Summary
Black-Scholes option pricing model은 정말 크게크게 성공한 공식이다. 덕분에 파생상품에 관련된 모든 곳에서 해당 공식을 사용하고 있다. 다만, 강력한 가정 때문에 사용에 조심해야 한다.
- 현실은 discrete hedging을 할 수 밖에 없다. 그런데 BSM은 continuous delta hedging을 가정하고, delta hedging으로 모든 risk를 제거할 수 있다고 가정하고 있다. discrete delta hedging으로는 모든 risk를 제거할 수 없다.
- BSM은 transaction cost가 없다고 가정하고 있지만, 현실에서는 transaction cost가 있고, dynamic delta hedging을 통해 continuous delta hedging을 해야 하는데 이렇게 되면 엄청난 비용을 발생시킬 것이다.
- option에서 가장 중요한 variable은 단연 volatility이다. BSM에서는 이를 시간과 기초자산의 함수로 가정하고, option expiry동안 constant하며 known deterministic function이라고 가정한다. 하지만 현실에서는 시간과 기초자산의 함수로 volatility를 모델링 할 수 없으며, constant 가정은 비현실적이며, volatility는 전혀 known하지도, deterministic하지도, modeling하기도 쉽지 않다.
- dividend 역시 deterministic function이라고 가정하는데, 이 역시 현실과 다르다. 현실에서는 dividend declaration을 통해 언제 얼마나 배당을 하게 될지 알지만, declaration 전에는 전혀 얼마나 언제 하게 될지 모른다.
- BSM은 underlying price의 path가 continuous하다고 가정한다. 그러나 이 역시 당연히 현실과 다르다. 현실에서는 급격하게 자산이 변동해서 바닥에 쳐박히거나 하늘로 솟구친다.
- BSM은 option의 거래가 기초자산의 가격에 영향을 주지 않는다고 가정하지만 이는 현실과 전혀 다르다.
- BSM은 American option이 value maximization을 할 수 있는 순간 언제든지 exercise 될 수 있다고 가정한다. 그러나 이는 optimal이 writer와 holder 모두에게 optimal한 지점인지 모호한 면이 있다.
- BSM은 수익률 간에 serial autocorrelation이 없다고 가정한다. 이는 그래도 꽤 괜찮은 가정이지만 조심할 필요가 있다.
그래도 BSM은 개선의 여지가 있는 모델이다.
- jump diffusion의 도입으로 discontinous를 modeling 할 수 있고 이는 그나마 unpredictable한 volatility를 모델링하는 데 도움이 될 것이다.
- jump diffusion은 또한 delta hedge로 모든 risk를 제거할 수 없다는 점을 모델에 반영할 수 있게 한다. 즉, unhedgeable을 모델에 구현하는 의미가 있다.
- 만약 기초자산에 대해 확실한 view가 있다면 기초자산 자체로 포지션을 구축하기 보다 option으로 포지션을 구축하는 것이 더 유리할 것이다. 왜냐하면 leverage effect가 있기 때문이다. 그래서 option을 활용한 trading 전략들을 모델링 할 수 있고, 모델을 통해 기대 수익을 도출할 수 있다.
- BSM을 활용해서 optimal static hedge 포인트를 찾을 수 있다.
- Why the Black-Scholes assumptions are wrong
- how to improve the Black-Scholes model
Introduction
The Black-Scholes option pricing model works well in practice, and is widespread. It had much impact on financial markets. Everyone working in derivatives is using the Black-Scholes option pricing model.
It is used confidently in situations for which it was not designed, usually successfully. The value of vanilla options are often not quoted in monetary terms, but in volatility terms with the understanding that the price of a contract is its Black-Scholes value using the quoted volatility. You use a model other than Black-Scholes only with extreme caution, and you will have to be pretty convincing to persuade your colleagues that you know what you are doing. The ideas of delta hedging and risk-neutral pricing have taken a formidable grip on the minds of academics and practitioners alike. In many ways, especially with regards to commercial success, the Black-Scholes model is remarkably robust.
Nevertheless, there is room for improvement.
Discrete hedging
Black-Scholes assumes
- Delta hedging is continuous
Consider first, the continuous-time world of stochastic calculus. When we derived the Black-Scholes equation we used the continuous-time Ito’s lemma. The delta hedging that was necessary for risk elimination also had to take place continuously. If there is a finite time between rehedges then there is risk that has not been eliminated.
Transaction costs
Black-Scholes assumes
- Delta hedging is continuous
- There are no costs in delta hedging
The buying and selling of assets exposes us to bid-ask spreads. In other markets, the cost can be so great that we cannot afford to hedge as often as we would like.
Overview of volatility modeling
Volatility is the single most important determinant of an option’s value that we don’t accurately know.
Deterministic volatility surfaces
Black-Scholes assumes
- Volatility is a known constant
If volatility is not a simple constant then perhaps it is a more complicated function of time and/or the underlying.
We see how to back out from market prices the ‘implied volatility surface’ and the resulting ‘local volatility surface.’
This technique is popular for pricing exotic options, yielding prices that are ‘consistent’ with traded prices of similar contracts.
Stochastic volatility
Black-Scholes assumes
- Volatility is a known constant (or a known deterministic function)
The Black-Scholes formulate require the volatility of the underlying to be a known deterministic function of time. The Black-Scholes equation requires the volatility to be a known function of time and the asset value. Neither of these is true. All volatility time series show volatility to be a highly unstable quantity. It is very variable and unpredictable. It is therefore natural to represent volatility itself as a random variable.
Stochastic volatility models are currently popular for the pricing of contracts that are very sensitive to the behavior of volatility. Barrier options are the most obvious example.
Uncertain parameters
Black-Scholes assumes
- Volatility, interest ares and dividends are known constants (or known deterministic functions)
So volatility is not constant. Nor, actually, it is a deterministic function of time and the underlying. It’s definitely unpredictable. Worse still it may not even b emeasurable.
Volatility cannot be directly observed and its measurement is very difficult. How then ca we hope to model it? Maybe we should not attempt to model something we can’t even observe. What we should do is to make as few statements about its behavior as possible; we will not say what volatility currently is or even what probability distribution it has. We shall content ourselves with placing a bound on its value, restricting it to lie within a given range. The probability distribution of the volatility within this range will not be prescribed. If it so desires, the volatility can jump from one extreme to the other as often as it wishes. This ‘model’ is then used to price contracts in a ‘worst-case scenario.’
The idea of ranges for parameters is extended to allow the short-term interest rate and dividends to be uncertain, and to lie within specified ranges.
Empirical analysis of volatility
Black-Scholes assumes
- Volatility is a known constant
Having determined a plausible stochastic differential equation model for the volatility from data, I suggest ways in which it can be used. It can be used directly in a stochastic volatility model, or indirectly in an uncertain volatility model to determine the likelihood of volatility ranges being breached.
Stochastic volatility and mean-variance analysis
Black-Scholes assumes
- Volatility is a known deterministic function of asset value and time
When volatility is itself stochastic we can derive a theory that is consistent be requires knowledge of a new function, the market price of risk. This function is only observable via option prices themselves and so we find ourselves with a circular argument; we can price options if we know their market values.
Asymptotic analysis of volatility
There is often a conflict between whether to use a scientifically accurate model, that is infuriatingly slow to crunch, or a fast but not so good model. It turns out that you can something have you cake and eat it if you exploit the relative largeness or smallness of parameters in a mathematical model.
Jump diffusion
Black-Scholes assumes
- The underlying asset path is continuous
It is common experience that markets are discontinuous, from time to time they ‘jump,’ usually downwards. This is not incorporated in the lognormal asset price model, for which all paths are continuous.
When I say ‘jump’ I mean 2 things. First, that the sudden moves are not contained in the lognormal model; they are too large, occurring too frequently, to be from a normally distributed returns model. Second, they are unhedgeable; the moves are too sudden for continuous hedging through to the bottom of the jump.
These discontinuities are not modeled by the lognormal random walk that we have been using so far. Jump-diffusion is an improvement on the model of the unerlying but introduces some unsatisfactory elements: risk elimination is no longer possible and we must price in an ‘expected’ sense.
Crash modeling
Black-Scholes assumes
- The underlying asset path is continuous
If risk elimination is not possible can we consider worst-case scenarios? That is, assume that the worst does happen and then allow for it in the pricing. But what exactly is ‘the worst’? The worst outcome will be different from different portfolios.
Speculating with options
If one has a view on the underlying, then it is natural to invest in options because of their gearing. But how can this view be quantified? One way is to estimate because of their gearing. But how can this view be quantified? One way to estimate real expected returns from an unhedged position. This together with an estimate of the risk in a position enables one to choose which option gives the best risk/reward profile for the given market view.
This idea can be extended to consider many types of model for the underlying, each one representing a different view of the behavior of the market. Furthermore, many trading strategies can be modeled.
Optimal static hedging
If the governing equation for pricing is non linear then the value of portfolio of contracts is not the same as the sum of the values of each components on its own. One aspect of this is that the value of a contract depends on what it is hedged with. As an extreme example, consider the contract whose cashflows can be hedged exactly with traded instruments: to place this contract we do not even need a model. In fact, to use a model would be suicidal.
The beauty of the non-linear equation is that fitting parameters to traded prices (as in the implied volatility surfaces or in ‘yield curve fitting’) become redundant. Traded prices may be right or wrong, we don’t much care which. All we care about is that if we want to put them into our portfolio then we know how much they will cost.
Imagine that we have a contract called ‘contract,’ and it has a value that we can write as
\[ V_{NL}(contract) \]
where \(V_{NL}\) means the value of the contract using whatever is our non-linear pricing equation (together with relevant boundary and final conditions). Now imagine we want to hedge ‘contract’ with another contract called ‘hedge.’ And suppose that is costs ‘cost’ to buy or sell this second contract in the market. Suppose that we buy \(\lambda\) of these hedging contracts and put them in our portfolio; then the marginal value of our original ‘contract’ is
\[ V_{NL}(contract + \lambda ~ hedge) - \lambda ~ cost \]
In this expression ‘contract + \(\lambda\) hedge’ should be read as the portfolio made up of the ‘union’ of the original contract and \(\lambda\) of the hedging contract. Since \(V_{NL}\) is non linear, this marginal value is not the value of the contract its own. We have hedged ‘contract’ statistically, we may hold ‘hedge’ until expiry of ‘contract.’ We can go one step further and hedge optimally. Since the quantity \(\lambda\) can be chosen, let us choose it to maximize the marginal value of ‘contract.’ That is, choose $\lambda$ to maximize the marginal value of contract. This is optimal static hedging. We can, of course, have as many traded contracts for hedging as we want, and we can easily incorporate bid-ask spread.
In the event that ‘contract’ can have all its cashflows hedged away by one or more ‘hedge’ contracts, we find that we are using our non-linear equation to value an empty portfolio and that the contract value is model independent.
The feedback effect of hedging in illiquid markets
Black-Scholes assumes
- The underlying asset is unaffected by trade in the option
In the Black-Scholes model it is assumed that moves in the underlying are exogenous; that some cosmic random number generator tells us the prices of all ‘underlyings.’ In reality, large trade in the underlying will move the price in a fairly predictable fashion.
I will try to quantify this effect, introducing the idea that a trade in the underlying initiated by the need to delta hedge can move the price of the underlying. Thus it is no longer the case that the underlying moves and the option price follows, now it is more of a chicken-and-egg scenario. We will see that close to expiry of an option, when the gamma is large, the underlying can move in a very dramatic way.
Utility theory
Black-Scholes assumes
- Delta hedging eliminates all risk
If, for any reason, we cannot perfectly delta hedge then we are left with some residual risk. We will therefore need a framework for valuing this risk.
Advanced dividend modeling
Black-Scholes assumes
- The underlying asset has deterministic dividends.
Dividend payment is a subtle subject for modeling, and can have a significant effect on the prices of derivatives.
Serial autocorrelation in returns
Black-Scholes assumes
- There is no serial autocorrelation in returns
Today’s return is random, and independent from what happened yesterday or any time in the past. That’s the usual assumption.
Summary
There are many faults with the Black-Scholes assumptions. Although it is easy to come up with any number of models that improve on Black-Scholes from a technical and mathematical point of view, it is nearly impossible to improve on its commercial success.
Further reading
