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2
votes
2answers
38 views

About the definition of a complete market

In Steven Shreve's book "Stochastic Calculus for Finance 2", Definition 5.4.8 says a market is complete if every derivative security can be hedged. What exactly does every derivative security mean? ...
0
votes
0answers
24 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. ...
1
vote
0answers
39 views

Exercise: interpretation of terms in black-scholes

I have following exercise: This is what I did: \begin{align} C(K)&= e^{-r\tau} \mathbb{E}^\mathbb{Q}[((S_T - K)^+] \\ &= e^{-r\tau}\mathbb{E}^\mathbb{Q}[((S_T - K)\mathbb{1}_{S_T>K}] \\...
1
vote
0answers
54 views

$\mathbb{P}$ and $\mathbb{Q}$ probability measure/distribution interpretations

I'm trying to understand probability distributions implied from market prices and was reading through this reference explaining the interpretation of $N(d_1)$ and $N(d_2)$ in the log-normal vol Black-...
2
votes
2answers
164 views

How to price a stock under Q and stochastic interest rates?

I am interested in pricing a stock under $\mathbb{Q}$ when I assume that $$dS(t) = \mu(S(t))dt + \sigma(S(t))dW(t)$$ where $W(t)$ is a Wiener process under $\mathbb{P}$ and $$dr(t) = a(b-r(t))dt +...
3
votes
1answer
120 views

How to price this basket option?

Underlying assets are three global stock index : Eurostoxx 50, HSI, KOSPI 200 Maturity: 36 months with advanced redemption date in every 6 months if prices of indexes satisfy given conditions at each ...
15
votes
4answers
2k views

How to estimate real-world probabilities

In the world of finance, Risk-neutral pricing allow us to estimate the fair value of derivatives using the risk free rate as the expected return of the underlyings. However, the behavior of ...
0
votes
0answers
22 views

Market Price of Risk in EDF Model

Consider Moody's Expected Default Frequency (EDF) model. To arrive from risk-neutral default probabilities to real-world default probabilities, we need to know ...
0
votes
2answers
175 views

The State-Price Deflator in a Binomial pricing model

This question comes from a Financial Economics exam and I'm very confused about a state-price deflator which doesn't seem to exist. I've included the whole question for completeness, but my actual ...
1
vote
2answers
63 views

Simulate drifted geometric brownian motion under new measure

I have a very fundamental question regarding simulation of DRIFTED geometric brownian motion. We have the standard Blackos Scholes model: $dS(t)=r S(t)dt+\sigma S(t) dW^{\mathbb{P}}(t)$, where $W^{\...
6
votes
2answers
181 views

Interpret simulation results ($P$ and $Q$ measures)

I am struggling in interpreting results of my simulations. I use Monte Carlo algorithm to simulate stock paths and calculate option price. The notation: $r$ is a risk free interest rate, $T$ is time ...
0
votes
0answers
30 views

Generating process for stock price paths in this paper?

I am reading Longstaff and Schwartz Valuing Aerican Options by Simulation because monte carlo simulations, especially their use in option pricing, is interesting to me. However, I am having some ...
6
votes
1answer
193 views

Obtaining risk-neutral probability from option prices

Suppose I have the following data (for the current stock and option prices of the Bank of America) Strike Last IV Probability 4 8 5.43 0.5813566 0.0000000 7 11 2.45 0.2868052 ...
1
vote
2answers
94 views

How to show that the exponential Vasicek model is not an affine term-structure model?

From the pricing formula, we know that the value at time $t\in [0,T]$ of a zero coupon bond maturing at time $T$ is $$ B(t,T)=E\left(\exp{\left(-\int_{t}^{T}r_sds\right)}\bigg|\mathcal{F}_t\right). $$...
3
votes
0answers
68 views

Interpolation of forward zeros-coupons bonds simulations for missing maturities (ESG data)

I have a set of economic scenarios simulated with Barrie and Hibbert ESG. The stochastic model for interest rates used is Libor Market Model Shifted. I am facing a problem with zeros-coupons prices. ...
2
votes
1answer
57 views

Option pricing: Risk neutral probability calculation

Let $u=1.3$ $d=0.9$ $r=.05$ $S(0)=50, X = \text{strike} = 60$. Assume binomial model Why isn't the risk neutral probability found by solving the following for $p$: $$E[S(T)]=p65+(1-p)45=S(0)(1+r)^T=...
4
votes
2answers
95 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, ...
2
votes
2answers
129 views

$E[F_T] = F_0 \ \rightarrow \ \text{or} \ \leftarrow \ p = \frac{1-d}{u-d}$?

From Ch 12 in Hull's OFOD, we compute the risk-neutral probabilities for a futures contract: Later in Ch 17, futures options are valued, and we have the same result: In relation to ...
6
votes
0answers
172 views

Estimating Parameters - Vasicek

The Vasicek model for the short rate $r_t$ is given by the SDE $$ dr_t = \alpha(\beta - r_t)dt + \sigma dW_t, $$ where $W_t$ is a Brownian motion under the physical measure. I'd like to compute bond ...
1
vote
0answers
50 views

Jabbour-Kramin-Young ABMC Binomial Parameterization

The JKY ABMC Model (taken from Jabbour, et al. 2001) parameterizes the binomial model (in a risk-neutral world) such that, $u = e^{r\Delta t} + e^{r\Delta t}\sqrt{e^{\sigma^2\Delta t} - 1}$ $d = e^{...
0
votes
1answer
64 views

completeness of the binomial model - proof

I am reviewing the steps of proof that the binomial model is complete and don't understand the marked in red transition. Could anybody explain this step? If $P^{**}$ is a risk-neutral measure, so ...
2
votes
3answers
72 views

How would I exploit arbitrage if risk-neutral pricing doesn't hold? (Option Pricing)

We are just learning about binomial option pricing, and how the up-factor and the down-factor must match the risk-neutral price. p * u + (1 - p) * d = continuous risk free rate compounded CRR ...
2
votes
1answer
78 views

Option analysis

Assume zero dividend and that the strike price for a European call option on a stock at a fixed maturity T and strike price K is given by C(K).Suppose that $C(K)=e^{-k}$ for all $K\geq 0$ ,then, I ...
4
votes
1answer
151 views

How to use the Girsanov theorem to prove $\hat{W_t}$ is a $\hat{\mathbb P}$-Brownian motion?

Let $T > 0$, and let $(\Omega, \mathscr F, \{\mathscr F_t\}_{t \in [0,T]}, \mathbb P)$ be a filtered probability space where $\mathbb P = \tilde{\mathbb P}$ (risk-neutral measure) and $\mathscr F_t ...
5
votes
2answers
196 views

How to derive this approximation of the risk-neutral expectation of the variance?

On the paper Bollerslev, Tauchen and Zhou (2009 RFS) the authors say about equation (15): The corresponding model implied risk-neutral conditional expectation $$E^Q_t(\sigma^2_{r,t+1})=E_t(\...
1
vote
1answer
56 views

How to change to risk neutral measure in a mean reversion process?

For example, in the Ornstein-Uhlenbeck process do I just replace the drift term with the risk free rate, like in the GBM case?
0
votes
0answers
25 views

Effect on variance of change of measure

My current understanding: (a) changing the probability measure of a diffusion process does not change the variance. (b) for a general stochastic process the variance may change. Please confirm whether ...
1
vote
2answers
93 views

Fourier Transform

In a notes on "Option Pricing using Fourier Transform": Price of plain vanila call is given by $$ C(t, S_t) = e^{-rT}\mathbb{E}^{\mathbb{Q}}[(S_T -K)^+|\mathcal{F}_0] = e^{-rT} \int_K^{\infty} (S_T -K)...
1
vote
1answer
49 views

How to Calculate Return Option with Forward Measure

I am trying to computing the price of an option at time $t$, with payoff $X = \frac{S_{T_2}}{S_{T_1}}$, at time $T_2$, where $t < T_1 < T_2$. Here how I compute it: Using the forward measure $...
2
votes
1answer
119 views

Relationship between risk-neutral probability and subjective probability

I recently came across a Paper by a paper of Rubinstein and Jackwerth (1997): http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.441.5214&rep=rep1&type=pdf where they assume that you ...
12
votes
1answer
441 views

Consistency of economic scenarios in nested stochastics simulation

I am interested in references on research regarding the consistency of economic scenarios in nested stochastics for risk measurement. Background: Pricing by Monte-Carlo: For pricing complex ...
16
votes
9answers
7k views

Why Drifts are not in the Black Scholes Formula

This question has puzzled me for a while. We all know geometric brownian motions have drifts $\mu$: $dS / S = \mu dt + \sigma dW$ and different stocks have different drifts of $\mu$. Why would ...
3
votes
1answer
239 views

Risk neutral drift vs real world

I was of the understanding that risk neutral drift was always the risk free rate. A section from Gregory's book on Credit Value Adjustment seems to say risk neutral drifts are typically estimated from ...
1
vote
2answers
79 views

does there need to be risk-neutral agents in the market to enforce risk-neutral pricing?

I'm trying to understand a fundamental link between mathematical finance and economics. I understand that risk-neutral pricing is free of arbitrage with replicating portfolio. Does risk-neutral ...
2
votes
2answers
160 views

Option with payoff $K^2/S^2$

Given the dynamics of the risky asset ( with dividend $q$ ), $$ \frac{dS_t}{S_t}=(\mu-q)dt + \sigma dW_t^P $$ Consider a european option with payoff, $$ P_0(S) = \begin{cases} 1, & \text{...
0
votes
0answers
69 views

how best to equalize individual pair risk in a portfolio of stock pairs?

I am building a portfolio of stock pairs in which each pair is individually hedged via beta/hedge ratio adjustment. I am looking for a method to ensure that I am taking the same risk in each pair that ...
0
votes
1answer
132 views

Bond price in Ho-Lee Model

I know Ho-Lee model and want to extract the price at $t$, of a European call option with strike price $K$ and exercise date $T$, on an underlying $S$-bond, but I don't know what way should I choose: ...
2
votes
2answers
196 views

Real world monte-carlo (P-measure)

Consider the 2 following approaches to pricing a security: Monte-carlo ($\mathbb{Q}$-measure) $\begin{equation} C = \frac{1}{N} \sum_{i=1}^{n} e^{-rT} max(S_i(t) - K, 0) \end{equation}$ Monte-carlo ...
1
vote
1answer
399 views

Provide a bond pricing differential equation and invoke Feynman-Kac Theorem

Grateful for any assistance. Consider the process: $dZ=r(t)Z\,dt$ , where $r(t)$ is stochastic interest rate and $Z=Z(r,t;T)$ is a zero coupon bond Price. Provide a bond pricing partial ...
2
votes
2answers
136 views

How to discretize a GBM under P- and Q-measures?

Under the P-measure, a geometric Brownian motion can be specified using the following SDE: $$dS_t=\mu S_tdt+\sigma S_tdW_t^P$$ and its Euler discretization is $$S_{t+\Delta t}=S_t + \mu S_t \Delta ...
2
votes
1answer
85 views

Is probability implied by binary FX options risk neutral or real world?

If we consider binary FX options in the market and estimate the market implied probabilities of certain FX rates occurring, would these resulting probabilities be risk neutral or real world? I hear ...
9
votes
1answer
248 views

What's Risk-Neutral in an Interest Rate Model?

In Shreve II, on p. 265 he states the Hull-White interest rate model as $$ dR(u) = \left( a(u) - b(u)R(u)\right) dt + \sigma(u)d\tilde{W}(u), $$ and then mentions "...$\tilde{W}(u)$ is a Brownian ...
2
votes
3answers
110 views

Unique risk neutral measure for Brownian Motion

For a standard geometric Brownian motion model of stock prices: $$ dS = a S dt + \sigma S dZ$$ we can transform the process to be under risk neutral measure: $$ dS = r S dt + \sigma S d \tilde{Z}$$ ...
3
votes
3answers
332 views

Physical or Real-world Probability Measure

For counterparty credit risk, in particular, for potential future exposure computation, people use the real-world probability measure to evolve the underlying risk factors. My question is that whether ...
1
vote
2answers
163 views

Intuitive Reasoning for Using Risk-Neutral Measure

Although we thoroughly covered risk-neutral pricing in university I never fully understood it in the context of continuous-time processes. But first of all, lets consider a discrete time example: ...
0
votes
1answer
55 views

probability that the stock price is below the strike price

How can I prove that under the risk-neutral probability: $\mathbb{P}[S_{t}<K]=-\frac{\partial{C}}{\partial{K}}(K,T)$ where $S_{t}$ is the stock price, K is the strike price, C is the call ...
0
votes
2answers
269 views

Does a forward price have a drift component in any measure?

Going by intuition, a forward price should already take into account the drift in the underlying price process. Further, assuming interest rates are deterministic, the stochasticity in the forward ...
0
votes
1answer
69 views

Good book about replicating portfolios

I want to know if anybody can suggest me a good textbook which explains in detail and in an understandable way how to create replicating portfolios of financial instruments like options "cash or ...
1
vote
0answers
35 views

Risk Neutral Measure [duplicate]

I have been looking for a good intuitive reasoning for introduction of the risk neutral measure and its uses in quantitative finance, but I have yet to find one. I was wondering if any one could ...
3
votes
5answers
736 views

Why the Black-Scholes formula can be used in the real world?

The BS formula is deduced using the risk neutral measure. Why can it be used in the real world?