Financial Engineering

5. The Black-Scholes Model

5.1 Assumptions

Recall the usual assumptions of no arbitrage and no market frictions.

Recap (No arbitrage assumption): It is not possible to build a portfolio π\pi such that at time t=0t = 0 the value is zero, i.e. π0=0\pi_0 = 0, and the value at some time in the future T>0T > 0 can be positive, i.e. P(πT>0)>0\P(\pi_T > 0) > 0, but not negative, i.e. P(πT≥0)=1\P(\pi_T \geq 0) = 1.
Recap (No market frictions assumption): We assume the following:
  • We can buy/sell any fraction of shares.
  • We can buy/sell unlimited amounts of shares.
  • There is no bid/ask spread.
  • There are no transaction costs.
  • There are no taxes.

The following assumptions add to the usual ones.

Definition 5.1 (Black-Scholes model assumptions): We assume the following:
  • The stock price StS_t follows the geometric Brownian motion given by  dStSt=μ dt+σ dWt\frac{\dd S_t}{S_t} = \mu \dd t + \sigma \dd W_t.
  • The drift μ\mu and volatility σ\sigma of the stock StS_t are constant.
  • The riskless bond price BtB_t is given by  dBtBt=r dt\frac{\dd B_t}{B_t} = r \dd t.
  • The risk-free interest rate rr is known and constant.
  • There are no dividends.
Note: The stock price StS_t is the only random factor in the Black-Scholes model.

Note also that choosing a geometric Brownian motion for the stock price StS_t implies that the stock price follows a log-normal distribution and the instantaneous returns follow a normal distribution. This is not necessarily true in real-world prices and fatter tails can be observed in empirical data. Nonetheless, the Black-Scholes model is the foundation for more realistic option pricing models.

Proposition 5.2 (Itô's Lemma in the Black-Scholes model): Let

\cb{S_t}_{t \in [0,T]}

be a geometric Brownian motion with drift μ\mu and volatility σ\sigma. Then for any twice continuously differentiable function f(t,St)f(t, S_t), we have  df(t,St)=(∂f∂t+μSt∂f∂St+12σ2St2∂2f∂St2) dt+σSt∂f∂St dWt. \dd f(t, S_t) = \pa{\frac{\partial f}{\partial t} + \mu S_t \frac{\partial f}{\partial S_t} + \frac{1}{2} \sigma^2 S_t^2 \frac{\partial^2 f}{\partial S_t^2}} \dd t + \sigma S_t \frac{\partial f}{\partial S_t} \dd W_t.
Proof: Let Xt=StX_t = S_t. We have  dXt= dSt=μSt dt+σSt dWt=μt dt+σt dWt \dd X_t = \dd S_t = \mu S_t \dd t + \sigma S_t \dd W_t = \mu_t \dd t + \sigma_t \dd W_t where μt=Stμ\mu_t = S_t \mu and σt=Stσ\sigma_t = S_t \sigma. Then  df(t,St)\dd f(t, S_t) follows directly from Itô's Lemma.

5.2 Derivation

Consider a call option with price CtC_t. The evolution of Ct(St,t)C_t(S_t, t) is random and depends on the evolution of its underlying stock price StS_t. Similar to the binomial model for discrete time, we use dynamic replication together with the no-arbitrage assumption, market completeness and the Law of One Price to derive the price of a contingent claim.

5.2.1 Black-Scholes PDE

Definition 5.3 (Self-financing portfolio): Consider a portfolio

\cb{\pi_t}_{t \in [0,T]}

composed of αt\alpha_t units of the underlying and βt\beta_t units of the riskless bond where

\cb{\alpha_t}_{t \in [0,T]}

and

\cb{\beta_t}_{t \in [0,T]}

are adapted processes. The portfolio is self-financing if π0=α0S0+β0B0\pi_0 = \alpha_0 S_0 + \beta_0 B_0 and  dπt=αt dSt+βt dBt \dd \pi_t = \alpha_t \dd S_t + \beta_t \dd B_t
Definition 5.4 (Risk-free portfolio): A portfolio

\cb{\pi_t}_{t \in [0,T]}

is risk-free if  dπt=rπt dt \dd \pi_t = r \pi_t \dd t
Note: A risk-free portfolio follows the dynamics of the risk-free bond and is therefore deterministic. In the context of the Black-Scholes model, this means that π\pi should not depend on WtW_t.

We construct a self-financing risk-free portfolio π\pi with αt\alpha_t units of stocks and γt\gamma_t units of options at time tt.  dπt=αt dSt+γt dCt=(αtμSt+γt∂Ct∂t+γtμSt∂Ct∂St+γt12σ2St2∂2Ct∂St2) dt+(αtσSt+γtσSt∂Ct∂St) dWt=(μ(αtSt+γtSt∂Ct∂St)+γt∂Ct∂t+γt12σ2St2∂2Ct∂St2) dt+σ(αtSt+γtSt∂Ct∂St) dWt\begin{align*} \dd \pi_t ={}& \alpha_t \dd S_t + \gamma_t \dd C_t \\ ={}& \pa{\alpha_t \mu S_t + \gamma_t \frac{\partial C_t}{\partial t} + \gamma_t \mu S_t \frac{\partial C_t}{\partial S_t} + \gamma_t \frac{1}{2} \sigma^2 S_t^2 \frac{\partial^2 C_t}{\partial S_t^2}} \dd t + \pa{\alpha_t \sigma S_t + \gamma_t \sigma S_t \frac{\partial C_t}{\partial S_t}} \dd W_t \\ ={}& \pa{\mu \pa{\alpha_t S_t + \gamma_t S_t \frac{\partial C_t}{\partial S_t}} + \gamma_t \frac{\partial C_t}{\partial t} + \gamma_t \frac{1}{2} \sigma^2 S_t^2 \frac{\partial^2 C_t}{\partial S_t^2}} \dd t + \sigma \pa{\alpha_t S_t + \gamma_t S_t \frac{\partial C_t}{\partial S_t}} \dd W_t \end{align*} As π\pi is defined to be risk-free, it follows that αtSt+γtSt∂Ct∂St=0\alpha_t S_t + \gamma_t S_t \frac{\partial C_t}{\partial S_t} = 0 thus αtγt=−∂Ct∂St=−Δt \frac{\alpha_t}{\gamma_t} = -\frac{\partial C_t}{\partial S_t} = -\Delta_t In other words, to hedge the risk of the option one needs to sell Δt\Delta_t units of the underlying at time tt. This is called Delta-hedging. Following this,  dπt\dd \pi_t depends solely on  dt\dd t and we write the risk-free condition as  dπt=(γt∂Ct∂t+γt12σ2St2∂2Ct∂St2) dt=rπt dt  ⟹   γt∂Ct∂t+γt12σ2St2∂2Ct∂St2=−rγt∂Ct∂StSt+rγtCt  ⟹   ∂Ct∂t+rSt∂Ct∂St+12σ2St2∂2Ct∂St2=rCt\begin{align*} & \dd \pi_t = \pa{\gamma_t \frac{\partial C_t}{\partial t} + \gamma_t \frac{1}{2} \sigma^2 S_t^2 \frac{\partial^2 C_t}{\partial S_t^2}} \dd t = r \pi_t \dd t \\ \implies ~ & \gamma_t \frac{\partial C_t}{\partial t} + \gamma_t \frac{1}{2} \sigma^2 S_t^2 \frac{\partial^2 C_t}{\partial S_t^2} = -r \gamma_t \frac{\partial C_t}{\partial S_t} S_t + r \gamma_t C_t \\ \implies ~ & \frac{\partial C_t}{\partial t} + r S_t \frac{\partial C_t}{\partial S_t} + \frac{1}{2} \sigma^2 S_t^2 \frac{\partial^2 C_t}{\partial S_t^2} = rC_t \end{align*} With that we have derived the Black-Scholes stochastic PDE.

Note: The Black-Scholes PDE does not depend on the fact that CtC_t is a call option and is therefore satisfied by all derivatives on the underlying SS. For each derivative we specify a final condition for time t=Tt = T to find the unique solution to the stochastic PDE, e.g. C(ST,T)=max⁡(0,ST−K)C(S_T, T) = \max(0, S_T - K) for a call and P(ST,T)=max⁡(0,K−ST)P(S_T, T) = \max(0, K - S_T) for a put option.

We also note another surprising fact.

Note: The Black-Scholes PDE does not involve the drift term μ\mu of the underlying.

5.2.2 Martingale Approach

Recap (Martingale): A martingale with respect to the measure P\P is a stochastic process

\cb{M_t}_{t \geq 0}

such that for every t≥0t \geq 0

  • \E_\P[\abs{M_t}] < \infty

  • \E_\P[M_t \mid \sigmaF_s] \peq M_s

Recap (Equivalent measures): Two measures P\P and Q\Q for the same σ\sigma-algebra F\mathcal{F} are equivalent if P(A)=0  ⟺  Q(A)=0\P(A) = 0 \iff \Q(A) = 0 for all A∈FA \in \mathcal{F}.
Theorem 5.5 (First Fundamental Theorem of Asset Pricing): The two statements are equivalent:
  1. The no-arbitrage condition holds.
  2. There exists a probability measure Q\Q equivalent to P\P such that the discounted price process of every tradeable asset is a martingale under Q\Q.
Note: Such measure Q\Q is called risk-neutral measure or equivalent martingale measure.
Example (Todo): The discounted stock price process

\cb{e^{-rt} S_t}_{t \in [0, T]}

and the discounted option price process

\cb{e^{-rt} C_t}_{t \in [0, T]}

are martingales under the risk-neutral measure Q\Q.

To define the risk-neutral measure Q\Q, we introduce the stochastic exponential.

Definition 5.6 (Stochastic exponential): Let

\cb{L_t}_{t \geq 0}

be a P\P-martingale. The stochastic exponential or Doléans exponential

\cb{\mathcal{E}(L)_t}_{t \geq 0}

is the solution of the stochastic differential equation  dE(L)t=E(L)t dLt\dd \mathcal{E}(L)_t = \mathcal{E}(L)_t \dd L_t, i.e. E(L)t=exp⁡Lt−L0−12[L]t \mathcal{E}(L)_t = \exp{L_t - L_0 - \frac{1}{2} [L]_t}
Proposition 5.7 (Novikov Condition): Let LtL_t be a P\P-martingale. Then E(L)t\mathcal{E}(L)_t is a P\P-martingale if and only if

\E_\P\bk{\exp{\frac{1}{2} [L]_T}} < \infty

.
Theorem 5.8 (Girsanov's Theorem): Let us assume the Novikov Condition holds and E(L)t\mathcal{E}(L)_t is a P\P-martingale. Then:
  1. We can define a probability measure Q\Q equivalent to P\P such that the Radon-Nikodym derivative is  dQ dP∣Ft=E(L)t\frac{\dd \Q}{\dd \P}\vert_{\sigmaF_t} = \mathcal{E}(L)_t.
  2. If LtL_t is continuous, for a Brownian motion WtW_t under measure P\P the process W~t=Wt−[W,L]t\tilde{W}_t = W_t - [W,L]_t is a Brownian motion under measure Q\Q.
  3. For every stochastic process XtX_t in LP1L^1_{\P} we have

    \E_\P\bk{\frac{\mathcal{E}(L)_T}{\mathcal{E}(L)_t} X_T \mid \sigmaF_t} = \E_\Q[X_T \mid \sigmaF_t]

    .

We start under P\P as follows:  dStSt=μ dt+σ dWt=r dt+σ(μ−rσ dt+ dWt)=r dt+σ dW~t \frac{\dd S_t}{S_t} = \mu \dd t + \sigma \dd W_t = r \dd t + \sigma \pa{\frac{\mu - r}{\sigma} \dd t + \dd W_t} = r \dd t + \sigma \dd \tilde{W}_t where we defined W~t=Wt+μ−rσt\tilde{W}_t = W_t + \frac{\mu - r}{\sigma} t. Let Lt=r−μσWtL_t = \frac{r - \mu}{\sigma} W_t be a P\P-martingale. Note that the Novikov Condition is satisfied since

\E_\P\bk{\exp{\frac{1}{2}[L]_{\P,T}}} = \E_P\bk{e^{\frac{(r-\mu)^2 T}{2 \sigma^2}}} < \infty

which means we can apply Girsanov's Theorem. Also note that

\tilde{W}_t = W_t - \frac{\mu - r}{\sigma} t = W_t - \bk{W, \frac{r-\mu}{\sigma} W}_{\P,t} = W_t - [W,L]_{\P,t}

Hence we have found an LtL_t for the Doléans exponential such that under the change of measure defined by  dQ dP∣Ft=E(L)t\frac{\dd \Q}{\dd \P}\vert_{\sigmaF_t} = \mathcal{E}(L)_t, the process W~t\tilde{W}_t is a Q\Q-Brownian motion. Under this new measure Q\Q, the expected returns

\E_\Q\bk{\frac{\dd S_t}{S_t}} = r d_t

are risk-free. Integration with Itô's Lemma gives the expression of StS_t w.r.t. W~t\tilde{W}_t, i.e. Stb=Stae(r−σ22)(tb−ta)+σ(W~tb−W~ta) S_{t_b} = S_{t_a} e^{(r - \frac{\sigma^2}{2})(t_b - t_a) + \sigma (\tilde{W}_{t_b} - \tilde{W}_{t_a})} where 0≤ta≤tb≤T0 \leq t_a \leq t_b \leq T.

Proposition 5.9 (Consequence of the FFTAP): The value of any derivative can be calculated by discounting its final payoff under the risk-neutral measure Q\Q, i.e.

C_t = e^{-r(T-t)} \E_\Q[C_T \mid \sigmaF_t]

Proof: As e−rtCte^{-rt} C_t is a Q\Q-martingale, we have

e^{-rt} C_t = \E_\Q[e^{-rT} C_T \mid \sigmaF_t]

and thus

C_t = e^{r(T-t)} \E_\Q[e^{-rT} C_T \mid \sigmaF_t]

.

The call option price can thus be written as

\begin{align*} C_t & = e^{-r(T-t)} \E_\Q[C_T \mid \sigmaF_t] \\ & = e^{-r(T-t)} \E_\Q[(S_T - K)^{+} \mid \sigmaF_t] \\ & = e^{-r(T-t)} \E_\Q[(S_T - K) \ind{S_T > K} \mid \sigmaF_t] \\ & = e^{-r(T-t)} \E_\Q[S_T \ind{S_T > K} \mid \sigmaF_t] - K e^{-r(T-t)} \Q(S_T > K \mid \sigmaF_t) \end{align*}

We need to calculate

\E_\Q[S_T \ind{S_T > K} \mid \sigmaF_t]

and

\Q(S_T > K \mid \sigmaF_t)

. We focus on

\Q(S_T > K \mid \sigmaF_t)

first:

\begin{align*} \Q(S_T > K \mid \sigmaF_t) & = \Q\pamid{S_t e^{(r - \frac{\sigma^2}{2})(T-t) + \sigma (\tilde{W}_T - \tilde{W}_t)} > K}{\sigmaF_t} \\ & = \Q\pamid{\tilde{W}_T - \tilde{W}_t > \frac{\ln{\frac{K}{S_t}} - \pa{r - \frac{\sigma^2}{2}}(T-t)}{\sigma}}{\sigmaF_t} \\ & = 1 - \Q\pamid{\frac{\tilde{W}_T - \tilde{W}_t}{\sqrt{T-t}} \leq \frac{\ln{\frac{K}{S_t}} - \pa{r - \frac{\sigma^2}{2}}(T-t)}{\sigma \sqrt{T-t}}}{\sigmaF_t} \\ & = 1 - \Phi\pa{\frac{\ln{\frac{K}{S_t}} - \pa{r - \frac{\sigma^2}{2}}(T-t)}{\sigma \sqrt{T-t}}} \\ & = \Phi\pa{\frac{\ln{\frac{S_t}{K}} + \pa{r - \frac{\sigma^2}{2}}(T-t)}{\sigma \sqrt{T-t}}} \\ & = \Phi(d_2) \end{align*}

where d2=ln⁡StK+(r−σ22)(T−t)σT−td_2 = \frac{\ln{\frac{S_t}{K}} + \pa{r - \frac{\sigma^2}{2}}(T-t)}{\sigma \sqrt{T-t}}. It remains to calculate

\E_\Q[S_T \ind{S_T > K} \mid \sigmaF_t]

. Let L~t=σW~t\tilde{L}_t = \sigma \tilde{W}_t be a Q\Q-martingale. Note that the Novikov Condition is satisfied since

\E_\Q\bk{\exp{\frac{1}{2}[\tilde{L}]_{\Q,T}}} = \E_\Q\bk{e^{\frac{\sigma^2 T}{2}}} < \infty

which means we can apply Girsanov's Theorem. Also note that E(L~)TE(L~)t=exp⁡σW~T−σW~0−12[σW~]Q,T−(W~t−σW~0−12[σW~]Q,t)=e−σ22(T−t)+σ(W~T−W~t)\begin{align*} \frac{\mathcal{E}(\tilde{L})_T}{\mathcal{E}(\tilde{L})_t} &= \exp{\sigma \tilde{W}_T - \sigma \tilde{W}_0 - \frac{1}{2} [\sigma \tilde{W}]_{\Q,T} - \pa{\tilde{W}_t - \sigma \tilde{W}_0 - \frac{1}{2} [\sigma \tilde{W}]_{\Q,t}}} \\ & = e^{-\frac{\sigma^2}{2}(T - t) + \sigma \pa{\tilde{W}_T - \tilde{W}_t}} \end{align*} thus

\begin{align*} \E_\Q[S_T \ind{S_T > K} \mid \sigmaF_t] &= \E_Q\bkmid{S_t e^{(r - \frac{\sigma^2}{2})(T-t) + \sigma (\tilde{W}_T - \tilde{W}_t)} \ind{S_T > K}}{\sigmaF_t} \\ & = S_t e^{r (T-t)} \E_\Q\bkmid{\frac{\mathcal{E}(\tilde{L})_T}{\mathcal{E}(\tilde{L})_t} \ind{S_T > K}}{\sigmaF_t} \\ & = S_t e^{r (T-t)} \E_{\Q^*}\bkmid{\ind{S_T > K}}{\sigmaF_t} \\ & = S_t e^{r (T-t)} \Q^*(S_T > K \mid \sigmaF_t) \end{align*}

The process Wt∗=W~t−[W~,σW~]Q,t=W~t−σtW^*_t = \tilde{W}_t - [\tilde{W}, \sigma \tilde{W}]_{\Q,t} = \tilde{W}_t - \sigma t is a Q∗\Q^*-Brownian motion. We write  dStSt=r dt+σ dW~t=(r+σ2) dt+σ dWt∗ \frac{\dd S_t}{S_t} = r \dd t + \sigma \dd \tilde{W}_t = (r + \sigma^2) \dd t + \sigma \dd W^*_t Integration with Itô's Lemma gives the expression of StS_t w.r.t. Wt∗W^*_t, i.e. Stb=Stae(r+σ22)(tb−ta)+σ(Wtb∗−Wta∗) S_{t_b} = S_{t_a} e^{(r + \frac{\sigma^2}{2})(t_b - t_a) + \sigma (W^*_{t_b} - W^*_{t_a})} thus

\begin{align*} S_t e^{r (T-t)} \Q^*(S_T > K \mid \sigmaF_t) &= S_t e^{r (T-t)} \Q^*\pamid{S_{t} e^{(r + \frac{\sigma^2}{2})(T - t) + \sigma (W^*_{T} - W^*_{t})} > K}{\sigmaF_t} \\ & = S_t e^{r (T-t)} \Q^*\pamid{\frac{W^*_{T} - W^*_{t}}{\sqrt{T - t}} > \frac{\ln{\frac{K}{S_t}} - \pa{r + \frac{\sigma^2}{2}}(T - t)}{\sigma \sqrt{T - t}}}{\sigmaF_t} \\ & = S_t e^{r (T-t)} \Phi\pa{\frac{\ln{\frac{S_t}{K}} + \pa{r + \frac{\sigma^2}{2}}(T - t)}{\sigma \sqrt{T - t}}} \\ & = S_t e^{r (T-t)} \Phi(d_1) \end{align*}

where d1=ln⁡StK+(r+σ22)(T−t)σT−td_1 = \frac{\ln{\frac{S_t}{K}} + \pa{r + \frac{\sigma^2}{2}}(T - t)}{\sigma \sqrt{T - t}}.

Theorem 5.10 (Black-Scholes Formula for call options): The Black-Scholes Formula for call options is Ct(BS)=StΦ(d1)−Ke−r(T−t)Φ(d2) C^{(\mathrm{BS})}_t = S_t \Phi(d_1) - Ke^{-r(T-t)}\Phi(d_2) where d1=ln⁡StK+(r+σ22)(T−t)σT−td_1 = \frac{\ln{\frac{S_t}{K}} + \pa{r + \frac{\sigma^2}{2}}(T - t)}{\sigma \sqrt{T - t}} and d2=ln⁡StK+(r−σ22)(T−t)σT−td_2 = \frac{\ln{\frac{S_t}{K}} + \pa{r - \frac{\sigma^2}{2}}(T-t)}{\sigma \sqrt{T-t}}.
Note:
  • We have d2=d1−σT−td_2 = d_1 - \sigma \sqrt{T - t}
  • For put options, the Black-Scholes Formula is Pt(BS)=Ke−r(T−t)Φ(−d2)−StΦ(−d1)P^{(\mathrm{BS})}_t = Ke^{-r(T-t)}\Phi(-d_2) - S_t \Phi(-d_1)

5.3 Greeks and Hedging

The call price

\cb{C_t}_{t \geq 0}

is a stochastic process. The Black-Scholes Formula implies the following: The option price Ct(BS)C^{(\mathrm{BS})}_t at time tt for a given strike price KK and maturity TT is a deterministic function Ct(BS)(St,t)C^{(\mathrm{BS})}_t(S_t, t) of the current stock price StS_t and time tt.

We introduce some useful identities for the derivation of the Greeks.

Proposition 5.11: We have Stϕ(d1)=Ke−r(T−t)ϕ(d2)S_t \phi(d_1) = Ke^{-r(T-t)}\phi(d_2).
Proof: Ct(BS)=StΦ(d1)−Ke−r(T−t)Φ(d2)  ⟹   StΦ(d1)=Ct(BS)+Ke−r(T−t)Φ(d1−σT−t)  ⟹    d dd1StΦ(d1)= d dd1Ke−r(T−t)Φ(d1−σT−t)  ⟹   Stϕ(d1)=Ke−r(T−t)ϕ(d1−σT−t)  ⟹   Stϕ(d1)=Ke−r(T−t)ϕ(d2)\begin{align*} & C^{(\mathrm{BS})}_t = S_t \Phi(d_1) - Ke^{-r(T-t)}\Phi(d_2) \\ \implies~& S_t \Phi(d_1) = C^{(\mathrm{BS})}_t + Ke^{-r(T-t)}\Phi(d_1 - \sigma \sqrt{T - t}) \\ \implies~& \frac{\dd}{\dd d_1} S_t \Phi(d_1) = \frac{\dd}{\dd d_1} Ke^{-r(T-t)}\Phi(d_1 - \sigma \sqrt{T - t}) \\ \implies~& S_t \phi(d_1) = Ke^{-r(T-t)}\phi(d_1 - \sigma \sqrt{T - t}) \\ \implies~& S_t \phi(d_1) = Ke^{-r(T-t)}\phi(d_2) \end{align*}
Proposition 5.12: We have ∂d1∂St=∂d2∂St=St−1σT−t\frac{\partial{d_1}}{\partial S_t} = \frac{\partial{d_2}}{\partial S_t} = \frac{S_t^{-1}}{\sigma \sqrt{T - t}}.

5.3.1 Delta

Definition 5.13 (Delta): The delta Δt\Delta_t of an option is the sensitivity of the option price VtV_t to changes in the underlying asset price StS_t: Δt=∂Vt∂St \Delta_t = \frac{\partial V_t}{\partial S_t}
Example (Delta in the Black-Scholes model): The delta Δt(BS Call)\Delta^{(\mathrm{BS}~\mathrm{Call})}_t of a call option is given by Δt(BS Call)=∂Ct(BS)∂St=Φ(d1)+Stϕ(d1)∂d1∂St−Ke−r(T−t)ϕ(d2)∂d1∂St=Φ(d1) \Delta^{(\mathrm{BS}~\mathrm{Call})}_t = \frac{\partial C^{(\mathrm{BS})}_t}{\partial S_t} = \Phi(d_1) + S_t \phi(d_1) \frac{\partial d_1}{\partial S_t} - K e^{-r(T-t)} \phi(d_2) \frac{\partial d_1}{\partial S_t} = \Phi(d_1) similarly, the delta Δt(BS Put)\Delta^{(\mathrm{BS}~\mathrm{Put})}_t of a put option is given by Δt(BS Put)=−Φ(−d1)=Φ(d1)−1\Delta^{(\mathrm{BS}~\mathrm{Put})}_t = -\Phi(-d_1) = \Phi(d_1) - 1.
Note:
  • We have −1<Δt(BS Put)<0<Δt(BS Call)<1-1 < \Delta^{(\mathrm{BS}~\mathrm{Put})}_t < 0 < \Delta^{(\mathrm{BS}~\mathrm{Call})}_t < 1
  • As StS_t increases, d1(St,t)d_1(S_t, t) increases and hence Δt(BS)\Delta^{(\mathrm{BS})}_t increases
  • As t→Tt \to T, we have d1(St,t)→ln⁡StK⋅∞d_1(S_t, t) \to \ln{\frac{S_t}{K}} \cdot \infty, and lim⁡t→TΔt(BS Call)(St)=lim⁡t→TΦ(d1(St,t))={1if St>K0if St<K \lim_{t \to T} \Delta^{(\mathrm{BS}~\mathrm{Call})}_t (S_t) = \lim_{t \to T} \Phi(d_1(S_t, t)) = \begin{cases} 1 & \text{if } S_t > K \\ 0 & \text{if } S_t < K \end{cases} i.e. the Δt(BS Call)\Delta^{(\mathrm{BS}~\mathrm{Call})}_t as a function of StS_t gets closer to a step function
Proposition 5.14 (Delta hedging): The portfolio πt(BS Δ)\pi^{(\mathrm{BS}~\Delta)}_t with π0(BS Δ)=C0−α0S0\pi^{(\mathrm{BS}~\Delta)}_0 = C_0 - \alpha_0 S_0, i.e. one call option and α0\alpha_0 shorted stocks at time t=0t = 0, whose value is independent of price fluctuations in the stock StS_t is given by πt(BS Δ)=Ct−αtSt\pi^{(\mathrm{BS}~\Delta)}_t = C_t - \alpha_t S_t with αt=Δt(BS Call)\alpha_t = \Delta^{(\mathrm{BS}~\mathrm{Call})}_t.
Proof: ∂πt(BS Δ)∂St=∂Ct∂St−αt=Δt(BS Call)−Δt(BS Call)=0\frac{\partial \pi^{(\mathrm{BS}~\Delta)}_t}{\partial S_t} = \frac{\partial C_t}{\partial S_t} - \alpha_t = \Delta^{(\mathrm{BS}~\mathrm{Call})}_t - \Delta^{(\mathrm{BS}~\mathrm{Call})}_t = 0.
Note: Practical issues with delta hedging are that rebalancing is costly and that trading quantities are discrete and not continuous as αt\alpha_t.

5.3.2 Gamma

Definition 5.15 (Gamma): The gamma Γt\Gamma_t of an option is the sensitvity of the delta Δt\Delta_t to changes in the stock price StS_t: Γt=∂Δt∂St \Gamma_t = \frac{\partial \Delta_t}{\partial S_t}
Example (Gamma in the Black-Scholes model): The gamma Γt(BS Call)\Gamma^{(\mathrm{BS}~\mathrm{Call})}_t of a call option is given by Γt(BS Call)=∂Δt(BS Call)∂St=∂Φ(d1)∂St∂d1∂St=St−1ϕ(d1)σT−t \Gamma^{(\mathrm{BS}~\mathrm{Call})}_t = \frac{\partial \Delta^{(\mathrm{BS}~\mathrm{Call})}_t}{\partial S_t} =\frac{\partial \Phi(d_1)}{\partial S_t} \frac{\partial d_1}{\partial S_t} = \frac{S_t^{-1} \phi(d_1)}{\sigma \sqrt{T- t}} equally, the gamma Γt(BS Put)\Gamma^{(\mathrm{BS}~\mathrm{Put})}_t of a put option is given by Γt(BS Call)=St−1ϕ(d1)σT−t\Gamma^{(\mathrm{BS}~\mathrm{Call})}_t = \frac{S_t^{-1} \phi(d_1)}{\sigma \sqrt{T- t}}.
Note:
  • We have Γt=∂Δt∂St=∂2Δt∂St2\Gamma_t = \frac{\partial \Delta_t}{\partial S_t} = \frac{\partial^2 \Delta_t}{\partial S_t^2}, i.e. gamma measures the curvature of the option price with respect to the stock price
  • The fact that Γt(BS Call)=Γt(BS Put)\Gamma^{(\mathrm{BS}~\mathrm{Call})}_t = \Gamma^{(\mathrm{BS}~\mathrm{Put})}_t is called put-call parity
  • Gamma is non-negative, i.e. Γt(BS)≥0\Gamma^{(\mathrm{BS})}_t \geq 0, hence the option prices are convex w.r.t. StS_t
  • As St→∞S_t \to \infty, ϕ(d1)St→0\frac{\phi(d_1)}{S_t} \to 0 hence lim⁡St→∞Γt(BS)=0\lim_{S_t \to \infty} \Gamma^{(\mathrm{BS})}_t = 0
  • As St→0S_t \to 0, ϕ(d1)∼e−ln⁡St2→0\phi(d_1) \sim e^{-\ln{S_t}^2} \to 0, thus ϕ(d1)St∼St→0\frac{\phi(d_1)}{S_t} \sim S_t \to 0 and lim⁡St→0Γt(BS)=0\lim_{S_t \to 0} \Gamma^{(\mathrm{BS})}_t = 0
  • Let St=KS_t = K and t→Tt \to T, then ϕ(d1)∼eT−t→1\phi(d_1) \sim e^{T-t} \to 1 and ϕ(d1)T−t∼(T−t)−12→∞\frac{\phi(d_1)}{\sqrt{T-t}} \sim (T-t)^{-\frac{1}{2}} \to \infty, hence

    \lim_{S_t \to 0} \Gamma^{(\mathrm{BS})}_t \mid_{S_t=K} = \infty

  • The latter property makes delta hedging impossible as an infinite quantity of stock StS_t would be required to hedge the option

We can relate an option price CC to its delta Δ\Delta and gamma Γ\Gamma via the Taylor expansion: C(S+ΔSt,t+Δt)=C(S,t)+∂C∂t(S,t)+ΔtΔS+ΓtΔSt22+O(Δt2)+O(ΔSt3) C(S + \Delta S_t, t + \Delta t) = C(S, t) + \frac{\partial C}{\partial t}(S,t) + \Delta_t \Delta S + \Gamma_t \frac{\Delta S_t^2}{2} + O(\Delta t^2) + O(\Delta S_t^3) Thus Delta-hedging equivaltes to approximating the option price by its Taylor while ignoring the convexity term.

5.3.3 Vega

Definition 5.16 (Vega): The vega Vt\Vega_t of an option is the sensitivity of the option price VtV_t to changes in the volatility σ\sigma: Vt=∂Vt∂σ \Vega_t = \frac{\partial V_t}{\partial \sigma}
Example (Vega in the Black-Scholes model): The vega Vt(BS Call)\Vega^{(\mathrm{BS~Call})}_t of a call option is given by: Vt(BS Call)=∂CtBS Call∂σ=Stϕ(d1)∂d1∂σ−Ke−r(T−t)ϕ(d2)∂∂σ(d1−σT−t)=Ke−r(T−t)ϕ(d2)T−t=Stϕ(d1)T−t\begin{align*} \Vega^{(\mathrm{BS~Call})}_t &= \frac{\partial C^{\mathrm{BS~Call}}_t}{\partial \sigma} \\ & = S_t \phi(d_1) \frac{\partial d_1}{\partial \sigma} - K e^{-r(T-t)} \phi(d_2) \frac{\partial}{\partial \sigma} (d_1 - \sigma \sqrt{T-t}) \\ & = K e^{-r(T-t)} \phi(d_2) \sqrt{T-t} \\ & = S_t \phi(d_1) \sqrt{T-t} \end{align*} equally, the vega Vt(BS Put)\Vega^{(\mathrm{BS~Put})}_t of a put option is Vt(BS Put)=Stϕ(d1)T−t\Vega^{(\mathrm{BS~Put})}_t = S_t \phi(d_1) \sqrt{T-t}.
Note:
  • Vega is strictly positive, i.e. Vt(BS)>0\Vega^{(\mathrm{BS})}_t > 0
  • As σ\sigma increases, d1d_1 increases and hence Vt(BS)\Vega^{(\mathrm{BS})}_t increases
  • As St→0S_t \to 0, ϕ(d1)∼St2→0\phi(d_1) \sim S_t^2 \to 0 and hence lim⁡St→0Vt(BS)=0\lim_{S_t \to 0} \Vega^{(\mathrm{BS})}_t = 0
  • As St→∞S_t \to \infty, ϕ(d1)∼e−ln⁡St2→0\phi(d_1) \sim e^{-\ln{S_t}^2} \to 0 which converges faster than St→∞S_t \to \infty, hence lim⁡St→∞Vt(BS)=0\lim_{S_t \to \infty} \Vega^{(\mathrm{BS})}_t = 0
Proposition 5.17 (Maximum Vega): The maximum vega Vt(BS)\Vega^{(\mathrm{BS})}_t w.r.t. the stock price SS is achieved at max⁡S>0Vt(BS)=Ke−(r+σ22)(T−t)\max_{S > 0} \Vega^{(\mathrm{BS})}_t = K e^{-(r + \frac{\sigma^2}{2})(T-t)}.
Proof: We calculate the derivative ∂Vt(BS)∂S\frac{\partial \Vega^{(\mathrm{BS})}_t}{\partial S}: ∂Vt(BS)∂S=ϕ(d1)T−t+Sϕ′(d1)T−t∂d1∂S=(1−d1σT−t)ϕ(d1)T−t\begin{align*} \frac{\partial \Vega^{(\mathrm{BS})}_t}{\partial S} &= \phi(d_1) \sqrt{T-t} + S \phi'(d_1) \sqrt{T-t} \frac{\partial d_1}{\partial S} \\ & = \pa{1 - \frac{d_1}{\sigma \sqrt{T-t}}} \phi(d_1) \sqrt{T-t} \end{align*} We need to find the solution to 1−d1σT−t=01 - \frac{d_1}{\sigma \sqrt{T-t}} = 0. 1−d1σT−t=0  ⟹   d1=σT−t  ⟹   ln⁡SK+(r+σ22)(T−t)σT−t=σT−t  ⟹   ln⁡SK=−(r+σ22)(T−t)  ⟹   S=Ke−(r+σ22)(T−t)\begin{align*} & 1 - \frac{d_1}{\sigma \sqrt{T-t}} = 0 \\ \implies~& d_1 = \sigma \sqrt{T-t} \\ \implies~& \frac{\ln{\frac{S}{K}} + \pa{r + \frac{\sigma^2}{2}}(T-t)}{\sigma \sqrt{T-t}} = \sigma \sqrt{T-t} \\ \implies~&\ln{\frac{S}{K}} = -\pa{r + \frac{\sigma^2}{2}}(T-t) \\ \implies~& S = K e^{-(r + \frac{\sigma^2}{2})(T-t)} \end{align*}

5.3.4 Theta

Definition 5.18 (Theta): The theta Θt\Theta_t of an option is the sensitivity of the option price VtV_t with respect to time: Θt=∂Vt∂t \Theta_t = \frac{\partial V_t}{\partial t}
Note:
  • Θt(BS Call)\Theta^{(\mathrm{BS~Call})}_t is always negative and Θt(BS Put)\Theta^{(\mathrm{BS~Put})}_t is only positive for ITM put options
  • Θt(BS)\Theta^{(\mathrm{BS})}_t is large and negative for ATM options
  • Θt(BS)\Theta^{(\mathrm{BS})}_t has a large magnitude as t→Tt \to T

5.3.5 Rho

Definition 5.19 (Rho): The rho Pt\Rho_t of an option is the sensitivty of the option price VtV_t with respct to the interst-rate: Pt=∂Vt∂r \Rho_t = \frac{\partial V_t}{\partial r}
Example (Rho in the Black-Scholes model): The rho Pt(BS Call)\Rho^{(\mathrm{BS~Call})}_t of a call option is given by: Pt(BS Call)=∂Ct(BS Call)∂r=Stϕ(d1)∂d1∂r−Ke−r(T−t)ϕ(d2)∂d2∂r+K(T−t)e−r(T−t)Φ(d2)=K(T−t)e−r(T−t)Φ(d2)\begin{align*} \Rho^{(\mathrm{BS~Call})}_t &= \frac{\partial C^{(\mathrm{BS~Call})}_t}{\partial r} \\ & = S_t \phi(d_1) \frac{\partial d_1}{\partial r} - K e^{-r (T-t)} \phi(d_2) \frac{\partial d_2}{\partial r} + K (T-t) e^{-r (T-t)} \Phi(d_2) \\ & = K (T-t) e^{-r (T-t)} \Phi(d_2) \end{align*} similarly, the rho Pt(BS Put)\Rho^{(\mathrm{BS~Put})}_t of a put option is given by Pt(BS Put)=−K(T−t)e−r(T−t)Φ(−d2) \Rho^{(\mathrm{BS~Put})}_t = -K (T-t) e^{-r (T-t)} \Phi(-d_2).
Note:
  • For call options, Pt(BS Call)≥0\Rho^{(\mathrm{BS~Call})}_t \geq 0 as the replicating strategy involves shorting bonds, hence if rr increases, the bond price BtB_t decreases and the call option price CtC_t increases
  • For put options, Pt(BS Put)≤0\Rho^{(\mathrm{BS~Put})}_t \leq 0 as the replicating strategy involves buying bonds, hence if rr increases, the bond price BtB_t decreases and the put option price PtP_t decreases