14 votes
Accepted

delta-hedging is failing

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 ...
Daneel Olivaw's user avatar
9 votes

Throwing a dice and risk neutral probability

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 ...
Cettt's user avatar
  • 1,446
8 votes
Accepted

Replicate a Portfolio with Given Payoff

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,...
LocalVolatility's user avatar
8 votes

Pricing and hedging caps and floors on illiquid emerging markets

It could be worse. You're not asked to price rate exotics like accreters that might need more inputs besides implied vol cube :) and you're only asked to make markets. I.e., if I understand the ...
Dimitri Vulis's user avatar
6 votes
Accepted

What Positions on an Underlier CANNOT be Hedged with Vanillas?

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 ...
Antoine Conze's user avatar
6 votes
Accepted

Replicating a square derivative with calls and puts

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)...
Gordon's user avatar
  • 21.1k
6 votes
Accepted

Replicating a portfolio with a certain payoff function

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) =...
Daneel Olivaw's user avatar
5 votes
Accepted

Quanto Total Return of a Foreign Asset into Domestic

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. ...
Gordon's user avatar
  • 21.1k
5 votes

Collateral replication argument

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 ...
Daneel Olivaw's user avatar
5 votes

Replicating the square of an option $C^2 (S,K,t,T)$

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] &...
Daneel Olivaw's user avatar
4 votes

Replicate a Portfolio with Given Payoff

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: ...
vonjd's user avatar
  • 27.4k
4 votes
Accepted

How do you price an option on fresh corn?

Ideally, you should have a futures market, so you can hedge your option using the corresponding future. That is actually the right instrument to hedge and replicate, not physical corn picked in ...
Juan Ignacio Gil's user avatar
3 votes
Accepted

Replicating the square of an option $C^2 (S,K,t,T)$

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. ...
Antoine Conze's user avatar
3 votes

Discontinuous derivative payoff approximation

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 ...
Ali's user avatar
  • 131
3 votes
Accepted

ETF Replication

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 ...
AlRacoon's user avatar
  • 6,447
3 votes
Accepted

Cash deposit in replicating portfolio for BS equation unnecessary?

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 $V_0$ ...
Daneel Olivaw's user avatar
3 votes

What Positions on an Underlier CANNOT be Hedged with Vanillas?

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 ...
James Spencer-Lavan's user avatar
3 votes
Accepted

Replicating an option

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 ...
Attack68's user avatar
  • 10.2k
3 votes
Accepted

CMS options, cash-settled/physically-settled swaptions

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 ...
Antoine Conze's user avatar
3 votes
Accepted

Understanding the relationship between the Black-Scholes formula and a replicating portfolio

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 ...
Quantuple's user avatar
  • 14.6k
3 votes

How to replicate a correlation swap using only vanilla options and underlying

That's impossible. Since neither the vanilla options nor the underlyings have any exposure to the correlation, no portfolio of these instruments can either.
dm63's user avatar
  • 17k
3 votes
Accepted

Black-Scholes Delta value at maturity?

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\...
Kevin's user avatar
  • 15.9k
3 votes
Accepted

Replication (binomial tree)

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 ...
piterbarg's user avatar
  • 940
3 votes

Replicating Bloomberg Barclays index and sub-index monthly total and excess returns using constituent-level index-data

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 ...
Helin's user avatar
  • 11.7k
3 votes

Is it possible to replicate the payoff of a portfolio of options taken from a set of strikes {K1}, given another set {K2} with the same underlying?

I assume that you want to minimize some error function of your replication. For simplicity, I will focus on the squared error integral below. Without loss of generality, let us assume that for some ...
Kermittfrog's user avatar
  • 6,554
3 votes

Pricing and hedging caps and floors on illiquid emerging markets

Just add to Dimitri's excellent answer, particularly regarding hedging: In terms of development, rates vol markets in EM tend to lag FX vol markets. So chances are there is some FX options trading ...
user35980's user avatar
  • 1,386
3 votes
Accepted

Hedging gamma, theta or other risks

In the Black-Scholes model Gamma and theta do not need to be hedged because the BS PDE says that they balance each other (I'll take $r = 0$): $$ \frac{\partial f}{\partial t} + \frac12 \sigma^2 S^2\...
Frido's user avatar
  • 1,854
2 votes
Accepted

Calculating the annual return on an option using a replicating porfolio

Your computation of $\Delta$ is correct. However, your computation of the cash amount is wrong. You choose the cash amount $\beta$ that you need to initially lend or borrow such that in the up state, ...
LocalVolatility's user avatar
2 votes

Signs for the assets in a portfolio and definition of portfolio value

To sell the entire portfolio you need to sell one bond and buy three stocks. You have to buy the three stocks to get out of your short position held in the portfolio. The price of the portfolio is the ...
Mats Lind's user avatar
  • 1,402
2 votes
Accepted

Binary option analytical formula

As you say, you simply differentiate with respect to $K$. Assuming your binary's maturity is $T$, note that in a Black-Scholes framework with constant risk-free rate $r$, by the Breeden-Litzenberger ...
Daneel Olivaw's user avatar

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