Chapter 9

Continuous-time security markets

In this chapter we consider financial markets in continuous time, contrary to the discrete time approach in Chapter 3. The multiperiod binomial model considered in Chapter 3 allowed the asset prices to take only one of two values in the next period, which certainly contradicts the actual behaviour of stock prices. In real financial markets, trading is not restricted only to take place at a limited number of time points; hence, it seems reasonable to model the financial market in continuous time where the prices are allowed to change at any time.1 The mathematical description of a stochastic process in continuous time presented in Chapter 8 will be used to model the price of securities.

9.1 From discrete to continuous time

Consider a financial market with n securities, e.g., stocks. The objective of this section is to derive, in an ad hoc manner, a formula for the wealth process of a self-financing portfolio in the n securities in continuous time. However, in order to obtain a formulation that is consistent with the definition of Itō stochastic integrals in Chapter 7, the discrete time model of Chapter 3 is restated. Let

1. hi(t) = the number of stocks of type i held in the period [t, t + k[.
2. V(t, h) = the value of the portfolio h at time t.
3. Si(t) = price for one stock of type i in period [t, t + k[.

The interpretation is as follows:

1. At time t, i.e., at the beginning of period t, we are equipped with the portfolio h(t−k) ∈ ℝn from period t − k.
2. At time t we observe the stock prices Si(t) ∈ ℝn for all i.
3. After observing the stock price, we decide the portfolio h(t) for the next period t.

At time t the value of the portfolio h(t − k) is given by

$V(t,h)=h^T(t-k)S(t)={\displaystyle \sum _{i=1}^n}{h}^i(t-k)S^i(t)(9.1)$

and the value of the new portfolio bought at time t is

$V(t,h)=h^T(t)S(t)={\displaystyle \sum _{i=1}^n}{h}^i(t)S^i(t).(9.2)$

If we assume that the portfolio is self-financing we have the following equality according to the Definition 3.7.

$h^T(t-k)S(t)=h^T(t)S(t).(9.3)$

By introducing the lag operator ΔX(t) = X(t) − X(t − k) formula (9.3) may be written as

$S^T(t)\Delta h(t)=0.(9.4)$

Since our aim is to obtain a model in continuous time, we consider ST (t + k)Δh(t + k) = 0 and let k → 0. We then get that

$S^T(t)\text{d}h(t)=0.(9.5)$

In Chapter 7 where Itō stochastic calculus was introduced, it was shown that the value of a certain stochastic integral depends critically on where the integrand is evaluated in the interval [t, t + k[. In the formulation above the integrand is evaluated at the right endpoint of the interval, which means that the results will be inconsistent with the definition of Itō integrals (and differentials). In order to obtain Itō differentials the integral must be evaluated at the left endpoint. By adding and subtracting S(t)Δh(t + k) in (9.4) we get

$\begin{array}{llll}0& =& S^T(t)\Delta h(t)+S^T(t)\Delta h(t+k)-S^T(t)\Delta h(t+k)& (9.6)\\ & =& S^T(t)\Delta h(t+k)+\Delta S^T(t+k)\Delta h(t).& (9.7)\end{array}$

For k → 0 we obtain

$0=S^T(t)\text{d}h(t)+\text{d}{h}^T(t)\text{d}S(t)(9.8)$

which might be interpreted as an infinitesimal budget restriction, since it says that one is only allowed to change the portfolio according to (9.8). Otherwise the portfolio (trading strategy) would not be self-financing. In order to obtain a stochastic differential equation for the wealth process V(t, h) we apply the Itō formula (8.10)

$\text{d}V(t,h)=h^T(t)\text{d}S(t)+S^T(t)\text{d}h(t)+\text{d}{h}^T(t)\text{d}S(t).(9.9)$

For self-financing trading strategies the last two terms cancel out and we get

$\text{d}V(t,h)=h^T(t)\text{d}S(t),V(t)={\displaystyle {\int }_0^t{h}^T(u)\text{d}S(u)}.(9.10)$

The interpretation of the equation is that changes in the total value are generated by changes of the asset prices S(t). It is important to understand that the differentials and the integral should be interpreted in the Itō sense, because if we instead had used the concept of, e.g., Stratonovitch integrals, the SDE for the wealth process would not be the same as (9.10).

Define the relative portfolio strategy as

$u^i(t)=\frac{h^i(t)S^i(t)}{V(t,h)},\text{for}\text{all}i=1,\mathrm{...},n(9.11)$

where

${\displaystyle \sum _{i=1}^n}{u}^i(t)=1(9.12)$

and where ui(t) denotes the fraction of the total wealth that is placed in asset i. By substitution of hi(t) into the wealth process (9.2) we get

$\text{d}V(t,h)=V(t,h){\displaystyle \sum _{i=1}^n}{u}^i(t)\frac{\text{d}{S}^i(t)}{S^i(t)}(9.13)$

which gives an expression of the wealth process we shall need in the following.

9.2 Classical arbitrage theory

In this section we consider a financial market consisting of a riskless asset with a deterministic price process Bt, and a stock with the stochastic price process St. The model we shall consider is the famous Black-Scholes model and we will derive the Black-Scholes formula for pricing European call options.

The riskless asset is assumed to follow the following differential equation

$\text{d}B(t)=rB(t)\text{d}t(9.14)$

where r is a constant interest rate. This asset is called the money market account. The solution is

$B(t)=B(0)exp\left({\displaystyle {\int }_0^t}r\text{d}s\right)=B(0)exp(rt).(9.15)$

The corresponding solution, if the short term interest rate was varying deterministically, is simply given by

$B(t)=B(0)exp\left({\displaystyle {\int }_0^t}r(s)\text{d}s\right).(9.16)$

The price process of the stock is assumed to be stochastic in the following way

$\text{d}S(t)=\alpha S(t)\text{d}t+\sigma S(t)\text{d}{W}_t(9.17)$

where α and σ are constants. Note that the price process has a deterministic drift of magnitude αS(t). In Example 8.10 we found that the solution to (9.17) is given by

$S(t)=S(0)exp\left((\alpha -\frac{1}{2}{\sigma }^2)t+\sigma W(t)\right).(9.18)$

It seems like the growth rate is reduced to $\alpha -\frac{1}{2}{\sigma }^2$ plus some random noise. However by computation of the expected value we get

$E[S(t)]=S(0)exp\left((\alpha -\frac{1}{2}{\sigma }^2)t\right)E[exp(\sigma W(t))]=S(0)exp(\alpha t)(9.19)$

because exp{σW(t)} is lognormally distributed with mean exp{σ2t/2}. Thus the solution gives a trajectory with random fluctuations around the deterministic growth curve S(0)exp{αt}.

Remark 9.1 (Lognormal distribution).

Assume that X(t) is normally distributed. Then E[exp(X(t))] = exp (E[X(t)] + Var[X(t)}/2).

The objective is now to find the “correct” price P(t) for a European call option at time t ≤ T. As we have seen in a previous chapter, the value at time t = T is given by P(T) = max[S(T) − K, 0] where K is the exercise price. However it is not obvious what the price of a call option should be at time t < T.

Since the solution of the SDE for the stock price is a Markov process it is natural to assume that the price of a European call option is a function of time and the actual stock price P(t) = F(t, S(t)). Similar to the approach taken in the discrete time framework we now want to price the call option by constructing a replicating trading strategy.

Since the price of the option is a function of the stock price, we can apply the Itō formula to obtain a stochastic differential equation for the price of the call option.

$\text{d}P(t)=[F_1+\alpha S(t)F_s+\frac{1}{2}{\sigma }^{2}S{(t)}^2{F}_{ss}]\text{d}t+\sigma S(t)F_s\text{d}W(t)(9.20)$

where

$F_t=\frac{\partial F(t,S(t))}{\partial t},F_s=\frac{\partial F(t,S(t))}{\partial S\left(t\right)},F_{ss}=\frac{{\partial }^{2}F(t,S(t))}{\partial S{\left(t\right)}^2}.(9.21)$

By defining

${\alpha }_P=\frac{F_t+\alpha S(t)F_s+\frac{1}{2}{\sigma }^{2}S{(t)}^2{F}_{ss}}{P(t)}\text{and}{\sigma }_P=\frac{\sigma S(t)F_s}{P(t)}(9.22)$

we get

$\text{d}P(t)={\alpha }_{P}P(t)\text{d}t+{\sigma }_{P}P(t)\text{d}W(t).(9.23)$

Consider a trading strategy in the stock with price process (9.17) and the option with the above price process. According to (9.13), the total wealth process of such a trading strategy is given by

$\text{d}V(t)=V(t)\left([u^1\alpha +u^2{\alpha }_P]\text{d}t+[u^1\sigma +u^2{\sigma }_P]\text{d}W(t)\right)(9.24)$

where u1 and u2 denote the fraction of the total wealth placed in the stock and the option. If u1 and u2 are chosen such that

$u^1\sigma +u^2{\sigma }_P=0(9.25)$

it is readily seen that the stochastic part of the wealth process (9.24) cancels, and the wealth process then becomes deterministic with the drift term (u1α + u2αp). Since we know that u1 + u2 = 1 we can express the relative trading strategy as

$\begin{array}{cc}{u}^1=\frac{S(t)F_s(t,S(t))}{S(t)F_s(t,S(t))-F(t,S(t))},& (9.26)\\ u^2=\frac{F(t,S(t))}{S(t)F_s(t,S(t))-F(t,S(t))}.& (9.27)\end{array}$

Since the stock price as well as the option price is a stochastic process, it is obvious that the relative trading strategy is a stochastic process as well. By choosing the trading strategy above we obtain a riskless return of u1α + u2αp. Since the market consists of a riskless asset with interest rate r the riskless return of the trading strategy above must be the same as the riskless asset; otherwise arbitrage is possible in the market. The argument goes like this: If u1α + u2αp < r you borrow money in the bank (the riskless asset) at the interest rate r, and reinvest the money in the trading strategy (u1, u2) with the riskless return u1α + u2αP. Thus the trading strategy is an arbitrage. If u1α + u2αp < r, a similar argument gives that the trading strategy is an arbitrage. This gives us the following restriction

$u^1\alpha +u^2{\alpha }_P=r.(9.28)$

By inserting u1 and u2 from (9.26) we get

$\frac{1}{2}{\sigma }^{2}S{(t)}^2{F}_{ss}(t,S(t))+S(t)r{F}_s(t,S(t))+F_t(t,S(t))-rF(t,S(t))=0.(9.29)$

We have now derived a parabolic partial differential equation (PDE) which the price process F(t,S(t)) for the call option must fulfil in order to exclude arbitrage possibilities in the market consisting of the riskless asset, the stock and the call option. The boundary condition is

$F(T,S(T))=max(S(T)-K,0)(9.30)$

where T denotes the time of exercise. Notice that the PDE and the boundary condition do not involve the drift parameter α of the stock price. Contrary to most PDEs it is possible to give an analytic solution, which we shall return to later. In the derivation of the PDE above we did not use the fact that the price process F(t, S(t)) was the price process of a call option. We only use it in the boundary condition at time T. This gives rise to the following generalization:

Theorem 9.1 (Arbitrage-free pricing).

Consider the financial market given by (9.14) and (9.17) and a contingent claim with payoff Φ(S(T)) at time T. If the market is free of arbitrage, the price Pt = F (t, S(t)) is given by the solution to the following PDE:

$\frac{1}{2}{\sigma }^{2}S{(t)}^2{F}_{ss}(t,S(t))+S(t)r{F}_s(t,S(t))+F_t(t,S(t))-rF(t,S(t))=0(9.31)$

with the boundary condition

$F(T,S(T))=\Phi (S(T)).(9.32)$

Proof. Follows from the above.

The theorem enables us to price contingent claims X = Φ(S(T)), where the value at the exercise price T is a function Φ(·) of the underlying stock S. The reader will now notice that the boundary value problem stated in Theorem 9.1 is similar to the Cauchy problem considered in Section 8.3 where some Feynman-Kac representation theorems were stated. According to Theorem 8.10 the solution of the boundary value problem is given by

$F(t,S(t))=e^{-r(T-t)}{E}^{\mathbb{Q}}[\text{Φ}(S(T)|S(t)=s)](9.33)$

where S(u) has the following dynamics

$\begin{array}{llll}\text{d}S(u)& =& rS(u)\text{d}t+\sigma S(u)\text{d}\tilde{W}(u)& (9.34)\\ S(t)& =s& \hfill (9.35)\end{array}$

where s denotes the value of the underlying stock at time t and $\tilde{W}(u)$ is a standard Wiener process.

9.2.1 Black-Scholes formula

We will now apply this representation theorem to derive the famous Black-Scholes formula for pricing European call options, where Φ(S(T)) = max(S(T) − K, 0). The solution to the SDE for the stock (9.34) is given by

$S(T)=s{e}^{\left((r-\frac{1}{2}{\sigma }^2)(T-t)+\sigma (\tilde{W}(T)-\tilde{W}(t))\right)}.(9.36)$

Since increments of the standard Wiener process are normally distributed with zero mean and variance T − t, the exponent $(r-\frac{1}{2}{\sigma }^2)(T-t)+\sigma (\tilde{W}(T)-\tilde{W}(t))$ has the distribution $N(r-\frac{1}{2}{\sigma }^2)(T-t),{\sigma }^2(T-t))$. Let ξ ∈ N(0, 1) be a standard Gaussian random variable, then the value of the stock at time t = T is given by

$S(T)=s{e}^{\left((r-\frac{1}{2}{\sigma }^2)(T-t)+\sigma \sqrt{T-t}\xi \right)}.(9.37)$

We now want to find which values of the random variable ξ give the call option a positive value at the time of expiry.

$\begin{array}{lll}& S(T)>K& (9.38)\\ \iff & s{e}^{\left((r-\frac{1}{2}{\sigma }^2)(T-t)+\sigma \sqrt{T-t}\xi )\right)}>K& (9.39)\\ \iff & \left(r-\frac{1}{2}{\sigma }^2\right)(T-t)+\sigma \sqrt{T-t}\xi >ln\left(\frac{K}{s}\right)& (9.40)\\ \iff & \xi >\frac{ln(K/s)-(r-\frac{{\sigma }^2}{2})(T-t)}{\sigma \sqrt{T-t}}=-d.& (9.41)\end{array}$

In this case we say that the call option is in the money. Later on it will become clear why the entity −d is introduced. The price of the call option at time t is given by

$\begin{array}{lll}F(t,S(t))& =& e^{-r(T-t)}{E}^{\mathbb{Q}}[max(S(T)-K,0)]\\ & =& e^{-r(T-t)}{\displaystyle {\int }_{-\infty }^{\infty }}max(S(T)-K,0)\frac{1}{\sqrt{2\pi }}{e}^{-\frac{{\xi }^2}{2}}\text{d}\xi \\ & =& e^{-r(T-t)}{\displaystyle {\int }_{-d}^{\infty }}(S(T)-K)\frac{1}{\sqrt{2\pi }}{e}^{-\frac{{\xi }^2}{2}}\text{d}\xi \\ & =& e^{-r(T-t)}\left[{\displaystyle {\int }_{-d}^{\infty }}(S(T)\frac{1}{\sqrt{2\pi }}{e}^{-\frac{{\xi }^2}{2}}\text{d}\xi -{\displaystyle {\int }_{-d}^{\infty }}K\frac{1}{\sqrt{2\pi }}{e}^{-\frac{{\xi }^2}{2}}\text{d}\xi \right].(9.42)\end{array}$

The last term is given by

$e^{-r(T-t)}{\displaystyle {\int }_{-d}^{\infty }}K\frac{1}{\sqrt{2\pi }}{e}^{-\frac{{\xi }^2}{2}}\text{d}\xi =e^{-r(T-t)}K\Phi (d)(9.43)$

where Φ(·) is the cumulative normal distribution function. By inserting the solution (9.37) into the first term in (9.42) we get

$\begin{array}{ll}{e}^{-r(T-t)}& {\displaystyle {\int }_{-d}^{\infty }}S(T)\frac{1}{\sqrt{2\pi }}{e}^{-\frac{{\xi }^2}{2}}\text{d}\xi \hfill \\ & =e^{-r(T-t)}{\displaystyle {\int }_{-d}^{\infty }}s{e}^{\left((r-\frac{{\sigma }^2}{2})(T-t)+\sigma \sqrt{T-t}\xi \right)}\frac{1}{\sqrt{2\pi }}{e}^{-\frac{{\xi }^2}{2}}\text{d}\xi \hfill \\ & =s{\displaystyle {\int }_{-d}^{\infty }}\frac{1}{\sqrt{2\pi }}{e}^{\left(-\frac{{\sigma }^2}{2})(T-t)+\sigma \sqrt{T-t}\xi -\frac{{\xi }^2}{2}\right)}\text{d}\xi \hfill \\ & =s{\displaystyle {\int }_{-d}^{\infty }}\frac{1}{\sqrt{2\pi }}{e}^{-\frac{1}{2}{\left(\xi -\sigma \sqrt{T-t})\right)}^2}\text{d}\xi \hfill \\ & =s\Phi (d+\sigma \sqrt{T-t}).(9.44)\hfill \end{array}$

From the calculations above we get the following theorem:

Theorem 9.2 (Black-Scholes formula for European call options).

In the Black-Scholes model the price of a European call option with strike price K and time of expiry T is given by

$P_t=F(t,S(t))=s\Phi (d+\sigma \sqrt{T-t})-e^{-r(T-t)}K\Phi (d)(9.45)$

where

$d=\frac{ln(s/K)+\left(r-\frac{{\sigma }^2}{2}\right)(T-t)}{\sigma \sqrt{T-t}}.(9.46)$

From this formula it is seen that the price of a European call option depends on

1. Time to expiry T − 1.
2. The actual value of the underlying stock s.
3. The strike price K.
4. The deterministic interest rate r.
5. The volatility parameter σ. Notice once again that the pricing formula does not depend on the drift parameter α.

In Figure 9.2 the price of the call option is plotted for some specific choice of the parameters. It is readily seen that the price of a call option with strike price K = 100 is increasing in the actual price of the underlying asset, which could be expected because as the actual price of the stock increases so does the probability that the stock price at the time of expiry exceeds the strike price. Notice furthermore that the price as a function of the actual stock price converges towards the payoff function max(S(T) − K, 0) as the time to expiry T − t decreases.

9.2.2 Hedging strategies

In the derivation of the partial differential equation for the Black-Scholes model, we constructed a relative portfolio (u1, u2) in the stock and the call option, which eliminated the stochastic part of the SDE for the price process of the call option. This trading strategy is called Delta-hedging in the financial literature. Hedging means reduction of the sensitivity of a portfolio to the movements of the underlying asset by taking opposite positions in different financial instruments. The delta is defined as $\Delta =\frac{\partial F(t,S(t))}{\partial S(t)}$. The idea behind Delta-hedging is that the writer of a call option can eliminate the risk associated with the call option by taking a position in the underlying asset (the stock). If we look at the absolute portfolios h1, h2 instead of the relative ones we immediately get from (9.26)

$\begin{array}{llll}{h}^1& =& S(t)F_s(t,S(t))=S(t)\Phi (h+\sigma \sqrt{T-t})& (9.47)\\ h^2& =& -F(t,S(t))=(S\Phi (h+\sigma \sqrt{T-t})-e^{-r(T-t)}K\text{Φ}(h))& (9.48)\end{array}$

where F (t, S(t)) denotes the price of the call option. By holding this portfolio the writer of the call option will automatically hold the correct amount (one or zero units) of the stock at expiry. This should be expected since delta-hedging is a risk-free strategy. If the option expires in-the-money (S(T) > K), the required asset has been bought over the lifetime of the option, firstly in setting the initial hedge and secondly in a series of transactions as S(t) change. Conversely, if the option expires out-of-the-money, the initial hedge is gradually sold. Since the delta-hedging is only instantaneously risk-free, the portfolio must be rebalanced continuously according to the movements of the underlying stock, which make this strategy inefficient from a practical point of view due to transaction costs in the underlying stock.

In delta-hedging the random component, the stock price, is eliminated. However one can be more subtle and hedge against the dependency of some of the parameters in the pricing formula. The following list is usually considered.

$\begin{array}{llll}\Delta & =& \frac{\partial F}{\partial S}=\text{Φ}(d+\sigma \sqrt{T-t}),& (9.49)\\ \Gamma & =& \frac{{\partial }^{2}F}{\partial S^2}=\frac{1}{\sigma S\sqrt{T-t}}{\text{Φ}}^{\prime }(d+\sigma \sqrt{T-t}),& (9.50)\\ \Theta & =& \frac{\partial F}{\partial t}=\frac{S{\text{Φ}}^{\prime }(d+\sigma \sqrt{T-t})\sigma }{2\sqrt{T-t}}-rK{e}^{-r(T-t)}\text{Φ}(d),& (9.51)\\ \rho & =& \frac{\partial F}{\partial r}=K(T-t)e^{-r(T-t)}\text{Φ}(d),& (9.52)\\ \nu & =& \frac{\partial F}{\partial \sigma }=S{\text{Φ}}^{\prime }(d+\sigma \sqrt{T-t})\sqrt{T-t}.& (9.53)\end{array}$

These are for obvious reasons called the Greeks. Hedging against any of these dependencies requires the use of other options as well as the call option and the stock. It is beyond the scope of this book to go any further into the subject of hedging.

9.2.2.1 Quadratic hedging

Hedging based on the Greeks (e.g., Δ or Δ − Γ strategies) may be accurate enough as long as then rebalancing is frequent and the asset paths are continuous. An alternative that is more appropriate for advanced models in general and jump processes in particular is to minimize the quadratic hedge error

$\{h_1,h_2\}=argminE\left[{(V(t+\Delta t)-h_{1}B(t+\Delta t)+h_{2}S(t+\Delta t))}^2|ℱ(t)\right](9.54)$

where we are deliberately vague regarding the probability measure; see Cont and Tankov [2004] for a discussion on using the ℙ measure or the ℚ measure.

It turns out that quadratic hedging and adaptive calibration can be done simultaneously, as shown by Lindström and Guo [2013], Wiktorsson and Lindström [2014]. This makes it computationally comparable to using Greeks, while in general providing better hedge strategies (especially when there are jumps in the asset dynamics).

9.3 Modern approach using martingale measures

In the following section we shall use some of the advanced techniques presented in Chapter 7 in order to derive a general pricing formula for contingent claims in the Black-Scholes model. In the derivation of the partial differential equation of Theorem 9.1, it was assumed that the price P(t) = F(t, S(t)) only depended on the time and the actual value of the underlying asset. By using a modern approach this assumption is not needed, which from a mathematical point of view is more satisfactory. However the primary object of deriving the pricing formula for contingent claims once again is that it demonstrates how the Girsanov measure transformation may be applied to a concrete financial model. Hopefully the reader will get a better understanding of the basic idea of measure transformations, which will be used in a subsequent chapter concerning the term structure of interest and bond pricing.

In the following the results are stated for the simple Black-Scholes model, although they are valid for a larger class of financial models. To begin with, the Black-Scholes model is restated. On the filtered probability space (Ω, ℱ, ℙ, {ℱ(t)}t≥0), we have the following two assets

$\begin{array}{llll}\text{d}B(t)& =& rB(t)\text{d}t,& (9.55)\\ \text{d}S(t)& =& \alpha S(t)\text{d}t+\sigma S(t)+\sigma S(t)\text{d}\tilde{W}(t),& (9.56)\end{array}$

where $\tilde{W}(t)$ is a Wiener process with respect to the probability measure ℙ, and B denotes a risk-free asset and S denotes a risky asset. As in discrete time we shall need the concept of martingale measures.

Definition 9.1 (Martingale measure).

We say that a probability measure ℚ is a martingale measure if

1. ℚ ~ ℙ i.e., the two measures are equivalent.
2. The discounted price process Z(t) = S(t)/B(t) is a martingale under the measure ℚ.

The set of martingale measures is denoted by ?. The denominator B(t) is called the numeraire.

In the Black—Scholes model it is easy to find the class of equivalent martingale measures. The stochastic differential equation of the discounted price process Z(t) = φ(S(t), B(t)) = S(t)/B(t) is found by applying the multivariate Itō formula stated in Theorem 8.5. The following derivatives are needed

$\begin{array}{l}\frac{\partial \phi }{\partial t}=0,\frac{\partial \phi }{\partial S(t)}=\frac{1}{B(t)},\frac{{\partial }^2\phi }{\partial S{(t)}^2}=0\\ \frac{\partial \phi }{\partial B(t)}=-\frac{S(t)}{B{(t)}^2},\frac{{\partial }^2\phi }{\partial B{(t)}^2}=-2\frac{S(t)}{B{(t)}^3},\\ \frac{{\partial }^2\phi }{\partial S(t)\partial B(t)}=-\frac{1}{B{(t)}^2}\end{array}$

which gives

$\begin{array}{lll}\text{d}Z(t)& =& \left[-\frac{S(t)}{B{(t)}^2}rB(t)+\frac{1}{B(t)}\alpha S(t)\right]\text{d}t+\frac{1}{B(t)}\sigma S(t)\text{d}\tilde{W}(t)\\ & =& (\alpha -r)Z(t)\text{d}t+\sigma Z(t)\text{d}\tilde{W}(t).(9.57)\end{array}$

Thus the discounted price process is again a geometric Brownian motion, where the drift is (α − r). In order to find the class of equivalent martingale measures we perform an absolute continuous measure transformation by applying the Girsanov theorem (8.15). Define a new measure ℚ where

$\begin{array}{llll}\text{d}L(t)& =& g(t)L(t)\text{d}W(t),& (9.58)\\ L(0)& =& 1.& (9.59)\end{array}$

According to the Girsanov theorem we have

$\text{d}\tilde{W}(t)=g(t)\text{d}t+\text{d}W(t)(9.60)$

where W(t) is a Wiener process with respect to the probability measure ℚ. By substituting $\text{d}\tilde{W}(t)$ into (9.57) we get the following dynamics of the Z-process under the ℚ-measure

$\text{dZ}(t)=(\alpha -r+\sigma g(t))Z(t)\text{d}t+\sigma Z(t)\text{d}W(t).(9.61)$

By choosing the Girsanov kernel g(t) to be

$g(t)=\frac{r-\alpha }{\sigma }(9.62)$

the drift term in the Z-process is removed; hence the process is a martingale under the measure ℚ. Since ℙ and ℚ are equivalent measures the ℚ measure is a martingale measure according to Definition 9.1.

Definition 9.2 (Wealth process).

For a given portfolio strategy h we define the wealth processes V(t, h) and VZ(t, h) as

$\begin{array}{cccc}V(t,h)& =& h^1(t)B(t)+h^2(t)S(t),& (9.63)\\ V^Z(t,h)& =& \frac{V(t,h)}{B(t)}=h^1(t)+h^2(t)Z(t).& (9.64)\end{array}$

Definition 9.3 (Self-financing portfolio strategy).

A portfolio strategy h is selffinancing if

$V(t,h)=V(0,h)+{\displaystyle {\int }_0^t}{h}^1(u)\text{d}B(u)+{\displaystyle {\int }_0^t}{h}^2(u)\text{d}S(u).(9.65)$

The self-financing condition is intended to formalize the intuitive idea of a trading strategy with no exogenous infusion or withdrawals of money, i.e., a strategy where the purchase of a new asset is financed solely by the sale of assets already in the portfolio.

Lemma 9.1 (Self-financing portfolio strategy).

1. A portfolio strategy h is self-financing if and only if

   $\text{d}{V}^Z(t,h)=h^2(t)\text{dZ}(\text{t})\text{.}(9\text{.66})$

2. If h is a self-financing portfolio, the wealth process VZ(t, h) is a ℚ-martingale.

Proof. Let h be a self-financing portfolio, then we want to prove (9.66). Since VZ(t, h) = e−rtV(t, h), we can apply the Itō formula and get

$\begin{array}{lll}d{V}^Z(t,h)& =& d[e^{-rt}V(t,h)]=-r{e}^{-rt}V(t,h)\text{d}t+e^{-rt}\text{d}V(t,h)\\ & =& -r{e}^{-rt}(h^1(t)B(t)+h^2(t)S(t))\text{d}t\\ & +& e^{-rt}(h^1(t)\text{d}B(t)+h^2(t)\text{d}S(t))\\ & =& (\alpha -r)h^2(t)e^{-rt}S(t)\text{d}t+h^2(t)\sigma e^{-rt}S(t)\text{d}W(t)=h^2(t)\text{d}Z(t).\end{array}$

The opposite implication is proved in a similar way.

By integration of (9.66) we get

$\begin{array}{lll}{V}^Z(t,h)& =& V^Z(0,h)+{\displaystyle {\int }_0^t}{h}^2(u)\text{d}Z(u)\\ & =& V^Z(0,h)+{\displaystyle {\int }_0^t}{h}^2(u)\sigma Z(u)\text{d}\tilde{W}(u).(9.67)\end{array}$

By taking conditional expectations under the measure ℚ on each side the stochastic integral cancels and we get

$E^{\mathbb{Q}}[V^Z(t,h)|ℱ(t)]=V^Z(0,h)(9.68)$

which shows that VZ (t, h) is a ℚ-martingale.

Definition 9.4 (Contingent claims).

A contingent claim with expiry date T is a ℱ(T)-measurable random variable.2

The set of contingent claims is called ?. Now define a subset ? consisting of contingent claims X ∈ ? such that

$\mathbb{P}(X\ge 0)=1,\text{and}\mathbb{P}(X>0)>0.(9.69)$

These two conditions ensure that X is non-negative and not all the probability mass is assigned to the event X = 0.

Definition 9.5 (Arbitrage strategies).

A trading strategy h ∈ ℋ is an arbitrage strategy if

$V_0(h)=0,and{V}_T(h)\in {?}^{+}.(9.70)$

The definition says that a trading strategy is an arbitrage strategy if the value of the portfolio initially is zero, and the value at time T is non-negative and with a positive probability greater than zero. We recognize that the definition is similar to the one stated in discrete time financial models (3.1). A model without any arbitrage strategies is free of arbitrage.

Theorem 9.3.

Assume that there exists a martingale measure ℚ. Then the model is free of arbitrage in the sense that there exist no arbitrage portfolios.

Proof. Assume that h is an arbitrage portfolio with ℙ(V(T, h) ≥ 0) = 1 and ℙ(V(T, h) > 0) > 0. Then since ℚ ~ ℙ we also have ℚ(VZ(T, h) ≥ 0) = 1 and ℚ(VZ(T, h) > 0) > 0 and consequently

$V(0,h)=V^Z(0,h)=E^{\mathbb{Q}}[V^Z(T,h)]>0(9.71)$

which contradicts the arbitrage condition V (0) = 0.

Since we have found an equivalent martingale measure in the Black-Scholes model, it immediately follows that it is free of arbitrage.

We have seen that the existence of a martingale measure implies the absence of arbitrage, and a natural question is whether there is a converse to this statement, i.e., if the absence of arbitrage implies the existence of a martingale measure. For models in discrete time with a finite sample space, we have seen that this is indeed the case. In continuous time there is not complete equivalence between the existence of a martingale measure and absence of arbitrage. Although it is somewhat unsatisfactory, the general consensus seems to be that the existence of a martingale measure is (informally) considered to be more or less equivalent to the absence of arbitrage.

Definition 9.6 (Complete markets).

1. A contingent claim X with expiry date T is said to be attainable if there exists a self-financing portfolio h, such that the corresponding wealth process has the property that

   $V(T,h)=X,\mathbb{P}-a.s.(9.72)$

2. The market is said to be complete if every claim is attainable.

Theorem 9.4.

If the martingale measure Q is unique, then the market is complete in the sense that every claim X satisfying

$\frac{X}{S_0(T)}\in L^1(\Omega ,ℱ(T),\mathbb{Q})(9.73)$

is attainable, with L1(Ω, ℱT, ℚ) being the class of stochastic variables with finite expected values under the measure ℚ. The class L1(Ω, ℱ(T), ℚ) is defined explicitly in Definition 8.3.

Proof. Omitted. See e.g. Duffie [2010].

Remark 9.2.

In the Black-Scholes model we found a unique martingale measure by Girsanov transformation of the objective probability measure ℙ; thus the model is complete.

9.4 Pricing

We now turn to the problem of determining a “reasonable” price process Π(t, X) for a contingent claim with a fixed date of expiry T. Assume that the market is free of arbitrage and the market is complete, i.e., a martingale measure exists and it is unique for the market consisting of the money market account B and the stock S. Then it seems reasonable to demand that the price process Π(·,X) should be chosen such that the extended market [B, S, Π(·,X)] is free of arbitrage possibilities. This can be obtained by requiring that the discounted price process Π(t,X)/B(t) is a martingale under ℚ, where ℚ is the martingale measure for the market [B, S]. Thus we have

$\frac{\Pi (t,X(t))}{B(t)}=E^{\mathbb{Q}}\left[\frac{\Pi (T,X(T))}{B(T)}|ℱ(t)\right]=E^{\mathbb{Q}}\left[\frac{X(T)}{B(T)}|ℱ(t)\right](9.74)$

and since B(T) is deterministic we get the following pricing formula

$\text{Π}(t,X(t))=B(t)E^{\mathbb{Q}}\left[\frac{X(T)}{B(T)}|ℱ(t)\right]=e^{-r(T-t)}{E}^{\mathbb{Q}}\left[X(T)|ℱ(t)\right].(9.75)$

Notice that this is the same pricing formula as (9.33) where the classical approach to arbitrage-free pricing was taken.

Example 9.1 (Binary option).

A so-called binary option (or digital option) is a claim which pays a fixed amount if the stock at certain dates lies within some prespecified interval. Otherwise nothing will be paid out. Consider a binary option which pays K $ to the holder at date T if the stock price at time T lies in the interval [α,β], i.e., the contingent claim X = 1[αβ](S(T))K where 1[·,·] is the indicator function. The arbitrage-free price of the option is determined by the pricing formula above. We get

$\text{Π}(t,X)={exp}^{-r(T-t)}{E}^{\mathbb{Q}}\left[1_{[\alpha ,\beta ]}S(T)K|ℱ(t)\right].(9.76)$

Since the solution S(T) under the equivalent martingale measure ℚ is given by

$S(T)=S(t)exp\left((r-\frac{1}{2}{\sigma }^2)(T-t)+\sigma (W(T)-W(t))\right)(9.77)$

we notice that the exponent $(r-\frac{1}{2})(T-t)+\sigma (W(T)-W(t))$ is normally distributed with mean $m_z(r-\frac{1}{2}{\sigma }^2)(T-t)$ and variance ${\sigma }_z^2={\sigma }^2(T-t)$

$S(T)=S(t)exp(m_z+{\sigma }_{z}z).(9.78)$

It is easily found that the option pays K $ if mz + σzz ∈ [log (α/S(t)), log(β/S(t))]. Thus we have

$\begin{array}{lll}\Pi (t,X)& =& e^{-r(T-t)}{\displaystyle {\int }_0^{\infty }{1}_{[\alpha ,\beta ]}(S(T))K\text{d}\mathbb{Q}(S(T))}\\ & =& e^{-r(T-t)}K{\displaystyle {\int }_{log\left(\frac{\alpha }{S(t)}\right)}^{log\left(\frac{\beta }{S(t)}\right)}\text{d}\mathbb{Q}(S(T))}\\ & =& e^{-r(T-t)}K{\displaystyle {\int }_{log\left(\frac{\alpha }{S(t)}\right)}^{log\left(\frac{\beta }{S(t)}\right)}}\frac{1}{\sqrt{2\pi {\sigma }_z}}{e}^{-\frac{(z-m_z)}{2{\sigma }_z}\text{d}z}\\ & =& e^{-r(T-t)}K\left[\text{Φ}\left(\frac{log\left(\frac{\beta }{S(t)}\right)-m_z}{{\sigma }_z}\right)-\text{Φ}\left(\frac{log\left(\frac{\alpha }{S(t)}\right)-m_z}{{\sigma }_z}\right)\right],(9.79)\end{array}$

where Φ(·) is the cumulative normal distribution function.

The standard Black & Scholes framework reduces the statistical problem to a single parameter, the volatility parameter σ. This can be estimated from either historical price data (the ℙ measure) or from options (which in general carries information about the ℚ measure) as the parameter coincides under both measures. Calibration in a more general context is discussed in Section 14.11.1.

It turns out, however, that the model fit is not perfect (it is still very good compared to most models in social science!), something that should be reasonable as few of the stylized facts reviewed in Chapter 1 were incorporated into the model. Model extensions (or rather classes of models) are briefly introduced in Section 9.5, but it is worth remembering that a good model is both able to capture the relevant empirical properties and is computationally convenient to work with (for pricing, hedging and calibration purposes). Efficient computational techniques are discussed in Section 9.6.

9.5 Model extensions

The risk-neutral framework is very general and applies for all extensions below. Modern pricing theory has developed in several overlapping directions; cf. Section 7.5.

• Stochastic volatility (Hull and White [1987], Heston [1993]). Adding an extra process means that the market becomes incomplete, but the completeness of the market can be recovered by introducing a liquid vanilla option. Some models, like the SABR model, even admit closed form expressions for the price (Hagana et al. [2002], Larsson [2012]).
• Jump models (Merton [1976], Kou [2002]) introduce random jumps to capture the small, but non-zero risk of large price movements that are hard to hedge. Cont and Tankov [2004] is a good starting point for further studies. Mathematics needed for jump processes was developed in Section 7.5.
• Local volatility models define the volatility as a complex function of time and the underlying asset (Derman and Kani [1994]) so that the market is complete, and still provides perfect fit to the market. The approach is similar to fitting splines the observations, as the fit to data (in sample) is perfect but there is a real risk for overfitting the data (Orosi [2010]). The overfitting may lead to models that provides good fit in sample, but also introduces arbitrage. Lindholm [2014] introduces a regularizaton through optimal control that forces the recovered local volatilty model to be free from arbitrage.
• Uncertain volatility models (Cont [2006], Lindström [2010], Lindström and Strålfors [2012], Lindstrom and Wu [2014]) tries to account to the imperfect-facing investors. Small, but consistent, improvements are found correcting for uncertainty.

Some models, like the class of affine jump diffusions (Duffie et al. [2003]), include both stochastic volatility and jumps, but still allow for computationally efficient methods for pricing (Fourier methods, see Carr and Madan [1999], Lindström et al. [2008], Hirsa [2013]).

9.6 Computational methods

There is a need to be able to compute prices, at least numerically, if we are going to use option prices in a statistical model. There are roughly speaking four main techniques used today

• Trees, typically binomial or trinomial tree (Cox et al. [1979]). Trees were initially introduced as an easy alternative to the continuous-time derivation of the Black & Scholes model, but have also found useful applications when considering path dependent options, or local volatility models.
• Monte Carlo methods are very general, and therefore well suited for computing expectations as convergence is ensured by the law of large numbers. The method is very easy to implement for European type contracts, but can also be used for path dependent contracts (least squares Monte Carlo) (see Longstaff and Schwartz [2001]). Monte Carlo methods are often the only feasible method for high-dimensional problems.
• PDEs or PIDEs (for jump processes) are the equivalent formulation due to the Feynman-Kac representation. Numerical methods for PDEs/PIDEs, such as finite difference or finite element methods, are generally more efficient than Monte Carlo or binomial/trinomial trees for low-dimensional problems (one or two dimensions), but they also work well for some multiasset options (Lötstedt et al. [2007]). The methodology can also handle path-dependent problems well; see Hirsa [2013] for many schemes.
• Fourier methods that use the characteristic function of the log-price process to obtain nearly analytical expressions. These approximations are magnitudes more accurate and faster to compute than any of the others listed here.

9.6.1 Fourier methods

The title of the section may be misleading as there are several similar methods, using different transform methods, including several standard (Lindström et al. [2008]), and Fast Fourier methods (Carr and Madan [1999]), fractional Fourier methods (Hirsa [2013]) and cosine methods (Fang and Oosterlee [2008]); see Hirsa [2013] for a general overview.

However, we will review some of the basic steps of the Fourier method below. Assume that the characteristic function (cf. Section 7.5) of the log price s(T) = logS(T) is known

$\psi (u)=E\left[e^{ius(T)}\right]={\displaystyle \int s^{ius}\text{d}\mathbb{Q}(\text{s})}.(9.80)$

The risk-neutral measure is absolutely continuous with respect to the Lebesgue measure in virtually every model we consider in this book, and we will therefore assume that we can use the density instead, dℚ(s) = q(s)ds.

It is known from Section 9.4 that the price of a European call option is given by

$C(k)={\displaystyle {\int }_k^{\infty }{e}^{-rT}(e^s-e^k)q(s)\text{d}s}(9.81)$

where k = log (K) is the log strike price. The option price is not a square integrable (which is required by the Parseval's theorem), but a modified version of the price is

$c(k)=e^{\alpha k}C(k)(9.82)$

where α is some positive number. It is then possible to compute the Fourier transform of modified price c(k) as

$\varphi (v)={\displaystyle \int e^{ivk}c(k)\text{d}k.}(9.83)$

This expression can be extend further accordingly

$\begin{array}{llll}\phi (v)& =& {\displaystyle \int e^{ivk}c(k)\text{d}k}& (9.84)\\ & =& {\displaystyle \int e^{ivk}{e}^{ak}}{\displaystyle {\int }_k^{\infty }{e}^{-rT}(e^s-e^k)q(s)\text{d}s\text{d}k}& (9.85)\\ & =& \frac{e^{-rT}\psi (v-(\alpha +1)i)}{{\alpha }^2+\alpha -v^2+i(2\alpha +1)v}.& (9.86)\end{array}$

It is also possible to compute the option price by applying the inverse Fourier transform to (9.83)

$C(k)=\frac{e^{-\alpha k}}{2\pi }{\displaystyle \int }{e}^{-ivk}\varphi (v)dv=\frac{e^{-\alpha k}}{\pi }{\displaystyle {\int }_0^{\infty }{e}^{-ivk}\varphi (v)dv.}(9.87)$

where the second equality holds as C(k) is real. Hence, call prices are given by inserting Equation (9.86) into (9.87) arriving at

$C(k)=\frac{e^{-\alpha k}}{\pi }{\displaystyle {\int }_0^{\infty }}{e}^{-ivk}\frac{e^{-rT}\psi (v-(\alpha +1)i)}{{\alpha }^2+\alpha +v^2+i(2\alpha +1)v}\text{d}v.(9.88)$

The integral can be computed using either Fast Fourier Transform (FFT) or related fast transforms (Hirsa [2013]) or Gauss-Laguerre quadrature methods (Lindström et al. [2008]) as both types of methods provide very accurate approximations with very limited computational efforts.

The modification, here parametrized by α, is needed due to Parseval, but it can also be seen that choosing α = 0 would introduce a singularity in (9.88). It can also be shown that the α parameter can dampen numerical oscillations in the integrand, leading to better numerical approximations (Lee [2004], Lindström et al. [2008]) for further details.

9.7 Problems

Problem 9.1

Consider a standard Black-Scholes model with the usual dynamics

$\begin{array}{llll}\text{d}S(t)& =& \alpha S(t)\text{d}t+\sigma S(t)\text{d}W(t),& (9.89)\\ \text{d}B(t)& =& rB(t)\text{d}t& (9.90)\end{array}$

and T2 with 0 < T1 < T2, and consider the contingent claim X = S(T2)/S(T1). The claim is to be paid out at T2.

1. 1. Compute the arbitrage-free price.
2. 2. Try to construct the replicating portfolio.

Problem 9.2

Consider a standard Black-Scholes model with the usual dynamics

$\begin{array}{llll}\text{d}S(t)& =& \alpha S(t)\text{d}t+\sigma S(t)\text{d}W(t),& (9.91)\\ \text{d}B(t)& =& rB(t)\text{d}t.& (9.92)\end{array}$

1. 1. Determine the arbitrage-free price of a put option X = max[K − S(T), 0], where K is the strike price and T is the time of expiry.
2. 2. Express the value of a put option in terms of a call option, the underlying stock and the money market account.

Remark. This formula is called the put-call parity.

Problem 9.3

Consider the standard Black-Scholes model. Fix the time of maturity T and consider a so-called butterfly defined by

$\begin{array}{cc}X=\{\begin{array}{cc}K-S(T)& \text{if}0<S(T)\le K,\\ S(T)-K& \text{if}K<S(T).\end{array}& (9.93)\end{array}$

This contract can be replicated using a portfolio, consisting solely of bonds, stock and European call options, which are constant over time.

1. 1. Determine this portfolio.
2. 2. Determine the arbitrage-free price of the contract.

1It may be argued that it is misleading to model prices in continuous time as stock exchanges are closed at night and during the weekends and that there is only a finite number of trades each day. However, as the financial markets are becoming more internationalized and computerized, there always exists an open stock exchange somewhere around the globe where the trading can be effected.

2This means that the value of the contingent claim, e.g., an option, is known at time T.