Pricing a Call Whose Strike Is Below Every Possible Future Price
Summary
The document considers a one-period security with finitely many possible terminal prices, all above a call’s strike. Since the call is certain to finish in the money, its payoff in every state equals the security price minus the strike. This makes the call payoff replicable by holding one unit of the security and borrowing the present value of the strike. Under continuous compounding for a one-period horizon, the resulting no-arbitrage relation is the current security price minus the discounted strike.
The answers also describe risk-neutral pricing: assign probabilities under which discounted traded asset values have the appropriate expected return, then value the option by its expected payoff under those probabilities. The replication argument gives the price directly and does not require knowing the terminal probabilities. The post’s initial uncertainty about the current security price and interest rate is resolved by treating them as observable inputs at valuation time. The result assumes the strike is below every possible terminal price and that the financing and trading assumptions permit the stated replication.
Key ideas
- When every possible terminal price exceeds the strike, the call payoff equals terminal asset value minus the strike in every state.
- One share combined with borrowing the discounted strike replicates that payoff.
- The call price is the current share price minus the present value of the strike.
- Risk-neutral expected payoff pricing gives the same value when the no-arbitrage assumptions hold.
- The conclusion relies on the strike being below all possible terminal prices.
Tags
Full text
# Price an option whose strike price is always lower than the future price of the security
# Price an option whose strike price is always lower than the future price of the security
Suppose it is known that the price of a certain security after one period will be one of the $m$ values $s_1,\ldots,s_m$. What should be the cost of an option to purchase the security at time $1$ for the price $K$ when $K < \min s_i$? (This problem is Exercise 5.3 in Sheldon M. Ross, An Elementary Introduction to Mathematical Finance, 3/e.)
I know that the value of the call option at time $1$ will be $s_i-K$ if the price of the security will be $s_i$ at time $1$. Then one unit of the security at time $1$ will be equal in value to one unit of the call plus a sure amount of $K$. Then if the interest rate is $r$ compounded continuously, the security price $S$ and the option price $C$ at time $0$ should satisfy $$S=C+Ke^{-r}.$$ However I am not sure how to calculate the exact cost of this option, given that neither the price of the security nor the interest rate is known. This chapter is mainly about arbitrage so I think this problem should be solved via arbitrage.
## Answer by Anna Taurogenireva (score 2, accepted)
https://quant.stackexchange.com/a/16149
That is the answer, $S-e^{-r}K$. It depends on the values $S$ of the security at time $0$, on $K$ and on the interest rate $r$. All of these you can assume know at time $0$, i.e. now.
## Answer by Arshdeep (score 2)
https://quant.stackexchange.com/a/41378
You use no-arbitrage theorem:
There exist probabilities on each outcome such that expected gain on every wager is 0.
Or, risk neutral pricing:
The expected return on each wager grows at the compounded discount rate. In particular, consider the wager of buying the stock today and selling tomorrow.
By either argument the 'risk neutral' probabilities associated with each possible outcome are such that expected gain on above wager is 0. Now price the option as expected payoff under these probabilities. You arrive at the answer as posted above.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.