1
$\begingroup$

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!

$\endgroup$
2
$\begingroup$

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.

$\endgroup$
  • 1
    $\begingroup$ it is nice and analytical (+1) $\endgroup$ – user16651 Jul 11 '16 at 13:09

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.