Initial/Boundary Conditions for a Butterfly Option?

Question

What are the initial and boundary conditions for a Butterfly Option? I want to write up a PDE program for it and I have a rough idea of what the payoff should be (is it just a call and a put at the strike price?) but if anyone can provide me with definitive answers then I'd greatly appreciate it. In particular, I'm after stuff like the time $T$ boundary condition (which is usually the option payoff and taken as the initial condition) which is written as $u(T,x)$, the boundary condition as $x \rightarrow 0$ i.e. $\lim_{x\rightarrow 0} u(t,x)$ (which I think should be equal to $0$) and the boundary condition as $x \rightarrow \infty$ i.e. $\lim_{x\rightarrow \infty} u(t,x)$

On a related note, I'm new to financial mathematics and every time I need to look for the conditions for options other than a call option I usually find it incredibly difficult (I have to google search everything for nearly an hour to find something relevant it seems). Does anyone have a resource which provides the initial and boundary conditions for a range of options?

Thanks in advance.

EDIT: Okay, a quick search showed me that the payoff to a Butterfly Spread is $(S - K_c)^+ + (K_{p} - S)^+ - (S - K_{atm})^+ - (K_{atm} - S)^+$ where $K_{atm} = \frac{K_c + K_p}{2}$, however, I still don't know what the boundary conditions are, can someone tell me what they are (and hopefully even how to derive them?) Thanks!

Answer 1

You are trying to write a program which solves the following pricing PDE (Black-Scholes assumed)

$$ \frac{\partial V}{\partial t}(t,S) + (r-q)S\frac{\partial V}{\partial S}(t,S) + \frac{1}{2}\sigma^2 S^2 \frac{\partial^2 V}{\partial S^2}(t,S) - rV(t,S) = 0 $$

where $V_0:=V(0,S_0)$ is the target option premium.

The terminal condition is that, at $t=T$ (expiry of the contract), the value of the option should be equal to its payoff $\phi (S_T) $ by absence of arbitrage opportunity: $$ V(T, S_T) = (S_T-(K-a))^+ - 2(S_T-K)^+ + (S_T-(K+a))^+,\ \ \forall S_T \in \mathbb{R}^+ $$ where the real parameter $a>0$ describes the "width" of your butterfly position.

To get the boundary conditions it is illuminating to remember that the solution of the above pricing PDE is, by Feynman-Kac formula:

$$ V(t,S_t) = \mathbb{E}^\mathbb{Q} \left[ e^{-r(T-t)} \phi(S_T) \vert \mathcal{F}_t \right] $$

where $\mathbb{Q}$ is a probability measure under which the value of any self-financing portfolio, discounted at the risk-free rate $r$, constitutes a martingale. Using the above formulation, we can heuristically(*) see that \begin{align} \lim_{S \rightarrow 0} V(t,S_t=S) &= \lim_{S_T \rightarrow 0} \mathbb{E}_t^\mathbb{Q}\left[ e^{-r(T-t)} \phi(S_T) \right] \\ &= e^{-r(T-t)} \phi(0) \\ &= 0 \\ \lim_{S \rightarrow \infty} V(t,S_t=S) &= \lim_{S_T \rightarrow \infty} \mathbb{E}_t^\mathbb{Q}\left[ e^{-r(T-t)} \phi(S_T) \right] \\ &= e^{-r(T-t)} \underbrace{\lim_{S_T \rightarrow \infty} \mathbb{E}_t^\mathbb{Q}\left[ (S_T-(K-a)) - 2(S_T-K) + (S_T-(K+a)) \right]}_{=0} \\ &= 0 \end{align}

Note that, instead of Dirichlet conditions, you could equivalently use Von Neumann conditions at boundaries. For instance here, the fact that Gamma is expected to vanish i.e. \begin{align} \lim_{S\rightarrow 0} \frac{\partial^2 V}{\partial S^2}(t,S_t=S) = 0 \\ \lim_{S\rightarrow \infty} \frac{\partial^2 V}{\partial S^2}(t,S_t=S) = 0 \\ \end{align}

(*) [REM] Although the limit is theoretically defined for the random variable $S=S_t$ we can "heuristically" propagate it to the argument $S_T$ of the payoff function. This is because under homogeneous diffusion models à la Black-Scholes $`\ S_T \propto S_t\ '$. I'm using quotes because this is not the rigorous way to write it, but you get the idea.