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## Hot answers tagged no-arbitrage-theory

17

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 cannot admit arbitrage. I provide you with two counterexamples to your statements. A volatility smile that is concave around the forward does not necessarily ...

12

In the derivatives context, "arbitrage free" means almost surely for the probability measure under consideration. This is in opposition with statistical arbitrage used at high frequencies for example. More precisely the assumption is that there is no $T\geq 0$ and self-financed portfolio $V$ such that $V_0 = 0$, $P(V_T < 0) = 0$ and $P(V_T > 0) > ... 11 This is an interesting question that I have asked myself. Below is my take. Let us consider an economy$(\Omega,\mathcal{F},P)$equipped with a filtration$(\mathcal{F})_{t \geq 0}$consisting on a traded asset$S_t$and a numéraire$N_tspecified by the following stochastic differential equations: \begin{align} \text{d}S_t&=\alpha(t,S_t)\text{d}t+\... 10 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 any time, you can hedge any derivative easily. If you can always add the amount of money you need, then hedging becomes trivial. Thus, one requires the self-... 9 This option is a perpetual one touch option. Its price depends on the model used; additional assumptions are required to get a model-independent price. Let us first consider 3 important example models for stock price S. Constant: S(t) \equiv 1. There is 0 probability that the perpetual one touch pays off, so its price is 0. Black-Scholes: S ... 7 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 this interval you need to make further assumptions/tweak inputs I agree with your conclusion that there no suitable probability measure. But I am not sure about ... 6 In three bullet points: Efficiency: the obtained prices maximize assumed utilities of different agents. In their paper "The Valuation of Option Contracts and a Test of Market Efficiency", Cohen, Black and Scholes compare the theoretical value of options to their market price. The efficiency is in this sense: can agents obtain more or less in practice than ... 6 Making money is not the only reasonable objective to trading. Another common reason is to manage/reallocate risk. For example, this is exactly the objective of liability-driven-investors, such as pension funds. They're specifically trying to match durations of their liabilities. It doesn't matter if pension fund managers believe there are no inefficiencies ... 6 Let T= \inf\{t>0: S_t = H\}. Then the option payoff is given by \mathbb{1}_{\{T < \infty\}}, and the value of the option is given by \mathbb{P}(T< \infty). We assume that the stock price process is a geometric Brownian motion, that is, for t>0 S_t = \exp\big(-\frac{1}{2}\sigma^2 t + \sigma W_t\big),$$where \{W_t, t \geq 0\} is a ... 6 I believe there is not a unique price if you can't short. Say, instead of buying the option you spent 0.5 on a half a unit of the asset S^2_1 This asset pays out [0.4, 0.6, 0.8] which first order stochastically dominates the option. So, no matter your probability beliefs about the states, in that setting you'd never pay 0.5 for the option which pays ... 6 Assuming that the only things that can happen on the period are 100 and 50, and we can buy a stock and a call option with strike 90, even without knowing the probabilities of these moves we can relate the price of the stock S and the option C If we buy 0.2 S and sell one call option C, we have a portfolio that will be worth 10 in either end-... 5 Suppose that the given condition is true. You want to construct an arbitrage portfolio to take advantage of this. Now, d is an interest rate, and the condition suggests that d is too high. So you will want to receive d in order to profit. If you could, you would borrow money at r and lend it to the stock broker or exchange to collect the interest ... 5 Sell 1 unit of S1,2,3 respectively, gain 3; buy 2 units of risk-free asset, cost 2. No matter which state appears, the future payoff/loss is 0 for sure, while you will gain 1 at the beginning. 5 You'll find here that in terms of European option prices, the absence of calendar arbitrage writes$$ \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)}, \forall k \in \Bbb{R}, \forall \, 0 < t_1 < t_2 \tag{1} $$where \tilde{C}(K,t) denotes the undiscounted European call price for strike K and time to ... 5 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 these vols to be compatible with specific IR models. For instance when calibrating a Hull & White model on a set of coterminal swaptions, it sometimes happens ... 5 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 Suppose the price process is a semimartingale S. Let K_0 represent the space of all claims generated by admissible trading strategies (self-financing and zero ... 4 The dynamics of the underlying stock process are obviously crucial to the derivative's price. Thus if you don't necessarily assume S_t to be log normally distributed (B&S-Model) you won't get the same price even if the market is arbitrage free. Example: Assume S_t=C \forall t \in \mathbb{R}^+ and r=0. Thus S_t is constant and the interest ... 4 The formation of asset price bubbles, such as the recent US housing market bubble, is perhaps the clearest indication that markets are not efficient. Hundreds of bubbles have been documented for all kinds of traded assets; see the tulip mania for an extreme case. Many practitioners also routinely use trading strategies such as momentum or reversion to the ... 4 I do this question to death in Concepts and ... If (discounted price of) everything is a martingale then every trading strategy is a martingale. Therefore any self-financing portfolio of initial value zero and has expectation zero. Therefore there are no arbitrages (since these have positive expectation and initial value zero). So there is no arbitrage in ... 4 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 $$\int_{-\infty}^\infty x f(x) \mathrm{d}x = \int_0^\infty x f(x) \mathrm{d}x.$$ Since f(x) \geq 0 for it to be a valid PDF, it follows that \begin{... 4 Let's focus on a European call option for the sake of the argument. Assume deterministic rates to keep notations uncluttered. Define \Bbb{Q} as the probability measure associated to the money market numéraire B_t.$$ C(K,T) = \frac{1}{B_T} \Bbb{E}^\Bbb{Q} \left[ (S_T-K)^+ \right] = \frac{1}{B_T} \int_K^\infty (S - K) q(S) dS $$Whence (Leibniz rule)$$ \... 4 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 and the same. You can think of the risk-neutral probabilities as those that give the arbitrage free prices of derivatives. Suppose the interest rate,r$, is ... 4 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$X$at time$0$is $$PV[S(T)|S(0)] = PV[S(T_0)|S(0)] = PV[(1-\delta) S(T_0^-)|S(0)] = (1-\delta) S(0)$$ b) on$t=0$you buy$1-\delta$units of the stock ... 4 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 coupon, in fact you don't even really need to know its value in terms of \$ prior to $T$, because all settlements will occur on $T$. As stated by @Matthew ...

4

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 strategy is not self-financing, then it must be either: Generating cash outflows for the bank. It is therefore uneconomical for the bank to trade this product; ...

4

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 put option on the same stock with strike price 90 dollars is priced at 9 dollar The difference between the two payoffs is equal to 10 dollars (90 strike puts ...

4

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 price $F_t$ with maturity $t$ is by definition the solution of the equation $$\tag{1} \mathbb E\left[e^{-\int_0^tr(s)\,ds}\right]F_t-\mathbb E\left[e^{-\int_0^tr(... 3 Consider a portfolio where I sell \frac{1}{H} in stock and use that to buy an option. This is a 0 cost portfolio. When I hit the barrier the price of this portfolio is also 0. Law of one price would suggest that this portfolio should be zero cost at all times. So the price of the option at any time must be$$ C_t = \frac{1}{H}*S_t  Also, the option ...

3

No this is not a risk free arbitrage. What you are talking about is modeling a stock price with GBM and it has nothing to do with Black-Scholes. Black-Scholes is an option pricing formula that assumes that stocks follow GBM (which is a bad assumption to begin with but we won't get into that). What you are talking about doing is taking on leverage. \$ E[S_T]=...

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