Antoine Conze
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Hull-White Extension of Vasicek Model
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$\theta(t) - a(t) r(t)$ is the risk neutral drift. The Hull & White models posits the dynamics $dr(t) = (\theta(t) - a(t) r(t)) dt + \sigma dW(t)$ under the risk neutral measure $P$ and then ...

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Local Volatility implementation
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The usual way is to fit a surface (e.g. smoothing splines) to the grid and to compute derivatives off the surface. Note however that the entire process tends to be more stable when applying the Dupire ...

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Bachelier model call: computation of delta of a call option
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Zero interest rate and drift so $S(T) = S(t) + \sigma (W(T)-W(t))$ and $\frac{d S(T)}{dS(t)} = 1. $ $$ C(t) = E_t[(S(T) - K)^+] $$ $$ \frac{dC(t)}{dS(t)} = \frac{d}{dS(t)} E_t[(S(T) - K)^+] = E_t[\...

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Term structure used in Geometric Brownian Motions under Risk Neutral Measure?
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It should be time dependent and set to the spot forward rate $= -\frac{\partial}{\partial t} \ln(\text{discount}(t))$ when simulating in continuous time. When discretizing the simulation use the ...

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312 views
Rebasing of Cap Volatilities
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This says that Gaussian volatility $\approx$ Log Normal volatility $\times$ ATM strike is constant across tenors, which would essentially hold if you assume that the basis between tenors is ...

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Martingale measure result application for interest rates under T-forward measure?
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Do not confuse the fixing date $T$ and the payment date $T^*$. In your example you are valuing a floating coupon that fixes on $T$ and pays $R(T, T, T^*)$ on $T^*$, and you are using the $T^*$ zero ...

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Spot Interest Rate at time $t$
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Assume the SDE $dr(t)=\mu(r(t),t)dt + \sigma(r(t),t) dB(t)$ is under the risk neutral measure and that is has a solution. By construction $P(t,T) = E[e^{-\int_t^T r(u) du}]$ under the risk neutral ...

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Importance of full value functions for option pricing
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Some numerical methods, e.g. finite difference schemes, enable you to compute the entire function $s \mapsto v(s)$ at once. This can be useful as no additional pass is required to compute the delta ...

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154 views
Put Call Parity confusion
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Being short a put simply means that you have sold the put, hence its payoff is from your point of view $-(K - S_T)^+$. When your are long a call and short a put your total payoff is $(S_T - K)^+ -(K - ...

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965 views
Is Libor a martingale under T-forward measure
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Your definition of Libor is invalid as you make it cover the period $t, T$. A Libor with tenor $\delta$ that fixes on $T$ (or to be accurate usually 2 days before $T$) covers the period $T, T+\delta$...

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convexity adjustment in YOY inflation swap , compared with TRS, and considering autocorrelation
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You get a convexity adjustment from forward correlations only if you model separately the forwards and they are not perfectly correlated on the time interval $[0, T_1]$, as is the case in inflation ...

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416 views
CMS convexity adjustment in a range accrual Monte Carlo
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If you have done your simulation under the payment date forward measure then you only need to take the expectation of the indicator of the swap rate being between $K_1$ and $K_2$. If you have done ...

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Forward and discount curves for cross currency swaps
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The EUR leg should be valued in EUR but in a manner consistent with GBP collateral: This means: a 3m EURIBOR forward curve consistent with GBP collateral a EUR discount curve consistent with GBP ...

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Implied volatility as price transform
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It is difficult to gain intuition by just looking at the price surface, and it is also easier to calibrate models on the volatility surface rather than on the price surface because with the later you ...

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321 views
Multiperiod return formulae with dividends
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The correct formula is to compute multi period gross returns as products of single period gross returns. Conceptually it is equivalent to calculating the return on a self-financing portfolio initially ...

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No-arbitrage bounds on Implied Volatility under Black-Scholes
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Simply make sure the forward variances remain non negative: $\Sigma(T_{i+1})^2 T_{i+1} - \Sigma(T_{i})^2 T_{i} \geq 0$ for all $i$.

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Why co-terminal swaptions are that important?
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Hull & White is often use to value Bermudan swaptions, given a market for European swaptions. The idea is, at given mean reversion speed, to calibrate the instantaneous volatility to the set of ...

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In search of double barrier out option on a BM
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This is a well tackled problem in the GBM case. See Geman/Yor (1996), Pricing and Hedging Double-Barrier Options: A Probabilistic Approach. Mathematical Finance, 6(4), p. 365-378 among other ...

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how to simplify Inflation year-on-year option to Zero-coupon option
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You can't readily map YY options payoffs into ZC options payoffs. To go from ZC to YY requires: a convexity adjustment for transforming the CPI forwards ratio into a YY forward CPI correlations ...

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Static hedge for up-and-out Digital Call
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\begin{equation*} \begin{split} \mathbb{1}_{S_T > K, \max_{[0,T]} S_t < H} &\approx \frac{(S_T - (K-\varepsilon))^+ - (S_T - (K+\varepsilon))^+}{2 \varepsilon} \mathbb{1}_{\max_{[0,T]} S_t &...

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Finite difference methods for (continuously) strike-resettable American options
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You need to add an auxiliary state variable that represents the current strike $K_t$, with dynamics $K_{t} = K_{t^-}$ if $S_t > 0.8 K_{t^-}$, $K_{t} = S_t$ if $S_t \leq 0.8 K_{t^-}$. You will get a ...

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Is it possible to model path-dependent clauses using finite difference methods?
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The usual approach to deal with path dependency in finite differences/lattices solvers is to capture the path dependency trough one or more auxiliary variable(s) that make the problem non path ...

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Volatility surface tenors
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See https://en.wikipedia.org/wiki/Foreign_exchange_date_conventions for details. In summary expiry = T+tenor for weekly tenors and expiry = ((T+2)+tenor)-2 for monthly and yearly tenors, with all the ...

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Risk-neutral density from spot prices?
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Do a quick search on this site to see how the risk neutral cumulative distribution function is related to the derivative of vanilla option prices with respect to strike. In your case you would have ...

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Pricing American style Asian option
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I don't think there is any good approximation to the american option $\max_{\tau}E^P\left[e^{-r \tau}(S_{\tau} - M_{\tau})^+\right]$ where $M_t = \frac{1}{t}\int_0^t S_u du$ is the running average, ...

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729 views
bootstrapping bloomberg
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This is not exactly an answer to your question, but I have found that for practical purpose it is best to use directly the discount factors (last column on the screen), which you can export to Excel ...

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Arbitrage-free calculation of flat term structure out of normal term structure for e.g. pricing european options
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Black-Scholes does not really require a constant interest rate. For a european option with maturity $T$ the only rate involved is the zero coupon rate for maturity $T$. The theory behind this comes ...

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Xccy without back notional exchange
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A standard XCCY minus an “XCCY without back notional exchange” is a currency forward struck at today’s spot. The difference will be positive or negative depending on how the forward FX compares to the ...

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fx vanilla option's forward delta in single currency
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Actually the forward delta is the option's sensitivity to the PV of the forward contract with same maturity so it is $$ \frac{1}{D}\frac{\partial C}{\partial F} = N(d_{+}) $$ For an option on ...

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316 views
Upper bound option price in volatility dimension
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This does not directly answer your question, but here is a suggestion: Most options, with the exception of barrier options, tend to behave linearly for extreme values of the model state variable(s). ...

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