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Questions tagged [no-arbitrage-theory]

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14
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2answers
5k views

Fundamental Theorem of Asset Pricing (FTAP)

In the spirit of canonical questions please state here versions of the FTAP in the following form (please only one theorem by answer) : Necessary definitions (or a direct link to definitions) ...
12
votes
7answers
2k views

What is the fair price of this option?

Without having to use Black-Scholes, how do I price this option using a basic no-arbitrage argument? Question Assume zero interest rate and a stock with current price at \$$1$ that pays no dividend. ...
11
votes
2answers
726 views

Is it possible to understand financial theory without mathematics?

I am trying to develop a short course on financial theory, covering the fundamentals of forward and options pricing, and 'efficient market' theory. I want to reduce the amount of mathematics to a ...
9
votes
2answers
315 views

Does numeraire have to be a tradable asset

I thought we create replicating portfolios using underlying and the numeraire i.e. the numeraire has to be a tradable asset (assuming simple binomial model). But I have seen some examples which ...
8
votes
2answers
2k views

Arbitrage Free Volatility Smile

When ATM implied volatility is higher than OTM put and call I believe that the volatility smile is no longer arbitrage free? Why is that? On the other hand, when ATM implied volatility is lower than ...
8
votes
4answers
572 views

Efficient Markets Paradox

Basically all Quant Finance theory is build on No-Arbitrage presumption and Efficient Markets Hypothesis. The known Grossman-Stiglitz Paradox says: if one can't make money from trading, one wouldn't ...
8
votes
1answer
982 views

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

Aaron Brown (in the book, The Poker Face of Wall Street, p. 196), discusses four approaches to deriving the same Black-Scholes-Merton option-pricing formula: Ed Thorp, Myron Scholes, Robert Merton, ...
8
votes
1answer
313 views

Non-arbitrage theory and existence of a risk premium

Consider a probability filtred space $(\Omega, \mathcal F, \mathbb F, \mathbb P)$, where $\mathbb F = (\mathcal F_t)_{0\leq t\leq T}$ satisfing the habitual conditions and isgenerated by $1 d $- ...
8
votes
2answers
570 views

Efficiency vs. Robustness - To use a constant or not in single factor time-series regression?

Arbitrage pricing theory states that expected returns for a security are linear combination of exposures to risk factors and the returns on these risk factors. Betas, or the exposures of the security ...
8
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2answers
2k views

How to check that an interest rate curve is arbitrage free

I have 2 interest rate curves (LIBOR 3M and OIS). I want to create stress scenarios for those two curves. Is it possible that some scenarios will make my term structure arbitrageable? How can I test ...
8
votes
1answer
151 views

Linear interpolation of local vol no arbitrage

We already know the equivalence between local vol, implied vol and option price and there ...
8
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1answer
367 views

Integral-differential equation for forward rates

I am struggling in this question: Let $P(t,T)$ denote the price of a zero-coupon bond (with marturity at time $T$) at time $t \in [0,T]$. As usual, at time $t$ for maturity $T$, the forward rate is ...
7
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1answer
1k views

Arbitragefree Pricing: Q vs. P

I read that the Fundamental Theorem of Asset Pricing states, that a market is arbitrage-free if and only if there exists an equivalent martingale measure Q, under which the discounted asset price ...
6
votes
1answer
193 views

FTAP a-la Harrison, Kreps and Pliska

I was reading the papers co-authored by Harrison, Kreps and Pliska, that initiated the formal research on the connection between pricing, martingale measures, arbitrage and completeness. I have some ...
6
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0answers
153 views

No arbitrage conditions for normal implied volatility

usually the term implied volatility refers to Black-Scholes implied volatility (also Log-Normal volatility): it is defined as a quantity which when plugged in the Black-Scholes formula returns the ...
6
votes
0answers
198 views

Arbitrage free price of a derivative when the price is collected over the lifetime of the derivative

Let $X_t$ be an american style financial derivative with random exercise time $T$ where $t$ and $T$ belongs to some finite set $A$. Buying this derivative requires the buyer to pay $p_t$ up to time $T$...
5
votes
1answer
292 views

Prove arbitrage opportunity

The continuously compounded interest rate is $r$. The current price of the underlying asset is $S(0)$ and the forward price with delivery time in 1 year is $F(0,1)$. Short selling of the stock ...
5
votes
1answer
971 views

Sufficient conditions for no static arbitrage

In Carr and Madan (2005), the authors give sufficient conditions for a set of call prices to arise as integrals of a risk-neutral probability distribution (See Breeden and Litzenberger (1978)), and ...
5
votes
2answers
2k views

Does No arbitrage(NA) imply efficient markets (EMH)?

The EMH states that stocks are traded at its fair values. This means there is no arbitrage strategy in efficient markets. However, if the market is no arbitrage, can we conclude the market is ...
5
votes
1answer
838 views

Proving that Absence of Arbitrage does not imply law of one price

I am trying to prove that the Absence of arbitrage statement (AOA) does not necessarily imply the law of one price (LOP). For the definitions of these concepts I am using Cochrane's book "Asset ...
5
votes
4answers
300 views

risk-neutral valuation implies no arbitrage?

It is known that in an arbitrage-free continuous time market, the price of every asset is evaluated as the corresponding price in the replicating strategy using risk-neutral valuation. I want to ...
5
votes
2answers
456 views

arbitrage in Heston model

Really struggling in this question: Consider a market with two assets $(B,S)$ whose price dynamics satisfy \begin{equation} dB_t = B_t r dt \end{equation} \begin{equation} \quad \quad \quad \quad \, ...
4
votes
3answers
395 views

arbitrage free volatility surface

Why is calendar spread arbitrage equivalent to $\partial_t \omega(k,t) \geq 0, \forall k \in \Bbb{R}$ where $\omega(k,t) = \sigma^2(k,t) t$ and $\sigma(k,t)$ represents the Black-Scholes implied ...
4
votes
2answers
237 views

What is the arbitrage opportunity in this simple one-period market?

I have a single period market, and three states, and I have 3 risky assets. I assume no interest. So I have three states $\Omega=\{\omega_1,\omega_2,\omega_3\}$. All assets start with the value 1, ...
4
votes
4answers
4k views

Simple value of a Forward contract at an intermediate time question

I am taking "Financial Engineering and Risk Management Part I" from Columbia University on coursera and I got a seemingly simple question wrong on the first quiz. This is all based on the no-arbitrage ...
4
votes
2answers
630 views

Equivalent (true) Martingale Measures and no-arbitrage conditions

I hope this is the correct site for this question, as it is rather theoretical... In their famous paper, Delbaen and Schachermayer proved that the No Free Lunch with Vanishing Risk condition is ...
4
votes
2answers
946 views

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

The title, and might I add, that this question is in relation to the Black-Scholes model and why the concepts are important for option pricing in general.
4
votes
1answer
379 views

Build a Synthetic Loan for Personal Finance

Suppose I am short of cash and want a loan for some mundane objective like travelling or buying a car. The interest rate for personal loan with my bank is too high. Is there any way in finance that ...
4
votes
2answers
404 views

How does this follow from the separating hyperplane theorem?

This is from Pliskas book in mathematical finance. I do not know what was best to write the question so I included the pages from the book. He has not written what form of the separating hyperplane ...
4
votes
2answers
1k views

law of one price, understanding

I am reading about mathematical finance, and I was tipsed to ask the quesiton on this site. It is about the "law of one price". Just first I'll make precise the model my book uses: I have a single ...
4
votes
0answers
52 views

Why does risk-neutral price processes do not, in general, compose all arbitrage-free price processes?

I was reading reviewing my mathematical finance notes and I came across a remark I cant understand fully Remark :Contrary to discrete time models, the risk-neutral price processes do not, in general, ...
4
votes
1answer
128 views

No-arbitrage in term-structure models

I am a bit confused about what the implication of "no-arbitrage" in popular term struchture models (such as affine term struchtre models or HJM models) are? Is it solely a restriction on the cross-...
4
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0answers
226 views

Show that in an arbitrage-free and non-redundant market a certain set is compact

Some notation: We consider a financial market with $d+1$ assets, the $0$-th asset is considered the risk-free asset, the others are the risky ones. The vector $\overline \pi \in \mathbb R^{d+1}$ ...
3
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2answers
499 views

Pricing when arbitrage is possible through Negative Probabilities or something else

Assume that we have a general one-period market model consisting of $d+1$ assets and $N$ states. Using a replicating portfolio $\phi$, determine $\Pi(0;X)$, the price of a European call option, with ...
3
votes
3answers
154 views

How to price a path dependent exchange option using?

Assume you have two stocks $S$ and $P$ so that at initial time $t = 0$: $S_0 > P_0$. You bought an option which pays off $S_T - P_T$ as long as $S_t > P_t$ through the time $0 < t < T$. ...
3
votes
1answer
300 views

How to price this option without using BS framework

We have a stock at price 1 dollar which pays no dividend. Also we assume zero interest rate. When the price hits $H$ dollars for the first time where $H>1$, we can exercise the option and receive 1 ...
3
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1answer
568 views

How to understand the no call or put spread arbitrage condition

The book Advanced Equity Derivatives Volatility and Correlation page 22 said To preclude arbitrage we must at least require: ...
3
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2answers
189 views

Do underlying assets have a no-arbitrage price?

Can it be shown that the Fundamental Theorem on Asset Pricing (FTAP) applies to underlying assets -- namely bonds, equities, and commodities? FTAP says that assets have no-arbitrage prices equal to ...
3
votes
1answer
66 views

How to derive no-arbitrage conditions w.r.t. the variance of a trinomial tree?

For a trinomial pricing tree, some notes say there are two no-arbitrage conditions: (1) $E[S(t_{i+1})|S(t_{i})]=e^{r{\Delta}t}S(t_{i})$ (2) $Var[S(t_{i+1})|S(t_{i})]=[S(t_{i})]^2\sigma^2\Delta{t}$ ...
3
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2answers
355 views

Dumb question: is risk-neutral pricing taking conditional expectation?

Dumb question: is risk-neutral pricing taking conditional expectation? $\tag{1}$ In trying to recall intuition for risk-neutral pricing, I think I read that we should price derivatives risk-neutrally ...
3
votes
1answer
303 views

Prove that a market is arbitrage free

The question is based on a one period model. Let a market be arbitrage free, and then let a security $X$ be added to it. Denote $P(X)$ as the price of this security at $t=0$. The security has the ...
3
votes
1answer
208 views

Equivalent Definitions of Self-Financing Portfolio

Consider a multi-period model with $t=0,...,T$. Suppose there is a bond with $B_0=1$ and $B_t=(1+R)^t$ and a stock with $S_0=s_0$ and $$ S_{t+1}=S_t\,\xi_{t+1}, $$ with $\xi_t$ iid random variables....
3
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1answer
259 views

Self-Frontrunning Arbitrage

If I have a large order to fill, shouldn't I always buy a derivative in the same direction to profit from the market impact? E.g. I sell 1 million shares and so I buy a put, which will hence almost ...
3
votes
1answer
611 views

Why is the volatility smile important

One thing I can't understand clearly is why there is so much focus on the volatility smile. Given my knowledge of the Black and Scholes model, this is what I get: People use the volatility smile as a ...
3
votes
0answers
203 views

Binomial model's Radon-Nikodym derivative

Related: Dumb question: is risk-neutral pricing taking conditional expectation? In the one-step binomial model... For $\frac{d \mathbb Q}{d \mathbb P}$, I think it's $\frac{d \mathbb Q}{d \mathbb P}...
3
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2answers
449 views

Arbitrage and dominant strategies

If there is no arbitrage there is no dominant trading strategy, but there may be arbitrage opportunities even if there are no dominant trading strategies. Could you explain this statement and bring ...
3
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0answers
219 views

Stochastic discount factor (aka deflator or pricing kernel) and class D processes

When (under what assumptions on the model) does a Stochastic Discount Factor need to be of Class D? What would be the implications if it was not? Is it connected to one of the no-arbitrage notions?
2
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1answer
108 views

Forward contract pricing of coupon paying security

PLease help me in understanding how to price forward contract for coupon paying security. For instance if we get into a contract to buy a security in next six month whose coupon due in next two month. ...
2
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1answer
109 views

Showing that a market model has arbitrage and describing martingales

This is an exercise which I came upon while studying an introduction to financial mathematics. Exercise : Consider the finite sample space $\Omega = \{\omega_1,\omega_2,\omega_3\}$ and let $\...
2
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1answer
75 views

Pricing and Arbitrage of Inverse Asset Claim

I'm working through the following little exotic exercise and have some questions and curiosity as to whether I'm on the right track Consider the claims $$Y_t=\frac{1}{S_t}$$ $$X=\frac{1}{S_T}$$ a) ...