Replicating an Increasing-Payout Claim with a Nonnegative Stock Position
Summary
The document considers a European claim whose terminal payout is a bounded, smooth, increasing function of the underlying asset. It asks whether the claim can be replicated by an admissible strategy whose stock holding is never negative. The accepted answer uses the self-financing portfolio relation and applies Ito’s lemma to the derivative value as a function of the stock price and time.
Assuming the value satisfies the Black–Scholes partial differential equation, the resulting change in claim value can be matched to the changes in the stock and bond. The stock position is identified with the claim’s derivative with respect to the stock price; monotonicity of the payout is used to infer a nonnegative position. This is a theoretical argument under the stated continuous-time model and regularity assumptions. The note does not establish the result for payouts outside those assumptions or discuss transaction costs, market frictions, or discrete rebalancing.
Key ideas
- A self-financing replicating portfolio matches the changes in the claim and its holdings.
- Ito’s lemma expresses claim-value changes in terms of the underlying and time.
- Under the Black–Scholes equation, the stock holding equals the claim’s price sensitivity to the stock.
- An increasing payout supports a nonnegative stock position under the stated assumptions.
- The derivation assumes a continuous-time model without discussed trading frictions.
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Full text
# Black Scholes model: condition of payout function
# Black Scholes model: condition of payout function
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 by emcor (score 3, accepted)
https://quant.stackexchange.com/a/15633
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.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.