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I am doing the following exercise from a math finance textbook but I got stuck at the end of the part 2. I found nothing on the internet concerning solutions of exercises from this textbook (called Introduction to stochastic calculus applied to finance by Lamberton & Lapeyre for those interested). At the end of the exercise (that I have copy/paste for the sake of completeness) I explain where I'm stuck.

If the way that I presented the exercise is bad don't hesitate to tell me, it's just that it's difficult for me to explain my problem if I don't provide all the necessary information before.

$\mathbf{Exercise}$
Super-replication in an incomplete market. We consider, here, an extended version of the Cox-Ross-Rubinstein model allowing the asset price to take three different values at each time step.

As for the Cox-Ross-Rubinstein model, let $S_{n}$ be the price at time $n$ of the risky asset, let $r$ be the riskless return over one period of time and $S_{n}^{0}=$ $(1+r)^{n}$ be the price of the riskless asset. Between two successive periods the relative price change can be $a, b$ or $c$, with $-1<a<b<c$ : $$ S_{n+1}=\left\{\begin{array}{l} S_{n}(1+a) \\ S_{n}(1+b) \\ S_{n}(1+c) \end{array}\right. $$ The initial stock price is denoted by $S_{0}$. The set of possible states is $\Omega=$ $\{1+a, 1+b, 1+c\}^{N}$, where each $N$-tuple represents the values of $S_{n+1} / S_{n}$, $n=0,1, \ldots, N-1$. We also assume that, for $n=1, \ldots, N, \mathscr{F}_{n}=\sigma\left(S_{1}, \ldots, S_{n}\right)$ is the $\sigma$-field generated by the random variables $S_{1}, \ldots, S_{n}$. We assume that $\mathbb{P}$ gives to each singleton in $\Omega$ a strictly positive probability. This assumption defines $\mathbb{P}$ up to an equivalent change of probability.

$\mathbf{Part\;I:\; Viability\; and\; completeness}$

Q1) At which condition on $a, b, c$ and $r$ is this model viable? We assume, in the sequel, that this assumption is fulfilled.

Q2) Assuming that $N=1$ and $r=0$, show, by constructing a contingent claim that cannot be replicated, that the model is incomplete

We will now prove that we are able to construct a super-replicating portfolio for every contingent claim with payoff $f\left(S_{N}\right), f$ being convex. More pecisely, a self-financing strategy $\phi=\left(\left(H_{n}^{0}, H_{n}\right), 0 \leq n \leq N\right)$ is a super-replicating strategy for the contingent claim with payoff $f\left(S_{N}\right)$ if and only if, by definition, its value $V_{n}(\phi)=H_{n}^{0} S_{n}^{0}+H_{n} S_{n}$ satisfies $V_{N}(\phi) \geq f\left(S_{N}\right)$ almost surely.

When such a super-replicating strategy exists, the super-replication price of the contingent claim is the smallest initial value of a super-replicating strategy, if such a minimal strategy exists.

$\mathbf{Part\;II:\; A\; lower\; bound\; for\; the\; super-replication\; price}$

We assume that there exists a super-replicating strategy $\phi=\left(\left(H_{n}^{0}, H_{n}\right), 0 \leq n \leq N\right)$ whose value at time $n$ is given by $V_{n}(\phi)$.

Q1) Show that if $\tilde{\mathbb{P}}$ is a probability equivalent to $\mathbb{P}$, under which $\left(\tilde{S}_{n}=\right.$ $\left.S_{n} / S_{n}^{0}, 0 \leq n \leq N\right)$ is a martingale, then $V_{0}(\phi) \geq \tilde{\mathbb{E}}\left(\frac{f\left(S_{N}\right)}{(1+r)^{N}}\right)$

Q2) Let $T_{n}=S_{n} / S_{n-1}$. We denote by $\mathbb{P}^{p_{1}, p_{2}, p_{3}}$ the probability on $\Omega$, such that $\left(T_{n}, 0 \leq n \leq N\right)$ is a sequence of independent random variables with

$$ \mathbb{P}^{p_{1}, p_{2}, p_{3}}\left(T_{n}=1+a\right)=p_{1}, \mathbb{P}^{p_{1}, p_{2}, p_{3}}\left(T_{n}=1+b\right)=p_{2}, \mathbb{P}^{p_{1}, p_{2}, p_{3}}\left(T_{n}=1+c\right)=p_{3}, \ $$ $p_{1}, p_{2}, p_{3}$ being positive real numbers such that $p_{1}+p_{2}+p_{3}=1$.

Prove that $\left(\tilde{S}_{n}, 0 \leq n \leq N\right)$ is a martingale under $\mathbb{P}^{p_{1}, p_{2}, p_{3}}$ if and only if $p_{1} a+p_{2} b+p_{3} c=r .$

Under which condition on $p_{1}, p_{2}, p_{3}$ is this probability equivalent to the initial probability $\mathbb{P}$ ?

Q3) Prove that the super-replication price $V_{0}$ is greater than

$$ \bar{V}_{0}=\sup _{\substack{p_{1}>0, p_{2}>0, p_{3}>0 \\ p_{1}+p_{2}+p_{3}=1 \\ p_{1} a+p_{2} b+p_{3} c=r}} \mathbb{E}^{p_{1}, p_{2}, p_{3}}\left(\frac{f\left(S_{N}\right)}{(1+r)^{N}}\right) $$

Q4) Prove that $$ \bar{V}_{0} \geq V_{\mathrm{CRR}}=\mathbb{E}^{p^{*}, 0,1-p^{*}}\left(\frac{f\left(S_{N}\right)}{(1+r)^{N}}\right), $$ where $p^{*}$ is such that $p^{*} a+\left(1-p^{*}\right) c=r$ (note that, using equations $p_{1}+p_{2}+p_{3}=1$ and $p_{1} a+p_{2} b+p_{3} c=r$, we can express $p_{1}$ as $\alpha\left(p_{2}\right)$ and $p_{3}$ as $\beta\left(p_{2}\right)$ and that $\left.\mathbb{E}^{\alpha\left(p_{2}\right), p_{2}, \beta\left(p_{2}\right)}\left(f\left(S_{N}\right)\right)\right)$ is a continuous function of $\left.p_{2}\right)$

Give an interpretation for $p^{*}$ and for $V_{\mathrm{CRR}}$ in a Cox-Ross-Rubinstein model with $d=1+a$ and $u=1+c$.

$\mathbf{Part\;III:\; Computation\; of\; a\; super-replication\; strategy}$

We will now show that we can construct a super-replication strategy with initial value $V_{\mathrm{CRR}}$ for a contingent claim with a convex payoff function $f$.

Let $v(n, x)$ be the price in the Cox-Ross-Rubinstein model with parameters $d=1+a$ and $u=1+c$, at time $n$ and for a current value $x$ of the asset. This price satisfies the recursive equations $$ \left\{\begin{array}{l} v(N, x)=f(x), x \in \mathbb{R}^{+} \\ v(n, x)=\frac{p^{*} v(n+1, x d)+\left(1-p^{*}\right) v(n+1, x u)}{1+r} \\ x \in \mathbb{R}^{+}, 0 \leq n<N \end{array}\right. $$ Let $\Delta(n+1, x)$ be the corresponding hedge between times $n$ and $n+1$, for a value $x$, defined by $$ \Delta(n+1, x)=\frac{v(n+1, x u)-v(n+1, x d)}{x(u-d)} . $$ Let $V_{n}$ be the value of the unique self-financing strategy with a quantity $\Delta\left(n, S_{n}\right)$ of risky asset at time $n$ and initial value $V_{0}=V_{\mathrm{CRR}}=v\left(0, S_{0}\right)$.

Q1) Assuming that $f$ is a convex function, prove that if $v$ is solution of $(1.5 .4)$, then, for all $n, v(n, .)$ is convex.

Q2) Let $\tilde{V}_{n}=V_{n} /(1+r)^{n}$ and $\tilde{S}_{n}=S_{n} /(1+r)^{n}$. Prove that

$$ \tilde{V}_{n+1}-\tilde{V}_{n}=\Delta\left(n+1, S_{n}\right)\left(\tilde{S}_{n+1}-\tilde{S}_{n}\right) . $$

Q3) Show, using equation (1.5.4), that for $\alpha=a$ or $c$,

$$ \frac{v(n+1, x(1+\alpha))}{1+r}=v(n, x)+\Delta(n+1, x)\left(\frac{x(1+\alpha)}{1+r}-x\right) $$ and deduce, using the convexity of $v$, that $$ \frac{v(n+1, x(1+b))}{1+r} \leq v(n, x)+\Delta(n+1, x)\left(\frac{x(1+b)}{1+r}-x\right) $$

Q4) Prove, by induction, that for all $0 \leq n \leq N, V_{n} \geq v\left(n, S_{n}\right)$ and deduce that $V_{\mathrm{CRR}}$ is the initial value of a super-replication strategy.

$\mathbf{End\;of\;the\;exercise}$

I'm stuck at the Q4) of the part 2. First, I don't understand why we can put $p_2=0$ since one condition for having an equivalent probability measure in this model is that $p_i>0,\quad\forall i\in\{1,2,3\}$. Now if we put $p_2=0$ I know that $p^{*}$ is the risk neutral probability measure from the CRR model but I cannot see what can be interpreted from that ? This leads that I don't understand what is meant by "give an interpretation of $p^{*}$ and $V_{CRR}$ for the Cox-Ross-Rubinstein model with d=1+c and u=1+a".

More generally, it is hard for me to see how we can make a link with the classical CRR model since the set of possible states is clearly no the same

Thank you a lot for your understanding

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  • $\begingroup$ Thank you nbbo2 for your help concerning my English mistakes ! $\endgroup$
    – coboy
    Jun 11 at 19:10

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