Replicating a First-Hit Option with a Fractional Share

# Price an option and find a replicating portfolio


# Price an option and find a replicating portfolio












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.

## Answer by Peter Carr (score 4, accepted)

https://quant.stackexchange.com/a/10342

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.

## Answer by Probilitator (score 2)

https://quant.stackexchange.com/a/10384

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.