# Tag Info

13

A digital call option (cash-or-nothing) can be replicated with two call options with different Strike. When we make the delta infinitely small and assume we have arbitrary strike prices. We get:

13

Regarding your 1st question, jumps are indeed unhedgeable. From a theoretical point of view, you might want to look at Merton's "Option pricing when underlying stock returns are discontinuous", the original paper that adapted Black-Scholes framework to include jumps. If you look at page 7, just after equation $(9)$: Unfortunately, in the presence of the ...

9

the information you provided is not sufficient to deduce risk neutral probabilities. You have to provide something like a price process from which risk neutral probabilities can be computed. Here are some examples: Example1: Consider a game where you pay 1 and you win 6 in case a six is thrown and 0 otherwise. So in financial mathematics terms we have a ...

7

Consider the case where we are interested in decomposing a continuous and piece-wise linear European payoff function $V \left( S_T \right)$ over $n$ intervals with $n + 1$ node points $S_i$ for $i = 0, 1, \ldots, n$. Without loss of generality, we assume that $S_0 = 0$ and write $V_i$ as short-hand for $V \left( S_i \right)$. We assume that the slope of the ...

6

A general hedging strategy Let assume that $S_1(t)$ and $S_2(t)$ are the price processes of your 2 stocks and that they follow a Geometric Brownian Motion (GBM): $$\forall \, i \in \{1,2\}, dS_i(t) =\mu_iS_i(t)dt + \sigma_iS_i(t)dW_i(t)$$ We assume both stocks have an instant correlation of $\rho$: $$dW_1(t)dW_2(t)=\rho dt$$ Let also $V(t)$ be the value ...

6

Any non path dependent European type payoff $f(S_T)$ can be replicated in a model independent way with vanilla calls and puts provided $f$ is twice differentiable (in the distribution sense). This is a consequence of the Carr-Madan formula.

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To see the exposure to FX risk and the difficulty for hedging, we assume constant interest rates and constant volatilities. Let $r_d$ and $r_f$ denote respectively the interest rates for USD and EUR. Moreover, let $X_t$ be the exchange rate at time $t$ from one unit USD to units of EUR. Finally, let $S_t$ be the price level of DAX at time $t$. We assume that,...

5

If I understand well, you have a market with 3 states: up, flat or down. You have 3 instruments: The stock The risk-free rate (50%) The option If you can create a portfolio today with these 3 instruments that can replicate de payoff of the option you have to price, then the law of one price tells you that the price of the option should be the price of ...

5

Note that \begin{align*} S_T^2 = 2\int_0^{S_T} k dk. \end{align*} Then \begin{align*} S_T^2 &= 2S_T^2-2\int_0^{S_T} k dk\\ &=2S_T\int_0^{S_T}dk-2\int_0^{S_T} k dk\\ &=2\int_0^{S_T} (S_T-k)dk\\ &=2\int_0^{\infty} (S_T-k)^+dk. \end{align*} For the partition $0=k_0 < k_1 < \cdots < k_n < \infty$, \begin{align*} S_T^2 &=2\int_0^{\...

5

I assume your trade $V(S,K,t,T)$ is European. Its payoff is: \begin{align} V(S,K,T,T)&=C^2(S,K,T,T) \\[3pt] &=\max(S_T-K,0)^2 \\[3pt] &=\boldsymbol{1}_{\{S_T\geq K\}}(S_T-K)^2 \\[3pt] &=\boldsymbol{1}_{\{S_T\geq K\}}f(S_T) \end{align} where $f(x)=(x-K)^2$. By Carr-Madan's static replication formula (see this question or this paper), we ...

4

I don't have much experience in the matter, but I've been doing some related literature research recently and I think these links can be helpful: A rather recent study from CME A (possible a bit biased) report by BlackRock A report by Lyxor (asset manager affialiated to Societe Generale)

4

There is no terminal $\mathcal{F}_T$ mesurable payoff $g$ such that $e^{-r(T-t)} E_t[g] = C(S_t, t, T, K)^2$, simply because $E_t[g]$ must be a martingale and $e^{r(T-t)} C(S_t, t, T, K)^2$ is not. So any deal that has npv $C(S_t, t, T, K)^2$ must involve a stream of intermediary payoffs $h(S_t,t) dt$, which you can solve for by plugging $V(S,t) = C(S, t, ... 3 Let me know whether this helps, but the author mentions a paper from Fujii and Takahashi; I have been looking for it on the internet and I have found what seems to be a version of it: Collateral Posting and Choice of Collateral Currency. I think they give a relatively transparent explanation$-$in terms of funding costs$-$of why the discount rate for ... 3 It is my understanding that a replicating portfolio for a put involves short selling stock and lending money. You cannot statically replicate an option. So this is not true in general, you'll need to re-balance your replicating portfolio (underlying + cash) dynamically if you want to replicate the option. This will imply sometimes buying stock and borrowing ... 3 That's impossible. Since neither the vanilla options nor the underlyings have any exposure to the correlation, no portfolio of these instruments can either. 3 You are correct that showing the self-financing condition for the BS-portfolio is not as straightforward as one may think: A portfolio$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 self-... 3 Gap risk contracts. These are daily-restriking putspreads that pay & cancel only if the underlying drops more than (say) 20% as measured vs yesterday's closing level. Contracts can range from as short as 6 months to 10 years. Cannot replicate that using Europeans. 3 I provide a general algorithm and an implementation in R to solve those kinds of problems in general: Financial Engineering: Static Replication of any Payoff Function. For your example: payoff <- data.frame(pi = c(0, 10, 30, 40, Inf), f_pi = c(30, 50, -10, 0, Inf)) payoff ## pi f_pi ## 1 0 30 ## 2 10 50 ## 3 30 -10 ## 4 40 0 ## 5 Inf Inf ... 3 In a cash settled swaption the payoff is settled using the cash annuity contractually computed using the swap rate. Thus is you work out the replication procedure you will find that CMS replication is exact when you replicate on cash settled swaption (at least when$\delta=0$, that is for CMS with fixing in arrears), because Hagan's "street approximation" is ... 3 The pricing of options is married with the concept of a hedging strategy that replicates the effect of the option. If you can only long or short a stock that will not replicate the greeks, it only creates delta. It is the commitment to the strategy that achieves it. For example if the price goes up and you are committed to buying more to increase your delta ... 3 The key point here is that the portfolio must be self-financing, namely the initial option premium$V_0$should be enough to allow you to hedge it throughout its life. If not, the option price$V_0$is either too low or too high. Because the option is written on the asset$S$, buying or selling$S$is how you neutralize the changes in value of the option: ... 3 1) Physical Replication would entail taking actual positions in the full or subset of instruments that comprise the ETF. This method would necessarily require a list of the holdings and weights of the ETF or the Index which the ETF attempts to track. Alternatively, in order to minimize the costs of replication, some will use an optimization approach by ... 3 We should be able to replicate the payoff exactly in each of the two regions$S_{T}\leq k_{1}$and$S_{T}\geq k_{2}$. From the first, $$a_{0}+a_{1}S_{T}+a_{3}(k_{2}-S_{T}) =S_{T}$$ so, matching coefficients,$a_{0}+a_{3}k_{2}=0$and$a_{1}-a_{3}=1$. From the second, $$a_{0}+a_{1}S_{T}+a_{2}(S_{T}-k_{1})=P_{0}$$ so, matching coefficients,$a_{0}-a_{2}k_{1}=P_{...

3

You simply take limits. Recall that in the Black-Scholes world $$d_1=\frac{\ln\left(\frac{S_t}{K}\right)+\left(r-q+\frac{1}{2}\sigma^2\right)(T-t)}{\sigma\sqrt{T-t}}.$$ As $t\to T$, we have $d_1\to\begin{cases} \infty & \text{if } S_t> K \\ 0 & \text{if } S_t=K \\-\infty & \text{if } S_t<K \end{cases}$. Thus, $\Delta=\Phi(d_1)e^{-q(T-t)}... 3 When the dividend is paid, the stock price on your tree should drop by the same amount. Ie if the dividend is 10 and the value of stock is 100 before the dividend at a node, you should change it to 90 and then continue building the tree from there. A cursory google brings quite a few results, eg this around slide 46 seems to explain it well 3 The devil's in the details. Here are a few things to check off the top of my head: Index constituents: The index is rebalanced only once a month, at the end of each month. We'd switch to the the constituents of the forward-looking "statistical universe," which becomes the "returns universe" at the start of the next month and is then used ... 2 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 ... 2 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 ... 2 The state price vector are the prices of securities which pay \$1 if and only if that state of the world occurs. This is just a question of being able to replicate the payoffs $$\begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix}, \begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix}, \begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix}$$ with payoff vectors $\vec{b} = [1,1,1]^T$ and \$\...

2

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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