Applied Computational Finance
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:
Applied Computational Finance
(√)
dS(t) = S(t) rdt + ν(t)dW(t) , √
dν(t) = κ(θ – ν(t))dt + σ ν(t)dZ(t), Here W and Z are correlated Brownian motions, that is,
S(0) = S0, ν(0) = ν0.
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.
S(u )du .
1
0
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.
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Applied Computational Finance