# 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 ...
• 5,989
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 ...
• 8,059

### 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 ...
• 15.7k

### 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 ...
• 346
Accepted

### 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 ...
• 2,013
Accepted

### 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 ...
• 226

### 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 ...
• 5,662

### 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 ...
• 3,016

### 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 ...
• 5,662
Accepted

### 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.
• 121
Accepted

### 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 ...
• 224
Accepted

### 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 ...
• 14.6k

• 747

### Arbitrage Free Volatility Smile

Neither situation is necessarily an arbitrage. Negative smile is consistent with a 'thin-tailed' density function , just as positive smile is consistent with a fat tailed density function . It's ...
• 16.9k

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

Absence of arbitrage is in general considered equivalent to the existence of a risk neutral probability measure. The measure is unique iif the market is complete (meaning any payoff can be ...
• 5,662
Accepted

### At some intermediate time $t$, does money actually change hands in the trading of a futures contract?

Forward contract: exchange is done at maturity. Future contract: margin call is paid/received every day throughout the life of the contract, thus resetting the NPV of the position to zero every day. ...
• 5,662
Accepted

### Covered Interest Rate Parity with FX Spot-Adjustment

A simple trick is being used to come up with the right discount factors. Since $D_T=\frac{1}{1+r_1}\frac{1}{1+r_2}\frac{1}{1+r_3}\cdots\frac{1}{1+r_T}$ and $D_S=\frac{1}{1+r_1}\frac{1}{1+r_2}$ we ...
• 9,372