 0 = < < < … < the increments V() = V() - V(), V() - V()… V() – V () 2, p.93The process , t ? 0 is a Levy process and the monitoring times and t are integersFor, to be defined as a levy process, it has to satisfy these characteristics of a Levy process. 1. ,2.

have independent increments for that do not overlap for any .3. have stationary increments for that do not overlap for any .4.

have a continuous probability of 2.4. LB_1 is very close to the true value of the arithmetic Asian OptionMonte Carlo is one of the most widely used methods of analysis of Asian Options. The control variate estimator has a normally distributed error with variance, ?2 as shown by the equation below 4, p9. The central limit theorem affirms that as the number of sample trials increases, n??, the standard estimator converges the distribution to a normal curve.Cn – C ? N (0, ?2C /n) 4, p.10In the equation above, the probability the interval approaches true value C approximates 1-?; it depicts the distribution of the error in the simulation estimate. Precision in unbiased estimates and attainment of narrower confidence interval is achieved by lowering the error hence the variance.

The result is closer approach to the true value. Monte Carlo simulations are done in Matlab where the statistical summary of a sample distribution is derived and analysed.The Matlab code used is as follows:%% Monte Carlo Simulation in Matlab % An example showing that the lower bound value LB_1 in Monte Carlo methods% is approximately equal to the Arithmetic Asian Option as is the case of % the Black-Scholes model:% Sample Function : m^2/nx% Generate nSamp samples from a normal distribution% % r= a+rand(n,1)*(b-a)% where a is the minimum value % while b is the maximum value for variables.%%nSamp=100000;b= 11;a= 1; n=(randn(nSamp,1)*5)+100;m=5+rand(nSamp,1)*(b-a);%gives a random number btw zero and one.St=m.^2.

/n;hist(St,100);%plot a histogram for 100 simulationsxlabel (‘Number of Simulations’,’fontsize’,20)ylabel (‘Asian Option Price’,’fontsize’,20)title (‘Standard Monte Carlo’,’fontsize’,20) %calculation of summary statisticsSt_mean= mean(St)St_standardDeviation= std(St)St_median= median(St) RESULTSSt_mean = 1.0867St_standardDeviation = 0.5877St_median = 1.0043 Figure 1: A histogram showing the standard Monte Carlo Simulations for pricing Asian Options.

The sample size n= 100000, sample mean u = 1.0867 and a standard deviation s=0.5877 together give a confidence interval of not exceeding 1.6 4, p.10.

Hence, fulfilling central limit theorem and the fact that the value of standard deviation should be low as possible for it to be close to the true value.