Price Limits for a Bull Call Spread

This question focuses on the arbitrage-free pricing bounds for a simple vertical option strategy, specifically a bull call spread constructed from two European calls with different strikes. The candidate must reason about how the spread's payoff profile at expiration restricts its fair price today, and how these bounds relate to the structure of its maximum gain and maximum loss. It is a common style of interview question for derivatives, trading, and structuring roles where candidates are expected to reason in payoffs rather than just memorizing formulae.

In answering, a candidate needs to use basic no-arbitrage arguments, payoff diagrams, and monotonicity properties of call prices with respect to strike. The interviewer is looking for comfort in moving between payoff at expiration and present value, and in arguing about inequalities rather than computing a single "fair" price. They also want to see awareness of static replication: how combining standard options creates a new instrument whose value must stay within the range implied by its possible outcomes. Clear, stepwise economic reasoning is more important than algebraic manipulation.

What it tests

The fundamental structure underlying option spreads, like the bull call spread, is that their price must always be bounded by the present value of their maximum and minimum possible payoffs at expiration. This is a direct consequence of the no-arbitrage principle: if the spread could be bought for less than its minimum possible payoff or sold for more than its maximum, riskless profit would be possible. The price of any spread is thus sandwiched between the discounted value of its best and worst outcomes, regardless of the specific strikes or premiums. This bounding occurs because options are linear in payoff but nonlinear in price, so combining them creates a new instrument whose price cannot escape the range set by its expiration payoffs. The pattern holds for any spread: the price must reflect the range of possible outcomes, discounted to present value, and never allow an arbitrage opportunity.