Risk Neutral Valuation

There are a few different but equivalent ways of viewing derivatives pricing. The first to be developed was the partial differential equations method, which was how the Black Scholes equation was originally derived. There’s lots of discussion of this on the wikipedia page, and I’ll talk about it at some stage – it’s quite intuitive and a lot of the concepts fall out of it very naturally. However, probably the more powerful method, and the one that I use almost all of the time in work, is the risk-neutral pricing method.

The idea is quite simple, although the actual mechanics can be a little intricate. Since the distribution of the underlying asset at expiry is known, it makes sense that the price of a derivative might be the expected value of the payoff of the option at expiry (eg. (S_t-K)^+ for a call option, where a superscript ‘+’ means “The greater of this value or zero”) over the underlying distribution. In fact, it turns out that this isn’t quite right: due to the risk aversion of investors, this will usually produce an overestimate of the true price. However, in arbitrage-free markets, there exists another probability distribution under which the expectation does give the true price – this is called the risk-neutral distribution of the asset. Further, [as long as the market is complete] any price other than the risk-neutral price allows the possibility of arbitrage. Taken together, these are roughly a statement of the Fundamental Theorem of Asset Pricing. In the case of vanilla call options, a portfolio of the underlying stock and a risk-free bond can be constructed that exactly replicate the option and could be used in such an arbitrage.

In this risk-neutral distribution, all risky assets grow at the risk-free rate, so that the ‘price of risk’ is exactly zero. Let’s say a government bond – which we’ll treat as risk-free – exists, has a price B and pays an interest rate of r, so that

dB = r B dt

Then, the stochastic process for the underlying stock that we looked at before

dS = \mu dt + \sigma dW_t

is modified so that mu becomes r, and the process is

dS = rdt + \sigma dW_t

 so the risk-neutral distribution of the asset is still lognormal, but with mu’s replaced by r’s:

S_t = S_0 e^{(r - {1\over 2}\sigma^2)t + \sigma\sqrt{t}z}

 

I’ve not provided the explicit formula yet, so I’ll demonstrate here how this can be used to price vanilla call options

C(F,K,\sigma,t,\phi) = \delta(t){\mathbb E^{S_t}}[(S_t-K)^+]

= \delta(t)\int^\infty_{0} (S_t - K)^+ p_S(S_t)dS_t

= \delta(t)\int^\infty_{K} (S_t - K) p_S(S_t)dS_t

= {\delta(t) \over \sqrt{2\pi}}\int^\infty_{x_K} (S_0 e^{(r-{1\over 2}\sigma^2)t + \sigma \sqrt{t}x} - K) e^{-{1\over 2}x^2}dx

= {\delta(t) \over \sqrt{2\pi}}\Bigr[S_0 e^{(r-{1\over 2}\sigma^2)t} \int^\infty_{x_K} e^{\sigma \sqrt{t}x -{1\over 2}x^2}dx - \int^\infty_{x_K} K e^{-{1\over 2}x^2} dx \Bigl]

= {\delta(t) \over \sqrt{2\pi}}\Bigr[S_0 e^{(r-{1\over 2}\sigma^2)t} \int^\infty_{x_K} e^{-{1\over 2}(x-\sigma\sqrt{t})^2 + {1\over 2}\sigma^2t}dx - \sqrt{2\pi} K\Phi(-x_K)\Bigl]

= {\delta(t) \over \sqrt{2\pi}}\Bigr[S_0 e^{rt} \int^\infty_{x_K - \sigma \sqrt{t}} e^{-{1\over 2}x^2} dx - \sqrt{2\pi} K \Phi(-x_K)\Bigl]

= \delta(t)\Bigr[S_0 e^{rt} \Phi(-x_K + \sigma \sqrt{t}) - K \Phi(-x_K)\Bigl]

= \delta(t)\Bigr[F \Phi(d_1) - K \Phi(d_2)\Bigl]

which is the celebrated BS formula! In the above, F = forward price = \inline S_0 e^{rt}\inline \Phi(x) is the standard normal cumulative density of x, \inline x_K is the value of x corresponding to strike S=K, ie.

x_K = {\ln{K \over F} + {1\over 2}\sigma^2t \over \sigma \sqrt{t}}

it is typical to use the variables d1 and d2 for the values in the cfds, such that

d_1 = {\ln{F \over K} + {1\over 2}\sigma^2t \over \sigma \sqrt{t}}

d_2 = {\ln{F \over K} - {1\over 2}\sigma^2t \over \sigma \sqrt{t}} = d_1 - \sigma \sqrt{t}

In reality, certain we have made certain assumptions that aren’t justified in reality. Some of these are:

1. No arbitrage – we assume that there is no opportunity for a risk-free profit

2. No transaction costs – we can freely buy and sell the underlying at a single price

3. Can go long/short as we please – we have no funding constraints, and can buy/sell an arbitrarily large amount of stock/options and balance it with an opposite position in bonds

4. Constant vol and r – we assume that vol and r are constant and don’t vary with strike. In fact, it’s an easy extension to allow them to vary with time, I’ll come back to this later

I’ll look at the validity of these and other assumptions in a future post.

If prices of vanillas have non-constant vols that vary with strike, doesn’t that make all of the above useless? Not at all – but we do need to turn it on its head! Instead of using the r-n distribution to find prices, we use prices to find the r-n distribution! Lets assume that we have access to a liquid market of vanilla calls and puts that we can trade in freely. If we look at their prices and apply the Fundamental Theorem of Calculus twice

C(t)= \delta(t) \int^\infty_{K} (S_t - K)p_S(S_t)dS_t

{\partial C \over \partial K} = -\delta(t) \int^\infty_K p_S(S_t)dS_t

{1\over \delta(t)}{\partial^2 C \over \partial K^2} =p_S(K)

 So the curvature of call prices wrt. strike tells us the local risk neutral probability! This means for each expiry time that we can see vanillas option prices, we can calculate the market-implied r-n distribution (which probably won’t be lognormal, telling us that the market doesn’t really believe the BS assumptions as stated above either). Once we know this, we can use it calibrate our choice of market model and to price other, more exotic options.

[Post script: It is worth noting that although this looks like alchemy, we haven’t totally tamed the distribution, because although we know the underlying marginal distribution at each expiry time, we still don’t know anything about the correlations between them. That is, we know the marginal distributions of the underlying at each time, but not the full distribution. For concreteness, consider two times \inline t_1 and \inline t_2. We know \inline P(S_{t_1}) and \inline P(S_{t_2}) but not \inline P(S_{t_1},S_{t_2}). To price an option paying off the average of the spot at these two times, knowing the first two isn’t enough, we need to know the second, as the expectation is \inline \int^\infty_0 \int^\infty_0 {1\over 2}(S_{t_1} + S_{t_2})P(S_{t_1},S_{t_2}) dS_{t_1}dS_{t_2}. To see the difference, from Bayes Theorem we have that \inline P(S_{t_1},S_{t_2}) = \inline P(S_{t_1}).\inline P(S_{t_2}|S_{t_1}). So, although we know how the spot will be distributed at each time, we don’t know how each distribution is conditional on the times before, which we’d need to know to price exactly – our modelling challenge will be to choose some sensible process for this that is consistent with the marginal distributions.]

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