MATH40082 Computational Finance Assignment No. 1: Monte Carlo Methods
MATH40082 (Computational Finance) Assignment No. 1: Monte Carlo Methods
Version 10747214
1 Background 1.1 Stock Options
The trader has calibrated a specialised risk neutral process for some underlying stock price. Given the current stock is S0, market prices indicate the risk-neutral distribution of the stock price at time t is given by:
St ∼N(f(S0,t),v2(S0,t)t) (1) 2
for some calibrated functions f and v .
Consider a European put option P(S,t) written on the underlying asset S. The terminal condition on
the contract is given by
and it has the analytic solution
P (S, T ) = max(X − S, 0). (2) √12
P(S0,t=0)= XN(z)+v(S0,T) T√ e−z /2 −f(S0,T)N(z) e−rT, (3)
2π where N(z) is the standard cumulative normal distribution and
X −f(S0,T) z=√. v(S0,T) T
To carry out a Monte Carlo valuation of the put option, we may use samples from a standard random
normal distribution to write the equation
φ ∼ N(0,1) (4) √ ST =f(S0,T)+v(S0,T) Tφ.
(5) Then we can simply average out the discounted payoff over n to get our approximation to the value of the
put option:
1.2 Path Dependent Options
P(S0,t = 0) ≈ n . (6)
Now assume that the risk neutral stochastic process follows the SDE dS = f (S, t)dt + v(S, t)dW.
The path dependent options you will be pricing depends on S(tk) which are the share prices at K +1 equally spaced sampling times t0, t1,..., tk = k∆t, ..., tK with t0 = 0, tK = T and
T ∆t = K .
Unlike the first part, the computation cannot proceed from t = 0 to t = T in one step and the resulting distribution of S will no longer be normal. Full details are given the the lecture notes - but you should use an Euler type scheme to write
√ Si(tk)=Si(tk−1)+f(Si(tk−1),tk−1)∆t+v(Si(tk−1),tk−1) ∆tφi,k−1
for k = 1, 2, ..., K to estimate the underlying asset values at each time. Here φi,k is a random draw from a Normal distribution. It is very important to make sure that each φi,k is independent for k = 0, 1, ..., K − 1 and fixed i, i.e. the random numbers along the same path.
For path dependent options, the payoff function can be written
g(S(t0),...,S(tk),...,S(tK))
so the value of the path dependent option can be approximated by
Asian Option
V(S(t0),t=0)≈e−rT ?g(Si(t0),...,Si(tk),...,Si(tK)). (7)
Assume that a discretely sampled Asian option has a payoff depending on the discretely sampled average given by
Then we can write
1 ?K A=K S(tk). k=1
g(S(t0), ..., S(tK )) = G(S(tK ), A),
where G(S, A) is the payoff function depending the type of option.
There are different classes of Asian option, resulting in different payoff conditions. In this coursework we
look at simple European style call or put options. A fixed strike call option will have the payoff G(S, A) = max(A − X, 0)
where X is the strike price and a floating strike call option would be G(S, A) = max(S − A, 0).
where A is sometimes called the average strike price. A fixed strike put option will have the payoff
G(S, A) = max(X − A, 0) where X is the strike price and a floating strike put option would be
G(S, A) = max(A − S, 0).
where A is the strike price. Lookback Option
The discretely sampled Lookback option has a payoff depending on the discretely sampled maximum or minimum given by
A = maxS(tk), k 2 or
Then we can write
A = minS(tk). k
g(S(t0), ..., S(tK )) = G(S(tK ), A),
where G(S, A) is the payoff function depending the type of option.
There are different classes of Lookback option, resulting in different payoff conditions. In this coursework
we look at simple European style call or put options. We can either have a floating strike S or a fixed strike X. For example a floating strike Lookback call option would give
G(S, A) = max(S − A, 0)
where A must be the minimum, and a floating strike Lookback put option would be
G(S, A) = max(A − S, 0).
where A must be the maximum.
A fixed strike call option will have the payoff
G(S, A) = max(A − X, 0)
where X is the strike price and A must be the maximum. and a fixed strike put option will have the payoff G(S, A) = max(X − A, 0)
where X is the strike price and A must be the minimum. Barrier Options
The discretely sampled knock-out barrier option will be knocked out (and return a value of zero) if the a barrier asset price B is crossed before the maturity date.
The option will be an “up” option if the knock out condition is on S > B, or a “down” option if the condition is on S < B.
Let the variable A be a binary variable such that
Then we can write
A = 0 if the barrier is crossed at least once. g(S(t0), ..., S(tK )) = A · G(S(tK )),
where G(S) is the payoff function depending the type of option.
So for example an up-and-out knockout barrier call option has the payoff
where
G(S) = max(S − X, 0)0 if S(tk)>Bforanyk=1,2,...,K
A = 1 otherwise
and a down-and-out knockout barrier put option will be
where
G(S) = max(X − S, 0) 0 if S(tk)<Bforanyk=1,2,...,K A = 1 otherwise
2 Tasks
2.1 Stock Options
You must value a put option on an underlying asset S where the distribution of the stock price at time T is given by where
αT βT αT βT √ 1 γ√ ST =S0(2−e −e )+θ(1+e −e )+σ 1−αT3(S0 +2θ) Tφ
φ ∼ N(0,1)
The stock price at time t = 0 is S0 = 2996.16, and the risk-free interest rate is r = 0.03. According to the contract, the option matures at T = 0.75 with a strike price of X = 3000. The market fitted parameters are θ = 2990, α = 0.02, β = 0.06, γ = 0.94, and the volatility of the option is σ = 0.24.
- Write a program that uses Monte Carlo simulation (6) to approximate the value of the European put option P and calculate the option value using the analytic solution (3) both with the parameters stated above. You should state in your report the value of the option using the analytic solution, and your best estimate for the value of the put option using a Monte Carlo approximation. You need only include the code in the appendix of your report. (Coding 3 marks)
- Plot a single figure showing your Monte Carlo approximate of the option value P(S0,t = 0) with increasing n (n = 1000, 2000, ..., 50000, or more!) alongside the exact value from the analytical formula. Comment on the appearance and accuracy of your result. (Understanding 6 marks)
- Write a new program (or programs) that use antithetic variables, moment matching and/or Halton sequences to approximate the put option P and investage how the accuracy of the approximation changes with different value of n and how long each calculation takes. You should produce at most 4 plots or tables of your results, and write about them in the text to justify how the methods rank against each other in terms of their efficiency. (Understanding 4 marks, Originality/Initiative 4 marks)
2.2 Path Dependent Options
Now assume that our underlying asset follows the risk neutral process:
dS = (αθ − βS)dt + σ(|S|)γ dW (8)
where W is a Wiener process and α, β, γ, θ, and σ are all constant parameters determined by the market. Your task is to price a discrete minimum fixed-strike lookback put option V with the following parameters. As before, the risk free interest rate is r = 0.03, the market parameters are θ = 2990, α = 0.02, β = 0.06, γ = 0.94 and volatility is σ = 0.24. The stock price is currently S0 = 2996.16, the option matures at T = 0.5 with K = 50 equally spaced observations after the initial time. The fixed strike price is X = 3000. We do not have an analytic formula to price this option.
- Write a program to approximate the path dependent option V using (7) with the model and parameme- ters as stated above. You should state in the report your best estimate for value of the option using a Monte Carlo approximation. You need only include the code in the appendix of your report. (Coding 2 marks)
- Use your program to generate approximations to the path dependent option V with different values of n. With the outputs from your calculations you should generate at most 2 plots or tables to investigate the convergence of your result. Comment on your results, what value are they converging towards? (Understanding 6 marks)
• Finally, estimate the value of the partial derivative ∂V at S0 and t = 0 using Monte Carlo simulations. ∂γ
You should be able to generate values of V(S0,t = 0;γ = 0.94) and V(S0,t = 0;γ = 0.94+dγ), so a simple finite difference formula can approximate the derivative:
∂V ≈V(S0,t=0;γ=0.94+dγ)−V(S0,t=0;γ=0.94) ∂γ dγ
What type of Monte Carlo simulation, what values of N, and what values of dγ give the most accurate estimate of the derivative? Explain any problems you may find using this formula. State your estimate for the value in the text, you should include up to 2 plots or tables of results to support your result. (Understanding 4 marks, Originality/Initiative 6 marks)
3 Instructions
The deadline for this assignment is 11am on Monday 20th March, and as part of a 15 credit course unit you should expect this may take up to 20 hours to complete. Unless you have an agreed extension on coursework deadlines with DASS reports handed in AFTER 11am Monday 20th March will be docked 4 marks plus an additional 4 marks each day thereafter until a mark of zero is reached. Reports handed in after 5pm Friday 31st March will be awarded a mark of zero and will not be marked.
In order that your report conforms to the standards for a technical report, you should use the following structure:
- MS Word, LaTeX, or similar, and must be submitted without your name, but with your univer- sity ID number online through the TurnItIn system.
- approximately 8 - 10 pages long (excluding appendices)
- be written in continuous prose
- give a brief introduction stating the problem you are solving and the parameters you are using (from the model or method),
- present your results in the form of figures and tables, using the order of items in the bullet points as a guide as to the order of your document
- absolutely NO screenshots of running code need to be included,
- do not include overly long tables – a table should never cross over a page,
- present the results for any methods you have implemented, there is no credit for a discussion of a method that has not been shown to be implemented by you (through results) for your problem
- refer to and discuss each of your results in the text, part of the marks available in each bullet point are for interpreting the results
- try to keep to the page limit, removing any unnecessary results from the main text
- number and caption your figures and tables and refer to them by their number (not their position in the text),
- number any equations to which you refer,
- use consistent internal (and external) referencing.
