# Tagged Questions

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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 ...
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 ...
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### Formal proof for risk-neutral pricing formula

As you know, the key equation of risk neutral pricing is the following: $\exp^{-rt} S_t = E_Q[\exp^{-rT} S_T | \mathcal{F}_t]$ That is, discounted prices are Q-martingales. It makes real-sense for ...
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### 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 ...
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### How to transform process to risk-neutral measure for Monte Carlo option pricing?

I am trying to price an option using the Monte Carlo method, and I have the price process simulations as an inputs. The underlying is a forward contract, so at all times the mean of the simulations is ...
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### 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 ...
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### St Petersburg lottery pricing & short investing horizons

I am a statistician (no solid background in finance). Please forward me to a book \ chapter \ paper to resolve the following general question. Suppose we have a stock with the following monthly return ...
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### Version of Girsanov theorem with changing volatility

Is there a version of Girsanov theorem when the volatility is changing? For example Girsanov theorem states that Radon Nikodym (RN) derivative for a stochastic equation is used to transform the ...
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### 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 ...
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### How to choose a risk-neutral measure when the market is incomplete?

I am more of a probabilist than a financial mathematician. I am currently working on the features of American put options under a particular stochastic volatility model. Like most stochastic ...
1k views

### Is Vasicek risk neutral?

I am a bit new to this, and am trying to understand the concepts of the risk neutrality in interest-rate models. What I can't seem to understand is why the Vasicek model is risk-neutral? Following ...
1k views

### What are the main flaws behind Ross Recovery Theorem?

Stephen Ross’ new paper claims that it is possible to separate risk aversions and historical probabilities if the Stochastic Discount Factor is transition independent using Perron-Frobenius Theorem. ...
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### 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 ...
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### How does one go from measure P to Q(risk-neutral) when modeling an asset paying dividends?

I am really having a terrible time applying Girsanov's theorem to go from the real-world measure $P$ to the risk-neutral measure $Q$. I want to determine the payoff of a derivative based an asset ...
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### 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 ...
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### Difference betweem martingale property and adapted filteration

What is the difference between a random process that is adapted to a filteration and one that had the martingale property. It seems the two notions are quite similar and would be helpful to construct ...
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### 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, ...
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### Financial Mathematics - Martingales example

Was hoping somebody could help me with the following question. Prove that under the risk-neutral probability $\tilde{\mathsf P}$ the stock and the bank account have the same average rate of growth. ...
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### When to use the real world drift and when the risk neutral one for a Monte-Carlo simulation?

Under what conditions should the drift be real world and when risk neutral when simulating Delta Hedging option pricing trading strategy any other? For 2. it should be risk neutral. For 1., it ...
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### 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 ...
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### 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 ...
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### Data Selection for Empirical Pricing Kernel Estimation (Stochastic Discount Factor)

I want to estimate an empirical pricing kernel for an index. Hence, I need to estimate a physical and risk neutral density. For estimating the physical density, only the index data in an observed time ...
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### 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. ...