# Questions tagged [stochastic-calculus]

A branch of mathematics that operates on stochastic processes.

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I am reading a paper here: https://pdfs.semanticscholar.org/5f91/2d46b02b03230a4ffaaa42d655b2b6147d56.pdf The following is my confusion. The paper has the following continuous time model for the price ...
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### Stochastic Interest Rates in Option pricing

My lecturer has written the slide below. The function B^T(t) is a zero coupon bond. I don't understand how V(t) can be a negative integral from 0 to ...
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### CIR Process from Ornstein–Uhlenbeck process

The wikipedia entry on the CIR Model states that "this process can be defined as a sum of squared Ornstein–Uhlenbeck process" but provides no derivation or reference. Can any one do that? I could only ...
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### Dynamic programming and Bellman equation to obtain the maximum

This is the problem of Marhsall (1992) "Inflation and Asset Returns in a Monetary Economy" and Balvers and Huang (2009) "Money and the C-CAPM" Suppose an endowment economy where the representative ...
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### How to replicate the future instantaneous short rate?

Suppose we have an interest rate model $R(t)=\alpha(t)d(t)+\sigma d\tilde{W}(t)$, where the brownian motion is under the risk neutral measure. Suppose $S(t)$ is the price at time $t$ for a contract ...
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### Compo/Quanto Adjustment & Multivariate Ito

Related to the issue that I have raised here, I am facing another question. As the rule here is 1 question / 1 post, I take the opportunity to ask it below: By exploring StackExchange, I noticed the ...
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### Proving an Identity between a pair of correlated Wiener processes

Suppose we have the following subordinated stochastic differential equations: $dR(t)=\mu dt+\sigma (Y(t))dW_{1}(t)$ $dY(t)=f(Y)dt+g(Y)dW_{2}(t)$, where $W_i$'s are standard Wiener process such that ...
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### Differential bond price stochastic rates

Suppose that the short rate follows the process $$dr(t) = a(t, r(t))dt + \sigma(t, r(t))dW(t)$$ If $B(t) = exp(-\int_0^t r(u) d u)$, can one still write the differential $dB(t)$ a-la-Ito? Thanks.
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### Compute dZ(t) : Ito's formula/lemma

We need to find dZ(t). I know I have to use Ito's formula. But I am confused because in the Ito's formula we have f(y,t) is a twice differentiable function with two variables But here Z(t) = 1/(2+x(t)...
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### Proof that $f$ is continuous if and only if it has 0 quadratic variation?

I understand that $f$ continuous $\Rightarrow Q(f) = 0$ where this is defined over a bounded interval [0,T] as then we may use uniform continuity and the mean value theorem. But I am not sure how the ...
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### Hedging a short position in the Lookback Option

SOLUTION I got the correct answer using this formula $X_2(HH)=(1+r)*[X_1(H)-\Delta_1(H)*S_1(H)]+\Delta_1(H)*S_2(HH)$ $(1+0.25)[2.24-(.06667*8)]+0.06667*16=3.20$
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In this paper The Interplay between Stochastic Volatility and Correlations in Equity Autocallables by Alvise De Col, Patrick Kuppinger (2017) https://papers.ssrn.com/sol3/papers.cfm?abstract_id=...
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### Idea of using logarithm for solving SDE in Black-Scholes model

In the Black-Scholes model they consider that the stock follows this stochastic differential equation: $$dS = \mu S dt + \sigma S\ dW$$ I was wondering, was it common at the time they work on this ...