Black Scholes model: condition of payout function

Question

Given:

Consider a two-asset, continuous time model (B,S) where $$dB_t = B_t r dt, \quad dS_t = S_t ( \mu dt + \sigma dW_t)$$ Clearly, the martingale deflator is: $$Y_t = e^{(-r - \frac{\lambda^2}{2})t - \lambda W_t}$$

There is a theorem that states the following:

For a claim with payout $\xi_T$, $T>0$, if $\xi_T>0$, $\mathcal{F}_T$-measurable and such that $\xi_TY_T$ is integrable, then there exists an admissible strategy $(\pi_t)_{t \in [0,T]}$ that replicates the European claim with payout $\xi_T$.

Problem:

Suppose that $\xi_T = g(X_T)$, where $g$ is a bounded smooth Borel function. (Then the hypothesis for $\xi_t$ in the above theorem are satisfied.)

Suppose further that $g$ has a bounded derivative and is an increasing function.

Show that there exists an admissible replicating strategy $(\pi_t)_{t \in [0,T]}$ such that $\pi_t \geq 0$ a.s. for all $t \geq 0$.

Answer 1

A portfolio $V_t(\alpha_t,\beta_t)$ (for stock $S_t$ and zerobond $B_t$) is self-financing iff:

$$V_t=\alpha_tS_t+\beta_t B_t$$

It further implies

$$dV_t=\alpha_tdS_t+\beta_tdB_t$$

To replicate a derivative $C(S_t,t)$ by a self-financing portfolio of stock and bond, set: $$dV_t=dC_t$$

The dynamics of $dC$ can be specified using Ito's Lemma on $C(S_t,t)$:

$$dC=\partial_tCdt+\partial_sCdS+\frac{1}{2}\sigma^2S_t^2\partial_{SS}Cdt=\partial_SCdS_t+(\partial_tC+\frac{1}{2}\sigma^2S_t^2\partial_{SS}C)dt$$

Next assume $C$ satisfies the BS-PDE:

$$\partial_tC+\frac{1}{2}\sigma^2S_t^2\partial_{SS}C=rC-rS_t\partial_S C$$

Inserting this into $dC$:

$$dC=\partial_SCdS_t+(C-S_t\partial_SC)rdt$$

Now we further have the bond-dynamics $dB_t=B_trdt$, so:

$$dC=\partial_SC\cdot dS_t+\left(\frac{C_t}{B_t}-\frac{S_t}{B_t}\partial_SC\right)\cdot dB_t$$

Finally, the coefficients before $dS_t$ and $dB_t$ are exactly the self-financing portfolio weights:

$$\left(\alpha_t=\partial_SC,\,\beta_t=\dfrac{C_t}{B_t}-\dfrac{S_t}{B_t}\partial_SC\right)$$

So the stockweight is $\pi_t=\partial_SC\geq 0$ since $C=g$ is an increasing function by OP.