An exercise about replicable random variables

Question

The following question is an exercise which I have in my course for Financial Mathematics:

Let $h:[0,\infty) \to [0,\infty)$ be twice differentiable with $h'' \geq 0$. Establish, with the help of Fubini's theorem, the representation

$$h(x) = h(0) + h'(0)x + \int_{0}^{\infty}(x - K)_{+}h''(K)dK.$$

Now assume that in a financial market model (T = 1, d = 1), without arbitrage and such that $S_1 \geq 0$ a.s., all call options are replicable. Is the random variable $f(S) := h(S_1)$ also replicable?

I would appreciate any help here. I am quite lost regarding this question, so I can unfortunately not offer any real approach by myself. Has anybody an idea regarding the two questions in this task?

Edit: just for clarification: We call a random variable f $"$replicable$"$ if $\exists a \in \mathbb{R}$ and there exists a trading strategy $H$ on risky assets such that $f = a + (H.S)_T \ $ $\mathcal{P}$-almost surely. Here, we call $(H.S)_T = \sum_{i=1}^{T}H_i(S_{i+1} - S_{i})$ the stochastic integral of $H$ with respect to $S$.

Answer 1

Stratos already gave you an answer for the first part; for replication, observe that your derivative payoff can be decomposed as:

$$h(S_1) = h(0) + h^\prime (0) S_1 + \int_0^\infty (S_1 - K)^+ h^{\prime \prime} (K) dK$$

Assuming interest rates are zero, we may statically replicate by holding $h^\prime (0)$ of the stock and the infinitesimal amount $h^{\prime \prime}(K)dK$ of call options is held with each strike $K$. This gives you a recipe of replication using the risky assets (i.e. stocks and options).

You'll find this referred to in the mathematical finance literature as the Carr-Madan formula.

Answer 2

Hint for the first part : by the fundamental theorem of calculus applied respectively to $h$ and $h'$,

$$\begin{align}h(x) - h(0) &= \int_0^x h'(t)\ dt\\ &=\int_0^x \left[h'(0) + \int_0^th''(K)\ dK\right]\ dt\\ &= h'(0)x + \int_0^x \int_0^t h''(K)\chi_{[0,t]}(K)\ dK\ dt\\ &= h'(0)x + \int_0^x \int_0^\infty h''(K)\chi_{[0,t]}(K)\ dK\ dt\\\end{align}$$

Where $\chi_{[a,b]}(u) := \mathbf 1\{a\le u\le b\}$ is equal to $1$ if $u\in[a,b]$ and $0$ otherwise.

Can you finish now ? (one more hint : $(x-K)_+ =\int_0^{x} \chi_{[?,?]}(t)\ dt $)