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10

I'll outline how you can estimate the (implied) real-world density function from (observed) option prices. Having found this real-world density, you can then compute all sorts of probabilities and quantify the market's expectation of future prices. Recall firstly that (European-style) options are priced as risk-neutral expectation of the discounted payoff. ...


7

In the context of option pricing, "implied volatility" always refers to the equivalent diffusion coefficient in the geometric Brownian motion (GBM) dynamics that is necessary to match an observed European plain vanilla price for a given strike and maturity. When talking about "model implied volatility smile", what is meant is that: You choose some pricing ...


4

1) A straigthforward application is to price any complex payoff at maturity using this. By that I mean a payoff that is such that the price of the option is $$P = e^{-r(T-t)}E[f(S_T)] $$ Which you can then calculate by integrating $f(S_T)$ w.r.t. to your density. One of the challenges though is to have a proper marks and inter/extrapolation for the ...


4

We know that $-1\le\rho_{imp}\le 1$ so perhaps the simplest approach is to try the possible values $\rho_{imp}=\{-1,-0.9,-0.8,\cdots,0.8,0.9,+1\}$, to calculate resulting $\sigma$ values, d± values, and $M_{quote}$ values, then see which of these is closest to the observed market price. If desired you can then search a finer grid between two adjacent assumed ...


4

It is hard to know what "inherent volatility" refers to, as this term is somewhat non-standard. I will interpret it as the long term equilibrium level of volatility $\bar{\sigma}$ to which all volatilities are expected to revert. Clearly a short term vol of 60 and a long term vol of 34 is a highly unusual situation. The market expect volatility to be very ...


4

Just wanted to point out a few small issues in your statement and maybe help with the conceptual model of these formulas. implied volatility is defined as the value of the parameter σ we need to input into the Black-Scholes formula in order to get the price observed in the market. That is actually backward. Implied Volatility is actually better ...


3

Let $\rho\triangleq\rho_{imp}$. Note that: $$\frac{\partial \sigma}{\partial \rho}(\rho)=-\frac{\sigma_0\sigma_1}{\sigma(\rho)}<0$$ Therefore $\sigma$ is monotonic in implied correlation. In addition, the Margrabe pricing function $M(\cdot)$ is also monotonic in volatility $\sigma$ thus you can find an unique solution to the equation: $$\tag{1}M_{\text{...


3

We are going to this operation using borrowed money (via repo). How much capital do you need to do this? How many dollars for how many years? At first thought you need to raise $(P+A_b$) dollars (the dirty price of the bonds) for $d_1/360$ years, but actually you need less because you will receive $I_c$ in cash when there are $d_2$ days left to go and can ...


3

It's because the model assumes that the market will maximize its Sharpe ratio and your weights don't do that. Essentially, your example assumes investors are irrational in their allocation. If you solve for the weights that maximize the Sharpe ratio, the implied returns will equal the given returns. In your example, the Sharpe Ratio reaches a maximum ...


2

Forward implied volatility smile is implied from forward start options. For example call options have payoff $$ g_{T+\theta} = \left( \frac{S_{T+\theta}}{S_T} -K\right)_+ $$ If you are in a stochastic volatility model this can be rewritten $$ g_{T+\theta} = \left( e^{ \int_T^{T+\theta} r - \frac{1}{2}\sigma_t^2 dt + \int_T^{T+\theta}\sigma_tdW^S_t } -K\...


2

There are two approaches. Price call and put options with various strikes. Plot their BS implied volatilities. Find the slope of the graph. Price a call and digital call with the requisite strike. Compute the implied volatility of the call. Use the fact that $ DC(model) = DC(BS) - skew \times callvega,$ to solve for the skew. (See eg Section 7.7 of my ...


2

I agree to the above answer. The implied volatility should be the same. However if you record the traded option price and derive the implied volatility then these trades should be at the same point in time. For example some rarely traded option could be traded at noon - say a call. Then the put is traded some hours later an you take the last traded price of ...


2

Implied volatility will depend on the price the option is trading at. If more people buy a certain strike than another, or the given option is more difficult to hedge then the implied volatility will not be the same due to a different price. A simple example would be a stock trading at 10000 USD, and a call option expiring in 30 days with a strike of 12000 ...


2

The answer is vol for specific delta.You can use ATM vol to back out ATM strike. Because Vols of strikes on the same expiry is a smile(smirk), not a flat line, you have to use different vols to back out corresponding strikes based on delta and other given option variables(underlying price, vol, t, r, q).


2

The reason for the bid and ask twisting is that you can think of a long AUD forward as three transactions: Borrow USD Sell USD, buy AUD spot Lend AUD As a result, there are three sources of bid/offer cost for a forward. In contrast, for an interest rate, it's just one transaction (borrow or lend). This is why they twist those equations. They are trying to ...


2

I'm not sure I'm following 100%, but here is the answer to what I think you are asking: Your line of thinking was common before the financial crisis. Most people assumed that 6M LIBOR was roughly equal to 3m LIBOR compounded to 6m (using fixing and 3m implied forward), and in fact swap curves were constructed using multiple fixings at the short end. ...


1

Some stocks in the index may be hard to borrow. If you do include the borrow fee rates, you may get a lower forward price (lower interest rate) than you expect from this calculation. By no arbitrage, the index rate will likely be close to the weighted average of its constituents. To get pure equity option rate pricing, you may want to search the trading ...


1

First, VRP is (loosely speaking) the difference between the implied and objective variance of future returns: $VRP_t = Var_t^P[R_{t+1}] - Var_t^Q[R_{t+1}]$, of which only the second, risk-neutral variance is observed at time $t$. Assuming that (1) investors have been correct on average about the future variance, and that (2) the premium is stationary, one ...


1

Yes you can use implied default intensities to price Bonds if you have a quoted CDS for the issuer of the Bond or for an issuer with roughly the same characteristics even tough that's not the best thing to do. You can also avoid the whole CDS dilema with a repo on the security itself, and in this case counterparty risk is fully taken by the repoer... @Edit:...


1

That said, option market makers are very well informed traders that take large risk and so some of their information is reflected in the IV inputs, for example, if the stocks 30 day HV is 15% and 20 days to expiration they input 45% as expected future Volatility (IV) then they expect a big move (up or down) in the underline. So to answer your question , one ...


1

It would be a reasonable statement with "highest volatility." But implied volatility is a particular estimate of the volatility parameter which jumps around greatly from trade to trade and is not a good estimator of volatility.


1

The quote is based on the FX quote to achieve FX parity for your given rates. To understand how it works you can go to {FXFA} which uses the same principle. If you go to help on FXFA you can check the model together with all the calculations.


1

They use interest rate parity under the assumption that they are backing out the implied rate versus either USD or EUR. For the USD and EUR, they are using the OIS swaps. So for the US, the ticker for the 3M OIS swap is USSOC curncy and for EUR, the 3M OIS swap ticker is EUSWEC curncy. There are many other OIS swap tenors on BBG. I believe BBG will ...


1

Every line in the market in your example defines a different option (with a strike, maturity and call/put flag). Everyone of this function has a different price given by the market. What we call implied volatility is just the number you have to input as $\sigma$ in the Black Scholes equation to get the price which is traded in the market.


1

It might be worth noting that in the standard Black-Scholes model the implied volatility $\sigma$ is assumed to be a positive non-zero constant. If this were the case then we could simply look at the stock $S$, look at the historical data, and then compute the log-returns on some arbitrary time scale, and then compute the standard deviation, and we would ...


1

It comes from options. A common way to do it is from ATM (at the money) put and call and the Black Scholes formula. There are also other ways that use a larger number of options and more complicated maths.


1

European Call/Put parity (Call - Put = Discount x (Forward - Strike)), which is a consequence of the nor arbitrage condition, implies that they should be priced using the same implied volatility.


1

Take a look at Hull's Appendix of the Volatility Smiles chapter. (Chapter 16 in my version). It gives a method to calculate the probability density function based on option prices: $$ g(K) = e^{rT} \frac{\partial ^2 c}{\partial K^2} $$ This result comes from the Breeden Litzenberger 1978 paper.


1

First off, volatility smiles are often drawn over a delta space. Since you're asking, I'll assume you're trying to draw a volatility smile over strike prices, log moneyness, or some similar metric. If you have neither the spot price nor any strike prices associated with your data, I don't believe it's possible to back out both of those values. Not ...


1

Look the first answer of this thread: How to derive the implied probability distribution from B-S volatilities? Also many papers in Dupire volatility have your formula derivation. For example, look at (10) in http://www.javaquant.net/papers/DupireLocalVolatility.pdf


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