BMAN 70141: Derivative Securities
Coursework Assignment: First Semester 2024/25
Short Summary:
In this coursework assignment, you are meant to value derivatives using binomial trees, Monte-Carlo simulations, the finite difference method, and the Longstaff-Schwartz least-squares approach. To this end, assume the following set of baseline parameters: The initial stock price (S0) is 35, the stock vola- tility is 0.45 (45% per annum), and the risk-free rate is 0.03 (3% per annum). Consider a European put option whose strike price is equal to 35, with a time-to-maturity of one year. The dividend yield is 0.04 (4% per annum). In some later tasks, you also encounter the “ equivalent” American option.
Task 1 (Black-Scholes) - 10% of marks:
Use the Black-Scholes (1973) formula to calculate the value of the option. Use put-call parity to cal- culate the value of the “equivalent” call option (same underlying, strike price, and maturity date).
Task 2 (Binomial Tree) - 10% of marks:
Use a simple one-step binomial tree to value the above European put option. Set u equal to the expo- nential of sigma times the square root of the maturity time divided by the number of time steps (i.e., exp(σ(T/n)1/2) and d equal to the exponential of minus one times sigma times the square root of the maturity time divided by the number of time steps (i.e., exp(-σ(T/n)1/2) .
After that, increase the number of time steps (n) and repeat the binomial tree value calculation. So, for example, use a binomial tree with two, then three, then four, then five, then six, etc. time steps to value the option. I’ll leave it up to you how many binomial trees you use in your calculations. I would expect that the average group does binomial trees with one-, two-, three-, four-, and five-time steps. But the sky is the limit, and the more binomial trees you calculate the better for your mark.
Don’t show each and every binomial tree you do in the coursework assignment. Showing one tree is certainly enough for illustrative purposes. Keep the page limit in mind!
Plot the value of the European put option (y-axis) against the number of time steps used in the binomial tree (x-axis). Also plot the Black-Scholes value (e.g., as horizontal line) in the graph. Do more time steps render the binomial tree value closer to the Black-Scholes (1973) value? Explain in intuitive language why this is the case (you would want to look at your textbook).
Task 3 (More Complicated Binomial Tree) - 10% of marks (NEW):
Now assume that you want to value an American floating look-back put option, whose maturity payoff is the difference between the maximum stock price to that date and the current stock price (see the slides 17 and 18 in the lecture on “More on Models and Numerical Procedures”). To do so, again use a binomial tree approach. As using a binomial tree approach to value American floating look-back calls quickly becomes rather complicated, I am okay with you relying on a smaller binomial tree than in the prior task. That said, I want you to use at least a three-step tree, and, as always, the larger your binomial tree, the better for your mark (assuming you do the calculations correctly).
Task 4 (Monte Carlo) - 20% of marks:
4a. Use Monte Carlo simulation to value the European put option assuming that the underlying asset value obeys a Geometric Brownian motion. To do so, you can directly simulate the value of the stock at maturity (as we did in the lecture) – you don’t need to simulate the whole path. I leave it up to you how many maturity stock prices you generate. The more, the better (as always).
4b. Use Monte Carlo simulation to value the European put option assuming that the underlying asset value and the risk-free rate of return obey correlated Geometric Brownian motions. To do so, you can directly simulate the values of two independent Brownian motions on the maturity date. You can then combine these to obtain the values of two correlated Brownian motions on the same date. Using the values of the correlated Brownian motions, you can then calculate the values of two correlated Geo- metric Brownian motions. I leave it up to you to determine what the current value and the mean drift and volatility of the risk-free rate of return are (you could estimate those moments using data) . I also leave it up to you to decide what the correlation between the underlying asset value and the risk-free rate of return is. But be reasonable. The current value should probably be close to 0.01, the mean drift close to zero, and the volatility should be really small. Motivated students could try to induce some mean reversion into the risk-free rate drift (google “mean-reverting process,” e.g.) .
Remark: When the interest rate is stochastic, the discount factor is no longer e-rT, where r is the interest rate and T the time-to-maturity. Rather, the input argument of the exponential function must now be minus one times the integral of the interest rate at time t times the differential of time, − ∫T0 r(t)dt. If you simulate interest rate paths at the daily frequency, you can easily approximate that integral by the finite sum: − ∑t0 r(t) (
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) (i.e., minus one times the sum of the scaled daily interest rates).
4c. Use Monte-Carlo simulation to value the exotic-option counterpart of the plain-vanilla European put option. The exotic option has a payoff of max(K-S(T),0) at maturity if and only if the stock price before maturity never dropped below 30. If it dropped below 30 before maturity, the option payoff at maturity is equal to zero. This type of option is called a down-and-output option. To value the exotic option, you have to simulate the whole path of the stock price before maturity. I leave the time step and other details up to you to decide. Keep the number of simulated stock price paths reasonable (e.g., 10-20,000 seems a good number). You could simulate the different stock price paths in the columns of an Excel tab, with the rows indicating the different points in time.
Motivated students could consider using one or two of the variance-reduction techniques discussed in the Monte Carlo lecture, in tasks 3a, 3b, and 3c. But this is not a requirement.
Motivated students could further search the academic literature for the closed-form. valuation solution for the exotic option, to study how accurate their Monte-Carlo simulation exercise is.
Task 5 (Alternative Stochastic Processes) - 20% of marks:
Choose one of the stochastic processes assumed by the alternative (non-Black-Scholes) option valua- tion models. For example, you could pick the mixed jump-diffusion or the constant elasticity of vari- ance (CEV) model process. Alternatively, why not let volatility vary stochastically over time, as in, for example, the Heston (1993) model? Select some reasonable parameter values for the process that you choose. Then repeat the Monte Carlo simulations in Task 3 using the new process, valuing first the plain-vanilla option and next the more complicated exotic option.
Carefully compare the values of the options under the new stochastic process with those derived under the standard Geometric Brownian motion process in Task 3. Explain why the values of the options are either higher or lower compared to those derived earlier.
Task 6 (Finite Difference Method) - 10% of marks:
Use the finite difference method to value the (plain-vanilla) American put option. I leave it up to you to decide whether you want to use the implicit or explicit method. Motivated groups may consider using both and comparing outcomes. I also leave it up to you how many stock price- and time-steps you use in your grid (but go for more than used in the lecture). I would, however, advise you to simulate the stock price, not the log of the stock price – that’s more intuitive and thus a lot easier.
Task 7 (Longstaff-Schwartz Least-Squares) - 20% marks:
Use the Longstaff-Schwartz least-squares method to value the (plain-vanilla) American put option. I again leave it up to you to decide how many simulations of the stock price paths you do and how many time steps you have. If you were able to programme, you could think about using a regression on not only the stock price, but also higher order terms (the stock price squared, cubed, etc.) – I didn’t manage to find out how to do so in Excel. In any case, if your Longstaff-Schwartz estimate deviates signifi- cantly from that obtained using the finite-difference method, don’t panic. It’s bound to do so unless you use a gigantic number of iterations and time steps and a flexible regression function.
Design & Presentation:
The coursework assignment should clearly discuss how you solved each task. Be specific, notrepeti- tive. The reader should be able to replicate your results following your discussions.
The length of the essay should probably be 8,000-10,000 words, approximately 15 to 18 pages of text in Times New Roman, font size 12, single-spaced. But I don’t really care about the length of the as- signment as long as it is concise and up-to-the-point. If you have interesting material to report – and you don’t keep repeating yourself, you can go above 10,000 words. But really only if the material is sufficiently interesting. If you “waffle” or repeat yourself,I will subtract marks.
The essay should be clearly structured and should have a professional design. Mind:
1. Each coursework assignment should have a front page, stating students’ IDs, the title of the assignment, the date the assignment was completed, and other relevant information. The main text must be consistently formatted, that is, avoid modifying the font, size, and colour. Use page numbers. You could have a short table of contents or an abstract at the beginning. At the end, you should have a reference list, which must be consistently formatted.
2. The coursework assignment must be clearly and logically structured. Having only six sections called Task 1, Task 2, Task 3, Task 4 , etc. is not appropriate. At the very least, you must have an introduction and a conclusion. The introduction should state what the reader can expect from the CWA (e.g., you could write: “In this CWA, we offer the solutions to six derivative valuation tasks”). Shortly describe the six tasks. State the base-case parameters (here!). The conclusion can be super-short. The main text can largely follow the six tasks.
3. In case of each task, first explain to the reader what you do (“In this section, we use the Black - Scholes (1973) formula to value…”). Then give the reader some brief background material (“The
Black-Scholes model is derived from the assumptions…”). You could also say something about how the closed-form solution (if any) is derived (“Using those assumptions, Black and Scholes (1973) then use contingent claims analysis to…”). But be non-technical – I don’t want to see mathematics here – and be short. We really only talk about a couple of sentences. These sentences are meant to “set the stage.” Far too many groups immediately jump into action, without indicting to the reader what they do, why they do it, and why it makes sense,
4. Numbers should be rounded to two or three digits , e.g. round 1.23456789 to 1.23 or 1.234.
5. While no requirement, students can produce up to ten tables or figures for this coursework as- signment if they deem this helpful. However, they should produce appealing tables with hori- zontal (no vertical) lines and indicate which cell contains what sort of information. You could, for example, prepare the tables in Excel and then use ‘copy/paste special’ to insert them as a picture into Word. Number both tables and figures, e.g. ‘Table 1: Total Exports’ or ‘Figure 3: Government Deficit’ . Write a short explanation, explaining what the table shows. Take a look at the table at the end of this document. That’s how a neat and appealing table should look like. If your table looks different from that table, you’re doing something wrong.
6. The coursework assignment should be kept as short and concise as possible.
Note that the items in boldface are the minimum style. requirements.