ECON5065 Applied Computational Finance
Applied Computational Finance
Coursework (50% of final mark)
The problems involve the development of functional Matlab code. 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. The short report and the final MATLAB code should be submitted
as a single ZIP folder on Moodle.
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,
dν(t) = κ(θ – ν(t))dt + σ
√
ν(t)dZ(t), ν(0) = ν0.
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
max(G[0,T] – K)+, G[0,T] = exp
(
1
T
∫ T
0
ln S(u)du
)
.
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] =
1
T
∫ T
0
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