Questions tagged [no-arbitrage-theory]
The no-arbitrage-theory tag has no usage guidance.
116
questions
7
votes
2answers
453 views
Why do we need the self-financing assumption in risk-neutral pricing?
A portfolio is self-financing if the purchase of a new asset must be financed by the sale of an old one.
\begin{align*}
x_t(1+R) + y_tS_t = x_{t+1} + y_{t+1}S_t
\end{align*}
This says that, at each ...
1
vote
1answer
36 views
some questions about pricing an asset or nothing put option with a strike price equal to St
I am working on a homework exercise where the aim is to price an asset or nothing put with K = St, offcourse the normal formula could be used St * N(-d1), but I was wondering if pricing the asset by ...
1
vote
1answer
73 views
Required adjustments for stressed yield curves
I was looking at Basel proposed interest rate shocks. Using the standard US Treasury Yield Curve for the period starting from September 2017 to August 2019, I was able to construct Steep and Flat ...
0
votes
1answer
60 views
Exercise on arbitrage-free process
Consider the following problem, from Bjork's Arbitrage Theory in Continuous Time:
Consider the standard Black-Scholes model. Derive the arbitrage free price process for the $T$-claim $\mathcal{X}$ ...
1
vote
1answer
81 views
Binomial model in Björk's Arbitrage Theory in Continuous Time
I am having some trouble with variable $Z$ introduced in chapter $2$ in Björk's text. In the beginning, it is the random variable that attains $u$ resp. $d$ with probabilities $p_{1}$ and $p_{2}$, i.e....
1
vote
0answers
55 views
Arbitrage pricing models
I have been reading Wu's Interest rate modeling and in his chapter on the HJM model he says that
With arbitrage pricing models, the prices of the basic instruments are treated as model inputs ...
1
vote
2answers
138 views
Risk-neutral pricing and statistical arbitrages
I'm studying the martingale approach to asset pricing. Dealing with the concept of risk-neutral probability, I came up with a question about the possibility of "arbitrages in expectation". I'll be ...
1
vote
1answer
41 views
Is the undiscounted value process of a Euro call option under Bachelier model a Martingale? [duplicate]
Assume that $c_t$ is the UNDISCOUNTED price process for a European call option in Bachelier model. In Bachelier model call option pricing formula the formulas is discussed. The undiscounted value ...
1
vote
1answer
67 views
Covered Interest Rate Parity with FX Spot-Adjustment
The Covered Interest Rate Parity for FX is often quoted simplistically as
$$
X_T \quad=\quad X_S \cdot \frac{D^{base}_T}{D^{quote}_T}
$$
where $X_t$ is the (projected) FX rate at time $t$ (denoted as $...
2
votes
1answer
106 views
Fair price of a coupon paying bond
Consider a coupon paying bond with a maturity of $3$ years, that pays coupon annually. Let $c$ be the coupon rate (percentage) and let $F$ be the face value. This means that the holder of the bond ...
1
vote
2answers
121 views
No-arbitrage and the sharpe ratio?
I'm reading a paper and it says that in a no-arbitrage market the sharpe ratio is the same for all bonds. I'm guessing that a difference in two bonds sharpe ratios would open the possibility of ...
0
votes
1answer
42 views
Infinite Binomial Pricing no arbitrage
How to price a contract that pays only 1 at the first stock price drop? The stock follows an infinite binomial with no arbitrage $d<R<u$ condition.
So the probability of the price going down is ...
1
vote
0answers
40 views
Proof of no arb condition after shifting SABR’s rho
Does anyone know of any paper or research where they shift SABR’s skew and rebuild the surface?
In particular, I would like to prove theoretically whether the no arbitrage condition hold for the ...
4
votes
0answers
60 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
61 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
125 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
320 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
848 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
72 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.
...
5
votes
1answer
176 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
132 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
195 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 ...
8
votes
1answer
251 views
Linear interpolation of local vol no arbitrage
We already know the equivalence between local vol, implied vol and option price and there ...
3
votes
1answer
392 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 ...
1
vote
1answer
332 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)...
2
votes
0answers
90 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
243 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 ...
9
votes
2answers
430 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 ...
2
votes
0answers
199 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
620 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
73 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
124 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
140 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 ...
9
votes
0answers
231 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
234 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
65 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
228 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
1k 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.
3
votes
2answers
397 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
638 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:
...
2
votes
1answer
85 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
156 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
625 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
505 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, ...
-3
votes
1answer
263 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 ...
3
votes
2answers
205 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
171 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
657 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
269 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 ...
