Questions tagged [no-arbitrage-theory]
The no-arbitrage-theory tag has no usage guidance.
3
votes
0answers
40 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, ...
1
vote
0answers
51 views
When does funding cost of a portfolio enter into the portfolio's present value?
This question comes from some confusion when reading Hull's book and from the general concept of no-arbitrage/self-financing portfolios in stochastic finance books. I am not fully seeing the ...
3
votes
1answer
50 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) $Var[S(t_{i+1})|S(t_{i})]=[S(t_{i})]^2\sigma^2\Delta{t}$
...
2
votes
0answers
97 views
Butterfly Arbitrage condition
I hope anybody can help me. According to Gatheral and Jacquier (https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2033323) no Butterfly Arbitrage can be expressed like this:
Define the function
$\...
4
votes
3answers
254 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 ...
2
votes
0answers
45 views
Prove unique arbitrage-free price implies attainable
I just read a Corollary in a finance course note:
Suppose the market is arbitrage free and $C$ is a contingent claim. Then $C$ is attainable if and only if it admits a unique arbitrage-free price.
...
4
votes
1answer
97 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-...
2
votes
1answer
102 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 $\...
1
vote
1answer
102 views
Is there an arbitrage strategy if short selling of a stock is allowed?
Consider a market with a risk-free asset such that $A(0) = 100, A(1) = 110, A(2) = 121$ dollars and a risky asset, the price of which can follow
three possible scenarios
Is there an arbitrage ...
7
votes
1answer
129 views
Linear interpolation of local vol no arbitrage
We already know the equivalence between local vol, implied vol and option price and there ...
2
votes
1answer
250 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 ...
0
votes
1answer
100 views
Basic Replication of European Call Option
I am looking at the very basics of replicating an option with a portfolio of risky and risk free assets. As such we can define a portfolio of $x$ no. of shares, $y$ bonds & $z$ options at time $(T)...
0
votes
0answers
34 views
Arbitrage and state price formulation
I must be missing something really obvious due to my temporary obtuseness. Can someone please help me see the obvious? :-P Thank you.
I am just browsing Darrell Duffie's Dynamic Asset Pricing Theory. ...
2
votes
0answers
51 views
Forward spot calculation for a dividend paying no-short sell ETF
I am trying to fit an implied volatility curve for options on the SSE 50 etf that has no borrow (no short selling allowed) and pays a single annual dividend. I originally thought I could use the ...
2
votes
0answers
145 views
Detecting butterfly spread arbitrage for American options through European option prices
It's easy to demonstrate that if European option prices are concave with strike, then an arbitrage exists. For example, the risk-neutral probability density is the second derivative of European put ...
8
votes
2answers
252 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 ...
0
votes
0answers
55 views
Why is it called Linear Multi-factor models (APT)?
In what way are no arbitrage sentral for the APT model? The model is derived from maximizing a von Neumann-Morgenstern utility function subject to a budget constrant. Where does the no arbitrage ...
1
vote
0answers
130 views
American put option in binomial model - arbitrage opportunity?
I'm sorry this must be an elementary question. I spent a good deal of time searching through webs including this site for the problem but I got none.
Here's the problem:
Say we have a binomial tree ...
0
votes
2answers
423 views
How does a Delta Hedged portfolio yield the Risk-free?
Here I'm considering the simple case of a dealer writing call options on a stock and hedging the short position with a "textbook" Delta Hedge, i.e. goes long on $N_c \times Delta$ stocks (where $N_c$ ...
1
vote
0answers
59 views
How to Calculate the Value of a Growing Perpetuity Using a State Price Matrix?
Summary
I wish to value perpetual cash flows through state contingent claims on real consumption, where the state of the economy is assumed to follow a finite markov chain (Similar to Banz and Miller ...
1
vote
1answer
116 views
At some intermediate time $t$, does money actually change hands in the trading of a futures contract?
Assuming that the asset underlying a futures contract pays no dividends or associated (storage, etc) costs, I have the following formula for the price $F_t$ of a futures contract at time $t$:
$$
F_t = ...
1
vote
3answers
126 views
Besides arbitrage opportunities, are there other properties that real world markets cannot have
The article "What is ... a Free Lunch?" nicely explains why market models with arbitrage opportunity are unlikely to describe financial markets of the real world.
Are there other properties of ...
5
votes
0answers
113 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 ...
1
vote
1answer
188 views
Replicating portfolio for claim on stock with discrete dividend
This is a practice question for an exam:
Consider a market consisting of a bank account with a constant
interest rate $r$ and a stock $S$. The stock pays a proportional dividend
of size $\...
1
vote
1answer
62 views
Pricing weighted/average stock price claim
In a market consisting of a bank account with a constant interest
rate r and a non-dividend paying stock S, consider a T-claim that pays
$X = S(T)/S(T_0)$ at time T, where $T_0 < T$.
a) ...
3
votes
0answers
191 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}...
4
votes
2answers
723 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.
2
votes
2answers
335 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
533 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:
...
1
vote
1answer
67 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) ...
0
votes
1answer
124 views
How to show arbitrage when a European option price is greater than the no-arbitrage price?
My example is:
Current price = 20,
If it goes up it'll be worth 22, if it goes down it will be worth 18
risk free rate: 12%, time = 3 months
Strike = 21
call option is worth 0.633
I know that if the ...
1
vote
1answer
451 views
Why does arbitrage free imply complete market?
Proposition 2.10 of Tomas Bjork's "Arbitrage Theory in Continuous Time" states that if the general binomial model is free of arbitrage then it is also complete i.e. every contingent claim has a ...
4
votes
1answer
337 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 ...
0
votes
0answers
58 views
How realistic are the scenarios outlined in my course?
I am currently taking a course in Financial Mathematics as part of my Maths degree. Many of the covered topics are quite basic, and revolve around potential arbitrage opportunities. For example, ...
-4
votes
1answer
204 views
Option price in a neutral risk world is the same as in the real world. I can not understand! [closed]
Good evening. I know there are several posts on the subject but unfortunately I can not fully understand this concept and I hope you can help me.
To price the option the fundamental assumption ...
4
votes
2answers
183 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 ...
0
votes
1answer
156 views
Arbitrage problem [closed]
Question
A share of non-dividend paying stock is trading at USD 30. The maturity
of both options is 1 year from now. A put with a strike of USD 28 is trading at USD 1 and call with a strike of USD 29 ...
1
vote
1answer
525 views
No-arbitrage theorem: a proof
The market is arbitrage free iff there exists an equivalent martingale measure for the discounted price process of the stock.
My course only provides me part of the entire proof that shows that ...
1
vote
0answers
232 views
Fair price and no arbitrage
The market is arbitrage-free iff there exists an equivalent martingale measure for the discounted price process of the stock.
So in a world with a finite amount of possible outcomes $\Omega$ that ...
0
votes
0answers
51 views
Concatenation property of a set of semimartingales
Consider as in (1, Definition 2.1) a convex subset $\mathcal{X}_1$ of the set of semimartingales $\mathbb{S}$ satisfying the following properties:
$X_0=0$
$X_t\geq -1$ for all $t\geq 0$
for all ...
9
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 ...
1
vote
1answer
117 views
What is the principle of determining an arbitrary option price
First I want to talk about one of my wrong ways of pricing an European call option. When I consider the simplest case of European call option, the first idea of determining the price is to calculate ...
1
vote
1answer
521 views
American Option Bounds with Dividend Yield
What are the upper and lower bound of American call and put options for an underlying with continuous dividend yield?
For European options, the bounds are known as
\begin{align*}
[S_te^{-d\tau}-Ke^{-...
2
votes
3answers
200 views
Understanding the HJM drift condition's dimensions
In an HJM model the forward rate dynamics follow
$$
df_t(T) =a_t(f_t(T))dt+b_t(f_t(T))dW_t
$$
where $W_t$ is a $d$-dimensional brownian motion, $b_t$ takes values in $\mathbb{R}^{d\times d}$ and $a_t$ ...
0
votes
0answers
334 views
Risk neutral pricing - Example from a book is correct?
I found the following example in a book on Model Risk, while trying to explain how risk-neutral pricing takes properly into account the risk involved in different investments.
The Example is this.
...
0
votes
2answers
351 views
Solving for r in the Black Scholes equation
Could you please correct which parts of my reasoning are wrong?
Let's suppose that I know for sure that my estimate for a stock volatility is right (I have a crystal ball) and that it will be for ...
0
votes
1answer
75 views
Mathematically: How does increasing the number of assets reduce idiosyncratic risk?
As part of an Asset Pricing Module I'm currently taking, whilst looking at APT Ross (1974), we looked at how according to this model, risk originates from both systematic and idiosyncratic asset ...
1
vote
1answer
185 views
arbitrage proof question
prove the condition $D<R<U$ is equivalent to the absence of arbitrage:
R = risk free investment rate of return. U and D are returns corresponding to the upward/downward price movements of a ...
4
votes
2answers
224 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, ...
0
votes
0answers
122 views
Deriving the yield curve from the HJM dynamics
If I know that my model follows a no-arbitrage HJM model:
\begin{equation}
df(\tau)
=
\left(\sigma(\tau)\int_0^{\tau}\sigma(u)du\right)dt
+\sigma(\tau)dW_{\tau}
\end{equation}
(where $\tau:=T-t$, ...
