Antoine Conze
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$\theta(t) - a(t) r(t)$ is the risk neutral drift. The Hull &amp; 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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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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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[\... View answer 1 answers vote 256 views 2 votes 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 ... View answer 1 answers vote 312 views Accepted answer 2 votes 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 ... View answer 1 answers votes 551 views Accepted answer 2 votes 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 ... View answer 1 answers votes 86 views Accepted answer 2 votes 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 ... View answer 2 answers votes 78 views Accepted answer 2 votes 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 ... View answer 1 answers votes 154 views 2 votes 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 - ... View answer 1 answers vote 965 views 2 votes 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... View answer 1 answers vote 819 views 2 votes 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 ... View answer 1 answers votes 416 views Accepted answer 2 votes 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 ... View answer 1 answers votes 6k views 2 votes 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 ... View answer 1 answers vote 182 views Accepted answer 2 votes 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 ... View answer 1 answers votes 321 views 2 votes 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 ... View answer 1 answers votes 91 views 1 votes 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. View answer 1 answers vote 1k views 1 votes Hull &amp; 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 ... View answer 1 answers vote 63 views Accepted answer 1 votes 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 ... View answer 1 answers votes 245 views 1 votes 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 ... View answer 1 answers votes 132 views 1 votes \begin{equation*} \begin{split} \mathbb{1}_{S_T &gt; K, \max_{[0,T]} S_t &lt; H} &amp;\approx \frac{(S_T - (K-\varepsilon))^+ - (S_T - (K+\varepsilon))^+}{2 \varepsilon} \mathbb{1}_{\max_{[0,T]} S_t &... View answer 1 answers votes 115 views Accepted answer 1 votes You need to add an auxiliary state variable that represents the current strike K_t, with dynamics K_{t} = K_{t^-} if S_t &gt; 0.8 K_{t^-}, K_{t} = S_t if S_t \leq 0.8 K_{t^-}. You will get a ... View answer 2 answers votes 331 views Accepted answer 1 votes 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 ... View answer 1 answers vote 220 views 1 votes 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 ... View answer 3 answers votes 282 views 1 votes 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 ... View answer 1 answers vote 106 views Accepted answer 1 votes 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, ... View answer 2 answers vote 729 views 1 votes 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 ... View answer 2 answers votes 85 views Accepted answer 1 votes 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 ... View answer 1 answers votes 217 views 1 votes 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 ... View answer 1 answers votes 1k views 1 votes 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 ...