Pricing and hedging of financial products are key competencies in financial innovation. The importance of these skills distinguishes the financial sector from other industry sec- tors. Pricing and hedging is much more demanding in the financial industry due to the temporal or forward looking properties of financial contracts. Issuing such contracts for the clients requires for the issuer to master uncertain or risky cash flows. Not all finan- cial services are forward looking. The pricing of pure adminstration services or advisory services (financial planning, pension planning) are not considered.
There are three approaches to pricing.
• The integratedeconomic approachusing a fully fledged economic model. Solving
the model, i.e. finding the optimal policies for all agents in the economy under several constraints such as individual budget constraints and the market clearing conditions delivers the optimal individual decisions and from market clearing, the price dynamics of the asset in the economy are specified.
• The no arbitrage approach. This approach starts with the assumption that arbitrage is not possible: An investment strategy starting with zero initial wealth ending with certainty with no loss and in at least one state with a gain defines an arbitrage strategy. Such strategies are ruled out. Using this assumption and that people like more to less money relative pricing of assets follows: Derivatives are priced given the exogenous base assets such as stocks or bonds. This relative approach is neither contradictional nor orthogonal to the first general economic approach. In fact, no arbitrage is a necessary condition that financial equilibria exist. We discuss this in the next chapter.
• If prices of base assets do not exist since markets are in a state of infancy in practice
apotpourriof different methods is used: Ad hoc rules, try-and-error approaches, signaling pricing and others apply.
We consider arbitrage pricing in some details in the next chapter. We therefore focus on general economic pricing. The difference between the first two approaches can be
considered as follows. While economists populate the first approach, in the second one many physicists, mathematicians and other non-economists started to work in the last two decades (’Quants’). Banks were and are willing to pay high salaries to quants; higher ones than to comparable jobs which required an economic curriculum. The interdepen- dence between these two groups is small. The research ’Econophysics’ is an example that quants and economists often are not communicating. While quants pretend that their research is closer to reality since in their opinion they use less ambiguous economic concepts (utility functions), many economists argue that econophysics has little to do with economics but a lot with the application of physical concepts to economics.
Since ricing of financial products requires models, one facesmodel riskindependent
whether one considers the general equilibrium or the no arbitrage approach. Model risk sources can be possible misspecification of model parameters (utility functions, correla- tions), omissions of price sensitive variables, misspecification of estimates or uncertainty. Consider a new derivative product where no market exists for mark-to-market but mark- to-model is required for pricing. Traditional pricing models of Black and Scholes (equity),
Black and LIBOR43market models (interest rates) or extensions of them apply. All these
models face model risk, i.e. there are not enough payoffs to span the possible states of the world (market incompleteness), trading is restricted (short positioning is not possible) are two examples, volatility is a constant in the model but a function in reality, etc.
Pricing is only one side of the medal withhedgingon the other side. Hedging applies
to investment and trading product innovations where the issuer of the product faces a liability to the investor. The idea of hedging is: The initial price of a product is chosen such that the bank can invest in a portfolio which generates the payoff of the product in any possible states at any future dates. If this is possible, one speaks about a perfect
hedge or replication. One assumes that the bank neither needs to inject additional
money to cover the liability nor can it withdraw cash (self-financing). For other products such as bank deposits hedging is different. First, there are no tradeable products such that a bank deposit can be replicated. Second, risk management is defined on the whole balance sheet. That is interest -, liquidity - and credit risk are considered jointly for the asset and liability side. Hedging then means that the risk figures of the balance sheet are calculated and compared with the risk tolerance expressed by Value-at-Risk, Greek (key rate delta, convexity) and liquidity figures (NSFR) on an daily, operational basis
43
London interbank offered rate (LIBOR) describes an interest rate which is published daily by the British Bankers Association. LIBOR is the average interest rate which banks in London are charging each other for borrowing. It’s calculated by Thomson Reuters for the British Banking Association (BBA). It is a used benchmark for short term, i.e. up to 1 year, interest rates. LIBOR is offered in ten major currencies GBP, USD, EUR, JPY, CHF, CAD, AUD, DKK, SED, and NZD. LIBOR has been a factor in the pricing of hundreds of trillions of dollars of loans, securities and assets There are many vanilla LIBOR based instruments which are actively traded both on exchanges and over the counter such as LIBOR futures, forward rate agreements. The significance of these instruments is that: (a) They allow professionals effectively hedge their interest rates exposure. (b) One can use them to synthetically create desired future cash flows and thus effectively manage assets versus liabilities. (c) They allow market participants express their views on future levels of interest rates.
and with economic capital and risk budgeting on annual basis.
Hedging of investment and trading products requires that the innovator has a strong technology, strong human skills and risk capacity. Without a fast performing and secure IT innovation in financial products is no longer feasible. First, an overview over the positions is missing which means a blind flight in the management of risks follows. Second, without a performing infrastructure traders will lose money in particular if their is high flow due to market panics or market exaggerations. Third, derivative houses are under permanent attack of specialized software, so called high frequency trading. Say a traders set bid-ask offers at 100-101. Then the high frequency machine observes this and offers 100.01-100.99.
The pricing and hedging of derivative offer some intellectual challenges. We consider
plain vanilla European call and put options on a liquid stock.44 The pricing of the options
follows from no arbitrage and mathematical reasoning in a perfect market: That is, there is a unique price formula for this options consistent with no arbitrage. Assuming zero dividend, the option price is driven by two parameters: The risk free interest rate and the volatility parameter. But it is not the historical volatility which matter - it is the
implied volatilityσim. By definition, this is the value which we put into the theoretical
pricing formula to equalize observable market price:
TheoPrice(σim) =MarketPrice.
This requires an option pricing model such as the Black and Scholes model for example. Why is volatility a key parameter in trading? First, trading prices of vanilla options is the same than trading volatility - there is one-to-one relationship in the Black and Scholes model. An increasing volatility means increasing option prices and vice versa. We note that implied volatility for a call option by definition leads to the correct market price of the call but that if one inserts the same value in another option type, say a digital option, a wrong result follows, even if the options have the same underlying, the same maturity. As a second remark we observe that implied volatility is not constant. This parameter is a function of the maturity and the moneyness, i.e. how deep the option is in- or out-of-the-money. Implied volatility is indeed a surface in the two dimensions ’maturity’ and ’moneyness’. Fixing one dimension, a volatility curve follows, see Figure 1.33.
Figure1.33 shows the bid and ask volatility curves as a function of the moneyness.
They show a typical shape for equity derivatives: A smile, i.e. U-shaped pattern, and a skew, i.e. a asymmetric smile. The figure shows that volatility increase with distance to the at-the-money (ATM) region, i.e. where actual stock price and strike price are close. This is typically the region of liquidity. The increase in volatility away from ATM indicates that uncertainty increases which make the option more expensive. Rebonato
44Plain vanilla means that the options do not possess a complicated structure such as trigger events,
barriers or other path dependent features. European means that the option can only be exercised at maturity.
Figure 1.33: Volatility curves for fixed maturities for call and put options on Standard and Poor’s 500 Index.
summarizes:
The implied volatility is the wrong number to put in the wrong formula to get the right price.
The traders use the calculated volatility curves from say Eurex options prices.
The Black and Scholes model (BSM) approach is widely used to price vanilla option. Even though volatility is not constant, the approach is consistent with market prices. But the model is no longer consistent for path-dependent options, i.e. options where the price not only depends on the probability distribution at expiry. For all path-dependent options BSM prices will not be in-line with observed market volatilities for relevant strikes and maturities. In order to bring prices for path-dependent options in-line with the market, option specific spreads are presently used on the vanilla option implied volatility surfaces.
Consider a barrier option. Then we face using BSM a problem which volatility we should plug into the analytical barrier option price formula - the implied volatility at the strike or barrier level in the volatility surface? One can choose one volatility and add/subtract a spread for the volatility at the other point. The problem of finding the single BSM volatility is exacerbated if the distance between the barrier and strike is large. The reason for the problem magnification is due to the curvature of the skew and often the large difference between strike and barrier volatility. If one consider other options, the problem accentuates, i.e. multiple barriers emerge, cash flows can enter, etc. The BSM then becomes unsuitable.
The Local Volatility Model (LVM) enable pricing of path-dependent option that is consistent with observed market volatility surfaces without the need of a volatility spread. The price is systematically determined by providing a set of volatility data instead of a single volatility value, as is the case with the BSM.
Local volatility models assume the following underlying price dynamics:
dS(t) =rS(t) dt+σ(St, t)S(t) dWS(t), (1.10)
where σ is a deterministic function of both time and the underlying’s price. This is
opposite to stochastic volatility models where there is an own stochastic dynamics for the volatility state variable. Hence, the same risk source drives both the underlying and the volatility in LVM.
The motivation for this model class is as follows. For vanilla options, implied volatil-
ity I(K, T) = IBS is a function of strike and maturity. Hence, the fair vanilla option
price is a function of K and T or in other words, I(K, T) determines the risk neutral
probability for vanilla option’s underlying value S(T) at maturity. The option value at
maturity is independent of the underlying’s path. If we consider a barrier option, not only
The implied volatility surface for vanilla options will not tell the right answer, since the volatility figure is of a global type, i.e. path independent. The path dependency requires
a volatility figure which is also path dependent, i.e. σ(St, t) as for the local volatility
model. Since this volatility is not a global figure but is only valid for givenStandt, the
expression "local volatility" is used.
Dupire (94) shows that, given a complete set of European option prices for all strikes and maturities, local volatilities are uniquely determined by the vanilla option prices and their derivatives. The Dupire equation describes the relationship between implied and
local volatility. The non-discounted risk-neutral value C = C(S0, K, T) of a European
call option is given by
C=
Z ∞
K
ϕ(ST, T)(ST −K) dST (1.11)
whereϕis the unknown probability density of the final spot price at maturity. Then,
∂C ∂T = σ2K2 2 ∂2C ∂K2 −rK ∂C ∂K,
which is the Dupire equation with initial condition
C(K,0) = (S(0)−K)+ .
For the proof see Appendix7.2. Implied local volatilityσLoc(K, T) is then defined as:
σLoc2 (K, T) = ∂C ∂T +rK ∂C ∂K K2 2 ∂2C ∂K2 . (1.12)
This is the local volatility function consistent with the given prices of options and their
sensitivities whereas the unknown density function φ has been eliminated. Equation
(1.12) holds for non-dividend paying stocks. With a continuous dividend stream d,
Dupire’s equation reads:
σLoc2 (K, T) = ∂C ∂T + (r−d)K ∂C ∂K +dC K2 2 ∂2C ∂K2 . (1.13)
It follows that Dupire’s equation is obtained byswitchingfrom the PDE in the variables
S, tto a PDE in the variables K, T.
There is a simple interpretation of Dupire equation in terms of static option strategies. That for, we set the interest rate equal to zero. Then,
σ2Loc(K, T) = ∂C ∂T K2 2 ∂2C ∂K2 .
But ∂C∂T is the infinitesimal version of
C(S, t, K, T+ ∆T)−C(S, t, K, T) ∆T
i.e. a long call position with maturityT+ ∆T and a short call with maturityT. In other
words, a calendar spread with strike K. Similar, ∂K∂2C2 is the infinitesimal version of
C(S, t, K+ ∆K, T)−2C(S, t, K, T) +C(S, t, K−∆K, T)
(∆K)2 .
But this is a butterfly spread with strike K. Hence, local variance is proportional to
the ratio of a calendar and a butterfly spread.
Dupire equation looks much like the Black-Scholes equation with t replaced by T
and S replaced by K. But whereas the Black-Scholes equation holds for any contingent
claim on S, Dupire equation holds only for standard calls and puts. This equation tells
you how to find σLoc(K, T)and hence build an implied local volatility tree from options
prices and their derivatives. You can then use that implied tree to value exotic options and to hedge standard options, knowing that you have one consistent model that values all standard options correctly rather than having to use several different inconsistent Black-Scholes models with different underlying volatilities.
Dupire’s approach requires a continuous set of options data for allK and T. Since
data are only available for a discrete set and options out- and in- the money are suffering from illiquidity several problems arise in the implementation of the approach. First, an interpolation between the discrete data points is needed. It turns out that different interpolation methods have a strong impact on the outcome. Besides this instability due to the interpolation, the calibration of the local volatility surface is also instable over time.
If we re-express Dupire equation in term of the original variable, by applying the chain rule and using the formula for the Greeks in Black and Scholes we get:
Proposition 1.8.1(Local variance in terms of Black-Scholes implied variance). For zero dividends and zero interest rates, implied local variance reads in term of Black-Scholes implied variance: σLoc2 (K, T) = 2 ∂IBS ∂T + IBS T−t K2 ∂2I BS ∂K2 −d1 √ T −t∂IBS ∂K 2 +I1 BS 1 K√T−t+d1 ∂IBS ∂K 2 (1.14) with d1 = ln(S/K) + (r+12IBS2 )(T−t) IBS √ T −t .
Using the forward price and the transformation
x= ln K F0,T , y(T, x) =IBS2 (K, T|S, t)(T −t),
the result (1.14) reads:
σLoc2 (x, T) = ∂y ∂T 1−xy∂y∂x+14 −14 −1y +xy22 ∂y ∂x 2 + 12∂x∂2y2 .
This proposition relates the two concepts: Implied local volatility and implied Black and Scholes volatility.
Proof. For further properties of LVM we refer to the literature, see Derman et al. (1994,
1996), Dupire (1996) and Gatheral (2006). There is in particular a relationship between local variance and instantaneous variance, i.e. the variance which is for example given in a stochastic volatility model such as Heston’s one. We only assume that the underlying process is a diffusion - but no further specification is assumed:
σ2Loc = EσT2|ST =K
.
Implied local volatility is equal to the risk neutral expectation of the instantaneous vari-
ance conditional on the final stock priceST being equal to the strike priceK. In this way,
(squared) local volatility can be thought of as the conditional, risk-neutral expectation of future variance. This is again analogous to forward rates which can be thought of as the risk-neutral expectation of future interest rates.
There are numerous advantages of using this model in practice. The most impor- tant of these is that it gives a description of volatility, without adding further sources of risk/Stochastic factor to the model. Therefore the model will still be complete. This is due to the LVM being a function only of the stock price and time. This leads to many nice features of the model. One of these features, and the second major advantage of the model, is that all European options can be fully hedged using only the underlying and risk less bonds (as was the case in the BSM). This means that many of the use- ful features of the BSM are preserved in the LVM. The third advantage of the model is that it is relatively simple since it only has one stochastic factor, making numerical implementation fairly easy. Finally, it is important to note that for a sufficiently smooth IVS, the model guarantees the existence of a unique LVM, meaning that knowledge of the IVS is equivalent to knowledge of the LVM. The model also has a few drawbacks. The first of these is that it gives wrong predictions of future volatility. This can be seen by comparing the model’s prediction today of volatility at some future date, to the model’s prediction of volatility at that future date after reconstructing the local volatility surface at that point. These two values will not necessarily be the same, meaning that the model’s prediction of stock price dynamics can be inconsistent. The local volatility