Valuing an American Call with a Payoff Change at an Intermediate Date

Summary

The document poses an option-pricing problem in a model where the underlying follows a driftless diffusion under the stated pricing measure. The American claim expires at a later date, but its exercise payoff changes at an intermediate date: before that date it is based on a fixed strike, and afterward it is based on the difference between the asset price and its value at the intermediate date. The question asks for the arbitrage price throughout the option’s life.

The author observes that after the payoff change, the claim resembles a European call struck at the intermediate-date asset value, and tentatively suggests taking a maximum of two call values at the change date. No answer, derivation, interest-rate assumptions, or numerical example is provided, so that suggested valuation is unverified. A complete solution would need to account for the American exercise opportunity and the conditional value of the later payoff; the document is best read as a setup for a stopping-time pricing exercise rather than as a settled pricing method.

Key ideas

• The claim’s exercise payoff changes at a specified intermediate date.
• After that date, the payoff depends on the asset price relative to its value at the change date.
• The author proposes a maximum of two call values at the intermediate date but gives no proof.
• A complete arbitrage valuation must account for early exercise and the later payoff’s conditional value.

Tags

# Price of an American call option


# Price of an American call option












I'm working through revision questions at the moment and we are asked to compute the price of an American call option.

Suppose that $dS_t = \sigma S_t dW^*_t, S_0 >0$

Let $0<U<T$ be fixed dates and let $K>0$ be a constant. Consider the American call option with expiration date $T$ and payoff process $(X_t)_{t\in[0,T]}$ given by the following expressions:

$X_t = g_1(S_t,t) = (S_t-K)^+ , \forall t \in [0,U]$

\

$X_t = g_2(S_t,t) = (S_t-S_U)^+ , \forall t \in (U,T]$

Find the arbitrage price of the option at time $t\in [0,T]$

We know that in time $t \in (U,T]$ the price is a European call with strike $S_U$, and the price at time $U$ is max$(C_u(K), C_u(S_u))$ (I think).

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