ECON5065 Applied Computational Finance
Applied Computational Finance
Coursework (50% of final mark)
The problems involve the development of functional Matlab code. This code should be included in the submission, ideally as a zipped file. All the problems bear equal weight. Please explain carefully the technical challenges faced while programming and comment on the final results obtained in a short report.
Topic : Pricing Asian Options under Heston’s Stochastic Volatility Model
We consider the price of an asset St whose dynamics under the risk-neutral measure is described by the following system of stochastic differential equations:
(√)
dS(t) = S(t) rdt + ν(t)dW(t) , √
S(0) = S0, ν(t)dZ(t), ν(0) = ν0.
dν(t) = κ(θ – ν(t))dt + σ
Here W and Z are correlated Brownian motions, that is,
dW(t)dZ(t) = ρdt,
r is the interest rate, κ, θ and σ are positive constants satisfying 2κθ ≥ σ2.
Problem 1: Use the formula derived in Theorem 4.1 of the article by Kim and Wee [3] to compute the prices of geometric fixed-strike Asian call options. The payoff function of the option is given as
(1∫T ) max(G[0,T] – K)+, G[0,T] = exp T ln S(u)du .
0
Use the following model parameters: S0 = 100, ν0 = 0.09, t = 0, r = 0.05, θ = 0.348, σ = 0.39, κ = 1.15, ρ = –0.64. In the analytical formula, use n = 10, 20, 30 terms in the infinite series expansion and use 105 as the upper bound in the infinite integral. Illustrate the results as in Table 1 of the article by Kim and Wee [3] for T = 0.5,T = 1.0,T = 2.0 with K = 90,100,110 used for each T value.
Problem 2: Using the parameter values as in Problem 1, use an appropriate discretisation scheme (for example, the Deelstra-Delbaen [1] discretisation scheme) to estimate the prices of arithmetic fixed-strike Asian call options via Monte Carlo simulation. The payoff function of the option is given as
max(A[0,T] – K) , A[0,T] = T
0
+ 1∫T
S(u )du .
Use different levels of discretisation step ∆t = 10–3,10–4,10–5 and illustrate the results in a table for T = 0.5, T = 1.0, T = 2.0 with K = 90, 100, 110 used for each T value. The results must be produced for number of sample paths 50, 000 and 100, 000.
1 Course coordinator: Dr. Ankush Agarwal
ankush.agarwal@glasgow.ac.uk
ECON5065 Applied Computational Finance
References
[1] Deelstra, G. and Delbaen, F. (1998) Convergence of discretized stochastic (interest rate) processes with stochastic drift term. Applied Stochastic Models in Business and Industry. 14(1), 77-84.
[2] Heston, S. (1993) A closed-form solution for options with stochastic volatility with applications to bonds and currency options. Review of Financial Studies. 6(2), 327-343.
[3] Kim, B. and Wee, I.S. (2014) Pricing of geometric Asian options under Heston’s stochastic volatility model. Quantitative Finance. 14:10, 1795-1809.
2 Course coordinator: Dr. Ankush Agarwal
ankush.agarwal@glasgow.ac.uk