mmencke
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Martingale measure and replicating portfolio in Risk Neutral Pricing of Defaultable Zero-Coupon Bonds
7 votes

The risk-neutral probability measure is defined in terms of its numeraire. For the usual risk-neutral probability measure the numeraire is the bank account, $\beta(t)$. If we have a tradeable asset $X(...

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Cash settled contracts price convergence at expiry
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5 votes

I think you can make the exact same argument for the cash-settled futures. The only difference is that you have to sell the underlying at expiry and delivery the cash instead of delivering the actual ...

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Why do we need to split market and default information into 2 separate filtrations?
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5 votes

I think you are absolutely correct if the hazard rate is deterministic, although I think you are forgetting a discounting factor in your example. But sometimes the hazard rate cannot be assumed to be ...

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Why does the definition of the riskless asset vary in discrete vs continuous time?
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4 votes

Let us say we have a yearly interest rate of $r$ that compounds over $n$ periods. With annual compounding that means $n=1$, with semi-annual compounding that means $n=2$ and with daily compounding ...

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Girsanov transform when drift coefficient is a function of the stock price
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3 votes

You forget the $e^{-rt}$ term in the diffusion: $$dS^*=-rS^*dt+e^{-rt}dS=(\mu e^{-rt}-rS^*)dt+e^{-rt}\sigma B^\mathbb{P}(t)$$ Using Girsanov we can write $B^\mathbb{P}(t)=B^\mathbb{Q}(t)+\phi(t)dt$, ...

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Quantlib ZeroCurve interpolation
2 votes

The data stored in the object is adjusted such that compounding is Continuous and frequency is NoFrequency. The C++ source code is available here: zerocurve.hpp. I think that the reason for this is ...

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Limit of digital call and put price when volatility goes to infitiny
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2 votes

Your intuition is correct for a symmetric distribution. However, the log-normal distribution, as is assumed in the Black-Scholes model, is an asymmetric distribution. I have illustrated the effect of ...

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Delta hedge error black-scholes by Mark Davis
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2 votes

On page 119 in Björk (3rd edition) we have the replicating portfolio (equations 8.20 and 8.21): Hold $\frac{\partial C}{\partial s}$ of the stock and $\frac{X_{t}-S_{t}\frac{\partial C}{\partial s}}{...

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Risk-neutral pricing to determine no-arbitrage price
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1 votes

To find the $S$-dynamics under $\mathbb{Q}$ we have to use Girsanov's theorem: $$dW_t^P=\varphi_t dt+dW_t^Q$$ Dynamics under $\mathbb{Q}$ is thus $$dS_t=a(b-S_t)dt+\sigma S_t(\varphi_t dt+dW_t^Q)=abdt-...

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Find the value of put option using a two-period binomial model
1 votes

I do not recall having seen that formula before and I cannot make much sense of it. I will try to go through the steps necessary to price the option. This is by no means the fastest way to calculate ...

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Is it possible to price a plain vanilla interest rate swap in Python and simulate the price using Hull White 1 Factor Model simultaneously?
1 votes

For a plain vanilla swap we can define the annuity $$A(t)\equiv\sum_{i=0}^{N-1} \tau_i P(t,T_{i+1})$$ and write the swap rate as $$S(t)\equiv\frac{P(t,T_0)-P(t,T_N)}{A(t)}$$ such that the swap price ...

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Sampling change in the driving brownian motion of a CIR process
1 votes

Edit: This is probably incorrect. The Quadratic Exponential scheme is the best one I have seen as it converges in distribution and is pretty fast, so nice choice there! When $\eta$ is constant you can ...

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Hull White 1 Factor Formulas with Time Dependent Variables
1 votes

In [Andersen & Piterbarg (2010)] on pages 415-417 the so-called General One-Factor Gaussian Short Rate Model is described, where the short rate is assumed to follow SDE $$ dr(t)=\kappa(t)(\theta(t)...

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Correlated Wiener Process
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1 votes

The Cholesky decomposition works in the following way: Suppose you have a vector of $n$ independent normal realisations: $Z$. Set the elements of $\Sigma$ to be $\rho_{i,j}$ where $i$ indicates the ...

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Valuation discount rate using risk free interest rate versus inflation rate
0 votes

To avoid arbitrage opportunities in your valuation model, you have to use the nominal risk-free interest rate as a discount rate: Let us consider a Zero Coupon Bond (ZCB) expiring in one year with a (...

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Pricing binary options
0 votes

I will assume that the interest rate is 0. The price of a binary option is then the same as the risk-neutral probability that the event will occur $$\mathbb{E}^{\mathbb{Q}}\left[\mathbb{1}_{S(T)\geq K}...

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