Option Pricing

A cash settled European option on underlying F pays \nu(F) at expiration for some function \nu\colon 𝑹\to 𝑹. Its forward value is E[\nu(F)].

If the option pays off in shares its forward value is E[F\nu(F)] =fE[(F/f) \nu(F)] = fE^*[\nu(F)] where f = E[F] and the corresponding probablility measures have Radon-Nikodym derivative dP^*/dP = F/f. If F > 0 then P^* is a probability measure since E[F/f] = 1. We call P^* the share measure.

A put with strike k has payoff \nu(F) = \max\{k - F, 0\} = (k - F)1(F\le k) where 1(F\le k) = 1 if F\le k and 1(F\le k) = 0 if F > k. Its forward value is \begin{aligned} p &= E[(k - F)^+] \\ &= E[(k - F) 1(F\le k)] \\ &= kP(F\le k) - fE[F/f 1(F\le k)] \\ &= kP(F\le k) - fP^*(F\le k) \\ \end{aligned}

Exercise. Show p = (k - f)P(F\le k) + f(P(F\le k) - P^*(F\le k))_.

Hint: -P(F\le k) + P(F\le k) = 0

Exercise. Show p = (k - f)P(F\le k) + \operatorname{Cov}(F,1(F\le k))_.

Hint: \operatorname{Cov}(a + X, Y) = \operatorname{Cov}(X,Y) = E[XY] - E[X]E[Y] if $a is constant.

Note \operatorname{Cov}(X, f(X)) < 0 if f is decreasing.

Suppose the underlying F can only take on values k_j with risk-neutral probabilities P(F = k_j) = q_j and put prices are available p_j = E[(k_j - F)1(F\le k_j)] = \sum_{i\le j} (k_j - k_i)q_i.

We have \begin{aligned} p_{j+1} - p_j &= \sum_{i\le j+1} (k_{j+1} - k_i)q_i - \sum_{i\le j} (k_j - k_i)q_i \\ &= 0 + \sum_{i\le j} (k_{j+1} - k_i)q_i - (k_j - k_i)q_i \\ &= \sum_{i\le j} (k_{j+1} - k_j)q_i \\ &= (k_{j+1} - k_j) \sum_{i\le j} q_i \\ \end{aligned} so p'_j = \Delta p_j/\Delta k_j = \sum_{i\le j} q_i and p'_j - p'_{j-1} = q_j.

Suppose the underlying F can only take on values k_j with risk-neutral probabilities P(F = k_j) = q_j and call prices are available x_j = E[(F - k_j1(F\ge k_j)] = \sum_{i\le j} (k_j - k_i)q_i.

We have \begin{aligned} p_{j+1} - p_j &= \sum_{i\le j+1} (k_{j+1} - k_i)q_i - \sum_{i\le j} (k_j - k_i)q_i \\ &= 0 + \sum_{i\le j} (k_{j+1} - k_i)q_i - (k_j - k_i)q_i \\ &= \sum_{i\le j} (k_{j+1} - k_j)q_i \\ &= (k_{j+1} - k_j) \sum_{i\le j} q_i \\ \end{aligned} so p'_j = \Delta p_j/\Delta k_j = \sum_{i\le j} q_i and p'_j - p'_{j-1} = q_j.