- the foundation of derivatives theory: delta hedging and no arbitrage
- the derivation of the Black-Scholes partial differential equation
- the assumptions that go into the Black-Scholes equation
- how to modify the equation for commodity and currency options
A very special portfolio
option value:
\[ V(S, t;, sigma, \mu;, E, T; r) \]
- S and t - variable
- \(\sigma\) and \(\mu\) are parameters associated with asset price
- E and T are parameters associated with the details of the particular contract
- r is a parameter associated with the currency in which the asset is quoted
\(\Pi\) denotes the value of a portfolio of long option & short in some quantity, Delta, of the underlying:
\[ \Pi = V(S, t) - \Delta S \]
Assume the underlying follows a lognormal random walk
\[ dS = \mu S dt + \sigma S dX \]
\[ d\Pi = dV - \Delta dS \]
\[ dV = \frac{\partial V}{\partial t}dt + \frac{\partial V}{\partial S}dS + \frac{1}{2}\sigma^2 S^2 \frac{\partial^2 V}{\partial S^2}dt - \Delta dS \]
The deterministic terms are those with dt, and the random terms are those with the dS.
Elimination of risk: delta hedging
\[ \Delta = \frac{\partial V}{\partial S} \]
any reduction in randomness is generally termed hedging, whether the randomness is due to fluctuations in the stock market. The perfect elimination of risk, by exploiting correlation between two instruments is generally called delta hedging.
Delta hedging is an example of a dynamic hedging strategy. This means that the perfect hedge must be continually rebalanced.
Delta hedging was effectively first described by Thorp & Kassouf (1967).
No arbitrage
\[ d\Pi = (\frac{\partial V}{\partial t} + \frac{1}{2}\sigma^2 S^2 \frac{\partial^2 V}{\partial S^2})dt \]
This change is completely riskless.
\[ d\Pi = r \Pi dt \]
This is an example of the no arbitrage principle.
The Black-Scholes equation
\[ (\frac{\partial V}{\partial t} + \frac{1}{2}\sigma^2 S^2 \frac{\partial^2 V}{\partial S^2})dt = r(V - S\frac{\partial V}{\partial S})dt \]
\[ \frac{\partial V}{\partial t} + \frac{1}{2}\sigma^2 S^2 \frac{\partial^2 V}{\partial S^2} + rS\frac{\partial V}{\partial S} - rV = 0 \]
This is the Black-Scholes equation.
The Black-Scholes equation is a linear parabolic partial differential equation.
- linear - if you have two solutions of the equation then the sum of these is itself also a solution
- parabolic
- Financial equations are also usually parabolic
- meaning that they are related to the heat or diffusion equation of mechanics
- such equations are relatively easy to solve numerically
no mention to drift term
- because it will be eliminated when portfolio is constructed
- The economic argument for this is that since we can perfectly hedge the option with the underlying we should not be rewarded for taking unnecessary risk; only the risk-free rate of return is in the equation. This means that if you and I agree on the volatility of an assert we will agree on the value of its derivatives even if we have differing estimates of the drift.
We can use the same the Black-Scholes argument to replicate an option just by buying and selling the underlying asset. This leads to the idea of a complete market. In a complete market* an option can be replicated with the underlying, thus making options redundant.
*complete market
https://en.wikipedia.org/wiki/Complete_market
- two conditions
- negligible transaction costs and therefore also perfect information
- Every asset in every possible state of the world has a price
- In such a market, the complete set of possible bets on future states of the world can be constructed with existing assets without friction.
- A state of the world is a complete specification of the values of all relevant variables over the relevant time horizon. A state-contingent claim, or state claim, is a contract whose future payoffs depend on future states of the world
- state-contingent claim examples: futures, options, etc.
- dynamically-complete market
- In order for a market to be complete, it must be possible to instantaneously enter into any position regarding any future state of the market.
The Black-Scholes Assumptions
- The underlying follows a lognormal random walk
- The risk-free interest rate is a known function of time
- There are no dividends on the underlying
- Delta hedging is done continuously
- There are no transaction costs on the underlying
- There are no arbitrage opportunities
There are many more assumptions but the above are the most important
The underlying follows a lognormal random walk
This is not entirely necessary. To find explicit solutions we will need the random term in the stochastic differential equation for S to be proportional to S. The ‘factor’ \(\sigma\) does not need to be constant to find solutions, but it must only be time dependent. As far as the validity of the equation is concerned it doesn’t matter if the volatility is also asset-price dependent, but then the equation will either have very messy explicit solutions, if it has any at all, or have to be solved numerically.
- It seems obvious random term in the stochastic differential equation for S is proportional to S when using derivatives… But it is doubt that there are other examples conflicting this.
Do we need the drift term \(\mu S\) to take this form, after all it doesn’t even appear in the equation? - There is a technically here that whatever the stochastic differential equation for the asset S, the domain over which the asset can range must be zero to infinity.
- because S is distributed lognormally.
It is possible to choose the drift so that the asset is restricted to line within a range; such a drift could leave us with arbitrage opportunities.
- Why? from Copilot
- The drift term in the Black-Scholes model represents the expected return of the underlying asset over time. It is usually denoted by \(\mu\) and is assumed to be constant. However, in reality, the drift term is not well known and may vary depending on the market conditions and the risk preferences of the investors.
- If the drift term is chosen arbitrarily, it could create arbitrage opportunities for investors who can exploit the difference between the actual and the assumed return of the asset.
- To avoid arbitrage, the drift term must be consistent with the risk-neutral measure, which is a mathematical construct that allows us to price options without knowing the real probabilities of the asset price movements. Under the risk-neutral measure, the drift term is replaced by the risk-free rate, which is the return of a riskless investment such as a government bond. This ensures that the option price reflects the fair value of the option in the market and does not depend on the subjective expectations of the investors.
- The risk-neutral measure is derived from the no-arbitrage principle, which states that there should be no way to make a risk-free profit in a frictionless market. The no-arbitrage principle is one of the key assumptions of the Black-Scholes model, along with the geometric Brownian motion of the asset price, the constant volatility, the European-style exercise, and the continuous trading. These assumptions simplify the mathematical derivation of the option price, but they may not hold in the real world, leading to deviations from the actual option prices observed in the market.
The risk-free interest rate is a known function of time
In practice, the interest rate is often taken to be time dependent but known in advance. We’ve also assumed that lending and borrowing rates are the same.
Delta hedging is done continuously
This is definitely impossible. Hedging must be done in discrete time. Often the time between rehedges will depend on the level of transaction costs in the market for the underlying; the lower the costs, the more frequent the rehedging.
There are no arbitrage opportunities
It is extremely important to stress that we are ruling out model-dependent arbitrage. This is highly dubious since it depends on us having the correct model in the first place, and that is unlikely. I am happier ruling out model-independent arbitrage, i.e., arbitrage arising when two identical cashflows have different values.
Final conditions
We must specify the option value V as a function of the underlying at the expiry date T. That is, we must prescribe V(S, T), the payoff.
call option
\[ V(S, T) = max(S - E, 0) \]
put option
\[ V(S, T) = max(E - S, 0) \]
binary call
\[ V(S, T) = \mathcal{H}(S - E) \]
binary put
\[ V(S, T) = \mathcal{H}(E - S) \]
where \(\mathcal H\) is the heaviside function, which is zero when its argument is negative and one when it is positive.
Options on dividend-paying equities
\[ d\Pi = \frac{\partial V}{\partial t}dt + \frac{\partial V}{\partial S} dS + \frac{1}{2}\sigma^2 S^2 \frac{\partial^2 V}{\partial S^2}dt - \Delta dS - D \Delta S dt \]
The last term on the right-hand side is imply the amount of the dividend per asset, DSdt, multiplied by the number of the asset held.
In reality, when dividend is announced, the information is implied with the derivatives lying the remain terms to its expiration. It can be interpreted as change in the underlying asset from not-dividend-paying equity to dividend-paying equity during the derivative term.
Currency options
In holding the foreign currency we receive interest at the foreign rete of interest. This is just like receiving a continuous dividend.
\[ \frac{\partial V}{\partial t}dt + \frac{1}{2}\sigma^2 S^2 \frac{\partial^2 V}{\partial S^2}dt + (r - r_f) S \frac{\partial V}{\partial S} - rV = 0 \]
Commodity options
The relevant feature of commodities requiring that we adjust the Black-Scholes equation is that they have a cost of carry. That is, the storage of commodities is now without cost. This means that just holding the commodity will result in a gradual loss of wealth even if the commodity price remains fixed. To be precise, for each unit of the commodity held an amount qSdt will be required during short time dt to finance the holding.
\[ \frac{\partial V}{\partial t}dt + \frac{1}{2}\sigma^2 S^2 \frac{\partial^2 V}{\partial S^2}dt + (r + q) S \frac{\partial V}{\partial S} - rV = 0 \]
cf) Why is backwardation normal in commodity market?
- To avoid backwardation in commodity market, investor can short the underlying asset simultaneously buying futures. However, there is no way to short the underlying asset, the tangible commodity, so that the impossibility makes the backwardation in commodity market. Of course, this assumes the market is in the normal conditions, not in turbulent conditions.
Options on futures
\[ F = e^{r(T_F - t)} S \]
\[ \frac{\partial V}{\partial t}dt + \frac{1}{2}\sigma^2 F^2 \frac{\partial^2 V}{\partial F^2}dt - r\nu = 0 \]
Some other ways of deriving the Black-Scholes equation
The Martingale approach
The value of option can be shown to be an expectation not a real expectation but a risk-neutral one. This is useful result, since it forms the basis for pricing by simulation. The partial differential equation can be derived from the expectation. The concepts of hedging and no arbitrage are obviously still used in this derivation.
The binominal model
The binomial model is a discrete time, discrete asset price model for underlyings and again uses hedging and no arbitrage to derive a pricing algorithm for options. In taking the limit as the time step shrinks to zero we get the continuous-time Black-Scholes equation.
CAPM/Utility
CAPM is a model for the behavior of risky assets and a principle and algorithm for defining and finding optimal ways to allocate wealth among the assets. Portfolios are described in terms of their risk and reward. If you include options in this framework then the possible combinations of risk and reward are not increased. This is because options are in a sense, just functions of their underlyings. This is market completeness. The risk and reward on an option and on its underlying are related and the Black-Scholes equation follows.
Further reading
