Asymptotics for short maturity Asian options in jump-diffusion models with local volatility (Q6576884)

The authors calculate the first-order asymptotics, as time-to-maturity goes to zero, of the price of Asian options in a jump-diffusion model with local volatility; that is, the price of the underlying is \(S(t) = \hat{S}(t)\cdot e^{X(t)}\) where\N\[\Nd \hat S(t) = \hat S(t)\cdot \left(\left(r-q -\log\left(\mathbb{E}\left[e^{X(1)}\right]\right)\right)dt + \sigma\left(\hat S(t)\right)dW(t)\right),\N\]\N\(X\) is a Lévy process independent of \(\hat S\) with measure \(\nu\), \(r\) is the risk-free rate, and \(q\) is the dividend yield. For a fixed strike \(K\), the prices of the Asian call and put options are \N\[\NC(T) = e^{-rT}\mathbb{E}\left[(A(T)-K)^{+}\right], ~ P(T) = e^{-rT}\mathbb{E}\left[(K-A(T))^{+}\right]\N\]\Nwhere \(A(T) = \frac{1}{T}\int_{0}^{T}S(t)dt\). For floating-strike options, the prices are \N\[\NC_{f}(T) = e^{-rT}\mathbb{E}\left[(\kappa S(T) - A(T))^{+}\right], ~ P_{f}(T) = e^{-rT}\mathbb{E}\left[(A(T) - \kappa S(T))^{+}\right]\N\]\NUnder some integrability assumptions on \(X\), the authors obtain\N\begin{itemize}\N\item for \(K > S(0)\), \N\[\N\lim_{T \rightarrow 0}\frac{C(T)}{T} = \int_{0}^{1}\int_{\log\left(\frac{K-tS(0)}{(1-t)S(0)}\right)}^{\infty}\left(tS(0)+e^{y}(1-t)S(0)-K\right)\nu(dy)dt;\N\]\N\item for \(K < S(0)\), \N\[\N\lim_{T \rightarrow 0}\frac{P(T)}{T} = \int_{0}^{\frac{K}{S(0)}}\int^{\log\left(\frac{K-tS(0)}{(1-t)S(0)}\right)}_{-\infty}\left(K-tS(0)-e^{y}(1-t)S(0)\right)\nu(dy)dt;\N\]\N\item for \(\kappa < 1\), \N\[\N\lim_{T \rightarrow 0}\frac{C_{f}(T)}{T} = \int_{1-\kappa}^{1}\int_{\log\left(\frac{t}{\kappa - (1-t)}\right)}^{\infty}S(0)\cdot \left(\kappa e^{y}-t-e^{y}(1-t)\right)\nu(dy)dt;\N\]\N\item for \(\kappa > 1\), \N\[\N\lim_{T \rightarrow 0}\frac{P_{f}(T)}{T} = \int_{0}^{1}\int^{\log\left(\frac{t}{\kappa - (1-t)}\right)}_{-\infty}S(0)\cdot \left(t+e^{y}(1-t)-\kappa e^{y}\right)\nu(dy)dt.\N\]\N\end{itemize}\NIn the special case when \(X\) is a compound Poisson process, the at-the-money -- that is, \(K = S(0)\) and \(\kappa = 1\) -- asymptotics are \N\[\N\lim_{T \rightarrow 0}\frac{C(T)}{\sqrt{T}} = \lim_{T \rightarrow 0}\frac{P(T)}{\sqrt{T}} = \lim_{T \rightarrow 0}\frac{C_{f}(T)}{\sqrt{T}} = \lim_{T \rightarrow 0}\frac{P_{f}(T)}{\sqrt{T}} =\frac{1}{\sqrt{6\pi}}\sigma(S(0))S(0).\N\]\NThe authors explicitly calculate these asymptotics for the Merton jump-diffusion model, a jump-diffusion model where jump sizes follow the double exponential distribution, and the variance gamma model. The analytical approximations to option prices following from the asymptotic formulas are contrasted to simulation-based prices for options with one week and one month time-to-maturity.