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You are correct that showing the self-financing condition for the BS-portfolio is not as straightforward as one may think: A portfolio $V_t(\alpha_t,\beta_t)$ (for stock $S_t$ and zerobond $B_t$) is self-financing iff: $$V_t=\alpha_tS_t+\beta_t B_t$$ It further implies $$dV_t=\alpha_tdS_t+\beta_tdB_t$$ To replicate a derivative $C(S_t,t)$ by a ...


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As always I recommend reading Rennie and Baxter for an introduction to option pricing that's not too technical and gives intuition about how it all works.


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I read the question as follows: You have one stock $S_0$ and after one period it either goes up to $S^+$ where the option takes the value $f^+$ or it goes down to $S^-$ where the option takes the value $f^-$. The bond grows from $B_0$ to $B_1 = B_0 \exp(r)$. Then you need to solve $$ a S^+ + b B_1 = f^+ \\ a S^- + b B_1 = f^- $$ for $a,b$ which are $2$ ...


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A unique state price vector does not have to exist for there to be no arbitrage. It sounds like the state price vector in question has infinitely many solutions. Try to reduce the price matrix to row echelon form and show that at least one state price vector exists.


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I think the title here is misleading. Let's go back to the BS world with $r=0$ to $a(S_t)=S_t \sigma.$ In that case, all you are saying is that you can replicate a call option by holding $N(d_1)$ units of stock at time $t.$ What does this have to do with the second equation? I am guessing that this is the price process of an asset of nothing option with ...


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if you let the implied vol depend on K you get two terms the first is $N(d_2) $ but you get a correction term which is the slope times the vega $$ \frac{\partial C}{\partial \sigma} \frac{\partial \sigma}{\partial K}.$$ (see eg my book)


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The concept of replication is indeed applied to IR products, after all they are also hedged in practice. However, in the equity world we start with the replicating portfolio and then arrive to the pricing formula. In contrast, for IR products we employ a convenient numeraire which helps us to arrive at the pricing formula directly (in a non-constructive and ...


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For forward libor models, one can hedge interest rate options by using bonds. (Note that forward libor is a tradeable security under the forward measure). See http://www.columbia.edu/~mh2078/market_models.pdf For affine yield models (like Vasicek or CIR models) the inverse problem is the most useful. Given an interest rate process, I can compute a ...


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Let $0 \leq T < U$. Consider a European call on a U-Bond (Zero-coupon bond maturing at time U) with time of maturity $T$. What you do is that you hedge the call option with the aid of the U-Bond and the T-Bond. I could go in to more details on how to do this in particular models, but I would basically just write the same things as in this book: Interest ...


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As Brian B states above the short answer includes Money market accounts, swaps and zero coupon bonds among other instruments. Lets say we have an interest rate derivative that we need to value via replication. Now if we think of what we mean by a replicating portfolio its clear that the main ingredient needed is to match the pay structure\payout of the ...



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