Replicating a Bounded Claim with a Self-Financing Portfolio
Summary
The document outlines how to construct a self-financing portfolio that replicates a bounded terminal claim when an equivalent martingale measure exists. It discounts the claim by the savings account and takes its conditional expectation under that measure, producing a martingale whose evolution can be represented as a stochastic integral with respect to Brownian motion.
The risky-asset holding is chosen to match that integral, using the asset’s volatility and price to convert the martingale exposure into shares. The remaining portfolio value is held in the savings account. The response checks that discounted wealth follows the desired martingale and that terminal wealth equals the claim. This construction relies on martingale representation and on the given diffusion structure; admissibility and the ability to divide by volatility and asset price also matter. The initial attempt’s argument that a martingale alone establishes attainability is not sufficient without these assumptions.
Key ideas
- Discount the terminal claim by the savings account and take its conditional expectation under the risk-neutral measure.
- Martingale representation identifies the Brownian exposure needed to track the claim’s discounted value.
- Choose the risky-asset holding to match that exposure, then place residual wealth in the savings account.
- The construction depends on assumptions that permit representation and a well-defined replicating position.
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Full text
# replicating self-financing portfolio for risk neutral measure
# replicating self-financing portfolio for risk neutral measure
Let the price process $S_{t}, 0 \leq t \leq T$, be a diffusion, and savings account be $\beta_{t}$ such that the Equivalent Martingale Measure $Q$ exists. Let $C_{T}=g\left(X_{T}\right)$ be the claim at time $T$, for a bounded function $g$. Show that this claim is attainable, and find a replicating self-financing portfolio for this claim.
My attempt:
The claim is attainable as $\frac{S_t}{\beta_{t}}$ is a $Q$ -martingale and an admissible strategy exists. Therefore $\frac{V(t)}{\beta(t)}$ is also a Q martingale.
The Self replicating portfolio can be found as $V_{t}=a_{t} s_{t}+b_{t} e^{r t}$
$d V_{t}=a_{t} d s_{t}+r b_{t} e^{r t} d t$ is self financing.
let $\left.b_{t}=\left(V_{t}-a_{t} s_{t}\right)\right) e^{-r t}$ then we have
$d v_{t}=a_{t} d s_{t}+r\left(V_{t}-a_{t} s_{t}\right) d t$
$d V_{t}=a_{t} d s_{t}+r V_{t} d t-r a_{t} s_{t} d t$
I am not sure how to carry on further for this question.
## Answer by ir7 (score 1, accepted)
https://quant.stackexchange.com/a/64141
With EMM $Q$, associated $Q$-Brownian motion $W$, filtration ${\cal F}$, and
$$d\beta_t = r_t \beta_t dt, \; \beta_t ={\rm e}^{\int_0^t r_u du},$$
consider martingale:
$$ M_t =E\left[{\rm e}^{-\int_0^T r_u du} C_T | {\cal F}_t\right]. $$
By martingale representation theorem, there is a process $N_t$ such that
$$ M_t = M_0 + \int_0^t N_u dW_u, $$
where $$ M_0=E\left[{\rm e}^{-\int_0^T r_u du} C_T\right].$$
With given
$$ d(\beta_t^{-1} S_t) = \sigma_t \beta_t^{-1} S_tdW_t$$
under $Q$, we have:
$$ dM_t = N_t dW_t = a_t \sigma_t \beta_t^{-1} S_t dW_t = a_t d(\beta_t^{-1} S_t) $$
for $a_t$ chosen to be
$$ a_t := \frac{N_t\beta_t}{\sigma_t S_t }.$$
Strategy $a_t$ and $b_t:= \beta_t^{-1}(M_t-a_tS_t)$
$$\Pi_t := b_t\cdot \beta_t + a_t \cdot S_t = \beta_t M_t$$
is admissible (under $Q$, $\beta_t^{-1}\Pi_t$ is a martingale) and self-financing as
$$ d(\beta_t^{-1}\Pi_t) = dM_t = a_t d(\beta_t^{-1} S_t). $$
We also note that:
$$ \Pi_T = \beta_T M_T = C_T. $$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.