ECON5065 Applied Computational Finance
Applied Computational Finance
Coursework (25% of final mark)
Students are required to work in groups of up to four students and submit their results by March 20, 2020 on Moodle. The problems involve the development of functional Matlab code. This code should be included in the submission, ideally as a zipped file. Both problems bear equal weight. Please explain carefully both the programming code and its theoretical background 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 paper by Kim and Wee (Quant. Fin. 2014) to compute the prices of geometric fxied-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 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 paper 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 the Deelstra-Delbaen 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
+ 1∫T
max(A[0,T] – K) , A[0,T] = T
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.
1 Course coordinator: Dr. Ankush Agarwal
S(u )du .
0
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