18 votes
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Arbitrage Free Volatility Smile

I generally agree with @dm63's answer: A convex (concave) smile around the forward usually indicates and leptokurtic (platykurtic) implied risk-neutral probability density. Both situations can or ...
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12 votes
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Arbitragefree Pricing: Q vs. P

In the derivatives context, "arbitrage free" means almost surely for the probability measure under consideration. This is in opposition with statistical arbitrage used at high frequencies for example. ...
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  • 3,836
11 votes
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Does numeraire have to be a tradable asset

This is an interesting question that I have asked myself. Below is my take. Let us consider an economy $(\Omega,\mathcal{F},P)$ equipped with a filtration $(\mathcal{F})_{t \geq 0}$ consisting on a ...
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10 votes

Why do we need the self-financing assumption in risk-neutral pricing?

You don’t just need self-financing in a risk-neutral world but it’s a much more fundamental principle. If you look at a portfolio that is not self-financing, i.e. you can inject or withdrawal funds at ...
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  • 13.9k
9 votes
Accepted

What is the fair price of this option?

This option is a perpetual one touch option. Its price depends on the model used; additional assumptions are required to get a model-independent price. Let us first consider 3 important example ...
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  • 1,865
7 votes

Pricing when arbitrage is possible through Negative Probabilities or something else

You cannot use negative probabilities in this context. When there is no unique probability measure, there can be no unique price. You only know that it is in [0, 0.6] range, if you want to tighten ...
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  • 346
6 votes

What is the fair price of this option?

Let $T= \inf\{t>0: S_t = H\}$. Then the option payoff is given by $\mathbb{1}_{\{T < \infty\}}$, and the value of the option is given by $\mathbb{P}(T< \infty)$. We assume that the stock ...
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  • 20.5k
6 votes

Efficient Markets Paradox

Making money is not the only reasonable objective to trading. Another common reason is to manage/reallocate risk. For example, this is exactly the objective of liability-driven-investors, such as ...
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  • 463
6 votes

What is the difference between market efficiency, market equilibrium, and no-arbitrage?

In three bullet points: Efficiency: the obtained prices maximize assumed utilities of different agents. In their paper "The Valuation of Option Contracts and a Test of Market Efficiency", Cohen, ...
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  • 10.6k
6 votes
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Pricing when arbitrage is possible through Negative Probabilities or something else

I believe there is not a unique price if you can't short. Say, instead of buying the option you spent 0.5 on a half a unit of the asset $S^2_1$ This asset pays out $[0.4, 0.6, 0.8]$ which first order ...
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  • 226
6 votes

Help reconciling incorrect reasoning in options pricing brain teaser

Assuming that the only things that can happen on the period are $100$ and $50$, and we can buy a stock and a call option with strike $90$, even without knowing the probabilities of these moves we can ...
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  • 2,856
6 votes
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How to derive forward price on stock with continuous dividend

When the dividend yield $q$ is constant one can in fact derive a very simple forward formula under no model assumptions on $S_t$ (see (4) below). Only no arbitrage arguments are needed: The forward ...
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  • 1,509
5 votes
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Prove arbitrage opportunity

Suppose that the given condition is true. You want to construct an arbitrage portfolio to take advantage of this. Now, $d$ is an interest rate, and the condition suggests that $d$ is too high. So you ...
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  • 224
5 votes
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What is the arbitrage opportunity in this simple one-period market?

Sell 1 unit of S1,2,3 respectively, gain 3; buy 2 units of risk-free asset, cost 2. No matter which state appears, the future payoff/loss is 0 for sure, while you will gain 1 at the beginning.
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  • 121
5 votes

arbitrage free volatility surface

You'll find here that in terms of European option prices, the absence of calendar arbitrage writes $$ \frac{\tilde{C}(k\, F(0,t_2),t_2)}{F(0,t_2)} \geq \frac{\tilde{C}(k \, F(0,t_1),t_1)}{F(0,t_1)}, \...
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  • 14k
5 votes

Free Arbitrage conditions in ATM swaption surfaces

There are no no-arbitrage conditions on ATM vols of swaptions with different expiries/tenors, because the underlying swaps forward rates are different instruments. There are conditions however for ...
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5 votes
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No free Lunch and weak-star topology

The context of weak$^*$ topologies and no free lunch is often the proof of the first fundamental theorem of asset pricing. All the ideas below are from Delbaen and Schachermayer (1994). Notation ...
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  • 13.9k
5 votes
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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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  • 14.1k
4 votes

risk-neutral valuation implies no arbitrage?

I do this question to death in Concepts and ... If (discounted price of) everything is a martingale then every trading strategy is a martingale. Therefore any self-financing portfolio of initial ...
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  • 6,763
4 votes

Efficient Markets Paradox

The formation of asset price bubbles, such as the recent US housing market bubble, is perhaps the clearest indication that markets are not efficient. Hundreds of bubbles have been documented for all ...
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  • 876
4 votes
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No-arbitrage theorem: a proof

Consider a random variable $X$ that has a probability density function (PDF) $f(x)$. $X$ being non-negative means that $f(x) = 0$ for $x < 0$. The expectation of $X$ is thus \begin{equation} \int_{...
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4 votes
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How to understand the no call or put spread arbitrage condition

Let's focus on a European call option for the sake of the argument. Assume deterministic rates to keep notations uncluttered. Define $\Bbb{Q}$ as the probability measure associated to the money market ...
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  • 14k
4 votes

How are the two concepts No arbitrage & Risk neutral probability related?

A market model is arbitrage-free if and only if it has a risk-neutral probability measure. This is the fundamental theorem of asset pricing. That is, in a securities model, the two concepts are one ...
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  • 51
4 votes
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Replicating portfolio for claim on stock with discrete dividend

a) From the no arbitrage condition, and without ressorting to a specific model $$ PV[S(T)|S(T_0)] = S(T_0) $$ $$ S(T_0) = (1-\delta) S(T_0^-) $$ $$ PV[S(T_0^-)|S(0)] = S(0) $$ Therefore the PV of $...
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4 votes

Does numeraire have to be a tradable asset

An obvious example is using the maturity $T$ zero coupon as numeraire, and a European option with premium paid at time $T$ hedged with maturity $T$ forward contracts. You do not need to trade the zero ...
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4 votes

Risk-neutral pricing and statistical arbitrages

What you say is perfectly true and there is no contradiction. Arbitrage means risk free profit , so your ‘statistical arbitrage’ is not arbitrage at all. It just says that if you take risk, your ...
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  • 14.1k
4 votes
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Why do we need the self-financing assumption in risk-neutral pricing?

In practice, the self-financing condition can be regarded as an economic consequence of market competition. Take the perspective of an investment bank trading in hedgeable derivatives. If the hedging ...
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4 votes
Accepted

How to Take Advantage of Arbitrage Opportunity of Two Options

I think it is far easier to understand by just drawing the payoffs. You have two put options: A European put option on a non-dividend paying stock with strike price 80 is priced at 8 dollars, and a ...
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4 votes

Payoff of a Butterfly spread under risk neutral measure is always positive for any t<T

Note that \begin{align*} K_2 = \frac{K_1+K_3}{2}. \end{align*} Then \begin{align*} &\ \max(S_T-K_1, \, 0) + \max(S_T-K_3, \, 0) \\ =&\ \max\big(S_T-K_1 + \max(S_T-K_3, \, 0), \, \max(S_T-K_3, \...
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  • 20.5k
3 votes

What is the fair price of this option?

Consider a portfolio where I sell $\frac{1}{H}$ in stock and use that to buy an option. This is a 0 cost portfolio. When I hit the barrier the price of this portfolio is also 0. Law of one price would ...
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  • 603

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