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13

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 ...


10

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 ...


10

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


7

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.


6

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 ...


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

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 ...


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.


5

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

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


4

Since American style options allow early exercise, put-call parity will not hold for American options (unless they are held to expiration). In practice, there is also a difference between calls and puts for European options as well. The full description is here: What causes the call and put volatility surface to differ?


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.


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 ...


2

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}) ...


2

Let's talk about your first equation: If you exercised your option early, you got this payoff. But if you are a rational investor you'd realize that this is less than what you would get if you would just sell your option itself. i.e. the payoff at time t will be more than S(t)-K because the option is worth more than that as it also has some time value. so ...


1

If $PV_{t, T}(\text{Divs}) \ge K\big(1-e^{-r(T-t)}\big)$, since $P_{Eur}(S_t, K, T-t) >0$, the identity \begin{align*} C_{Eur}(S_t, K, T-t) = P_{Eur}(S_t, K, T-t) + (S_t-K) -PV_{t, T}(\text{Divs}) +K\big(1-e^{-r(T-t)}\big), \end{align*} implies that \begin{align*} C_{Eur}(S_t, K, T-t) > (S_t-K). \end{align*} That is, it is not rationale to exercise the ...


1

Let $\{X_t \mid t \ge 0\}$ be the foreign exchange rate rate from $£$ to $\$$. Moreover, let $C(X_0, K, T)$ and $P(X_0, K, T)$ be the prices of the respective call and put options with strike $K$ and maturity $T$. Then \begin{align*} \frac{1}{X_0}P(X_0,\, K,\, T) = K C\left(\frac{1}{X_0},\, \frac{1}{K},\, T \right). \end{align*} Based on the given condition, ...


1

What a difficult problem. The first line gave $165 e^{-rt} -3 S e^{-dt} = 15$ [since 50+55+60 = 165]. In the second line we want to evaluate $110 e^{-rt} -2 S e^{-dt} $. We notice that this is exactly two thirds of the left side of the above, because 110 is two thirds of 165 and 2 is two thirds of 3. So we take two thirds of the right hand side of the first ...


1

In this case, call option is deep in the money while put option is deep out of money. As maturity is very near, any change in stock price would have equivalent impact on the call option price. Decline in one dollar in stock price lead to almost one dollar decline in call option price. Whereas for put, its worth would increase but put is still deep out of ...


1

Futures payoff is indeed $S_t-F_0$, but the $t$ in question is the maturity date of futures. In this derivation $t$ denotes maturity date of the option, which is always before the futures maturity. Therefore, on the day of option maturities, the futures did not expire yet, but the value of the futures position is $F_t-F_0$ (in mark-to-market sense, you can ...


1

the call version pays $$ I_{S_T > K } S_T $$ the put version pays $$ -I_{S_T < K } S_T $$ Subtract to get a pay-off $$ S_T. $$ (ignoring the probability zero event of $S_T=K.$) So the prices subtract to give $S_0.$


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

@RemusStanescu Question 2) was answered quite intuitively but incorrectly by Freddy (he'd be right if he focused on conditional expectations rather than probabilities: indeed, P(s>110) < P(s<90) assuming lognormal dynamics for the underlying stock.) This follows from its negative skewness, which is key to your question. First note that call and put ...


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

In the Put-Call parity r is assumed to be risk-free interest rate. In reality, the interest rate the is rate at which interest is paid by a borrower for the use of money that they borrow from a lender. Its behavior is similar to price in the market , which price fluctuation depends on the news in the market. It is usually higher than risk-free interest ...


1

In the Put-Call parity you assume that a risk-free rate $r$ exists, but that's not the case in reality; there is no using risk-free rate. But I can't tell you why it's decreasing, but it's not surprising that it's not constant. What's more surprising though, is that $e^{-rt}>1$ is should be smaller. Could you plot $r$ only?



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