Call Spread Black–Scholes Terminal and Spot Boundary Conditions

Summary

The document identifies boundary conditions for valuing a European call spread with lower strike K₁ and higher strike K₂ under the Black–Scholes framework. At maturity, the payoff is the difference between the two call payoffs. As the underlying price approaches zero, the spread value approaches zero; as the price becomes very large, it approaches the discounted difference between the strikes.

The response clarifies that these are terminal and limiting spot-price conditions, rather than two interchangeable conditions of the same type. It also notes that a numerical PDE implementation must specify whether its boundary is absorbing or reflecting. The exchange does not develop a numerical scheme or address the poster’s question about whether an intermediate-time price plot should look symmetric. In general, symmetry does not follow just from symmetric strikes, since the option value also depends on time, volatility, rates, and the pricing model.

Key ideas

• At maturity, a call spread pays the lower-strike call payoff minus the higher-strike call payoff.
• The spread value tends to zero when the underlying price tends to zero.
• At very high underlying prices, the spread value tends to the discounted strike width.
• A numerical boundary must specify whether it is absorbing or reflecting.

Tags

# Boundary Conditions for Call Spread


# Boundary Conditions for Call Spread












I was just wondering if someone could verify whether these are the two boundary conditions for a Call Spread Black-Scholes PDE.

The first one I have is:

$max(S_{T} - K_{1}, 0) - max(S_{T}-K_{2},0)$

While the second boundary condition I have is:

$S_{t} - K_{1}e^{-r(T-t)} + K_{2}e^{-r(T-t)} - S_{t} = (K_{2}-K_{1})e^{-r(T-t)}$

Is this correct? Thanks in advance

EDIT: Rather than create a new question, I thought I should ask it here: Should a call spread at time $t_{0}$ always take on a symmetric shape? I have the graph of a call spread PDE at time $t_{0}$ for spot values between 0 to 20, with zero interest rate, and with strikes $K_{1} = 9$, $K_{2} = 11$. Does the shape of this call spread look okay, or does it have to be perfectly symmetrical? Thanks!

## Answer by M. Jeunesse (score 5, accepted)

https://quant.stackexchange.com/a/26114

Time $T$ boundary condition is correct $u(T,x)=(x-K_1)^+-(x-K_2)^+$.

Time $x\to 0$ boundary condition is known and is equal to $0$.

Time $x\to\infty$ boundary condition is also known and is correct $\lim_{x\to\infty}u(t,x)=(K_2-K_1)e^{-r(T-t)}.$

You need to be precise if you want your boundary be "absorbing" or "reflecting".

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