Hedging in a finite-state model (Binary Trees)

Today’s post is about hedging in a world where there are only a few outcomes. It demonstrates a lot of principles that are important in asset pricing, and it’s also a frequent topic for interview questions so almost got included as one of those!

A very basic approximation for a stock is a binary tree, as in the diagram shown below. The stock is worth 100 today, and after some time (say a year) it will be worth either 110 with probability p or 90 with probability (1-p). In this very basic model, we assume no other values are possible – the stock either does well and goes up by 10, or it does badly and goes down by 10. Furthermore, we assume that there is a risk-free yearly interest rate r at which we can deposit money in the bank and receive (1+r) times our initial deposit in a year.

The two possibilities for a stock's evolution in the binary tree model
The two possibilities for a stock’s evolution in the binary tree model

We’re going to try and price options in this model. Let’s say we’ve got a call option to buy the stock in a year’s time for 100. How much is this worth? Because of the simple state of the world, we can enumerate the two possibilities – either the stock is worth 110, in which case the call option to buy it for 100 is worth 10, or the stock is worth 90, and the option is worthless. We’ll try and simulate that payoff using a combination of cash-in-the-bank and the stock itself, and then by no-arbitrage the price must be the same today.

Our portfolio consists of \alpha stocks S and \beta cash-in-the-bank. The value of the cash will certainly be (1+r)\beta in a year, while the stock will be 110 \alpha or 90 \alpha depending on the state, and we want the sum of these two to equal the option payoff (in either the up or the down scenario), which gives us two simultaneous equations

    \[(1+r) \beta + 110 \alpha = 10\]

    \[(1+r) \beta + 90 \alpha = 0\]

so

    \[\alpha = 0.5\]

    \[\beta = {-45 \over (1+r)}\]

This says that a portfolio containing half of a stock and minus 45 pounds in the bank will exactly replicate the payoff of the option in either world state. The value of this portfolio today is just 0.5*100 – 45/(1+r), and I’ve plotted that as a function of r below.

The price of a call option on a stock in the two-state world given above
The price of a call option on a stock as a function of the risk-free rate in the two-state world given above

This gives meaningless prices if r > 0.1 (the price of the option must be between 0 and 10/(1+r) as these are the discounted values of the least/most that it can pay out at expiry). What does this mean intuitively? Well, if r > 0.1 then we have an arbitrage: 100 in the bank will *always* yield more than the stock at expiry, so we should short the stock as much as possible and put the proceeds in the bank. If r < -0.1 the situation is exactly reversed

The important point about all of this was that the option price DOESN’T depend on the relative chances of the stock increasing to 110 or falling to 90. As long as both of these are strictly greater than zero and less than one, then ANY set of probabilities will lead to the same price for the option. This is the discrete analogue of a result I presented before (here) – that the expected growth of the stock in the real world doesn’t matter to the option’s price, it is the risk-free rate that affects its price. Indeed, it is possible to derive the continuous result using a binary tree by letting the time period go to zero, I won’t attempt it here but the derivation can be found in many textbooks (eg. on wikipedia).

Things really get interesting when we try to extend the model in a slightly different way. Often banks will be interested in models that have not just a diffusive component, but also a ‘jump’ component, which gives a finite chance that a price will fall suddenly by a large amount (I’ll present one such model in the near future). Unlike a straight-forward Black-Scholes model, because these jumps happen suddenly and unexpectedly they are very difficult to hedge, and are meant to represent market crashes that can result in sudden sharp losses for banks.

In our simple tree model, this can be incorporated by moving to a three-branch model, shown below

The three possibilities for a stock's evolution in the three-branch tree model
The three possibilities for a stock’s evolution in the three-branch tree model

We have the same two branches as before, but an additional state now exists in which the stock price crashes to 50 with probability q. In this case, again trying to price an option to buy the stock for 100 gives three simultaneous equations

    \[(1+r) \beta + 110 \alpha = 10\]

    \[(1+r) \beta + 90 \alpha = 0\]

    \[(1+r) \beta + 50 \alpha = 0\]

Unlike before, we can’t find a single alpha and beta that will replicate the payoff in all three world states, as we have three equations and only two unknowns. Consequently, the best we will be able to to is sub-replicate or super-replicate the payoff. That is, find portfolios that either always pay equal to or greater than the option, or always pay less than or equal to the option. These will give bounds on the lowest and highest arbitrage-free option prices, but that’s as far as arbitrage-free prices will take us (in fact ANY of the prices between this limit is arbitrage-free) – choosing an actual price will require knowing the probabilities of p and q and deciding on our personal risk-premium.

In the simple three-state model, lower and upper bounds can be calculated by ignoring the second and third equation respectively above, and they give the limits shown in the graph below. Once again, as r \to 0.1 they converge, but note that a case where r < -0.1 is now possible, as the ‘crash’ option means that the stock can still pay out less than the bank account

The limiting values given by the no-arbitrage requirement on the price of a call option as a function of the risk-free rate in the three-branch tree model given above
The limiting values given by the no-arbitrage requirement on the price of a call option as a function of the risk-free rate in the three-branch tree model given above

This is what’s called an incomplete market. The securities that exist in the market aren’t sufficient to hedge us against all future possible states of the universe. In this case, we can’t uniquely determine prices by risk-neutral pricing – sice we can’t hedge out all of the risk, risk preferences of investors will play a part in determining the market-clearing prices of securities. Most models that are used in the industry are incomplete in this way – I’ll be looking at some examples of this soon which will include models involving jump processes and stochastic volatility.

Forwards vs. Futures

I’ve covered Forwards and Futures in previous posts, and now that I’ve covered the basics of Stochastic Interest Rates as well, we can have a look at the difference between Forwards and Futures Contracts from a financial perspective.

As discussed before, the price of a Forward Contract is enforceable by arbitrage if the underlying is available and freely storable and there are Zero Coupon Bonds available to the Forward Contract delivery time. In this case, the forward price is

F(t,T) = S(t) \cdot {1 \over {\rm ZCB}(t,T)}

In this post I’m going to assume a general interest rate model, which in particular may well be stochastic. In such cases, the price of a ZCB at the present time is given by

{\rm ZCB}(t,T) = {\mathbb E}\Big[ \exp{\Big\{\int_t^T r(t') dt'\Big\} } \Big]

Futures Contracts are a bit more complicated, and we need to extend our earlier description in the case that there are interest rates. The basic description was given before, but additionally in the presence of interest rates, any deposit that is in either party’s account is delivered to the OTHER party at the end of each time period. So, taking the example from the previous post, on day 4 we had $4 on account with the exchange – if rates on that day were 10% p.a., over that day the $4 balance would accrue about 10c interest, which would be paid to the other party.

Let’s say we’re at time s, and want to calculate the Futures price to time T. Our replication strategy is now as follows, following the classic proof due to Cox, Ingersall and Ross but in continuous time. Futures Contracts are free to enter into and break out of due to the margin in each account, so entering X Futures Contracts at time t and closing them at time t+dt will lead to a net receipt (or payment if negative) of \inline {\rm X}\cdot\big[ H(t+dt,T) - H(t,T)\big]. From t+dt to T, we invest (borrow) this amount at the short rate and thus recieve (need to pay)

{\rm X}\cdot\big[ H(t+\tau,T) - H(t,T)\big]\cdot\prod_t^T \big( 1 + r(t)\tau \big)

and now moving to continuous time

{\rm X}\cdot\big[ H(t+dt,T) - H(t,T)\big]\cdot\int_t^T e^{ r(t)}\ dt

We follow this strategy in continuous time, constantly opening contracts and closing them in the following time period [I’m glossing over discrete vs. continuous time here – as long as the short rate corresponds to the discrete time step involved this shouldn’t be a problem], and investing our profits and financing our losses both at the corresponding short rate. We choose a different X for each period [t,t+td] so that \inline {\rm X}(t) = \int_s^t \exp{\{r(t')\}}dt'. We also invest an amount H(s,T) at time s at the short rate, and continually roll this over so that it is worth \inline H(s,T)\cdot \int_s^T \exp{\{r(t)\}}dt at time T

Only the final step of this strategy costs money to enter, so the net price of the portfolio and trading strategy is H(s,T). The net payoff at expiry is

H(s,T)\cdot \int_s^T e^{r(t)}dt + \sum_s^T {\rm X}\cdot[H(t+dt,T)-H(t,T))]\cdot\int_t^T e^{r(t)}dt

= H(s,T)\cdot \int_s^T e^{r(t)}dt + \sum_s^T \int_s^t e^{r(t)}dt\cdot[H(t+dt,T)-H(t,T))]\cdot\int_t^T e^{r(t)}dt

= H(s,T)\cdot \int_s^T e^{r(t)}dt + \int_s^T e^{r(t)}dt \cdot \sum_s^T [H(t+dt,T)-H(t,T))]

= H(s,T)\cdot \int_s^T e^{r(t)}dt + \int_s^T e^{r(t)}dt \cdot [H(T,T)-H(s,T))]= H(T,T) \cdot \int_s^T e^{r(t)}dt

And H(T,T) is S(T), so the net payoff of a portfolio costing H(s,T) is

= S(T) \cdot \int_s^T e^{r(t)}dt

How does this differ from a portfolio costing the Forward price? Remembering that in Risk-Neutral Valuation, the present value of an asset is equal to the expectation of its future value discounted by a numeraire. In the risk-neutral measure, this numeraire is a unit of cash B continually re-invested at the short rate, which is worth \inline B(t,T) = e^{\int_t^T r(t')dt' }, so we see that the Futures Price is a martingale in the risk-neutral measure (sometimes called the ‘cash measure’ because of its numeraire). So the current value of a Futures Contract on some underlying should be

H(t,T) = {\mathbb E}^{\rm RN}\big[ S(T) | {\cal F}_t \big]

ie. the undiscounted expectation of the future spot in the risk-neutral measure. The Forward Price is instead the expected price in the T-forward measure whose numeraire is a ZCB expiring at time T

F(t,T) = {\mathbb E}^{\rm T}\big[ S(T) | {\cal F}_t \big]

We can express these in terms of each other remembering F(T,T) = S(T) = H(T,T) and using a change of numeraire (post on this soon!). I also use the expression for two correlated lognormal, which I derived at the bottom of this post

\begin{align*} F(t,T) &= {\mathbb E}^{T}\big[ F(T,T) | {\cal F}_t \big] \\ &= {\mathbb E}^{T}\big[ S(T) | {\cal F}_t \big] \\ &= {\mathbb E}^{T}\big[ H(T,T) | {\cal F}_t \big] \\ &= {\mathbb E}^{RN}\big[ H(T,T) {B(t)\over B(T)} {{\rm ZCB}(t,T)\over {\rm ZCB(T,T)}}| {\cal F}_t \big] \\ &= {\rm ZCB}(t,T){\mathbb E}^{RN}\big[ H(T,T) {1\over B(T)}| {\cal F}_t \big] \\ &= {\rm ZCB}(t,T){\mathbb E}^{RN}\big[ H(T,T)\big] {\mathbb E}^{RN}\big[ e^{-\int_t^T r(t')dt'} \big] e^{\sigma_H \sigma_B \rho} \\ &= H(t,T) \cdot e^{\sigma_H \sigma_B \rho} \\ \end{align*}

where \inline \sigma_H is the volatility of the Futures price, and \inline \sigma_B is the volatility of a ZCB – in general the algebra will be rather messy!

As a concrete example, let’s consider the following model for asset prices, with S driven by a geometric brownian motion and rates driven by the Vasicek model discussed before

{dS \over S} = r(t) dt + \sigma_S dW_t

dr = a \big[ \theta - r(t)\big] dt + \sigma_r \widetilde{dW_t}

And (critically) assuming that the two brownian processes are correlated according to rho

dW_t \cdot \widetilde{dW_t} = \rho dt

In this case, the volatility \inline \sigma_B is the volatility of \inline {\mathbb E}\big[ e^{-\int_t^T r(t')dt'}\big], and as I discussed in the post on stochastic rates, this is tractable and lognormally distributed in this model.

We can see that in the case of correlated stochastic rates, these two prices are not the same – which means that Futures and Forward Contracts are fundamentally different financial products.

 

For two standard normal variates x and y with correlation rho, we have:

\begin{align*} {\mathbb E}\big[ e^ {\sigma_1 x} \big]& = e^ {{1\over 2}\sigma_1^2 } \end{align*}

and

\begin{align*} {\mathbb E}\big[ e^ {\sigma_1 x + \sigma_2 y} \big]& = {\mathbb E}\big[ e^ {\sigma_1 x + \sigma_2 \rho x + \sigma_2 \sqrt{1-\rho^2}z} \big]\\ & = {\mathbb E}\big[ e^ {(\sigma_1 + \sigma_2 \rho) x + \sigma_2 \sqrt{1-\rho^2}z} \big]\\ & = \big[ e^ {{1\over 2}(\sigma_1 + \sigma_2 \rho)^2 + {1\over 2}(\sigma_2 \sqrt{1-\rho^2})^2} \big]\\ & = \big[ e^ {{1\over 2}\sigma_1^2 + {1\over 2}\sigma_2^2 + \sigma_1 \sigma_2 \rho} \big]\\ & = {\mathbb E}\big[ e^ {\sigma_1 x }\big] {\mathbb E} \big[ e^{\sigma_2 y}\big] e^{ \sigma_1 \sigma_2 \rho} \end{align*}

Digital Options

Today I’m going to talk about the valuation of another type of option, the digital (or binary) option. This can be seen as a bit of a case study, as I’ll present the option payoff and the analytical price and greeks under BS assumptions, and give add-ons to allow pricing with the MONTE CARLO pricer. I’ve also updated the ANALYTICAL pricer to calculate the price and greeks of these options.

Digital options are very straight-forward, they are written on an underlying S and expire at a particular date t, at which point digital calls pay $1 if S is greater than a certain strike K or $0 if it is below that, and digital puts pay the reverse – ie. the payoff is

P_{\rm dig\ call} = \Big\{ \ \begin{matrix} \$1 \quad S \geq K\\ \$0 \quad S < K\end{matrix}

We can calculate the price exactly in the BS approximation using the same method that I used to calculate vanilla option prices by risk-neutral valuation as follows

C_{\rm dig\ call}(0) = \delta(0,t)\ {\mathbb E}[ C_{\rm dig\ call}(t) ]

where \inline C_{\rm dig\ call}(t') is the price of the option at time \inline t', and we know that this must converge to the payoff as \inline t \to t', so \inline C_{\rm dig\ call}(t) = P_{\rm dig\ call}

= \delta(0,t)\ \int^{\infty}_{-\infty} P_{\rm dig\ call} \cdot e^{-{1\over 2}x^2} dx

\inline P_{\rm dig\ call} is zero if \inline S = S_0 e^{(r - {1\over 2}\sigma^2)t + \sigma \sqrt{t} x} < K which corresponds to \inline x < {\ln{K \over S_0} - (r-{1\over 2}\sigma^2)t \over \sigma \sqrt{t}} = -d_2

= \delta(0,t)\ \cdot \$1 \cdot \int^{\infty}_{-d_2} e^{-{1\over 2}x^2} dx

= \$ \delta(0,t)\cdot \Phi(d_2)

where

d_1 = {\ln{S\over K}+(r+{1\over 2}\sigma^2)t \over \sigma \sqrt{t}} \quad ;\quad d_2 = {\ln{S\over K}+(r-{1\over 2}\sigma^2)t \over \sigma \sqrt{t}}

Since we have an analytical price, we can also calculate an expression for the GREEKS of this option by differentiating by the various parameters that appear in the price. Analytical expressions for a digital call’s greeks are:

\Delta = {\partial C \over \partial S} = e^{-rt}\cdot {\phi(d_2)\over S\sigma \sqrt{t}}

\nu = {\partial C \over \partial \sigma} = -e^{-rt}\cdot {d_1 \ \phi(d_2)\over \sigma}

\gamma = {\partial^2 C\over \partial S^2} = -e^{-rt}\cdot {d_1 \ \phi(d_2)\over S^2 \sigma^2 t} = -e^{-rt}\cdot {d_1\ \phi(d_1)\over S K \sigma^2 t}

{\rm Vanna} = {\partial^2 C \over \partial S \partial \sigma} =e^{-rt}\cdot {\phi(d_2)\over S \sigma^2 \sqrt{t}}\Big( d_1 d_2 - 1 \Big)

{\rm Volga} = {\partial^2 C \over \partial \sigma^2} = e^{-rt}\cdot {\phi(d_2)\over \sigma^2}\cdot \Big( d_1 + d_2 - d_1^2 d_2 \Big)

where

\phi(d_1) = {1\over \sqrt{2 \pi}}\ {e^{-{1\over 2}d_1^2} }

Holding a binary put and a binary call with the same strike is just the same as holding a zero-coupon bond, since we are guaranteed to receive $1 wherever the spot ends up, so the price of a binary put must be

e^{-rt} = e^{-rt}\cdot \Phi(d_2) + C_{\rm dig\ put}(t'=0)\quad \quad \quad \quad [1]

C_{\rm dig\ put}(0) = e^{-rt} \cdot \Phi(-d_2)

Moreover, differentiating equation [1] above shows that the greeks of a digital put are simply the negative of the greeks of a digital call with the same strike.

Graphs of these are shown for a typical binary option in the following graphs. Unlike vanilla options, these option prices aren’t monotonic in volatility: if they’re in-the-money, increasing vol will actually DECREASE the price since it makes them more likely to end out-of-the-money!

Price and first-order greeks for a digital call option.
Price and first-order greeks for a digital call option.
DigitalGreeks2
Second-order greeks for a digital call option. Greeks for digital puts are simply the negative of these values

One final point on pricing, note that the payoff of a digital call is the negative of the derivative of a vanilla call payoff wrt. strike; and the payoff of a digital put is the positive of the derivative of a vanilla put payoff wrt. strike. This means that any binary greek can be calculated from the corresponding vanilla greek as follows

\Omega_{\rm dig\ call} = -{\partial \Omega_{\rm call}\over \partial K}

\Omega_{\rm dig\ put} = {\partial \Omega_{\rm put}\over \partial K}

where here \inline \Omega represents a general greek.

If you haven’t yet installed the MONTE CARLO pricer, you can find some instructions for doing so in a previous post. The following links give the header and source files for binary calls and puts which can be dropped in to the project in your C++ development environment

These will register the option types with the option factory and allow monte carlo pricing of the options (so far, all of the options in the factory also have analytical expressions, but I’ll soon present some options that can only be priced by Monte Carlo).