Pricing a Capped Average-Strike Call and Solving for Implied Volatility

Summary

The document poses a pricing problem for a call whose terminal strike depends on the average stock price over a future interval. The average is a continuous-time arithmetic average, and the strike is formed by scaling that average before applying lower and upper caps. The payoff is the positive part of terminal stock price minus this bounded, stochastic strike.

The underlying is specified with a volatility-driven diffusion under the risk-neutral measure. The question asks for an efficient pricing approach and for a way to recover implied volatility from observed market prices. However, the excerpt contains no proposed pricing method, numerical results, or solution procedure. It therefore identifies the contract features and computational goals, while leaving simulation or approximation choices, volatility inversion, and practical modeling assumptions unresolved.

Key ideas

• The call's strike is a scaled arithmetic average of the stock over a future interval, subject to lower and upper caps.
• The payoff depends jointly on the terminal stock price and the bounded stochastic strike.
• The underlying dynamics are specified under the risk-neutral measure with volatility as a model parameter.
• The document asks for fast valuation and implied-volatility calculation but does not provide a solution.

Tags

# Average Strike Option with bounds


# Average Strike Option with bounds












I'm looking to price a call option with an exotic feature. The price I'm trying to calculate at time $t=0$ is \begin{equation} C = E^\mathbb{Q}[(S_T-K_T)^+] \end{equation}

where $S_t$ is the stock price with dynamics \begin{equation} dS_t =\sigma S_t dW_t \end{equation}

The strike price $K_T$ is a stochastic variable given by \begin{equation} K_T = \min\{K_{max}, \max\{K_{min}, \lambda A\}\}, \quad \lambda < 1 \end{equation}

i.e., $K_T\in[K_{min},K_{max}]$, where $A$ is the average price during some future interval $[t_1,t_2]$ \begin{equation} A = \frac{1}{t_2-t_1}\int_{t_1}^{t_2} S_t dt \end{equation}

What is a good and fast way to calculate this and also solve for the implied volatility from market prices?

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