12

It comes down to the definition of LIBOR: London Interbank Offer Rate -> Every business day, a panel of large banks are asked by the BBA[*] (British Bankers Association) at what rate they would lend cash (unsecured) in a certain currency to another bank of that panel for a certain maturity, and that for a range of currencies and maturities. e.g. Currency: ...


11

There are two parts to this question: 1) Is OIS a good risk-free proxy? and 2) Why is OIS used to discount cash flows of derivatives. First, overnight indexed swaps, in the US, are indexed to the Fed funds effective rate, which in turn tracks the Fed funds target rate. The Fed Fund target rate is directly set by the Federal Reserve, while the Fed Funds ...


7

Let's take your questions in turn - If volatility isn't a concern, an investor is concerned with $$ E[\log S_T] = \log S_0 + (\mu - \tfrac{1}{2}\sigma^2)T $$ when deciding to purchase a stock, yes? i.e. the expected value of logarithmic returns is the correct measure of the performance of a stock, not log of the expected value? Yes, for long-term ...


5

So you can get depo and swap rates from markit daily, at links like this: http://www.markit.com/news/InterestRates_<cncy>_<yyyymmdd>.zip i.e. http://www.markit.com/news/InterestRates_USD_20170105.zip and there's a spec for it here - though that's from 2009 so may be out of date, maybe you can find a more up to date one someone on their site, ...


5

I think you have a little misunderstanding. OIS just means the rate for fed funds. Usually people are referring to "FEDL01 Index" on Bloomberg. That's the VWAP of trades for the previous day in Fed Funds with participants lending to each-other. That's all in the past. That tells you nothing about the future. The Fed Funds futures settle to the average ...


4

First of all, if you are new in quantitative finance, I suggest to read the Hull'book, that's the basic resource for someone who wants to get fundamentals of the topic. Your evaluation is correct if you assume that linear relationship, but on real prices anything is linear; so, it depends on what you're looking for: If you have to conclude a project work at ...


4

The current price of future access to any asset is its current forward price. This is true for any asset and true for whatever currency you use to measure price. Once you have the forward prices it is clear how to discount: Call the period $T$ (e.g. 3 months) the forward price of you asset $P(T)$ and the current rate for exchanging your asset for currency i....


4

No, this is not the same. For example, consider the scenario $$ \begin{align*} r_A &= 10\% \quad\quad \sigma_A = 10\% \\ r_B &= 1.5\% \quad\quad \sigma_B = 1\% \\ \end{align*} $$ If $r_f=1\%$, $$ \text{SR}_A=0.90 \quad\quad \text{SR}_B=0.50 $$ then $A$ has the higher sharpe. Now if $r_f=0\%$, $$ \text{SR}_A=1.00 \quad\quad \text{SR}_B=1.50 $$ then $...


4

Perhaps some big picture background is useful. The Black-Scholes formula was originally developed through a dynamic hedging argument, that by trading a stock and a riskless bond in continuous time, one can perfectly replicate the payoff of an option. If one believes that two equivalent payoffs should have the same price (i.e. the Law of One Price), then the ...


4

Hint: If these 2 stocks have perfect negative correlation (correlation: -1), then you can construct a risk free portfolio. What would the return on that risk free portfolio be?


4

There are many proxies for the risk free interest rate. For most purposes you may need a short term risk free rate, but there are in general no significant differences which one you chose. Treasury bill rates are commonly used for studies on the US-equity market. For European countries, many researchers use 1-month or 3-month EURIBOR rates for empirical ...


4

Since pretty much all trades (at least interbank) are collateralized nowadays, you would follow the principle of CSA discounting and use the interest rate on the collateral as a discount rate. Typically it's an overnight rate, for example SONIA in GBP, EONIA in EUR, Fed Funds in USD (broad switch to SOFR hasn't yet taken place I think), so you would use OIS ...


4

The standard way to think about this is that at time $t$ the riskless asset gives you known return of $r_{f,t}$ over a short time period. However, this rate may itself be time-varying and stochastic so that we don't know its futures values, say $r_{f,t+s}$. E .g. a common assumption is that the rate follows an Ornstein Uhlenbeck process (implying that the ...


3

A non academic answer: In the real world, when dealers or professional counterparties trade options with each other, the option premium is not funded by the dealer's unsecured borrowing. Rather, options and other derivatives are usually subject to Collateral Agreements whereby 'safe' collateral is posted to cover exposure between counterparties. If cash ...


3

user233051 notes that ^IRX is indeed the official discount rate of the US Treasury. So to answer his question we need to exactly understand how the Treasury computes the discount rate. My answer is based on www.treasury.gov pages here, here and here. The official way of calculating the discount rate $d$ is $d = \frac{100-P}{100}\frac{360}{n}$ where $P$ is ...


3

Well, I do not think there is a large difference: Given you deposit money at a Bank the value of this deposit changes according to $$\frac{dB_t}{B_t} = r dt$$ which simply means there is no uncertainty with respect to this evolution (instead of incorporating a risky component $dW_t$. If you really want to interpret the risk-less asset as a bond you are ...


3

You are absolutely right that no one would like to replicate return of risk free assets when such instrument is easily available in the market and can be bought directly. So, why financial managers put their time and energy in creating such risk free portfolio? The application of creating risk free portfolio is mostly used in pricing derivative securities. ...


3

A few points can be noted. The CIR model is usually for a short, or instantaneous, spot rate $r_t$, which is the forward rate over an infinitesimal interval. That is, \begin{align*} r_t = \lim_{\Delta \rightarrow 0}\frac{1}{\Delta}\left(\frac{1}{P(t, t+\Delta)}-1 \right), \end{align*} where $P(t, u)$ is the price at time $t$ of a zero-coupon bond with ...


3

You are supposed to create a new portfolio using the tangency portfolio $P_t$ and the risk-free rate $r$. You know that the volatility of the tangency portfolio $\sigma_{P_t}=0.20$. You also know that the risk-free asset has: No risk: $\sigma_r=0$ Is not correlated with anything $\rho_{P_t,r} = 0$ So you're asked to create a portfolio with a higher risk, ...


3

For many purposes we need a short term risk free rate. T-bill rates are ideal for this. Most Euro bonds have maturities measured in years, they cannot be considered "short term" or "money market" rates. Also, Eurobonds are issued by a variety of issuers. Although generally highly rated, they may differ somewhat as to default probability. In other words they ...


3

well generally only the discrete bonds associated to the ends of the forward rates are modelled. to make these be martingales the drifts of the rates are chosen to make them driftless. for an extension to all bonds, see http://ssrn.com/abstract=1461285


3

The first statement is kind of clear. If all investors are risk-neutral, they simply do not care about risk and do not pay more or less regardless how risky an asset is. As a consequence, the return of all assets is the risk-free rate. Regarding the second statement. What does risk-neutral pricing really do? We change the probabilities from the real world, $...


3

This holds due to a change of measure. There is the real-world $\mathbb{P}$ and the risk-neutral world $\mathbb{Q}$. (I am going to assume constant interest rate $r$) The first fundamental theorem of asset pricing states that if there are no arbitrage strategies in a market, then there exists at least one probability measure $\mathbb{Q}\sim\mathbb{P}$ such ...


3

It’s not entirely risk-free. Nothing in life is. The comet could hit etc. The difference is suppose I had a 100m OIS swap line open with Lehman, margined overnight. OIS settles at 1.81% vs 1.80%; and I’m paying. I’m owed 1bp on 100m that I’m not going to get, equals 10 grand. No tears required. If I had lent 100m to them, my lawyers and ops people would ...


3

I would use SONIA. That's the official RFR for the UK. See this BoE link: https://www.bankofengland.co.uk/markets/transition-to-sterling-risk-free-rates-from-libor


2

Since Ecuador uses the US dollar, the appropriate rate to use for discounting is the US dollar risk-free rate (i.e. the zero coupon rate bootstrapped from the overnight swap curve). The US dollar is the natural numeraire to use for valuing securities priced in US dollars. For example, say that Ecuador five year notes with a coupon of 10.75% are currently ...


2

The risk-free rate used in the valuation of options must be the rate at which banks fund the cash needed to create a dynamic hedging portfolio that will replicate the final payoff at expiry. Dealers borrow and lend at a rate close to LIBOR, which is the funding rate for large commercial banks. The LIBOR swap curve is therefore the rate to be used when ...


2

For American options there is no parity rule, as I stated in the comments. However, there is the following disequality: $$S_0 - D - K \leq C - P \leq S_0 - K e^{-rT}$$ where $C$ and $P$ are prices of American call and put respectively, $S_0$ is the spot price today, $K$ is the strike price, $D$ is present value of the cash dividend (not as percentage), $r$ ...


2

By buying all the state contingent claims you ensure that you will receive 1 USD in the next period (since one of the states will occur), and that is the definition of a risk free security: something that is guaranteed to pay off 1. The price at which it sells today is lower than 1, and that discounting defines the Risk Free rate. If the risk free basket of ...


2

You can refer to Sharpe's paper. If you are computing an ex post Sharpe Ratio, you should calculate the excess return for each period as the return of the fund over the risk free rate return over that same period. Note that if, for example, each period is a month, you need to calculate the monthly risk free rate (and not use the annualised yield). You then ...


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