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15

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 ...


12

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


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 ...


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

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.


6

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 ...


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 ...


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.


3

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}) ...


3

First, we have $P(t)+S(t)=C(t)+B(t,T)\cdot K$, Then, $\frac{\partial P(t)}{\partial S(t)} + \frac{\partial S(t)}{\partial S(t)} = \Delta^{\text{put}}_{t}+1$ and $\frac{\partial C(t)}{\partial S(t)} + \frac{\partial [B(t,T)\cdot K]}{\partial S(t)} = \Delta^{\text{call}}_{t}+0$. Finaly, $\Delta^{\text{call}}_{t}-\Delta^{\text{put}}_{t}=1$. This relationship ...


3

Put-call parity says that the difference between a call and a put is equivalent to the difference in the current stock price (adjusted down for dividends) and the strike price discounted at the risk-free rate. $$Call - Put = S_0*e^{-div} - K*e^{-rt}$$ So, if you want to have 120 dollars in the future, you would need to need to have $120 worth of "K" or ...


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

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 ...


2

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 ...


2

Let's call R the riskless security (100 today, 120 at time T). And call S the stock = 50, and either 70 or 30 at time T. One way to look at it is: A] Consider: buy 2 call options (C), short the stock (S), invest 50 (proceeds from S) in R. At time T: S=70: 2C=40, buy back S=-70, proceeds from R=60. net: 30 S=30: 2C=0, buy back S=-30, proceeds from ...


2

If the riskless security cost $100$ at time $t=0$ and $120$ at time $T$ then the risk free rate, $r$, is $20\%$. So that, $r=0.2$. Denote the initial stock price as $S_0$ and price of the call option as $c$. Suppose that at time $t=0$ you buy one stock and sell $\Delta$ options. Your portfolio value at time $t=0$ is $$P_0 = -\Delta\times c + S_0$$. At time ...


2

The intuitive explanation is given in @Alex C's comment. You should stick to that if you understand it. Yet, if you are more comfortable with a mathematical approach: Payoff of being long a forward contrat with maturity $T$: $(S_T - X)$. Interpretation: at time $T$, you pay a certain price $X$ in exchange for which you receive the underlying $S_T$ Payoff ...


2

Call-put parity writes (to see this, notice that $(S_T-K)^+ - (K-S_T)^+ = S_T - K $ and take the discounted risk-neutral expectation $E^{\mathbb {Q}} [. \vert \mathcal {F}_0 ]$ on both sides): $$ C(K,T) - P(K,T) = DF ( F(0,T) - K ) $$ with $DF = e^{-rT} $ the discount factor, and $F(0,T)$ the fair forward price given by $$ F(0,T) = (S_0 - D^*)e^{rT} $$ ...


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

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

It costs 0.03 dollars for the option to (sell 1 pound/buy 1.5 dollars. Now divide everything by 1.5: It costs 0.02 dollars for the option to (sell 2/3 pound / buy 1 dollar). Now convert to pounds at spot rate: It costs 0.0133 pounds for the option to (sell 2/3 pound / buy 1 dollar). Done


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, ...



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