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I'm starting to teach myself quantitative finance and I've got several questions (marked in bold) regarding the replicating portfolio of a security in the binomial model. I'm following, among others, the classical book "Stochastic Calculus for Finance I: The Binomial Asset Pricing Model".

First, I'll start with some notation so that there is no confusion. As always, let $d$, $u$, $r$ be the down-factor, up-factor, and risk-free interest rate, respectively, verifying that $d < 1 + r < u$. Then, if $V_{n}$, $0 \leq n \leq N$, is the value at time $n$ of a security that has a unique payoff $V_N$ at maturity, we know that the discounted process $\dfrac{V_n}{(1+r)^n}$ is a martingale under the risk-neutral probability measure $\widetilde{\mathbb{P}}$, where the probability of heads is $p := \dfrac{1+r-d}{u-d}$, and thus we can easily compute the value of each $V_n$ via

$$V_{n} = \widetilde{E}_{n}\Big(\frac{V_{N}}{(1+r)^{N-n}}\Big).$$

If $d = d_{n}$, $u = u_{n}$, and $r = r_{n}$ are not constant numbers but an adapted stochastic process, the result is the same but now the discounted process is given by $\dfrac{V_n}{(1+r_{0})·\dots·(1+r_{n-1})}$, and the risk-neutral pricing formula still holds true provided that

$$ \widetilde{\mathbb{P}}(w_{n+1} = H|w_{1},\dots,w_{n}) := p_{n} := \dfrac{1+r_n-d_n}{u_n-d_n}, $$

$$ \widetilde{\mathbb{P}}(w_{n+1} = T|w_{1},\dots,w_{n}) = 1- p_{n}. $$

To show the previous result, one usually constructs the following portfolio: suppose that $V$ is, for instance, an European call. Start with $X_{0}$ wealth, buy $\Delta_{0}$ shares of the underlying, and invest (or borrow) the remaining money at the risk-free rate $r$. At time $1$, sell the portfolio and reinvest the money doing the same strategy. At time $n+1$, the value of the replicating portfolio is given by

$$X_{n+1} = \Delta_{n}S_{n+1} + (1+r)(X_{n}-\Delta_{n}S_{n}).$$

Now that the notation is clear, my first question is the following:

1) Do we need to replicate the derivative security using the underlying security $S$? I'm aware of the hedging benefits of combining a derivative and its underlying, but since here the goal is to construct a replicating portfolio, could it be constructed trading with another security? What are the advantages of using the underlying over the rest of securities? The only benefit I see is that you only need to model the prices of one stock.

Now suppose that we want to price zero-coupon bonds using the binomial model. Assume that the interest rates form an stochastic adapted process, in such a way that 1 dollar invested at time $n$ yields $(1+r_{n})$ at time $n+1$. Let $B_{n,m}$ be the value at time $n$ of a zero-coupon bond that pays $1$ dollar at time $m$. Since the risk-neutral pricing formula also applies here, we can easily conclude that $$B_{n,m} = \widetilde{E}_{n}\Big(\frac{1}{(1+r_{n})·\dots·(1+r_{m-1})}\Big).$$ However, here goes another question:

2) How would one construct a replicating portfolio in this case? In what securities does it make sense to trade?

Finally, I see that in the book that I mention at the beginning, a portfolio process is constructed by trading in the zero-coupon bonds and the money market via the following equation: enter image description here

where $\Delta_{n,m}$ is the number of zero-coupon bonds of maturity $m$ held by the investor between times $n$ and $n+1$. I understand that this portfolio process, properly discounted, is a martingale, and hence there can't be arbitrage when trading in the zero-coupon bonds and the money market. My final question is:

3) How is this formula related to the proof that $B_{n,m} = \widetilde{E}_{n}\big(\frac{1}{(1+r_{n})·\dots·(1+r_{m-1})}\big)$?

If some of my questions are not clear enough, please let me know. Thanks a lot!

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1 Answer 1

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Here are some answers, hope it helps:

Regarding question 1, Do we need to replicate the derivative security using the underlying security $S$? I think the answer is related to your other question: What are the advantages of using the underlying over the rest of securities? The reason is that it is the underlying for which the derivative has a known sensitivity. Imagine you'd like to hedge a derivative on $S$ using an underlying $H$. How would you do it? How is an increase in value of the derivative linked to the performance or evolution of $H$? In the case of $S$ you know (or can estimate) that, is just the partial derivative (assuming no jumps).

2) How would one construct a replicating portfolio in this case? In what securities does it make sense to trade? Since these are adapted processes (the rate from $t$ to $t + \Delta t$ is known t and it is suppossed to be a rate at which any market player can capitalize, then you can just replicate it by buying zero coupon bonds or via a bank account.

Question 3 is related to this, and just comes from the compounded capitalization/borrowing formula. If the rate from $t_0$ to $t_1$ is $r_0$, the price of a zero coupon bond paying 1 unit at time $t_1$ is

$$ \dfrac{1}{1 + r_0 (t_1 - t_0)}$$,

now for a ZCB paying at time $t_2$, if the rate from $t_1$ to $t_2$ is $r_1$, following the same argument

$$ \dfrac{1}{(1 + r_0 (t_1 - t_0))(1 + r_1 (t_2 - t_1))},$$

and it follows directly. Note that in the book they're missing the $\Delta t$ term (maybe it is just taken into account in the $r$ factor).

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  • $\begingroup$ Thank you so much for your time! But I still can't see why I can't use another stock $H$ different from the underlying $S$, at least in the binomial model. If you start with $X_{0}$ wealth and follow the self-financing strategy defined above, replacing $S$ with $H$, I can impose X_{N} = V_{N} at maturity and determine all the $\Delta$'s and $X's$ so that the long or short position in V is hedged, I can't see any mathematical obstruction, at least in the binomial model. $\endgroup$
    – user_12345
    Commented Mar 1, 2023 at 19:26
  • $\begingroup$ As for questions 2 and 3, If interest rates are deterministic, your argument is obviously true, but if not, the answer should be the expected value under the risk-neutral measure of the quantities you mention. In order to construct the replicating strategy, I can't see why the bonds & money market strategy that I describe is useful, since you need to know the $B_{n,m}$'s, and that is precisely what you want to find. $\endgroup$
    – user_12345
    Commented Mar 1, 2023 at 19:34
  • $\begingroup$ Trading in the stock and money market isn't useful, since the delta's would be 0 and you'd be left with $X_{n} = (1+r_{n-1})\dots(1+r_0)X_{0}$, but that can't replicate a constant payoff unless the interest rates are constant. I guess the answer is found by trading in the stocks and derivatives markets. $\endgroup$
    – user_12345
    Commented Mar 1, 2023 at 19:38

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