Asian options and running averages

Definition Fix the valuation date $0$, a payment date $T > 0$, fixing dates $0 \le t_1 < t_2 < \dots < t_n \le T$ and weights $w_j > 0$ with $\sum_j w_j = 1$. The discrete arithmetic average-price Asian call with strike $K > 0$ pays $(A - K)^{+}$ at $T$, where

and the corresponding put pays $(K - A)^{+}$. The continuously averaged contract uses $\frac{1}{T}\int_0^T S_u \, du$ in place of $A$. In a model with a constant rate $r$, money-market account $B_t = e^{rt}$ and a pricing measure $\mathbb{Q}$ under which the discounted values of traded assets with dividends reinvested are martingales, the time-$0$ value of the call is

It is finite because $0 \le (A - K)^{+} \le A$ and $\mathbb{E}^{\mathbb{Q}}[S_t] = S_0 e^{(r - q)t}$ for a price with continuous dividend yield $q$. A contract is fully specified only when the start of the averaging window, the fixing dates, the weights, the payment date and any fixings already recorded are all given. Contracts that pay $(S_T - A)^{+}$, with the average as the strike, are average-strike Asian options. This lesson treats the average-price form.

Definition The discrete arithmetic average is $A = \sum_j w_j S_{t_j}$ with fixed dates and positive weights adding to $1$. The continuous arithmetic average over $[0, T]$ is $A_T/T$, where

is the running integral, measured in currency units times years. For continuous price paths the equally weighted averages over the grids $t_j = jT/n$ converge to $A_T/T$ path by path as $n \to \infty$, since they are Riemann sums, but for each fixed $n$ the discrete average is a different random variable. In a model with $\mathbb{E}^{\mathbb{Q}}[S_t] = S_0 e^{(r - q)t}$, linearity gives the exact means

and $\mathbb{E}^{\mathbb{Q}}[A_T/T] = S_0 \bigl(e^{(r - q)T} - 1\bigr)/\bigl((r - q)T\bigr)$ for $r \ne q$, with the value $S_0$ when $r = q$. In the Black-Scholes model the variance is

where $c_{jk} = \operatorname{Cov}^{\mathbb{Q}}(S_{t_j}, S_{t_k})$ is

The average itself is a weighted sum of dependent lognormal variables. It is not lognormal in general, and the arithmetic Asian call has no closed-form price of Black-Scholes type in this model, so it is computed numerically, for example by Monte Carlo simulation or by a pricing equation in an augmented state.

Definition A payoff paid at $T$ is path dependent if it is a function of the price path $(S_u)_{0 \le u \le T}$, or of prices at several dates, that cannot be written as $g(S_T)$ for a function $g$ of the final price alone. In a Markov model such as the Black-Scholes model, the time-$t$ value of a European claim $g(S_T)$ is a function of $t$ and $S_t$. The time-$t$ value of a path-dependent claim $X$ is still

but in general it depends on the path up to $t$ and not only on $S_t$, so two paths that reach the same price at $t$ can have different values. In the Black-Scholes model with $\sigma > 0$, an Asian payoff with at least two fixing dates after the valuation date is path dependent.

Definition Assume that under $\mathbb{Q}$

with constants $r$, $q$ and $\sigma > 0$, and consider continuous averaging from time $0$ with payoff $g(A_T) = (A_T/T - K)^{+}$. State augmentation adds the running integral $A_t$, with $dA_t = S_t \, dt$, as a second state variable. The pair $(S_t, A_t)$ is a Markov process, because

and the ratios $S_u/S_t$ for $u \ge t$ are independent of $\mathcal{F}_t$. Hence the value at $t$ is $V(t, S_t, A_t)$, where

and $\mathbb{E}^{\mathbb{Q}}_{t,S,A}$ is the expectation given $S_t = S$ and $A_t = A$. Suppose $V$ is continuously differentiable once in $t$ and $A$ and twice in $S$. Itô’s lemma for the pair, with $d[A]_t = 0$ and $d[S, A]_t = 0$ because $A$ is continuous with finite variation, gives the drift of the $\mathbb{Q}$-martingale $e^{-rt} V(t, S_t, A_t)$, and setting that drift to zero gives the augmented pricing equation

for $t < T$, $S > 0$ and $A \ge 0$, with $V(T, S, A) = (A/T - K)^{+}$. The terminal condition uses the average, never the final price $S$. The equation has no second derivative in $A$, because $A$ has no Brownian part of its own. For discrete fixings the state is the running weighted sum $A$ of the fixings recorded so far. Between fixing dates $V$ solves the Black-Scholes equation in $t$ and $S$ with $A$ as a parameter, at each fixing date $V(t_j^{-}, S, A) = V(t_j, S, A + w_j S)$, and at the payment date $V = (A - K)^{+}$ with every fixing included. Fixings recorded before the valuation date enter the initial value of $A$ and are not regenerated by the model.

Definition For positive fixings at dates $0 \le t_1 < \dots < t_n \le T$ with weights $w_j > 0$ adding to $1$, the geometric average is

which for equal weights is $\bigl(\prod_j S_{t_j}\bigr)^{1/n}$. Assume $S_t = S_0 \exp\bigl((r - q - \sigma^2/2)t + \sigma W^{\mathbb{Q}}_t\bigr)$ under $\mathbb{Q}$ with constant $r$, $q$ and $\sigma$, and value the contract at time $0$. The log fixings are jointly normal with $\operatorname{Cov}^{\mathbb{Q}}(\ln S_{t_j}, \ln S_{t_k}) = \sigma^2 \min(t_j, t_k)$, so $\ln G$ is normal with mean $m$ and variance $v$,

where $\bar{t} = \sum_j w_j t_j$ is the average fixing time. For equal weights, $\bar{t}$ is the plain average of the dates and $v = (\sigma^2/n^2) \sum_{j,k} \min(t_j, t_k)$. Then $\mathbb{E}^{\mathbb{Q}}[G] = e^{m + v/2}$. If $K > 0$ and $v > 0$, which holds when $\sigma > 0$ and $t_n > 0$, the geometric Asian call paying $(G - K)^{+}$ at $T$ has the exact value

To derive it, write $Y = \ln G \sim N(m, v)$. Then $\mathbb{E}^{\mathbb{Q}}\bigl[K \mathbf{1}_{\{Y > \ln K\}}\bigr] = K \Phi(d_2)$, and completing the square in $\mathbb{E}^{\mathbb{Q}}\bigl[e^{Y} \mathbf{1}_{\{Y > \ln K\}}\bigr]$ gives $e^{m + v/2} \Phi(d_1)$. If $v = 0$, the average $G = e^{m}$ is deterministic and $C_G = e^{-rT}(e^{m} - K)^{+}$. The formula is exact for the geometric average only. It is not a formula for the arithmetic contract, whose average is not lognormal.

Definition For positive numbers $x_1, \dots, x_n$ and weights $w_j > 0$ with $\sum_j w_j = 1$,

with equality exactly when all $x_j$ are equal. The logarithm is strictly concave, so Jensen’s inequality for the distribution with mass $w_j$ at $x_j$ gives $\ln\bigl(\sum_j w_j x_j\bigr) \ge \sum_j w_j \ln x_j$, and the exponential preserves order. Applied path by path to the same fixing dates and weights, it gives $A \ge G$. The function $x \mapsto (x - K)^{+}$ is nondecreasing with slope at most $1$, and $x \mapsto (K - x)^{+}$ is nonincreasing, so

Taking discounted $\mathbb{Q}$ expectations gives, for calls and puts with the same strike, dates and weights,

These orderings use only positive fixings and the positivity of discounted expectation, not the lognormal law, and they need matching fixings and weights. The geometric payoff also serves as a control variate when the arithmetic call is computed by Monte Carlo. With discounted payoffs $Y_i$ and $X_i$ of the arithmetic and geometric calls on the same simulated path $i$, the estimator

has expectation $C_A$ for every fixed $\beta$, because the exact $C_G$ centers the adjustment. A coefficient fitted on an independent pilot sample keeps this property, while one estimated from the same sample can add a small finite-sample bias, a point taken up in Variance reduction and unbiased estimators.