# Superhedging in Cox-Ross-Rubinstein model revisited

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 $$-1 : $$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 $$\left\{\begin{array}{l} v(N, x)=f(x), x \in \mathbb{R}^{+} \\ v(n, x)=\frac{p^{*} v(n+1, x d)+\left(1-p^{*}\right) v(n+1, x u)}{1+r} \\ x \in \mathbb{R}^{+}, 0 \leq n Let $$\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

• Thank you nbbo2 for your help concerning my English mistakes ! Jun 11 at 19:10