Pricing the Maximum of a Money-Market Account and a Stock

Summary

The document poses a no-arbitrage valuation problem for a payoff equal to the larger of a risk-free money-market account and a dividend-paying stock at expiry. The stock follows a geometric Brownian motion under the risk-neutral measure, while the account grows deterministically at the risk-free rate. The question is how to handle the expectation of the maximum in the pricing formula.

The answer rewrites the payoff as the money-market account plus the positive part of the stock’s excess over that account. This identifies the excess term as a call option on the stock with a strike equal to the account value at expiry. Consequently, the original claim can be valued as cash plus a call, using standard option-pricing methods. The document provides this payoff decomposition but does not walk through the subsequent calculation or discuss parameter assumptions beyond the stated model.

Key ideas

• The payoff maximum can be decomposed into the money-market account and a call payoff.
• The equivalent call’s strike is the deterministic account value at expiry.
• This identity converts an expectation involving a maximum into a familiar option-pricing problem.
• The result relies on the stated risk-neutral pricing setup and does not provide a full valuation derivation.

Tags

# Determining the No Arbitrage price of max[B(T), S(T)]


# Determining the No Arbitrage price of max[B(T), S(T)]












Following is given,

$dB(t)=rB(t)dt$

$dS(t)= (r-\delta)S(t)dt+\sigma S(t)dW(t)$

where, $r$ is the risk-free interest rate, $\delta$ the continous dividend yield $\sigma$ is the stock asset volatility and $W$ brownian motion.

By solving the SDE that is the money account B and applying Itô's lemma on the stock dynamic S I get,

$B(T)=e^{r(T-t)}$ and

$S(T)=S(t)e^{(r-\delta-\sigma^2/2)+\sigma(W(T)-W(t))}$

I know further that the No Arbitrage price is defined as,

$\Pi(t;X)=1/B(T)*E^Q_t[max[B(T),S(T)]]$

Question

Can I somehow cancel out B(T) since it is deterministic? How do I calculate the expectation of a maximum with a brownian motion W?

I am new to probability theory, financial mathematics and stochastic calculus. Would appreciate step-wise guidance. Thank you!

## Answer by user39119 (score 8, accepted)

https://quant.stackexchange.com/a/50216

$\max(B_T,S_T)=\max(0,S_T-B_T)+B_T,$ so this is just a call option (with strike $B_T$) plus $B_T.$

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

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