13
votes
Accepted
Prove that the butterfly condition is always greater than zero
You generally can't simply subtract two inequalities as you did in your attempt. Here are two approaches to solve your problem:
No-Arbitrage Argument
Assume that the initial value of the Butterfly ...
- 6,084
11
votes
Accepted
Conceptual explanation of the relationship between gamma and vega plotted against delta for a European call option
Gamma and vega have the same general shape , peaking at ATM and tapering to the tails. But gamma concentrate as the option gets closer to expiry (when vega is small). For options a long way from ...
- 1,113
11
votes
Do basket options have a closed form valuation formula?
I'm not completely certain from your question, but I'm going to assume you have a basket of $n$ stocks with prices $S_0(t)$ to $S_n(t)$, and you want to price an option with payoff at $C(\tau)$ at ...
- 3,056
9
votes
Option time value is Nd1-Nd2
That's nice. Starting from
$$C = e^{-r T}N_2 (F-K) + (N_1 - N_2) S$$
we can substitute $F= e^{r T}S$ (no dividend case) so we get
$$C = e^{-r T}N_2(e^{r T} S-K) + (N_1 - N_2) S =$$
$$= N_1 S -e^{-r T}...
- 11.6k
7
votes
Option pricing and mean reversion
From the SDE
\begin{align*}
\frac{dS_t}{S_t}= k(\theta-\ln S_t) dt + \sigma dW_t,
\end{align*}
where $\{W_t,\, t\ge 0\}$ is a standard Brownian motion, we obtain that
\begin{align*}
d(e^{kt}\ln S_t) = ...
- 21.2k
7
votes
Accepted
How do I prove that a certain price is price of European option in Black-Scholes framework
Yes it is actually just substituting it into the Black Scholes PDE. If the PDE is satisfied, $V(t,S(t)),t\ge 0$ is a martingale and hence $V(t,S(t)) = E_t (V(T,S(T))$ so that $V(t,S(t))$ is the ...
- 940
6
votes
Dependency of an option price on time till expiry
You've tagged this with 'black-scholes' but you don't have to make the assumptions of the Black-Scholes-Merton model to understand why the option price with time to expiry.
Consider this example: ...
- 8,581
6
votes
Accepted
Compute the price of a derivative
If you plot the function $f$, you see that you have a bear spread. You can build such vertical spreads either with call or put options. For example consider a portfolio selling one put option with ...
- 16.2k
6
votes
Accepted
Valuation of non-deliverable option
Without loss of generality, for a European call option with payoff $(S_T-K)^+$ at expiry $T$, whether the option is settled in cash or rather the strike/asset are exchanged should in theory have no ...
- 8,169
5
votes
A simple question: Cost of delta hedging when a call option is sold
You should go back to the derivation of the Black-Scholes equation (see this answer for example). The main point is that you can cancel the risk of the derivative over an infinitesimal time period $dt$...
- 3,946
5
votes
Can increase in volatility reduce the price of a deeply in-the-money European put?
If you hold an option, you're always vega long, i.e. if volatility increases, your position increases as well - regardless of moneyness and the option type (put or call). Note firstly that by the ...
- 16.2k
5
votes
Compute the price of a derivative
It would be much easier to start by writing the payoff using indicator functions. For example,
\begin{align*}
f(S_T) &= 3 \mathbb{I}_{S_T \le 30} + (33-S_T) \mathbb{I}_{30<S_T < 35} -2 \...
- 21.2k
5
votes
Accepted
Can strike prices of options be negative?
If the underlying asset cannot be negative, then an option on it with a negative strike would be meaningless. A call would always be in-the-money with no chance of being worthless, and a put would ...
- 1,511
5
votes
Proof European call price is always less than stock price. (proof verification)
What you need to note is the following:
\begin{align*}
S_T - \max(S_T-K, \,0) &= S_T + \min(K-S_T, \,0)\\
&=\min(K, \, S_T) >0.
\end{align*}
- 21.2k
5
votes
Why are there so many S&P 500 call options selling with strike @1000?
I'm also currently working on analyzing option-implied RNDs. I'm no expert but a couple of comments:
In addition to volume, you want to look at the open interest of the different strikes to conclude ...
- 760
5
votes
What is the intuition behind a positive theta for European long puts?
It’s just the effect of interest. If you are long a deep ITM European put, it is worth the PV of K minus the stock price. But one day later the PV of K has grown a bit. That’s it. It’s the opposite ...
- 17.5k
5
votes
Accepted
Why is this inequality strict for arbitrage argument for European call?
It is because to show the existence of arbitrage, it suffices to show that there is no chance of losing money,and a positive chance of making money. Arbitrage does not imply you are certain to make ...
- 17.5k
4
votes
Prove that the butterfly condition is always greater than zero
it's a model-free result. The pay-off is non-negative everywhere and positive somewhere.
Since it's non-negative everywhere, if its price was negative there would be a clear arbitrage.
We have to ...
- 6,993
4
votes
Accepted
How to make the arbitrage if intrinsic value is greater than European call value
how to construct the portfolio (St−K)+ or how to make this arbitrage
If you have this scenario on your hands then you construct the portfolio by putting as much capital as you can into the trade. It'...
- 4,368
4
votes
How to make the arbitrage if intrinsic value is greater than European call value
The intrinsic value of a call is the price of the underlying minus the strike (S0-K), so if you find a european call whose value is less that that you would:
Sell (...
- 1,511
4
votes
Is there any useful links for option pricing (american + asian + european) using R
Below is an example of how you could plot a "call" option value with RQuantLib:
...
- 1,704
4
votes
Valuation of Bermudan option as maximum of relevant European options
You are wrong. Using the maximum of the prices of the European options is equivalent to choosing (and making that choice final) on $t=0$ the date $t_i$ on which you will exercise. As such a choice ...
- 5,702
4
votes
A simple question: Cost of delta hedging when a call option is sold
Based on the inputs from other users, this is another non-rigorous proof of why the cost of delta-hedging is equal to the option price. This approach might be useful for students who use John Hull for ...
- 277
4
votes
Accepted
Why do we need to calibrate vega?
It seems like he is assuming that the shorter term volatilities change more than the longer term ones and the relatively sensitivity is proportional to $1 / \sqrt{T}$. Thus, this hedge is not against ...
- 6,084
4
votes
How to calculate implied correlation via observed market price (Margrabe option)
We know that $-1\le\rho_{imp}\le 1$ so perhaps the simplest approach is to try the possible values $\rho_{imp}=\{-1,-0.9,-0.8,\cdots,0.8,0.9,+1\}$, to calculate resulting $\sigma$ values, d± values, ...
- 9,402
4
votes
Compute the price of a derivative
Here's another way to do it, that I think is useful if you don't recognize/have knowledge of specific option spreads/techniques. This might help you on exams or other problems, although recognizing ...
- 666
4
votes
Accepted
Monte Carlo option pricing with R
Your code looks fine and it is encouraging that both MC simulations yield similar results. Please look at this simplified code for the analytical part of the Monte Carlo simulation. As you know, $$S_T=...
- 16.2k
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