19 votes
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 ...
LocalVolatility's user avatar
13 votes
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 ...
Daneel Olivaw's user avatar
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 ...
Kevin's user avatar
  • 15.2k
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 ...
Yulia V's user avatar
  • 346
7 votes
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 ...
Kurt G.'s user avatar
  • 2,003
6 votes
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 ...
BKay's user avatar
  • 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 ...
StackG's user avatar
  • 2,996
5 votes
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.
Brownian3's user avatar
  • 121
5 votes
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 ...
jwg's user avatar
  • 224
5 votes
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 ...
Quantuple's user avatar
  • 14.5k
5 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 ...
Antoine Conze's user avatar
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)}, \...
Quantuple's user avatar
  • 14.5k
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 ...
Antoine Conze's user avatar
5 votes
Accepted

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 ...
Kevin's user avatar
  • 15.2k
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 ...
dm63's user avatar
  • 16.5k
4 votes
Accepted

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_{...
LocalVolatility's user avatar
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 ...
febstar's user avatar
  • 51
4 votes
Accepted

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 $...
Antoine Conze's user avatar
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 ...
dm63's user avatar
  • 16.5k
4 votes
Accepted

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 ...
Daneel Olivaw's user avatar
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 ...
Magic is in the chain's user avatar
4 votes

STIR topics: Implied FX-OIS Basis and FX Forward/Swap Pricing

Implied FX-OIS basis should be pretty simple to "compute", it is the classical "Cross-currency" basis observed in FX Swaps & FX Forwards, that can be backed out when plugging ...
Jan Stuller's user avatar
  • 5,998
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, \...
Gordon's user avatar
  • 21k
4 votes

Is negative forward variance an arbitrage?

Let $$ V_t^{T_1,T_2}=\frac{(T_2-t)V_t^{T_2}-(T_1-t)V_t^{T_1}}{T_2-T_1} $$ be our forward variance where $t<T_1<T_2$, $V_t^{T_1}$ is the ATMF implied vol as seen at time $t$ for slice at maturity ...
fwd_T's user avatar
  • 747
3 votes

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 ...
dm63's user avatar
  • 16.5k
3 votes

How to price a path dependent exchange option using?

I solved it the following way, just want make sure I'm not missing something obvious. Set up a portfolio $PF$ consisting of long $S$ and short $P$ at time $t = 0$. Choose arbitrary time $0 < t <...
meg1806's user avatar
  • 33
3 votes

Forward contract pricing of coupon paying security

(This answer is broadly in line with the comment of Amsh. I added it because Amsh his 1 line solution says substract the PV (present value) of the div; However, the example below shows that one should ...
mbison's user avatar
  • 1,558
3 votes

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 ...
Antoine Conze's user avatar
3 votes
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. ...
Antoine Conze's user avatar
3 votes
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 ...
Alex C's user avatar
  • 9,332

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