14. Option Pricing

Notation. \(S_t\) underlying stock price; \(K\) strike (exercise) price; \(r\) continuous risk-free discount rate; \(\delta\) continuous dividend rate; \(\mu\) stock-price drift; \(\sigma\) volatility; \(T\) exercise date.

Assumption. The stock price \(S_t\) follows a geometric Brownian motion (14.1):

where \(\{B_t\}\) is a standard Brownian motion under the real measure \(\mathbf P\).

Result. With dividend rate \(\delta\), the European call option with time-\(T\) payoff \(\max\{S_T-K,0\}\) has time-\(t\) price \(C_t(S_t)\) given by (14.2):

Frequently used Greeks. Here \(N(\cdot)\) is the standard normal c.d.f., \(N(d_1)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{d_1}e^{-\frac12 z^2}dz\).

• Delta \(\Delta\equiv\dfrac{\partial C_t}{\partial S_t}\), and \(\Delta=e^{-\delta(T-t)}N(d_1)\) is an exact result (not an approximation); proof below.
• Gamma \(\Gamma\equiv\dfrac{\partial^2 C_t}{(\partial S_t)^2}\), also called convexity.
• Theta \(\theta\equiv\dfrac{\partial C_t}{\partial t}\).
• Vega \(v\equiv\dfrac{\partial C_t}{\partial\sigma}\).
• Rho \(\rho\equiv\dfrac{\partial C_t}{\partial r}\).

The shape of the option price \(C_t(S_t)\) in \(S_t\) is shown in Figure 14.1.

Figure 14.1: European call option price. The curve \(C_t(S_t)\) is convex; as \(S_t\to\infty\) it approaches an asymptote with slope \(e^{-\delta(T-t)}\), because \(\lim_{S_t\to\infty}\Delta=e^{-\delta(T-t)}\underbrace{N(d_1)}_{\to 1}\). The asymptote crosses the horizontal axis at the point \(A\) found by setting \(N(d_1)=N(d_2)=1\) in (14.2) and \(C_t(S_t)=0\), i.e. \(S_t=Ke^{-(r-\delta)(T-t)}\).

For the asymptote intercept \(A\): \(e^{-\delta(T-t)}S_t-Ke^{-r(T-t)}=0\ \Rightarrow\ S_t=Ke^{-(r-\delta)(T-t)}\), using the Leibniz integral rule (14.5):

All three routes yield the same Black-Scholes PDE (14.6):

Route 1: no-arbitrage replication. \(C_t(S_t)\) can be replicated by a dynamically traded portfolio of "risk-free bond (lend/borrow) + underlying stock." See Table 14.1 for the trading strategy.

Table 14.1: Replicating European Call Option

The net inflow at \(t\) is zero (zero-cost portfolio). Summing the third column gives \(f_t\) (14.7):

By Itô's formula (formula 3 in He 2019d): for \(dX_t=R_t dt+A_t dB_t\), (14.8):

By (14.1) and (14.8), (14.9):

Substituting (14.9) into (14.7): the random term \(dS_t\) cancels, the portfolio is riskless; the zero-cost no-arbitrage condition forces \(f_t=0\) (14.10):

which rearranges to the BS PDE (14.6).

Route 2: risk-neutral measure. Change from \(\mathbf P\) to the risk-neutral measure \(\mathbf Q\), then invoke Feynman-Kac to obtain (14.6).

Constructing \(\mathbf Q\) (Girsanov / Adding Drift). Let \(dM_t=mM_t\,dB_t\) (\(\{B_t\}\) standard BM under \(\mathbf P\)), and define \(\mathbf Q\) by (14.11):

Then \(\{W_t\}\) is standard BM under \(\mathbf Q\), with (14.12):

Under \(\mathbf P\), \(dS_t=\mu S_t\,dt+\sigma S_t\,dB_t\) (14.13). Choose (14.14):

Substituting into (14.13) reverse-engineers (14.15):

i.e. under \(\mathbf Q\), \(\{S_t\}\) is still a diffusion but with drift \(r-\delta\) instead of \(\mu\).

Construct an SDF \(\Lambda^{\mathbf H}_t\). Under an arbitrary measure \(\mathbf H\) define (this SDF is artificial; no uniqueness is claimed)

where \(\mu^{\mathbf H}\) is the drift under \(\mathbf H\) and \(\{B^{\mathbf H}_t\}\) is standard BM under \(\mathbf H\). To show \(\Lambda^{\mathbf H}_t\) is a valid SDF pricing the stock, it suffices that (14.16):

By §1.3.3, (14.16) is equivalent to (14.17):

Verifying (14.17) for both the risk-free asset (\(D_t/S_t=r\), \(S_t=1\)) and the stock (\(dS_t=\mu^{\mathbf H}S_t dt+\sigma S_t dB^{\mathbf H}_t\)) shows it holds, so (14.18):

is a valid continuous SDF pricing the stock and the risk-free asset under any \(\mathbf H\).

Special case: under \(\mathbf Q\) the SDF degenerates to \(e^{-rt}\). Plug \(\mu^{\mathbf Q}=r-\delta\), \(dB^{\mathbf Q}_t=dW_t\) into (14.18):

so risk-free \(r\) discounts risky cash flow under \(\mathbf Q\) — exactly why \(\mathbf Q\) is the risk-neutral measure.

Replication and pricing. The call is replicated by \(\frac{\partial C_t}{\partial S_t}\) shares plus \(\frac1r\left[\frac{\partial C_t}{\partial t}+\frac12\sigma^2 S_t^2\frac{\partial^2 C_t}{(\partial S_t)^2}\right]\) units of risk-free bond; since \(\Lambda^{\mathbf H}\) prices both, it prices the linear combination \(C_t\) (14.21):

Setting \(\mathbf H=\mathbf Q\) (\(\Lambda^{\mathbf Q}_t=e^{-rt}\)) gives (14.22):

(14.22) also yields a Monte Carlo method: under \(\mathbf Q\) simulate many \(S_T\) paths via \(dS_t=(r-\delta)S_t dt+\sigma S_t dW^{\mathbf Q}_t\), average \(F(S_T)\), and multiply by \(e^{-r(T-t)}\).

Feynman-Kac. Any \(C_t(S_t)\) satisfying (14.22) must also satisfy

which is exactly the BS PDE (14.6).

Alternative derivation without Feynman-Kac. Using the risk-neutral property \(C_t=\mathbb E_t^{\mathbf Q}[e^{-r\,dt}(C_t+dC_t)]\) and the Taylor expansion \(e^{-r\,dt}\approx 1-r\,dt\) gives (14.23):

Taking the \(\mathbf Q\)-expectation of (14.20) under \(dS_t=(r-\delta)S_t dt+\sigma S_t dW_t\) gives (14.24):

Substituting (14.24) into \(rC_t\,dt=\mathbb E_t^{\mathbf Q}[dC_t]\) gives the BS PDE (14.6).

Route 3: assume CAPM holds for all traded assets (including the option). Let the option be written on an underlying object \(x(t)\) (tradable or not, e.g. "the temperature of Chicago"), with price \(V(x,t)\). Assume (14.25):

The option's dividend rate is \(\delta\) (payment \(\delta V\,dt\) over \([t,t+dt]\)). Write \(V_t,V_x,V_{xx}\) for the partials. By Itô (14.26):

Computing covariance and variance from (14.26), (14.25), define the option's market beta (14.28):

Imposing CAPM on the option, \(\mathbb E_t[r_V]-r\,dt=\beta_V\big(\mathbb E_t[\frac{dP_M}{P_M}]-r\,dt\big)\), and substituting (14.27), (14.28) yields (14.29):

With zero option dividend \(\delta=0\), rewrite as (14.30):

If the underlying \(x\) is tradable (with \(x(t)\) the ex-dividend price and dividend rate \(m\)), applying the risk-premium relation to \(x\) shows \(\mu^\star=r\) (consistent with risk-neutral pricing, §14.1.3); substituting into (14.30) gives (14.31):

which is the BS PDE (14.6) (zero dividend). The implicit zero-dividend assumption on \(x\) is the only source of difference between (14.31) and (14.6).

Route 4: solve the PDE directly. Starting from (14.6), write \(x\equiv S_t\), \(\phi(t,x)\equiv C_t\), \(m\equiv r-\delta\), \(\tau\equiv T-t\). Rewrite as

with boundary condition \(\phi(T,x)=(x-K)_+\). With \(\tau=T-t\) (14.32):

Next set \(y=\ln x\) (\(x=e^y\)); via (14.33), (14.34) this gives a constant-coefficient PDE in \(y\) (14.35):

Under zero dividend, longing one European call and shorting one European put generates the same cash flow as buying one share and selling \(e^{-r(T-t)}K\) of risk-free bond. No-arbitrage gives put-call parity (14.51):

Difference: European can only be exercised at \(T\), American at any \(t\).

Zero dividend: by (14.51), \(C_t=P_t+S_t-e^{-r(T-t)}K\geq S_t-e^{-r(T-t)}K\geq S_t-K\). So a European call always exceeds the exercise payoff \(S_t-K\) at any \(t\). Hence even though an American call can be exercised anytime, early exercise is never better than selling it at \(C_t\); the American call degenerates to a European call, sharing the same pricing function (14.2)–(14.4).

Positive dividend: the opportunity cost of holding a call is the forgone future dividend cash flow. If the dividend is high enough, that cost makes holding the actual stock attractive enough that the total payoff of early-exercising the American call (\(S_t-K\) plus discounted future dividends) can strictly exceed holding to expiration — so by no-arbitrage the American call can be strictly more expensive than the European call.

Using (14.2)–(14.4) in (14.51): \(P_t=C_t-S_t+e^{-r(T-t)}K\). Substituting and applying \(N(d)-1=-N(-d)\) gives the European put (14.52):

Assumptions. Stock price and volatility follow

with \(\mathrm{Corr}(dW_{S,t},dW_{\sigma,t})=\rho\), zero dividend, risk-free rate \(r\). Two European calls exist: option 1 price \(C(t)=C(S,\sigma,t;K,T)\), option 2 price \(\hat C(t)=C(S,\sigma,t;\hat K,\hat T)\). Write \(C_t,C_S,C_\sigma,C_{SS},C_{\sigma\sigma},C_{S\sigma}\) for the partials (\(\hat C\) likewise with hats).

Two random sources → need two options to hedge. By second-order Taylor/Itô (14.55):

Construct a Delta-neutral and Vega-neutral portfolio (dynamically trading risk-free bond + underlying), see Table 14.2.

Table 14.2: Delta-Neutral and Vega-Neutral Position

Summing the third column (14.57), substituting (14.55), (14.56) gives (14.58). Choosing \(\alpha=\hat C_\sigma\), \(\beta=-C_\sigma\) cancels \(dW_{\sigma,t}\) (while \(dW_{S,t}\) is already hedged by the stock); (14.58) becomes (14.59):

(14.59) has no randomness, so no-arbitrage forces \(f_t=0\). Each option's bracket divided by its \(C_\sigma\) (or \(\hat C_\sigma\)) must be equal, denoted \(-m^\star(S(t),t)\) (shorthand \(-m^\star\)) (14.60). Rewriting gives the PDE each option satisfies (14.61), (14.62):

(14.61), (14.62) can be obtained by applying Itô's lemma to the processes under \(\mathbf Q\) (14.63), (14.64):

Assumptions. Stock price and short rate follow

with \(\mathrm{Cov}(dW_1,dW_2)=\mu\,dt\) (\(\mu\) constant).

Goal. Show the call written on \(S(t)\) with price \(C(S(t),r(t),t;K,T)\) satisfies the PDE

where \(m^\star(S(t),r(t),t)\) is some function.

Two random sources (\(S\) and \(r\)) → need two options. Let \(C(S,r,t)\) be option 1 and \(\hat C(S,r,t)\) option 2; by Itô (14.65), (14.66):

Construct a Delta-neutral and Rho-neutral portfolio (see Table 14.3).

Table 14.3: Delta-Neutral and Rho-Neutral Position

Summing the third column gives \(f_t\) (14.67); substituting (14.65), (14.66) gives (14.68). Choosing \(\alpha=\hat C_r\), \(\beta=-C_r\) kills the \(dW_2(t)\) randomness, so (14.68) becomes (14.69). With no randomness, no-arbitrage forces \(f_t=0\) (14.70); rewriting the LHS gives (14.71):

(14.71) can be obtained by applying Itô's lemma to the \(\mathbf Q\)-processes (14.72), (14.73):

where \(-m^\star(S(t),r(t),t)=C_t+\frac12 C_{SS}\sigma^2 S^2+\frac12 C_{rr}n^2+C_{Sr}\rho\sigma Sn+SC_S r-Cr\). The structure exactly parallels the stochastic-volatility model (§14.2.4): the stock's \(\mathbf Q\)-drift is \(r\) (tradable), the rate's drift is not.

Since the BS PDE (14.6) holds for any derivative on \(S_t\), we can use it to price exotic payoffs.

Example 14.1 (Power Payoff Derivative). Constant risk-free rate \(r\), stock \(\frac{dS(t)}{S(t)}=\mu(S,t)\,dt+\sigma\,dW(t)\) (\(\sigma\) constant), dividend rate \(\delta\). Find the time-\(t\) price \(V(S(t),t)\) (shorthand \(V\)) of a derivative paying \(S^N(T)\) (\(N\geq1\)) at \(T\).

By (14.6) (valid for any derivative) (14.74), with boundary condition (14.75):

Guess \(V=\lambda_1 S^N e^{\lambda_2(T-t)}\). The boundary gives \(\lambda_1=1\); substituting into (14.74) yields \(\lambda_2=N\left(r-\delta-\frac12\sigma^2\right)-r+\frac{N^2}{2}\sigma^2\), so (14.76):

Example 14.2 (Rainbow Option). Two zero-dividend stocks, under \(\mathbf P\)

with \(\sigma_{SX}\,dt=\mathrm{Cov}(\sigma_S dW_S,\sigma_X dW_X)\) and \(\sigma_S,\sigma_X,\sigma_{SS},\sigma_{XX}\) constants.

(a) Find the PDE for the European option \(C(S,X,t)\) with payoff \(C(S(T),X(T),T)=\max\{0,S(T)-X(T)\}\).

Construct a Delta-neutral portfolio (Table 14.4: long 1 option, short \(C_S\) shares of \(S\) and \(C_X\) shares of \(X\), lend the rest at \(r\)), sum to \(f_t\) (14.77); substitute Itô (14.78); no-arbitrage \(f_t=0\) gives the PDE (14.80):

(b) Simplify and solve. Hint: \(C\) is homogeneous of degree one in \((S,X)\), i.e. \(C(\lambda S,\lambda X,t)=\lambda C(S,X,t)\) for \(\lambda>0\) (bundling \(\lambda\) original options must, by no-arbitrage, be worth exactly \(\lambda C\)). By Euler's theorem (14.81):

Substituting into (14.80) simplifies to (14.82). Differentiating (14.81) w.r.t. \(X\) and \(S\) gives (14.83), (14.84), which together imply \(S^2 C_{SS}=X^2 C_{XX}\) (14.85). Substituting into (14.82) with \(\sigma^2\equiv\sigma_S^2+\sigma_X^2-2\sigma_{SX}\), \(\tau\equiv T-t\), the PDE collapses to (14.86):

with boundary condition (14.87) \(C(S(T),X(T),T)=\max\{0,S(T)-X(T)\}\).

Guess. (14.86) coincides with the ordinary European-option PDE (14.6) when \(r=\delta=0\). So plug \(r=\delta=0\), \(K=X\) into the BS formula (14.2)–(14.4) to guess (14.88)–(14.90):

Verify (14.88)–(14.90) satisfies both the PDE (14.86) and the boundary (14.87). Boundary: when \(S(T)>X(T)\), \(d_1,d_2\to+\infty\) and \(C\to S(T)-X(T)\); when \(S(T)

Theorem 14.1 (Factor Model and SDF). Suppose there are \(K\) factors (14.94):

with \(\{W_k(t)\}\) independent standard BMs under the real measure. Define \(\{M(t)\}\) (14.95):

with \(r\) the continuous risk-free rate. Suppose there are \(N\) stocks \(\frac{dS_i(t)}{S_i(t)}=m_i\,dt+n_i\,dB_i(t)\) (\(i=1,\dots,N\)). Then the following two sets of conditions are equivalent:

(1) \(M(t)\) is a valid SDF pricing all stocks \(S_i\); (2) all stocks can be priced by a \(K\)-factor model with factors \(F_1,\dots,F_K\).

In §14.1.3 the risk-neutral measure \(\mathbf Q\) was specific to one stock (shifting its drift to \(r-\delta\)). Under some assumptions a single risk-neutral measure that applies to all stocks can be constructed.

Theorem 14.2 (Uniform Risk Neutral Measure). Suppose \(K\) factors \(\frac{dF_k(t)}{F_k(t)}=\mu_k\,dt+\sigma_k\,dW^{\mathbf P}_k(t)\), with \(\{W^{\mathbf P}_k\}\) independent standard BMs under \(\mathbf P\); each stock \(\frac{dS_i(t)}{S_i(t)}=m_i\,dt+n_i\,dB^{\mathbf P}_i(t)\). Let \(r\) be the continuous risk-free rate and \(\rho_{ik}\) the correlation of stock \(i\) with factor \(k\). If all stocks can be priced by this \(K\)-factor model and \(\rho_{ik}=\rho_k\) for all \(i\), then a single risk-neutral measure \(\mathbf Q\) exists for all stocks.

The option price (e.g. a European call) can also be written as a beta representation like (1.23).

Theorem 14.3 (Beta Representation of Option). Suppose \(K\) factors \(\frac{dF_k(t)}{F_k(t)}=\mu_k\,dt+\sigma_k\,dW^{\mathbf P}_k(t)\) (independent standard BMs), each stock \(\frac{dS_i(t)}{S_i(t)}=m_i\,dt+n_i\,dB_i(t)\), \(r\) the continuous risk-free rate. If stock \(i\) has a beta representation (14.102):

then any option on stock \(i\) with price \(C(S_i(t),t)\) also has a beta representation (14.103):