Running-Maximum State in Black–Scholes Barrier Option Pricing
Summary
The document sets up an up-and-in call under Black–Scholes, with a payoff activated when the asset’s running maximum reaches an upper barrier. It reasons that the price may depend on time, the current asset price, and the running maximum, then applies Itô’s formula to this augmented state. Because the maximum is continuous and nondecreasing, the author observes that its quadratic variation and covariation with the asset price vanish. The resulting argument appears to impose both a Black–Scholes PDE and a zero derivative with respect to the maximum.
The key issue is that the running maximum changes only when the asset reaches a new high, so its finite-variation term contributes at the barrier boundary; the price need not be independent of the maximum before activation. A complete treatment must specify the state-space boundary and matching conditions, while care is needed around the payoff’s discontinuity and smoothness. The document is a question rather than a worked solution: it gives no boundary conditions, numerical method, or price comparison, and its tentative smoothness assumptions are unresolved.
Key ideas
- The option value can be represented using time, spot price, and the running maximum as state variables.
- The running maximum has finite variation and zero quadratic variation, but its changes still affect the pricing argument.
- The maximum matters before the barrier is reached because it determines how close the option is to activation.
- A valid PDE formulation needs appropriate boundary and matching conditions for the barrier state.
- The document poses the modeling problem without deriving a complete solution.
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Full text
# Black-Scholes equation for barrier options
# Black-Scholes equation for barrier options
I would like to write down the PDE for the price of an up-and-in call option under the Black-Scholes model as follows. The payoff of the option at expiry $T$ is
$$C_T := \max(S_T-K,0)1_{M_T \geq L}$$
where $M_t = \sup_{u\leq t}S_u$ and $L > K > 0$. The price of the option at time $t < T$ is given by
$$C_t = e^{-r(T-t)}E[\max(S_T-K,0)1_{M_T \geq L}\mid \mathcal{F}_t]$$ where $\mathcal{F}_t = \sigma(S_u: u\leq t)$. It is a well-known fact that the vector-valued process consisting of Brownian motion and its running maximum is Markov. I am assuming that this applies to geometric Brownian motion and its running maximum as well (I haven't checked this though). If that is the case, then $$C_t = f(t,S_t,M_t)$$ for some measurable function $f$. I don't know whether this function is smooth enough to apply Ito's lemma but again assuming that this is the case one obtains $$dC_t = \left(f_t(t,S_t,M_t) + f_S(t,S_t,M_t)rS_t + \frac{1}{2}f_{SS}(t,S_t,M_t)\sigma^2S_t^2\right)dt + f_M(t,S_t,M_t)dM_t + f_S(t,S_t,M_t)\sigma S_tdW_t^Q$$ The terms with $f_{MM}$ and $f_{MS}$ do not appear because $M$ is a continuous non-decreasing process. So its quadration variation as well as its covariation with $S$ are zero.
If $C$ is a self-financing traded asset, then $e^{-rt}C_t$ must be a martingale. This translates to $$f_t(t,S,M) + f_S(t,S,M)rS + \frac{1}{2}f_{SS}(t,S,M)\sigma^2S^2 = rf(t,S,M)$$ and $$f_M(t,S,M) = 0$$ The latter condition kind of makes sense. If $M \geq L$, then the option has become an ordinary vanilla call so at least on $M \geq L$ the pricing function is constant in $M$. If $M < L$, then the barrier is not reached yet but I am not convinced that on $M < L$ $f$ would be constant in $M$. Furthermore, if it were constant, this would induce discontinuity at $M = L$ and I am not sure if Ito's lemma would be applicable in the first place then.
My question is how can I make this approach to pricing of barrier options work? If that is not possible, then I would like to know why and specifically what it is that I am missing which is blocking this path.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.