Lecture 7 Overture to continuous models Before rigorously deriving the acclaimed Black-Scholes pricing formula for the value of a European option, we developed a substantial body of material, in continuous time. Then, we developed the risk-neutral asset pricing theory in discrete time. We can use our discrete techniques to see what form our results must take in the continuous world. It is easy to believe that we should be able to use a discrete model with very small time periods to approximate a continuous model. The Black-Scholes model is based in the lognormal model (geometric Brownian motion). With this in mind, we choose our approximation to have constant growth rate and constant noise. The model is parametrized by the time period, δt, and three fixed constant parameters, v, σ and the riskless rate r. The cash bond has the form B t = e rt, which does not depend on the interval size. The stock price process follows the nodes of a binomial tree. If the current value of the stock is s, then over the next time period it moves to the new value { s exp(vδt + σ δt), if up, s exp(vδt σ δt), if down, Suppose our belief is that the jumps are equally likely to be up or down. So under the market measure, P [up jump] = /2 = P [down jump] at each time step. For a fixed time t, set N to be the number of time periods until time t, that is N = t/δt. Then S t = S exp(vt + σ t( 2X N N N )) where X N is the total number of the N separate jumps which were up jumps. To see what happens as δt (or equivalently N ) we call on the Central limit Theorem. Theorem (Central Limit Theorem) Let ξ, ξ 2,... be a sequence of independent identically distributed random variables under the probability measure P with finite mean µ and finite non-zero variance σ 2 and let S n = ξ +... + ξ n. Then S n nµ nσ 2 converges in distribution to an N(, ) random variable as n. Now X N is the sum of N independent random variables {ξ i } i N taking the value + with probability /2 and otherwise. This means E[ξ i ] = /2 and var[ξ i ] = /4 so that by the Central Limit Theorem, the distribution of the random variable (2X N N)/ N converges to that of a
normal random variable with mean zero and variance one. In other words, as δt gets smaller (and so N gets larger), the distribution of S t converges to that of a lognormal distribution. More precisely, in the limit, log S t is normally distributed with mean log S + vt and variance σ 2 t. This is what happens under the original measure P. What happens under the martingale measure, Q, that we use for pricing? Under the martingale measure, the probability of an up jump is q = exp(rδt) exp(vδt σ δt) exp(vδt + σ δt) exp(vδt σ δt) which is approximately 2 ( δt( v + 2 σ2 r )). σ So under the martingale measure, Q, X N is still binomially distributed, but now has mean Nq and variance Nq( q). Thus, under Q, (2X N N)/ N has mean that tends to t(v + 2 σ2 r)/σ and variance that approaches one as δt tends to zero. Again using the Central Limit Theorem the random variable (2X N N)/ N converges to a normally distributed random variable, with mean t(v+ 2 σ2 r)/σ and variance one. Under Q then, S t is lognormally distributed with mean log S + (r 2 σ2 )t and variance σ 2 t. This can be written S t = exp(σ tz + (r 2 σ2 )t). where, under Q, the random variable Z is normally distributed with mean zero and variance one. If our discrete theory carries over to the continuous limit, then in our continuous model the price at time zero of a European call option with strike price K at time T will be the discounted expected value of the claim under the martingale measure, that is E Q [e rt (S T K) + ] where r is the riskless rate. Substituting, we obtain E Q [(S exp(σ T Z 2 σ2 T ) Kexp( rt )) + ]. We ll derive this pricing formula rigorously later and we will also show that this equation can be evaluated as S Φ( log S K + (r + 2 σ2 )T σ ) Ke rt Φ( log S K + (r 2 σ2 )T T σ ) T where Φ is the standard normal distribution function, Φ(z) = Q[Z z] = 2 z 2π e x2 /2 dx
The Girsanov Theorem In order to price and hedge in the Black-Scholes framework we shall need two fundamental results. The first will allow us to change probability measure so that the discounted asset prices are martingales. Recall that in our discrete time world, once we had such a martingale measure, the pricing of options was reduced to calculating expectations under that measure. In the continuous world it will no longer be possible to find the martingale measure by linear algebra. Nonetheless, before stating the continuous time result, we revert to our binomial trees for guidance. Suppose that, under the probability measure P, if the value of an asset at time iδt is known to be S i then its value at time (i + )δt is S i u with probability p and it is S i d with probability p. As we saw before, if we let Q be the probability measure under which the probability of an up jump is q = ( d)/(u d) and of a down jump is (u )(u d), then the process {S i } i N is a Q martingale. We can regard the measure Q as a reweighting of the measure P. For example, consider a path S, S,..., S i through the tree. Its probability under P is p N(i) ( p) i N(i), where N(i) is the number of up jumps that the path makes. Under Q its probability is L i p N(i) ( p) i N(i) where L i = ( q p )N(i) ( q p )i N(i). Evidently L i depends on the path that the stochastic process takes through the tree and can itself be thought of as a stochastic process adapted to the filtration {F i } i N. Moreover, L i /L i is q/p if S i /S i = u and is ( q)/( p) if S i /S i = d, so that E P [L i F i ] = L i (p q p + ( p) q p ) = L i. In other words, {L i } i N is a (P, {F i } i N ) martingale with E[L i ] = L =. If we wish to calculate the expected value of a claim in the Q-measure, it is given by E Q [C] = E P [L N C]. Notation: We have obtained the Radon-Nikodym derivative of Q with respect to P. It is customary to write L i = dq dp F i. We have shown that the process of changing to the martingale measure can be viewed as a reweighting of the probabilities of paths under our original measure P according to a positive, mean one, P-martingale. This procedure of reweighting according to a positive martingale can be extended to the continuous setting. Our aim now is to investigate the effect of such a reweighting on the distribution of the P-Brownian motion. Later this will enable us to choose the right reweighting so that under the new measure obtained in this way the discounted stock price is a martingale. 3
Theorem(Girsanov s Theorem) Suppose that {W t } t is a P Brownian motion with the natural filtration {F t } t and that {θ t } t is an {F t } t -adapted process such that. E[exp( 2 T θ 2 t dt)] < Define L t = exp( θ s dw s 2 and let P (L) be the probability measure defined by P (L) [A] = L t (ω)p(dω). A θ 2 sds) Then under the probability measure P (L), the process {W (L) t } t T, defined by is a standard Brownian motion. W (L) t = W t + θ s ds, Notation: We write dp (L) dp F i = L t. (L t is the Radon-Nikodym derivative of P (L) with respect to P.) Remark. The condition E[exp( 2 T θ 2 t dt)] <, known as Novikov s condition, is enough to guarantee that {L t } t is a (P, {F t } t )-martingale. Since L t is clearly positive and has expectation one, P (L) really does define a probability measure. 2. Just as in the discrete world, two probability measures are equivalent if they have the same sets of probability zero. Evidently P and P (L) are equivalent. 3. If we wish to calculate an expectation with respect to P (L) we have E P(L) [φ t ] = E[φ t L t ]. Moreover generally, E P(L) [φ t F s ] = E P [φ t L t L s F s ]. 4
This will e fundamental in option pricing. Example Let {X t } t be the drifting Brownian motion process X t = σw t + µt, where {W t } t is a P-Brownian motion and σ and µ are constants. Find a measure under which {X t } t is a martingale. 5