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Questions tagged [replication]

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25
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3answers
7k views

Why hold options when you can dynamically replicate their payoff?

When holding vanilla options, you can cancel out, theoretically, all risk with dynamic (delta) hedging. Then you earn the "risk free rate of return". Why would you make such a portfolio when you can ...
18
votes
2answers
928 views

Duality between constant rebalanced portfolio (CRP) and corresponding derivative

One of the greatest achievements of modern option pricing theory is finding corresponding dynamical trading strategies in linear instruments with which you can replicate and by that price derivative ...
9
votes
2answers
14k views

How to replicate a digital call option

Call Option S=100 K=100 Payoff=1 (option is not available) How can i replicate this (payoff) with calls and puts with strike prices with multiples of 5$ Thanks for help
9
votes
4answers
2k views

Replicating portfolio and risk-neutral pricing for interest rate options

For equity options, the pricing of options depends on the existence of a replicating portfolio, so you can price the option as the constituents of that replicating portfolio. However, I am not seeing ...
8
votes
4answers
1k views

Software for decomposing structured products into plain vanilla products

Nowadays structured products (or packages) with complex payoff diagrams are omnipresent. Do you know of any software, add-ons, apps, code whatever, that enables you to enter a payoff diagram or a ...
8
votes
1answer
282 views

Replicating a portfolio with a certain payoff function

Assume there are two stocks $S_1$ with price $p_1(t)$ and $S_2$ with price $p_2(t)$ where $t$ indicates time. Assume, there is a hypothetical derivative $D$, which is such that, price of $D$ at a time ...
8
votes
1answer
9k views

What is a self-financing and replicating portfolio?

I try to understand the derivation of the Black-Scholes equation based on the "constructing a replicating portfolio". From mathematical point of view it looks simple. We assume that: Stock prices is ...
8
votes
1answer
3k views

What is the replicating portfolio of swaptions for a constant maturity swap (CMS)?

How do you replicate the payoff of a constant maturity swap rate? That is, if the payoff of a contract pays the 5-year swap rate every year for 10 years, how would you replicate this payoff using ...
8
votes
1answer
781 views

How to run an asset replication regression?

I am doing extensive research on portfolio replication and was hoping to get some help with some problems I am encountering. I am running a regression between 2 assets that I believe replicate ...
8
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1answer
968 views

How to calculate equivalent futures position?

Let's say I have the following two positions: Buy ATM SPX call, expires in 1 month Sell ATM SPX put, expires in 1 month This creates a synthetic futures position. How do I calculate how many ...
7
votes
1answer
1k views

delta-hedging is failing

and thank you for answering me ! While I was recently testing a delta-hedging on a few products, I got a P&L result of 20% for some of them. First, I thought that the implementation was ...
7
votes
2answers
126 views

How to properly assess the costs of replicating an index via futures contracts?

I would like to validate this sentence, coming from a WSJ article: The cost of holding a Eurostoxx 50 future, for example, has climbed from an average of 0.07% of the contract value since 1998, ...
6
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3answers
2k views

Variance swap replication and variance vega

Noob here. I've been trying to gain a better understanding of variance swaps and what better way than to replicate it with a portfolio of better understood instruments. I have read the GS 1999 ...
6
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2answers
3k views

Which objective function should I choose to minimize tracking error?

Let say I have $n$ assets and their returns over $m$ periods which are represented by a matrix $X \in \mathbb{R}^{m \times n}$, and I have some other asset with return over the same period which is ...
4
votes
2answers
187 views

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

Given a vanilla options market, i.e. $C(S,K,t, T)$ for all strikes $K$, is it possible to replicate $C^2 (S,K,t,T)$? So I am looking for a self-financing portfolio which has a price equal to $C^2(S,K,...
4
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1answer
1k views

Replicating strategy in the Black-Scholes model

I have a two-asset Black-Scholes model for a financial market: $dB_t=B_t r dt$ $dS_t=S_t(\mu dt+\sigma dW_t)$ I introduce a European claim $\xi=max(K,S_T)$ with maturity $T$, for some fixed $K$. I ...
3
votes
2answers
245 views

What Positions on an Underlier CANNOT be Hedged with Vanillas?

Say I have infinite precision of strikes $K$ (continuous world $dk$) and expirations $T$ (continuous $dT$) all with liquidity (so no practical limitations). What positions in an underlying can't be ...
3
votes
1answer
153 views

Replicating a square derivative with calls and puts

I have a derivative that pays off $S_T^2$ at time $T > 0$ with $S_T$ denoting the price of a non dividend-paying stock at $T$. I came across a question about how one can statically replicate this ...
3
votes
1answer
710 views

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

I'm self-studying and I'm considering the below example. The specific example is not especially relevant, but I included it for reference. I'm trying to understand the relationship between a ...
3
votes
2answers
207 views

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

Assume I have two assets A and B that are positively correlated most of the time. I'm trading a strategy based on this correlation. Is there a way to protect myself in the event that the correlation ...
3
votes
3answers
482 views

Replication of a call option by cash-or-nothing digital option

I am so stuck on this question: Consider a two-asset model where asset 0 is cash, so that the price of asset 0 is $B_t=1$ for all $t \geq0$. Asset 1 has prices given by $dS_t = a(S_t) dW_t$, where the ...
3
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2answers
1k views

Black-Scholes Equation - Riskless portfolio derivation

The following is a summary of the derivation of the Black-Scholes equation as given on wikipedia (http://en.wikipedia.org/wiki/Black-Scholes_equation#Derivation) - I have a question regarding the ...
3
votes
1answer
170 views

Binary option expression

Given r=0, σ(K)=const Binary=lim┬(ε→0)⁡〖((C(K,σ(K))-C(K+ε,σ(K+ε))))/ε〗 What is the analytical expression for the binary option value? σ(K)=const Therefore, Binary=lim┬(ε→0)⁡〖((C(K)-C(K+ε)))/ε〗 ...
3
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0answers
47 views

How to Compute the payoff of Var Swaps, which I have replicated

I used Derman(1999) method, to calculate the fixed Kvar for Variance Swaps using actual option price data. The first Pic Shows the outcome. (ignore the 0s). Now the profit and loss of short var swaps ...
2
votes
4answers
690 views

easy one step option replication

I have the following question. Consider that you have a stock currently trading at 100\$. In one month it can jump to 120\$ with probability 99% and go to 80\$ with probability 1%. How much is call ...
2
votes
1answer
3k views

How to replicate this option?

I have a question I am not sure how to approach: Suppose interest rates is 50%, a stock worth \$1 today can be worth \$2, \$1, \$0.5 next year. If the option that pays \$1 only when S = \$2 is ...
2
votes
1answer
653 views

CMS options, cash-settled/physically-settled swaptions

CMS options are traditionaly replicated using a theoritical "continuous" strip of swaptions (see for instance Hagan's paper "Convexity Conundrums : Pricing CMS Swaps, Caps and Floors"): In the paper,...
2
votes
1answer
211 views

Binary option analytical formula

Given $r=0$, $\sigma(K)=\text{const}$ and: $$ \text{Binary} = \lim_{ε → 0} \frac{(C(K,\sigma (K))-C(K+ε,\sigma(K+ε)))}{ε} $$ I have to find the analytical expression for the above. Since $σ(K)=\...
2
votes
1answer
196 views

Collateral replication argument

I'm trying to follow the replication argument in the first page of the following paper http://www.math.columbia.edu/~fts/Collateralized%20trade%20pricing%20made%20simple%20v1a.pdf One can however ...
2
votes
1answer
162 views

Calculating the annual return on an option using a replicating porfolio

I am self-studying and encountered the following problem: My idea was to calculate the price of the put using a replicating portfolio, then use the formula: $$Pe^{\gamma h} = S\Delta e^{\alpha h} + \...
2
votes
2answers
710 views

What is the instantaneous P&L of a Variance Swap?

What is the instantaneous P&L of a variance swap. Is it $(\sigma^{2}_{t}-\sigma^{2}_{implied})dt$?
2
votes
1answer
85 views

Profit and Loss on delta-hedged portfolio

The overnight profit formula from a textbook (possibly Derivative Markets by McDonald) is the following: $$\Delta _{t}(S_{t+h}-S_{t})-(V_{t+h}-V_{t})-(e^{rh}-1)(\Delta_{t}S_{t}-V_{t}),$$ where Delta ...
2
votes
1answer
81 views

How bad off are we when we use the “regular delta replication” strategy in an FX market on a Quanto?

See this question for context: https://quant.stackexchange.com/questions/32725/dynamic-hedge-of-quanto-options#= In there, I expressed interest in how well the usual strategy of replicating an ...
2
votes
1answer
199 views

Which studies should be replicated?

In psychology voting on which studies should be replicated is established on a website. For economics, including financial economics, the ReplicationWiki (that I founded) offers a voting option but it ...
2
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0answers
35 views

Infinite Horizon Barrier Option Paradoxe [duplicate]

I've came across this question which is puzzling me. Imagine that interest rates are zero and that you observe a stock $S_t$ whose value today $S_0$ is equal to 1\$. We consider the derivative that ...
2
votes
0answers
422 views

interview question : replication strategy of a betting game

Here is a question I found in a book I am not able to finish. Your help will be much appreciated! I also included where I have been so far. Q: Team A plays team B in a series of 7 games, whoever wins ...
2
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0answers
46 views

Reference Request: Trader Replication

I am looking for any reference where the following problem was addressed: given the list of trades of a trader teach an AI to replicate that trader's strategy. There are several well-known results ...
2
votes
0answers
147 views

replicating strategy three step binomial

I am having some trouble setting up a replicating strategy for a call option with a three step binomial model (discrete). I have no trouble doing this in a two step binomial model by backward ...
2
votes
0answers
533 views

Simple Forward Interest Rate Proof

Just trying to check my logic here: Let $Z(t,T)$ be a Zero-Coupon Bond with maturity $T$ bought at time $t$, $S_m$ be the spot interest rate for time $m$ and $S_n$ for time $n$ respectively, where $n ...
1
vote
1answer
165 views

Quanto Total Return of a Foreign Asset into Domestic

Say we have a product that pays the following at expiry $T$: $$\text{Payoff}_{in\ USD} = \text{Notional}_{in\ USD} \cdot \frac{DAXLevel_{in\ EUR}\ at\ t=T}{DAXLevel_{in\ EUR}\ at\ t=0}$$ i.e. it ...
1
vote
1answer
66 views

Cash deposit in replicating portfolio for BS equation unnecessary?

The book on Option Valuation Methods that I currently study (Higham 2013) constructs a replicating portfolio $\Pi = A(S,t)S + D(S,t)$ for deriving the BS PDE, where $D$ is a cash deposit. $D$ does not ...
1
vote
1answer
141 views

Replicating an option

When we replicate a portfolio of cash and stock for a call option, shouldn't the replicating portfolio's greeks be equal to options greeks? Is that true? If it is, how is it that a portfolio of cash ...
1
vote
1answer
554 views

Replicate a Portfolio with Given Payoff

Looking for a convincing general strategy [not trial and error] to solve these kind of questions: Any help will be super helpful! Thanks a bunch! Replicate a portfolio on an underlying asset $S$ ...
1
vote
1answer
77 views

Discontinuous derivative payoff approximation

Consider a derivative of digital type which pays this kind of payoff at time $T$: \begin{align*} g(S_T,k) &= \begin{cases} P_0,~S_T>k \\ S_T, ~S_T\leq k \end{cases} \end{...
1
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2answers
99 views

ETF Replication

I have a question regarding the ETF replication methods. I know there are two main methods, namely physical and synthetic replications, but I would like to understand how an ETF trader can : ...
1
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1answer
59 views

Static hedge forward swap using zero coupon swaps

I'm trying to create a static hedge for a forward swap using two spot starting zero coupon swaps (to prove that there is no convexity adjustment needed). Here are the instruments - Paying fixed in ...
1
vote
1answer
76 views

Replication of the portfolio in single step binomial model

I would be grateful if anyone would comment how to construct this: Assume $S_{i}^k$ is a stock price at time level $i$ and at price level $k$. Assume option is written on $S$ with a a payoff $f_{T}^{...
1
vote
1answer
2k views

Determine state price vectors?

I have 3 states with two assets, stocks and bonds. The bond has a payoff of 1 in every state of the world. And the stock has a current price of $S_0 = 100$ and ...
1
vote
1answer
64 views

Pricing weighted/average stock price claim

In a market consisting of a bank account with a constant interest rate r and a non-dividend paying stock S, consider a T-claim that pays $X = S(T)/S(T_0)$ at time T, where $T_0 < T$. a) ...
1
vote
1answer
129 views

Why does this delta hedge work, and what to do in more general circumstances?

In the simple Black-Scholes model, we can replicate an option by investing its $\Delta$ in the underlying, and keeping that portfolio self-financing via the bank account. I have two questions. I don'...