28 votes
Accepted

Arbitrage opportunity interview question

A similar question for put option has been discussed in this question: Finding Arbitrage in two Puts. Basically, the call option payoff is a convex function of the strike. Then the call option price ...
  • 20.5k
24 votes

Carr-Madan Formula

For a sufficiently smooth function $f$, positive constant $a$, and $x>0$, Note that, \begin{align*} f(x) -f(a) &= \int_a^{x} f'(v) dv \\ &= \int_a^{x} \big[f'(v) -f'(a) + f'(a) \big] dv \\ &...
  • 20.5k
12 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 ...
12 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 ...
  • 2,856
9 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 ...
  • 1,018
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) = ...
  • 20.5k
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 ...
  • 910
6 votes

Carr-Madan Formula

The main interest of the formula is that it allows you, at least theoretically, to replicate any European option with payoff $f(\cdot)$ using only Call and Put options. As simple examples, consider $...
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: ...
  • 7,711
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 ...
  • 14k
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 ...
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 $dt$...
  • 3,856
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 \...
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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 ...
  • 1,111
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*}
  • 20.5k
5 votes

What is wrong with this method of european option pricing?

Firstly, there is nothing wrong with the fast Fourier transform approach from Carr and Madan (1999). However, there is a whole range of reasons why there is research about other numerical approaches. ...
  • 14k
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 ...
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 ...
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4 votes
Accepted

Isn't Black's approximation for American options inconsistent?

There is a logical fallacy in your argument. The price of a European call expiring 1 day before a dividend payment may well be greater than that of a call expiring after it. In other words, ...
  • 14.1k
4 votes
Accepted

How does financial institutions value European options in practice?

Practitioners tend to wear Black-Scholes glasses when dealing with European options: to them, quoting a certain option price today $V(S_0;T,K)$ is equivalent to quoting the forward price of the ...
  • 14.1k
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: ...
  • 1,644
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'...
  • 3,880
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 (...
  • 1,111
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 ...
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 ...
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 ...
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, ...
  • 9,167
4 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 ...
  • 14k
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 ...
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