The COS method for option valuation under the SABR dynamics

ABSTRACT

In this paper, we consider the COS method for pricing European and Bermudan options under the stochastic alpha beta rho (SABR) model. In the COS pricing method, we make use of the characteristic function of the discrete forward process. We observe second-order convergence by using a second-order Taylor scheme in the discretization, or by using Richardson extrapolation in combination with a Euler–Maruyama discretization on the forward process. We also consider backward stochastic differential equations under the SABR model, using the discretized forward process and Fourier-cosine expansion for the occurring expectations. For this purpose, we extend the so-called BCOS method from one to two dimensions.

Keywords:

• COS method
• Euler–Maruyama scheme
• backward stochastic differential equation
• SABR
• Richardson extrapolation

2010 AMS Subject Classifications:

• 60E10
• 65C30
• 91G60

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1 We omit the mean reverting term for the Heston variance.

2 From now on, we will refer to this scheme as the Euler scheme.

3 If desired, it is possible to choose $\mathrm{\Delta }t$ non-constant.

4 We left out the discount term $\exp (-r\mathrm{\Delta }t)$ for convenience.

5 For the COS method, we observe smooth, monotonic convergence for the Euler scheme, this behaviour is only observed when ${\theta }_2=1$ for the BCOS method

6 Sometimes additional assumptions are required to complete a market [8,32], such as the tradability of volatility swaps in Example 4.5.

7 In this paper, we assume a deterministic risk-free interest rate, which implies that the risk-free measure and the forward measure coincide.

8 Note that we transform $X_t^1$ and $X_t^2$, before applying the theory of Section 3.1.