# Tag Info

Accepted

### 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 ...
Accepted

### 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 ...

### 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 ...

### 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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### 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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### 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 ...

### 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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### 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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### 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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### 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 ...

### 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 ...