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I got stuck on the following question whilst learning about basic option pricing.

A stock is valued at \$75 today. An option will pay \$1 the first time the stock reaches \$100 in value, which it is assumed will happen with probability one at some point in the future. Find the price of the option and construct a replicating portfolio.

At first glance, this seems like some sort of continuous time problem, but I'm hoping that there's a simpler way to do things. How should one approach this sort of question?

Edit: For simplicity, let's assume there's no interest in this scenario.

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It looks like some kind of perpetual to me. Do you have more information e.g. about the dynamics of the process ? To the price: it should be $E[ exp(-r \cdot \tau)]$ If we assume the interest rate to be constant. $\tau$ would be the stoppingt time when the process hits the 100 boundary. If there was no interest, then you don't have time value of money. The price should be 1 in that case – Probilitator Feb 22 '14 at 22:36
There was no other information given. To obtain your result $\mathbb{E}[\exp{(-r\tau)}]$, did you assume the price was a martingale and use a result about bounded stopping times? (Or am I reading too much into this...) – sourisse Feb 22 '14 at 23:18
Yes the expectation is supposed to be under the Risk-neutral measure. One would have to check the boundedness of $\tau$. In the abscence of an interest rate you won't have time value of money. So you won't care when the 1$ is paid. The perfect hedgie would be buying 1% of the stock today and just hold it (contrary to the 1 I wrote in my previous comment) – Probilitator Feb 23 '14 at 7:04
where did you find this question? - I like it ^^ – Probilitator Feb 25 '14 at 9:13
I found it here. – sourisse Feb 26 '14 at 0:43
up vote 4 down vote accepted

Assume the stock pays no dividends before 100 dollars is hit. Interest rates can be arbitrary. Buy 1/100 of a share for 75 cents. Hold until $100 is hit then sell. The payoff of 1 dollar is replicated for an upfront cost of 75 cents. The arbitrage-free value of the option is 75 cents.

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this would only work in a world without interest rates I think. For in that case the expected return under the risk neutral measure should be zero. – Probilitator Feb 23 '14 at 15:38
Hm, I'm a little confused about why this strategy isn't an arbitrage. For example, suppose $\tau$ is the first time the stock hits \$100. Let's say that at time $t=0$ I borrow \$0.75 at zero interest from some riskless asset MoneyMart and use this to buy 1/100 of a stock. Then, at time $t=\tau$, I can sell this 1/100 of a stock for \$1, pay off the \$0.75 I owe to MoneyMart, and still have \$0.25 leftover. Have I made any assumptions that aren't allowed here? – sourisse Feb 24 '14 at 16:15
...Or are we treating the stock as a riskless asset because we know that it hits \$100 almost surely? – sourisse Feb 24 '14 at 16:22
the answer only makes sense if you don't allow borrowing - makes sense in a zero interest rate environment (thus no short position in the bank account). If you allow borrowing you will have an arbitrage as you have observed correctly. Also above exercise doesn't say anything about absence of arbitrage. – Probilitator Feb 25 '14 at 9:15
The strategy I proposed was not meant as an arbitrage. Perhaps I should have clarified that the only 2 assets in the economy are the target option and its underlying stock. If you add a third asset which is the ability to borrow at zero interest, then I agree that this 3 asset economy has arbitrage. My 2 asset economy is arbitrage-free and the payoff from the option is spanned by that ability to scale the position in the stock under the given assumption that the barrier is eventually hit. – Peter Carr Mar 1 '14 at 15:34

First let's recapitulate:

  • The market is free of arbitrage if (and only if) there exists a martingale measure;
  • The market is complete if and only if the martingale measure is unique;
  • In an arbitrage-free market, not necessarily complete, the price of any attainable claim is uniquely given, either by the value of the associated replicating strategy, or by the risk neutral expectation of the discounted claim pa yoff under any of the equivalent (risk-neutral) martingale measures.

It is hard to make an assumption on the existence of an equivalent martingale measure if the market dynamics are not given (e.g. if you don't know what stochastic processs drives the underlying asset)

Showing that an equivalent martingale measure exists depends on the setting. A lot has been researched here. I can recommend the following paper that gives a decent overview.

Let $S_t$ be the stock process. If $r=0$ and if there is an equivalent martingale measure $Q$ than $S_t exp(-rt)$=$S_t$ must be a martingale (due to $r=0$ we have no discounting). Thus $\mathbb{E}^Q[S_t]=S_0$.

Let $P_t$ be the porflio we use to hedge the claim. For us to create an arbitrage $P_0=0$ and $\mathbb{P}(P_T\geq 0)=1$ at some time $T$ in the future must hold. If we were to finance $w$-shares of the stock by borrowing our portfolio would be $P_0=wS_0 - wS_0=0$. At every time $t$ in the future the expected return will be $\mathbb{E}^Q[wS_t - wS_0]=0$. Now $S_t$ is a martingale. This means that $\forall t , \mathbb{P}(S_t<S_0)>0$. For if $\mathbb{P}(S_t<S_0)=0$ for some $t$ it would follow that $\mathbb{E}^Q[S_0]<\mathbb{E}^Q[S_t]$ and $S_t$ would not be a martingale.

This means that you always have a positive probility of loss nomatter how long you keep your stock (denoted by $\forall t , \mathbb{P}(S_t<S_0)>0$) Thus the arbitrage you constructed above can not exist.

Also note that in above setting the price your instrument would be $\mathbb{E}^Q[0.01 \cdot S_\tau]=0.01 \cdot S_0=0.01 \cdot 75=0.75$. I used optional sampling here (with $\tau$ being the stopping time of $S_t$ reaching $100$).

Alose note that I use $\mathbb{E}^Q[0.01 \cdot S_\tau]$ for $0.01 \cdot S_t$ is the porfolio that replicates the payout and as you know price of the instrument equals price of hedging etc.

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