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27

Life Without a Risk-Neutral Measure How would we price assets without the measure $\mathbb Q$? Well, we would start with some version of the Euler equation $P_t=\mathbb{E}_t[M_{t+1}P_{t+1}]$, where $M$ is the stochastic discount factor (SDF). This equation holds under very weak assumptions (law of one price) and uses real-world probabilities. So, we take the ...


17

We created the SABR model because we realized that (a) option values were nonlinear in the volatility, and (b) volatilities are stochastic. This means that if one had an option (or portfolio of options) which have positive gamma in the volatility dimension, on average we'd make money from fluctuations in the volatility, and we'd lose money with negative vol-...


16

Let's relabel this as What (TF) is SABR? Alpha, Beta and Rho are the point of the model. So explaining them is explaining the model. A model of two processes Unlike earlier models in which the volatility was modelled as a constant (Vasicek, Hull-White, etc), SABR assumes that as well as the price of the thing being stochastic, so is its volatility. That is, ...


13

I've been using QuantLib for quite a while. Let me share some experience: QuantLib is a highly sophisticated quantitative framework. It can do much and much more than a simple pricing of European option. For example, in your example, you could have changed the payoff to binary payoff or giving a monte-carlo pricing engine (rather than AnalyticEuropeanEngine)...


12

In the derivatives context, "arbitrage free" means almost surely for the probability measure under consideration. This is in opposition with statistical arbitrage used at high frequencies for example. More precisely the assumption is that there is no $T\geq 0$ and self-financed portfolio $V$ such that $V_0 = 0$, $P(V_T < 0) = 0$ and $P(V_T > 0) > ...


12

A popular open-source option for the numerics in .NET is Math.NET (https://github.com/mathnet/mathnet-numerics). It has both managed implementations and allows you to use the optimized MKL native libraries. This use of .NET as a front-end to an optimized native library is quite common. Meta.Numerics (http://www.meta-numerics.net) is an alternative open-...


12

I provided an answer, based on an elementary approach, to an exactly same question yesterday. However, that question has disappeared, even though I like to keep a record for what I wrote. I would suggest that people do not delete their questions as they may be helpful for others. Here, I re-post that answer. We assume that, under the risk-neutral measure, ...


11

As @ilovevolatility explains, the seminal reference for this matter is Geman, El Karoui & Rochet (1995). We assume none of the assets are dividend paying, and they are strictly positive. There are two potential options. You are considering a market with only assets $X$ and $N$. Then Assumption 1 of their paper would apply, which is related to the two ...


11

Intro: Great answer given by KeSchn above. I would like to contribute an additional perspective. My experience with and my understanding of the Risk Neutral measure is entirely based on "no arbitrage" and "replication / hedging" arguments. The way I would like to explain this view is via the following three-step construction: (i) First, I ...


9

I recommend reading Cao, Hansch, and Wang (2004) "The Informational Content of an Open Limit Order Book". They present a simple model for an order-book price called the weighted price ($\mbox{WP}$): $$ \mbox{WP}^{n_1 - n_2} = \frac{\sum_{j=n_1}^{n_2} (Q_j^d P_j^d + Q_j^s P_j^s)}{(Q_j^d + Q_j^s)} $$ Where: $n$ is the order book level $Q_j$ is the size at ...


9

Having traded these options for a number of years I have some insight. It’s my belief that those that make a living specifically out of these options do have tree-style models that take into account early exercise. On the other hand , those that have occasional use of these options (such as interest rate derivatives dealers who might use them to hedge otc ...


8

We actually managed to come up with the answer to this question ourselves but wanted to share the answer since it might be relevant to others as well. The calculation depends on what method is used to calculate the cost. There is the FIFO, LIFO and the average cost method, see: http://www.accounting-basics-for-students.com/fifo-method.html If FIFO or LIFO ...


8

The model of choice depends on the purpose of the exercise. In general there are two types of models: Equilibrium models: These are general used use for "fitting" the spot curve to the discount function available in the market. So different models will give you different yield curves. One can use this information to see the relative value of implementing a ...


8

See this excellent paper by @MarkJoshi which defines/discusses the use of power numeraires. Starting from a dynamics specified under the risk-neutral measure $\mathbb{Q}$ \begin{align} &\frac{dS_t}{S_t} = (r-q) dt + \sigma dW_t^{\mathbb{Q}}\\ \iff& S_T\ \vert\ \mathcal{F}_t = S_t e^{(r-q-\frac{\sigma^2}{2})(T-t) + \sigma(W_T-W_t)} \tag{EQ.0} \end{...


8

Julien Guyon was so kind as to explain the story behind the cover and gave me permission to share it with the rest of the community: There's no direct link between the contents of the book and the cover page. I love Josef Albers' art and we decided to choose one of Albers' artworks to illustrate the cover page. The lithograph somehow suggests nonlinearity, ...


7

$$\frac{1}{(1+r_{02})^2} = E\left(\frac{1}{1+r_{12}}\right)\frac{1}{1+r_{01}}$$ Indeed, in the pricing measure, the distribution of $r_{12}$ has to be such that this relation holds. If you look at drift derivations for the LIBOR market model, a lot of work goes into making this sort of equation hold.


7

At least two ways to price this: Use Carr-Madan Use $S^2$ as a (power) numeraire, in which case you can price the payoff $(S_T - 1)_+$ under the power numeraire measure. EDIT: Put-call symmetry. Maybe I can get another -1 for my answer. Is the purpose of answering questions here to do homework for someone else or to stimulate further study and generate ...


7

We don't model the prices, we model the returns. The stock prices aren't explicitly modelled as log-normal, but rather this is a consequence of the actual model used to describe the returns. The core of the model used in the Black-Scholes model is to assume a geometric Brownian motion for the change in the price $S_t$ where over some small time increment $\...


7

There are two parts to your question which I try to answer separately. The first one is about what calibration actually is whereas the second question deals with risk-neutral pricing. As an example, we can use any model. I continuously refer to the stochastic volatility model from Heston (1993) as an example for equity options. Any thoughts equally apply to ...


6

Independently if it makes economically sense or not, negative interest rates have become a reality for Europe which can no longer be neglected. (Even LIBOR became negative in the last months.) One common but wrong solution was to set the rate simply to zero. (One must - by the way take care - that this "solution" is not automatically applied by correction ...


6

fixedLegBPS is the basis-point sensitivity of the fixed leg, that is, how much its NPV changes when the fixed rate changes by one basis point: it's calculated as the NPV corresponding to a fixed rate of 1 bps. Since the NPV of the fixed leg is linearly proportional to the fixed rate, you can write the equation targetNPV : fixedRate = BPS : 1 basis point ...


6

And don't forget that there are wrappers as eq RQuantLib which I use on the command-line here: edd@max:~$ r -l RQuantLib -e 'print(EuropeanOption("call", 47, 40, 0.05, 0.0, 4/12, 0.2))' Concise summary of valuation for EuropeanOption value delta gamma vega theta rho divRho 6.4728 0.8899 0.0307 4.5139 0.7372 11....


6

Here are at least three mistakes in your code: p += s0 * exp(...) should be p *= exp(...). Your volatility and rates are per annum, so divide the days by 365 (or 255) in your function asset_price. In asset_price you multiply by days inside the loop. However, the loop is already iterating over the days - so you don't take two steps of one day but two steps ...


6

The first equation expresses the option price as a discounted expected value of the payoff contingent on an asset price $S \geqslant 0$. Without loss of generality, we assume that the probability density function has support in $[0,\infty)$, and rewrite as $$\begin{align} P_{t,T}(K) &=e^{-r(T-t)} \int_{-\infty}^{\infty}\left(K-S\right)^+ q_T^S(S)\,dS \\...


6

You are not giving the constructor a discountCurve. The constructor is: ql.ForwardRateAgreement(valueDate, maturityDate, position, strikeForward, notional, iborIndex, discountCurve=ql.YieldTermStructureHandle()) So you should add a the spotCurveHandle as the last parameter: fra = ql.ForwardRateAgreement(ql.Date(7, 5, 2018), ql.Date(15,12,2020), ql.Position....


6

I believe the other answers are nearly exhaustive; but here's a bit of intuition I'd like to add: Think of the decision (= equilibrium price) of a market as: Decision = f(probabilities, risk aversion) where probabilities are the chances of various events happening, and risk aversion is the taste preference of the market. Now it turns out that the 'iso-curve' ...


6

Let me try to answer, this topic is much deeper than my answer 1. Why are these models like this unpopular? First, these models produce marginal distributions that does not fit the market, which means they cannot reproduce vanilla option prices traded in the market SV models, e.g. Heston model, may fit to a few vanilla prices, they cannot fit the entire ...


6

I am not sure if you can classify it like that. Mind you, I never wrote a book. I'll write what I know below and you can decide if the classification makes sense or not. 1 ) STIR: as the term indicates - short term like Eurodollar frequently modelled with Black or Bachelier (normal) model. HW1F is also a short rate model. 2 ) HJM is a framework (M is not ...


5

Generally, there are few or no zero-coupon instruments traded in the market, especially for longer maturities. However, pricing of many derivatives relies on having a zero curve, so it becomes necessary to construct one using available instruments. Aside from derivatives, one can use a zero curve fitted to liquid bonds to price new or less liquid issues.


5

To add to Student T's answer, which I second: the complex setup starts making sense (and its cost gets amortized) once you start keeping the instruments around instead of throwing them away after the pricing. For instance, once the option above is built, you can change the market price of the underlying (or its volatility, or the risk free rate) by just ...


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