Boundary condition for Asian Option under Black-Scholes model

Question

I am looking at Kemna and Vorst's paper:

A PRICING METHOD FOR OPTIONS BASED ON AVERAGE ASSET VALUES. see http://www.javaquant.net/papers/Kemna-Vorst.pdf

Let $\text{d}S_t = S_tr\text{d}t + S_t\sigma\text{d}W_t$. Let $t_0 \leq t \leq T$, define $A(t)=\frac{1}{T-t_0}\int^T_{t_0}S_\tau\text{d}\tau$.

The Asian option has pay off $(A(T)-K)^+$. Let $C(s,a,t)$ be the time $t$ price the Asian option with $S(t)=s, A(t)=a$. This paper claims on the top of page 5 that

$\lim\limits_{s\rightarrow\infty}\frac{\partial C(s,a,t)}{\partial s}=\frac{T-t}{T-t_0}e^{-r(T-t)}$, but this is not what I arrived at.

Here is my heuristic/non-rigorous derivation. When $S(t)$ is sufficiently large, then you will almost certainly be in the money. Then

the value of the option should approximately be $C(a,s,t) = e^{-r(T-t)}\bigg((a-K)+\mathbb E\bigg(\frac{1}{T-t_0}\int^T_tS_\tau\text{d}\tau\bigg)\bigg)= e^{-r(T-t)}\bigg((a-K)+\bigg(\frac{s}{r(T-t_0)}(e^{r(T-t)}-1)\bigg)\bigg)$.

(This agrees with (15) in the paper, even)

so I calculate the derivative to be $\frac{1}{r(T-t_0)}(1-e^{-r(T-t)})$

What is stated the paper seems to be the time derivative of my answer, see (13). Did I make a mistake or there is a mistake in this classical paper?

Answer 1

Your analysis is correct. From the risk neutral process

$$dS_t = rS_tdt + \sigma S_tdW_t$$

we get

$$\mathbb{E}(S_\tau|\mathbb{F}_t) = S_te^{r(\tau-t)}$$

and

$$\mathbb{E}(\int_{t}^{T}S_\tau d\tau|\mathbb{F}_t) = \frac{S_t}{r}[e^{r(T-t)}-1]$$

Hence, as $S \rightarrow \infty$

$$C(S,A,t) \sim \frac{S_te^{-r(T-t)}}{r(T-T_0)}[e^{r(T-t)}-1] = \frac{S_t}{r(T-T_0)}[1-e^{-r(T-t)}] \texttt{ ---EQ (1)}$$

and $$\lim_{S \rightarrow \infty} \frac{\partial C}{\partial S}=\frac{e^{-r(T-t)}}{r(T-T_0)}[e^{r(T-t)}-1]. \texttt{ ---EQ (2)}$$

Most likely, the authors used the truncated Taylor approximation

$$e^{r(T-t)}-1 \approx r(T-t)$$

to obtain

$$\lim_{S \rightarrow \infty} \frac{\partial C}{\partial S}=\frac{T-t}{T-T_0}e^{-r(T-t)} .$$

So it is not clear if there was oversight or intent in the expression appearing in the paper.

However both forms of the boundary condition are valid, more or less. There is no closed-form solution for the Asian option without some form of approximation that probably makes the difference in the two forms of the boundary condition irrelevant. Furthermore if the solution is derived by solving the PDE numerically then the application of a far-field boundary condition must be implemented at the boundary of a truncated domain -- numerical error again will be more significant than any discrepancy in the boundary condition.