Early Exercise Valuation for an Asian Average-Price Contract

Summary

The document asks how to value early exercise for a contract paying the running arithmetic average of an asset price minus a fixed strike. The response says there is no closed-form value for the American exercise problem in the stated setting. It recommends an American Monte Carlo method or a two-dimensional finite-difference PDE, tracking both the asset price and its running average. The average’s dynamics depend on the gap between the current asset price and the average, making the contract state-dependent.

For zero interest rates, the response rewrites the average relative to the asset price and changes to the stock risk-neutral measure. This produces a one-dimensional process for the price-to-average ratio and a corresponding one-dimensional PDE scheme. The reduction is specific to the zero-rate case and assumptions shown; the document provides no numerical implementation or comparison of approximation accuracy. It also does not specify all practical details needed to choose exercise boundaries or calibrate the model.

Key ideas

• An American contract on a running average generally has no closed-form valuation in the stated model.
• Monte Carlo with early exercise or a two-dimensional finite-difference PDE can approximate its value.
• The joint state can be represented by the asset price and its running average.
• When the interest rate is zero, a change of measure and the average-to-price ratio reduce the problem to one dimension.

Tags

# Pricing an Asian style forward contract with early exercise feature


# Pricing an Asian style forward contract with early exercise feature












Is there an analytic way to price or approximate a contract with payout $A_t - K$, where $A_t$ is the running average price of the underlying asset from $[0, t]$ and $K$ is (fixed) strike.

If this is an European style contract, then I think we can replicate it using put-call parity. What if it is American (early exercise)? How to price/approximate the value of the early exercise option in such contract?

## Answer by Antoine Conze (score 3, accepted)

https://quant.stackexchange.com/a/43683

Welcome to Quant SE. Unfortunately there is no closed form formula for computing the american contract value $\max_{\tau}E^P\left[e^{-r\tau}(A_{\tau} - K)\right]$, so you have to resort to an american monte carlo method or a 2 dimensional PDE finite differences scheme for the joint dynamics $$ dS_t/S_t = (r - q) dt + \sigma dW_t \\ dA_t = d\left(\frac{1}{t} \int_0^t S_u du\right) = \frac{S_t-A_t}{t} dt $$ In the case where $r=0$ the problem reduces to computing $$ \max_{\tau}E^P\left[A_{\tau} \right] - K = \max_{\tau}E^P\left[S_{\tau}m_{\tau} \right] - K = S_0\max_{\tau}E^{\tilde{P}}\left[e^{-q\tau}m_{\tau} \right] - K $$ where $m_t=A_t/S_t$, $\tilde{P}$ is the stock risk neutral measure, and the dynamics for $m_t$ under $\tilde{P}$ is $$ d m_t=\left(\frac{1-m_t}{t}+qm_t\right)dt+\sigma m_t d\tilde{W}_t $$ and you can resort to a 1 dimensional PDE finite differences scheme.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.