8
votes
Boundary for European Put Option
Let's carefully distinguish which exercise type we consider.
European-style call option
$$ \max\{S_0-Ke^{-rT},0\}\leq C_E \leq S_0.$$
European-style put option
$$\max\{Ke^{-rT}-S_0,0\}\leq P_E\leq ...
- 15.1k
7
votes
Accepted
Interpretation and intuition behind the Put-Call symmetry under the Heston Model
This is a consequence of transforming a Put on $S_T$ with strike $K$ into a Call on $(K S_0)/S_T$ with strike $S_0$ under the stock measure. The new set of parameters $r_p$, $q_p$, $\kappa_p$, ... etc ...
- 5,622
7
votes
Is it possible to have only one volatility surface for american options (that fits both calls and puts)?
Usually, there is only one vol surface (I have never seen or heard of anyone using two). Almost certainly the most advanced commercially available vol surfaces are built by voladynamics. They also ...
- 7,689
6
votes
Accepted
Implying risk-free rates using Put/Call parity
Once upon a time, all option contracts ceased trading on the third Friday of every month. There was no after hours trading for the underlying. When the exchanges closed, everything was done. This ...
- 4,339
6
votes
Accepted
Violation of the call-put parity
On 10/24/17, Wells Fargo announced that they would pay a dividend of 0.39 to holders of record on 11/3/17. Thus, if you buy the stock after this date (through the exercise of the call) you do not get ...
- 16.2k
6
votes
How to prove Gamma is the same for a European call and European put with the same inputs?
Put-call parity says that a call and put (worth $C$ and $P$ respectively) with the same strike $K$ have the following relationship with the spot rate $S$, risk-free rate $r$, and time to maturity $T$ -...
- 5,808
5
votes
Accepted
Put-Call Parity Application
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)} +...
- 106
5
votes
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Relationship between forward and option prices
At the heart of the (relative) pricing theory is the concept of no arbitrage and replication. I'll focus on equities here because as stated in the comments it may be more complicated for commodities.
...
- 14.5k
5
votes
Implying risk-free rates using Put/Call parity
Elaborating on my comment: consider a 100 point in the money collar, one day before expiration. You are effectively claiming the price of this is 99.5. But if you call the pit and get a two way ...
- 16.2k
5
votes
Accepted
Question about the vega of a stock
Vega is the partial derivative of the option price (as a function of parameters -- current stock price $S_t$, strike price $K$, implied volatility $\sigma$, etc.) with respect to $\sigma$ -- holding ...
- 3,595
5
votes
Accepted
Put-Call Parity with dividends
Diviends or not, the put-call parity (with European options) always hold:
$ C(S,K) - P(S,K) = F - K*DF $
In the RHS, dividends will impact the forward $F$ (higher dividends imply lower forward). So ...
- 624
5
votes
Accepted
How to prove no-arbitrage when a long butterfly is strictly positive?
I am not sure why the question you link to does not provide an answer. I’ll try to answer it but it is really similar to what has already been said there. Bottom line is: if the value $K$ is reachable ...
- 7,865
5
votes
Accepted
Black and scholes option pricing
Let us start by considering a bear spread strategy, consisting on long a European put with strike $K_2$ and short another European put with strike $K_1$. Then the payoff of this portfolio at expiry $T$...
- 7,865
4
votes
Accepted
Put-Call Parity on Currency and Binomial Trees
You have forgotten the combinatorial factors for binomial probabilities on your terms. You need $$ {n\choose k} p^n(1-p)^{n-k},$$ not just $$ p^n(1-p)^{n-k}.$$ The second term should have a factor of $...
- 605
4
votes
Accepted
Put Call Symmetry
The Black-Scholes symmetry formula is valid only under Black-Scholes as its name suggests. It works only for a lognormal $S$. For other models, you can find symmetry relations but they will be ...
- 2,180
4
votes
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Put-call parity for cash settled swaptions
The market standard formula approximation for cash settled swaptions applies Black/shifted Black/Bachelier around the forward swap rate so that with this formula parity between payer and receiver ...
- 5,622
4
votes
Is there an advantage trading options based on deep in the money Open Interest Volume ratio
I don't mean to suggest such a large topic, but it would certainly be worth reading about delta-hedging with regards to your question.
Since such a large percentage of options are delta-hedged, the ...
- 395
4
votes
Accepted
Should Put/Call Parity result in Zero Return or the Risk-Free Rate?
You should see the risk free rate as the return on the strategy. That’s because you actually have to invest money , namely usd 19000 minus usd 38, for the one month period. Hence, there is no ...
- 16.2k
4
votes
Accepted
Proving the put call parity
There are usually two ways to write proofs of equalities (like put-call parity) in quantitative finance.
By replication,
by constructing arbitrage.
Both of these are actually the same, since the ...
- 165
4
votes
How to prove no-arbitrage when a long butterfly is strictly positive?
as it was stated correctly in the question all long butterfly options have to have a non-negative premium in order for No-arbitrage to hold.
So we can say that:
No-Arbitrage holds implies All ...
- 1,436
4
votes
Accepted
Put Call derivation using two approaches [ Some confusion of getting different results ]
The put-call parity from CME, C - P = F - K, is not correct. I think CME is making it simple. You need to discount the right-hand side. Then, you will get the same ...
- 1,143
3
votes
Early execise of American Call on Non-Dividend paying stock.
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 ...
- 31
3
votes
Relationship between forward and option prices
In developed equity markets you have at least those three entities: Stocks, futures and options.
As far as my experience goes you indeed often hedge options with futures.
What is more interesting ...
- 27.3k
3
votes
Accepted
At the money put and call having the same price
One thing to notice is that indeed $E^Q[S_T]=S_0$, by construction, even though the stock price can only drop down to 0, but it can go up to $2\,S_0, 3\,S_0, 100\,S_0$, ...
Thus, implicitly, there ...
- 335
3
votes
Accepted
How far the spot price is likely to go from the current level in three months if its volatility is 15.7%
Keep in mind that there is nothing dynamic about this at all...it is only a snapshot and it is only a 1 sigma range.
High side:
...
- 4,728
3
votes
Construct option and stock portfolio
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-...
- 719
3
votes
Accepted
Construct option and stock portfolio
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 ...
- 456
3
votes
Difficulty understanding put-call parity for currency options
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 ...
- 16.2k
3
votes
Risk of Put-Call-Parity in practice
Financing Risk
e.g. You sell the stock short and buy the Call and sell the Put. Lets say you can only get a 1w repo borrow on the stock yet your options are 3m options. You have the risk after that ...
- 31
3
votes
Decreasing value of the Put option with increasing Time to maturity
This situation can arise with some non-vanilla options. For example, a digital put option, which pays $1$ if the underlying price $S$ is below a strike $K$ at expiry, can exhibit "negative theta".
...
- 3,595
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