- May 17, 2020

Final Exam FE621 Posted May 11, 2020, Due May 18, 2020 at 9:00am Name: • Please note: for the problems using data we are not interested in all the minor steps performed when analyzing the data. However, you should explain your broad reasoning and hand in a final report. • Do not include any failed attempts at modeling just to give volume to the final report. • You need to argument with graphs or selected numerical results all the conclusions you draw. Put important plots and tables within the report and relegate any non essential ones to an appendix at the end of the document. • Please add supporting arguments (e.g., output) in the appendix. The whole report should be constructed as a regular journal article. • The final write-up (excluding the appendix) should not be more than 10-15 pages. • Communication with other students either physical or virtual is strictly forbidden. For instructor’s use only Problem Points Score A 100 B 100 C 100 Bonus 1 5 Bonus 2 25 Total 300 1 Problem A: Asian Option Pricing using Monte Carlo Control Variate. The payoff of an arithmetic Asian call option is:( 1 N + 1 N∑ i=0 Sti −K ) + . Its value may be computed using straight Monte Carlo simulations. However, in order to obtain a small standard error, the number of simulations must be very high. To solve this computationally extensive problem, we will use the payoff of a geometric Asian call option as the control variate:( N∏ i=0 Sti ) 1 N+1 −K + The idea is to use the known analytic price of the geometric Asian and the distance between MC simulations to obtain an approximate for the analytical formula for the arithmetic Asian price. In this problem we consider r = .25%, σ = 0.4, S0 = 100, and assume the goal is to price an arithmetic Asian call option with strike K = 100 and maturity T = 5. We also assume the asset follows the standard log-normal/geometric Brow- nian motion model: S(∆t) = S(0)e((µ− σ2 2 )∆t+(σ √ ∆t)) (a) The price of a geometric Asian option in the Black-Scholes model is given by: Pg = e −rT (S0eρTN(d1)−KN(d2)) where: ρ = 1 2 ( r − 1 2 σ2 + σˆ2 ) σˆ := σ √ 2N + 1 6(N + 1) such that σˆ is adjusted sigma and N is the total number of trading days (T ∗ 252). d1 := 1√ T σˆ ( ln ( S0 K ) + ( ρ+ 1 2 σˆ2 ) T ) d2 := 1√ T σˆ ( ln ( S0 K ) + ( ρ− 1 2 σˆ2 ) T ) Use the above formula to price this geometric Asian call option. 2 (b) Implement a Monte Carlo scheme to price an arithmetic Asian call option (P sima ). Use M = 1, 000, 000 simulations. Record the answer, a confidence interval and the time it takes to obtain the result. (c) Implement a Monte Carlo scheme to price a geometric Asian Call option (P simg ). (d) Using M = 10, 000 simulations and the same exact random variables create: • numbers Xi which are M replications for the arithmetic Asian Option price • numbers Yi which are M replication for the geometric Asian Option price Finally calculate b∗ such that: b∗ = ∑M i=1(Xi −X)(Yi − Y )∑M i=1(Xi −X)2 Note that b∗ is actually the slope of a regression line Y = a+bX+ε. Please also record the price of the arithmetic P sima and the geometric P sim g . (e) Calculate the error of pricing for the geometric Asian: Eg = Pg − P simg (f) Calculate the modified arithmetic option price (P ∗a ) as: P ∗a = P sim a − b∗Eg Compare with the results in (b). Comment. Vary the value of M in part (d). What do you observe. 3 Problem B. Using fractional Brownian motion in finance. For this problem we will look at a process called the fractional Brownian motion (fBm). For a simple introduction please see [1]. The formal definition and properties of the fBM are given on page 6 in the thesis [1]. For a deeper understanding of the calculus needed to work with fBm please see Eulalia Nualart notes [2]. You will not need to have a deep understanding of the topic for this problem. Just the definitions provided are fine. (a) Simulate a fBM using the Davies and Harte method, see section 2.1.3 in [1] and pay special attention to the algorithm presented on page 17. Plot four sample paths, each having different values of the Hurst parameter: H ∈ {0.2, 0.5, 0.6, 0.9}. Consider T = 1 with n = 2000 time steps. You may use any package or toolbox that returns the FFT of a given vector. Please describe the way in which the sample path changes with H. Bonus 1 Use the python package described in [3]. This is a package written by one of our Ph.D. students which implements several methods of generating fBm paths. This part is optional as it needs python and we do not know if python code is accessible from other languages. Check the answers you obtained in part (a) with the package results. (b) At the beginning of 2000’s the observed dependence structure of log–returns was modeled using an fBm process. Specifically, consider a regular geometric Brownian motion model, with Wt replaced by an fBm Zt, having E[Z2t ] = t2H : dSt = St(rdt+ σdZt), S(0) = S0. Please note that when H = 12 the fBm becomes a standard Brownian mo- tion. Thus, this model is an extension of the Black-Scholes-Merton settings. Price a European call option under this model, with S0 = K = 100, T = 1, r = 5%, and σ = 20%. Use a Monte Carlo simulation with an Euler discretization. Consider at least 1 million paths and four different values of the Hurst parameter, i.e. H ∈ {0.2, 0.5, 0.6, 0.9}. Compare with the result obtained using the classical Black–Scholes model. Please describe your observations. Bonus 2 Implement or use an implementation to estimate the Hurst parameter using the Rescaled range analysis (R/S method). This is described in section 3.1.7 of the thesis [1], but it is a straightforward methods that even has a wikipedia page. There are numerous packages in many programming languages that implement this method but you can also implement it if you wish. Use the function you created in (a) or the Bonus 1 function to generate fBm paths with a known H. Use the R/S implementation to estimate H. Compare the values (true and estimate) to verify your code is correct. Download historical (daily) closing prices for an equity picked by you for 24 months. Report the estimated Hurst parameter value and comment. 4 Problem C. New Inverse Transform Scheme for Simulating Random Variables. In class, we have discussed the inverse transform method for simu- lating a random variable whose cumulative distribution function (CDF) is avail- able. Except in some special cases, the CDF is not invertible in closed-form, and numerical methods have to be employed. In this problem, we shall consider a new method to simulate the random variable. (a) Given the CDF F (·) for the random variable X, the inverse transform method is as follows: simulate a standard uniform random variable U , then X = F−1(U) will have the desired distribution, and thus is a valid sam- ple. Consider the exponential random variable X with probability density function (pdf) given by f(x) = λe−λx, x > 0. Take λ = 0.6, use the inverse transform method to simulate 100 samples from the exponential distribu- tion, and then plot the histogram of the simulated sample and compare against the exact pdf f(x) = λe−λx on the same graph. (b) Consider the standard normal distribution X ∼ N (0, 1), and use the inverse transform method to simulate 100 samples from it. Plot the histogram of the simulated samples and compare against the exact pdf of the normal distribution. Hint: the cdf of the standard normal distribution can be found in most programming languages, or you can numerically approximate the cdf by approximating the integral of the pdf through a quadrature method of your choice. (c) (Bonus) We consider the new approach1, for the applications to computing implied volatility. You do not need to understand all the details of the derivation. In this question, you are asked to implement the following formula: X = F−1(U) = lim ε→+0 1 2 √ piε ∫ −∞ e− (F (s)−y)2 4ε sf(s)ds, (1) where f(·) is the pdf. Consider the standard normal distribution X ∼ N (0, 1), and implement the above expression (1) to simulate samples of X. In the implementation, you shall take a ε > 0 that is very small and close to zero. For example, you can take ε = 0.01. For the numerical integration above, you can choose an appropriate quadrature method, such as the trapezoidal rule, or the Gauss-Hermite quadrature rule, etc. 1It is proposed in https://www.researchgate.net/profile/Zhenyu_ Cui/publication/340570747_A_Closed-form_Model-free_Implied_Volatility_ Formula_through_Delta_Sequences/links/5e912d3f92851c2f5294bdb1/ A-Closed-form-Model-free-Implied-Volatility-Formula-through-Delta-Sequences.pdf 5 References [1] Dieker, Ton. Simulation of fractional Brownian motion. M.Sc. theses, University of Twente, Amsterdam, The Netherlands (2004). [2] Nualart, Eulalia Lectures on Malliavin calculus and its applications to finance Lecture Notes, 2009 http://www.math.wisc.edu/~kurtz/ NualartLectureNotes.pdf [3] Flynn, Christopher fbm 0.3.0, a Python package 2016 https://pypi. org/project/fbm/ 6