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12

This is not a trivial question. Here's a relevant excerpt (an appetizer, really) from Hull's book (7th Edition, P. 75): It is natural to assume that the rates on Treasury bills and Treasury bonds are the correct benchmark risk-free rates for derivative traders working for financial institutions. In fact, these derivative traders usually use LIBOR rates ...


8

If your trades are collateralized/margined, you should use the rate paid on your collateral/margin.


7

To answer a question with a question - are you assuming proportional or constant dividends? :) The general consensus of the market is that dividends are somewhere between proportional (fixed yield) and constant (fixed dollar). The carry embedded into the forward prices at different strikes reflects that consensus, in fact you can establish the ...


6

A simple intuitive answer why the OTM Call is more expensive than the OTM Put is because of the skewness of the log-normal distribution. Think about it, what is the probability that the stock price is above 110 at expiration and what is the probability it is below 90? This should answer your question. Written in probability terms: The median of the ...


5

Dividends are the key. For simplicity, let's include a single dividend at the time of expiration, and assume that the options are European and expire ex. (There is really no reason not to assume that an option on a market index is European. EDIT: not quite true; that's discussed here.) $S+P = e^{-rt}K+C + e^{-rt}D$ This is a certain fixed dividend, but ...


5

In John Hull's Option's, Futures and Other Derivatives, it states in the chapter "Properties of Stock Options" that from put-call parity, it follows for American options that $$ S_0 - K \le C - P \le S_0 - K e^{-rT} $$ where $C$ and $P$ are the American call and put prices. In the book, the derivation is left as an exercise.


5

I think you might use the relevant OIS-rate like EONIA or Fed Fund Rate, at least this is the current fad when discounting interest rate swaps.


4

If you are trying to arbitrage the put-call parity, then use your collateral interest rate for the options side, and your cost of funds on the stock side of the equation. Yes, that's right, 2 different interest rates. Also, don't forget to incorporate bid-ask spreads. If you are trying to turn a put into a call for your own book, you don't actually need ...


4

I think you should put YOUR attainable interest rate. Because it is your view of how much the forward is worth. So, on the offer the rate at which you are indifferent is computed with the interest rate at which you borrow. And if you go short, the interest at which you can put your have your money fructified. No what rate can your money be fructified at ? ...


4

The put call parity is given as follows: $$c_t-p_t = S_t - \frac{X}{e^{r(T-t)}}$$ If you assume $r=0$, you get $$c_t-p_t = S_t - X$$ So, $c_t \neq p_t$. The rationale behind it is much more financial than mathematical. You have to look at the payoff on both side of the equation, and you see that both portfolio will give the same payoff at time $T$ (the ...


3

You compare apples and oranges here. You can't possibly compare the profit generated involving S(t) on one side and S(T) on the other side. at time t you do not know what the stock will be worth at time T. Merton made the statement in the context of deciding whether to exercise the call option at any time before expiration OR to simply sell the call ...


3

Put-call parity is a model free relationship, i.e. it makes no assumptions regarding the underlying. The underlying can be any trade-able asset. So it should hold in your case.


3

There is an interesting article entitled American Put Call Symmetry from the mid 90s that might be what you want.


2

The word "premium" in forward premium is more akin to risk premium than it is to option premium. In fact, the forward premium may be negative, whence it is called a forward discount. The premium/discount is merely the difference between the spot and forward prices, which may be due to interest rates and/or interest rate differentials and cost of carry ...


1

The put-call parity equation: $$c-p = S_0 - Ke^{-rT}$$ can be seen as a equality in cash flows--namely, buying a call and selling a put have equivalent cash flows to the underlying stock price less the strike price of the options. Taking this into $t=0$ means the current price of the call less the current price of the put must equal the present value of the ...


1

I think this is where your logic goes wrong: $(C_t − P_t − S_t)e^{r(T−t)} + K$ With reference to the above equation, you are saying that "...To that money that we owe, we add the money that we owe to the contract buyer.." Yes, $(C_t − P_t − S_t)e^{ r(T−t)}$ is the money that we owe, but $K$ is not referring to money that we also owe the contract buyer. ...


1

A logical way of answering this question is proof by contradiction. First note that if it is not optimal to exercise an american option prior to expiry, then the option should have the same value as the european option. So assume that it is not optimal to exercise the option prior to expiry, i.e. assume that the american option has the same value as the ...


1

Look at it the way you would have to realize it, whatever your position in the market is, as the price of a synthetic bond that pays $K$ at time $T$: \begin{array}{c}Ke^{-r(T-t)} & = & S & + & P & -& C \\ (\text{bid}) &=& (\text{bid}) &+& (\text{bid}) &-& (\text{ask})\\ (\text{ask}) &=& (\text{ask}) ...



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