3
$\begingroup$

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 $t \geq 0$.

$\endgroup$
1
  • $\begingroup$ Wow that looks like an exercises set problem to me. @emcor gave a great answer so I'm not closing this but next time you should indicate how you attempted to solve this and where exactly your were stuck. $\endgroup$
    – SRKX
    Nov 27, 2014 at 3:58

1 Answer 1

3
$\begingroup$

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.

$\endgroup$
0

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge that you have read and understand our privacy policy and code of conduct.

Not the answer you're looking for? Browse other questions tagged or ask your own question.