Option Pricing in Continuous Time

Stochastic Foundations for Finance

Lorenzo Naranjo

Fall 2024

The Replicating Portfolio Approach

Consider a derivative V written on a non-dividend paying stock S with maturity T that pays F(S) at maturity.
The binomial model implies that the derivative can be replicated by buying (or selling) \alpha_{t} units of the stock and \beta_{t} units of a bond with face value K and maturity T, respectively.
If we call V the value of such replicating portfolio, we have that at time t < T: V_{t} = \alpha_{t} S_{t} + \beta_{t} B_{t}.
In order to replicate the derivative, we want to make sure that the value of the portfolio at time t = T is equal to the payoff of the derivative, that is: V_{T} = H_{T}
For example, for a European call option H_{T} = \max(S_{T} - K, 0).

The Replicating Portfolio is Self-Financing

At time t+\Delta t, the value of the replicating portfolio is: V_{t + \Delta t} = \alpha_{t} S_{t + \Delta t} + \beta_{t} B_{t + \Delta t}, which implies that: \Delta V_{t} = \alpha_{t} \Delta S_{t} + \beta_{t} \Delta B_{t}.
The new composition of the portfolio at time t+\Delta t is chosen such that: V_{t + \Delta t} = \alpha_{t} S_{t + \Delta t} + \beta_{t} B_{t + \Delta t} = \alpha_{t + \Delta t} S_{t + \Delta t} + \beta_{t + \Delta t} B_{t + \Delta t} which shows that the portfolio is self-financing, i.e., no new funds are added or withdrawn from the portfolio.

Replication in Continuous-Time

As \Delta t \rightarrow 0, we have that: \begin{aligned} dV & = \alpha dS + \beta dB \\ & = \alpha dS + \beta (r B dt) \\ & = \alpha dS + (\beta B) r dt \\ \end{aligned}
And since V = \alpha S + \beta B \Rightarrow \beta B = V - \alpha S, we can conclude that: dV = r (V - \alpha S) dt + \alpha dS

Applying Ito’s Lemma

We will assume for the moment that V is a smooth function of S and t, that is, V = V(S,t).
Then, Ito’s Lemma implies that: \begin{aligned} dV & = \frac{\partial V}{\partial S} dS + \frac{1}{2} \frac{\partial^{2} V}{\partial S^{2}} (dS)^{2} + \frac{\partial V}{\partial t} dt \\ & = \left(\frac{1}{2} \sigma^{2} S^{2} \frac{\partial^{2} V}{\partial S^{2}} + \frac{\partial V}{\partial t}\right) dt + \frac{\partial V}{\partial S} dS \end{aligned}
Therefore: \left(\frac{1}{2} \sigma^{2} S^{2} \frac{\partial^{2} V}{\partial S^{2}} + \frac{\partial V}{\partial t}\right) dt + \textcolor{blue}{\frac{\partial V}{\partial S} dS} = r (V - \alpha S) dt + \textcolor{blue}{\alpha dS}

The Delta of the Derivative

First, the previous equation shows that replication works if and only if: \alpha = \frac{\partial V}{\partial S}
This is a fundamental relationship in derivatives pricing.
It states that the number of shares needed to replicate the derivative is equal its sensitivity to the underlying asset.
We call this quantity the delta (\Delta) of the derivative.
Also, note that by choosing \alpha equal to the delta of the derivative, it really does not matter what drift we have for the stock.
We will use this fact in a moment to define the risk-neutral probabilities in continuous-time.

The Fundamental Partial Differential Equation (PDE)

Second, it must be the case that: \frac{1}{2} \sigma^{2} S^{2} \frac{\partial^{2} V}{\partial S^{2}} + \frac{\partial V}{\partial t} = r \left(V - S \frac{\partial V}{\partial S}\right)
Therefore: \frac{\partial V}{\partial t} + \frac{1}{2} \sigma^{2} S^{2} \frac{\partial^{2} V}{\partial S^{2}} + r S \frac{\partial V}{\partial S} - r V = 0 subject to V_{T} = H_{T}.
This is the celebrated Black-Scholes partial differential equation (PDE) which allowed the authors to compute their influential formula in 1973!
Solving PDEs, in general, is very hard so we will resort to a different approach to price European call and put options.

The Risk-Neutral Pricing Approach

The replicating approach is insensitive to the drift of the stock.
As a matter of fact, the drift might even change based on whose thinking about the asset.
Since the previous reasoning is silent about the drift and the type of investor pricing the asset, we can assume in our reasoning that all investors are .
Even if this is not true in real markets, such assumption would not affect of the replicating-portfolio argument.

A Risk-Neutral World

In a world populated by risk-neutral investors, the price today of any non-dividend paying asset is equal to the expected payoff at maturity discounted at the risk-free rate, that is: X = e^{-rT} \operatorname{E}(X_{T})
Therefore, the drift of a non-dividend paying stock is the risk-free rate: dS = r S dt + \sigma S dz
The same is true for all derivatives written on the stock: dV = \underbrace{\left(r S \frac{\partial V}{\partial S} + \frac{1}{2} \sigma^{2} S^{2} \frac{\partial^{2} V}{\partial S^{2}} + \frac{\partial V}{\partial t}\right)}_{=rV} dt + \left(\sigma S \frac{\partial V}{\partial S}\right) dz
We recover the same equation as before!

Pricing a European Call Option

Consider a European call option written on a non-dividend paying stock with maturity T and strike price K.
The price of the call should then be: \begin{aligned} C & = e^{-r T} \operatorname{E}\left((S_{T} - K) \large\mathbb{1}_{{S_{T} > K}} \right) \\ & = e^{-r T} \operatorname{E}\left(S_{T} \large\mathbb{1}_{{S_{T} > K}} \right) - e^{-r T} \operatorname{E}\left(K \large\mathbb{1}_{{S_{T} > K}} \right) \\ & = S \operatorname{\Phi}(d_{1}) - K e^{-r T} \operatorname{\Phi}(d_{2}) \end{aligned} where \begin{aligned} d_{1} & = \frac{\ln(S/K) + (r + \frac{1}{2} \sigma^{2}) T}{\sigma \sqrt{T}} \\ d_{2} & = d_{1} - \sigma \sqrt{T} \end{aligned}

Call Premium vs. Spot Price

Reconciling Both Pricing Approaches

It is tedious but straightforward to prove that: \alpha = \frac{\partial C}{\partial S} = \operatorname{\Phi}(d_{1}) \tag{1}
Also, we have that for a European call option: C = \alpha S + \beta B = S \operatorname{\Phi}(d_{1}) - K e^{-r T} \operatorname{\Phi}(d_{2}) which because of (1) implies that: \beta = -\operatorname{\Phi}(d_{2})

Call Delta

Hedging the Call

Our analysis so far implies that to replicate a European call option, we need to go \operatorname{\Phi}(d_{1}) shares of stock and \operatorname{\Phi}(d_{2}) risk-free bonds with face value K and maturity T.
The call is therefore a levered position in the underlying asset whose delta is given by \operatorname{\Phi}(d_{1}).
- Since 0<\operatorname{\Phi}(d_{1})<1, the delta of the call for a non-dividend paying asset is bounded between 0 and 1.
As we saw in the previous slide, for a given spot price, the delta of the call represents the slope coefficient of the tangency line at that point.

Pricing a European Put Option

Consider now a European put option with the same characteristics as the previous call.
According to put-call parity, it must be the case that: ` C - P = S_{0} - K e^{-r T}
Hence, \begin{aligned} P & = C - (S - K e^{-rT}) \\ & = S \operatorname{\Phi}(d_{1}) - K e^{-r T} \operatorname{\Phi}(d_{2}) - (S - K e^{-rT}) \\ & = K e^{-r T} (1 - \operatorname{\Phi}(d_{2})) - S (1 - \operatorname{\Phi}(d_{1})) \\ & = K e^{-r T} \operatorname{\Phi}(-d_{2}) - S \operatorname{\Phi}(-d_{1}) \end{aligned}

Put Premium vs. Spot Price

Put Delta

Hedging the Put

We can use put-call parity to compute \alpha for the put: \alpha = \frac{\partial P}{\partial S} = \frac{\partial (C - S + K e^{-rT})}{\partial S} = \operatorname{\Phi}(d_{1}) - 1 = -\operatorname{\Phi}(-d_{1}) < 0
The fact that we also have P = \alpha S + \beta B also implies that: \beta = \operatorname{\Phi}(-d_{2}) > 0
Therefore, to replicate a European put option, we need to go \operatorname{\Phi}(-d_{1}) shares of stock and \operatorname{\Phi}(-d_{2}) risk-free bonds with face value K and maturity T.

Finishing In-The-Money

Remember that we showed that: \operatorname{P}(S_{T} > K) = \operatorname{E}\left(\large\mathbb{1}_{{S_{T} > K}}\right) = \operatorname{\Phi}(d_{2}) which also implies that: \operatorname{P}(S_{T} < K) = 1 - \operatorname{P}(S_{T} > K) = 1 - \operatorname{\Phi}(d_{2}) = \operatorname{\Phi}(-d_{2})
Therefore, the risk-neutral probability that the call will expire in-the-money is equal to \operatorname{\Phi}(d_{2}) whereas the risk-neutral probability that the put finishes in-the-money is given by \operatorname{\Phi}(-d_{2}).

Example 1 Consider a non-dividend paying stock that currently trades for $100. The risk-free rate is 4% per year, continuously compounded and constant for all maturities. The instantaneous volatility of returns is 25% per year. Consider at-the-money call and put options written on the stock with maturity 9 months. Then, \begin{aligned} d_{1} & = \frac{\ln(100/100) + (0.04 + 0.5(0.25)^{2})(0.75)}{0.25\sqrt{0.75}} = 0.2468 \\ d_{2} & = 0.2468 - 0.25\sqrt{0.75} = 0.0303 \end{aligned} Therefore, \operatorname{\Phi}(d_{1}) = 0.5975 and \operatorname{\Phi}(d_{2}) = 0.5121, which implies that: \begin{aligned} C & = 100(0.5975) - 100e^{-0.04(0.75)}(0.5121) = \$10.05 \\ P & = 100e^{-0.04(0.75)}(1 - 0.5121) - 100(1 - 0.5975) = \$7.10 \end{aligned}

The Impact of Volatility

Option Premium vs. Volatility

One of the most important determinants of option prices in the Black-Scholes model is volatility.
We can show that for European call and put options: \frac{\partial C}{\partial \sigma} = \frac{\partial P}{\partial \sigma} = S \operatorname{\Phi}^{'}(d_{1}) \sqrt{T} > 0. \tag{2}
Hence, both European call and put options increase in value as volatility increases.
Moreover, this also implies that there is a one-on-one relationship between option value and volatility, i.e., we can use volatility to quote prices and vice-versa.
The volatility that matches the observed price of an option is called the implied volatility.

Example 2 Consider a non-dividend paying stock that currently trades for $100. The risk-free rate is 5% per year, continuously compounded and constant for all maturities. An ATM European call option written on the stock with maturity 12 months trades for $16. We can check that \sigma=34.66\% prices the call correctly: \begin{aligned} d_{1} & = \frac{\ln(100/100) + (0.05 + 0.5(0.3466)^{2})(1)}{0.3466\sqrt{1}} = 0.3176 \\ d_{2} & = 0.3358 - 0.3466\sqrt{1} = -0.0290 \\ \end{aligned} Therefore, \operatorname{\Phi}(d_{1}) = 0.6246 and \operatorname{\Phi}(d_{2}) = 0.4884, which implies that C = 100(0.6246) - 100e^{-0.05(1)}(0.4884) = \$16.00

How Can We Compute the Implied Volatility?

Unfortunately, it is not possible to solve analytically for the implied volatility.
For a call option, for example, it involves solving numerically for \sigma: C_{0} = C(\sigma_{\mathit{imp}})
Alternatively, we could tabulate the price of a call option for different values of \sigma (using the same parameters as the previous example):

\sigma	0.05	0.10	0.15	0.20	0.25	0.30	0.35	0.40
C	5.28	6.80	8.59	10.45	12.34	14.23	16.13	18.02

We can see that \sigma=35\% gives a price of \$16.13 for the call, which is quite close to the true implied volatility of 34.66%.